}(y,w)\in\mathbb{R}^{2}\setminus\operatorname{supp}\gamma_{\theta}^{\dagger}.$
(16)

By Theorem 49 we have the exact reconstruction $u_{t}=u^{\dagger}_{t}$ for all
$t\in{\mathcal{T}^{\prime}}$. Thus, by choice of $\theta$ we obtain

$\mathrm{Mv}_{t}\gamma_{\theta}=\mathrm{Rd}_{\theta}u_{t}=\mathrm{Rd}_{\theta}u^{\dagger}_{t}=\mathrm{Mv}_{t}\gamma^{\dagger}_{\theta}$

and therefore $0=\mathrm{Mv}_{t}\xi$ for all $t\in{\mathcal{T}^{\prime}}$.
Using the previously defined function $q$ we can calculate

$0=\frac{1}{|{\mathcal{T}^{\prime}}|}\sum_{t\in{\mathcal{T}^{\prime}}}\int_{\mathbb{R}}\tilde{q}_{t}\>\mathrm{d}\mathrm{Mv}_{t}\xi=\int_{\mathbb{R}^{2}}q\>\mathrm{d}\xi=\int_{\mathbb{R}^{2}}q\>\mathrm{d}\xi_{a}+\int_{\mathbb{R}^{2}}q\>\mathrm{d}\xi_{s}=\sum_{i=1}^{N}|\beta_{i}|+\int_{\mathbb{R}^{2}}q\>\mathrm{d}\xi_{s}.$

If $\xi_{s}=0$, then also
$\lVert\xi_{a}\rVert_{\mathcal{M}}=\sum_{i=1}^{N}|\beta_{i}|=0$, so there is
nothing left to show. On the other hand, $\xi_{s}\neq 0$ leads to a
contradiction: Since $|q|<1$ outside the support of
$\gamma^{\dagger}_{\theta}$ by (16), we get the inequality

$\lVert\xi_{a}\rVert_{\mathcal{M}}=\sum_{i=1}^{N}|\beta_{i}|=-\int_{\mathbb{R}^{2}}q\>\mathrm{d}\xi_{s}<\lVert\xi_{s}\rVert_{\mathcal{M}}.$

By optimality of $\gamma$ we get

$\lVert\gamma^{\dagger}_{\theta}\rVert_{\mathcal{M}}=\lVert\gamma^{\dagger}\rVert_{\mathcal{M}}\geq\lVert\gamma\rVert_{\mathcal{M}}=\lVert\gamma_{\theta}\rVert_{\mathcal{M}}=\lVert\gamma^{\dagger}_{\theta}+\xi\rVert_{\mathcal{M}},$

and from $\gamma^{\dagger}_{\theta}+\xi_{a}\perp\xi_{s}$ it follows that

$\lVert\gamma^{\dagger}_{\theta}+\xi\rVert_{\mathcal{M}}=\lVert\gamma^{\dagger}_{\theta}+\xi_{a}\rVert_{\mathcal{M}}+\lVert\xi_{s}\rVert_{\mathcal{M}}\geq\lVert\gamma^{\dagger}_{\theta}\rVert_{\mathcal{M}}-\lVert\xi_{a}\rVert_{\mathcal{M}}+\lVert\xi_{s}\rVert_{\mathcal{M}}>\lVert\gamma^{\dagger}_{\theta}\rVert_{\mathcal{M}}.$

Combining the last two inequalities yields the desired contradiction. ∎

######  Theorem 51 (Exact reconstruction of arbitrary snapshots).

Again as in Theorem 49 let
$\varnothing\neq{\mathcal{T}^{\prime}}\subset{\mathcal{T}}$ be a subset of
times $t$ at which the static problem
$\mathrm{P}_{\text{stat}}^{t}(f^{\dagger}_{t})$ reconstructs exactly, and let
$(u,\gamma)$ solve $\mathrm{P}_{0}(f^{\dagger})$. Further, fix some
$t\in\Sigma$. If $\Theta^{\prime}$ from Theorem 50 is such that
$u^{\dagger}_{t}$ is uniquely determined by
$(\mathcal{H}_{\Theta}{\mbox{\LARGE$\llcorner$}}\Theta^{\prime}){\otimes}_{\theta}\mathrm{Rd}_{\theta}u_{t}^{\dagger}$,
the Radon transform restricted to the directions in $\Theta^{\prime}$, then
$u_{t}=u^{\dagger}_{t}$.

###### Proof.

This follows immediately from the relation
$\mathrm{Rd}_{\theta}u_{t}=\mathrm{Mv}_{t}\gamma_{\theta}=\mathrm{Mv}_{t}\gamma^{\dagger}_{\theta}=\mathrm{Rd}_{\theta}u^{\dagger}_{\theta}$,
which holds for all $t\in\Sigma$ and by Theorem 50 for
$\mathcal{H}_{\Theta}$-almost all $\theta\in\Theta^{\prime}$. ∎

######  Remark 52 (Exact reconstruction for
$\Pi$-$\mathrm{P}_{0}(f^{\dagger})$).

With obvious modifications of the above proofs it is straightforward to see
that one may also replace $\mathrm{P}_{0}(f^{\dagger})$ with the alternative
model $\Pi$-$\mathrm{P}_{0}(f^{\dagger})$. Then one arrives at the analogous
conclusion that for the solution $(u,\gamma)$ of
$\Pi$-$\mathrm{P}_{0}(f^{\dagger})$

  1. 1.

we have $u_{t}=u_{t}^{\dagger}$ for all $t\in{\mathcal{T}^{\prime}}$ and all
$t\in\Sigma$ for which $u_{t}^{\dagger}$ is uniquely determined by the
projections
$\\{\mathrm{Rd}_{\theta}u_{t}^{\dagger}\,|\,\theta\in\Theta^{\prime}\\}$, and

  2. 2.

we have $\gamma_{\theta}=\gamma_{\theta}^{\dagger}$ for all
$\theta\in\Theta^{\prime}$.

The above reconstruction results depend on the set $\Theta^{\prime}$ for which
reason we analyse it in more detail in the next section. In particular, we
show that $\Theta^{\prime}$ coincides with $\Theta$ up to a
$\mathcal{H}^{d-1}$-nullset, which for instance implies that for
$\Theta=S^{d-1}$ and $\mathcal{H}_{\Theta}$ the (scaled) $(d-1)$-dimensional
Hausdorff measure both $\gamma^{\dagger}$ and $u^{\dagger}$ are exactly
reconstructed. This fact is discussed in the next section, while the
subsequent section considers the opposite case of a finite set $\Theta$.

###  3.2 Coincidences and ghost particles in projections

In this section we show that $\Theta^{\prime}$ from Theorem 50 contains
$\mathcal{H}^{d-1}$-almost all directions of $\Theta$. To this end we consider
projections $\mathrm{Rj}_{\theta}(S)$ of particle configurations
$S=\\{(x_{i},v_{i})\;|\;i=1,\dotsc,N\\}$, which themselves represent particle
configurations in one-dimensional space. Their sets of ghost particles with
respect to ${\mathcal{T}^{\prime}}$ are
$G(\mathrm{Rj}_{\theta}(S),{\mathcal{T}^{\prime}})$, and we furthermore
abbreviate

$L_{i,t}(\theta)\coloneqq\\{(x,v)\in\mathbb{R}\times\mathbb{R}\;|\;x+tv=\theta\cdot
x_{i}+t\theta\cdot v_{i})\\}=\mathrm{Rj}_{\theta}(L_{i,t}).$

We will separately consider coincidences and ghost particles.

######  Lemma 53 (Coincidences in projections).

Let ${\mathcal{T}^{\prime}}$ be a finite set of times and let the particle
configuration $S$ be free of coincidences at those times. Then also
$\mathrm{Rj}_{\theta}(S)$ is free of coincidences at those times for
$\mathcal{H}^{d-1}$-almost all $\theta\in S^{d-1}$.

###### Proof.

For a $\theta\in S^{d-1}$ and indices $i,j\in\\{1,\dotsc,N\\}$, $i\neq j$, the
projected particles $(\theta\cdot x_{i},\theta\cdot v_{i})$ and $(\theta\cdot
x_{j},\theta\cdot v_{j})$ take the same position at a time
$t\in{\mathcal{T}^{\prime}}$ if and only if $\theta$ fulfills the equation

$\theta\cdot(x_{i}-x_{j}+t(v_{i}-v_{j}))=0.$

Therefore we can write

$\\{\theta\in S^{d-1}\;|\;\text{$\mathrm{Rj}_{\theta}(S)$ has a coincidence at
a time in ${\mathcal{T}^{\prime}}$}\\}\\\
=\bigcup_{t\in{\mathcal{T}^{\prime}}}\bigcup_{i\neq j}\\{\theta\in
S^{d-1}\;|\;\theta\cdot(x_{i}-x_{j}+t(v_{i}-v_{j}))=0\\}.$

We have $x_{i}-x_{j}+t(v_{i}-v_{j})\neq 0$ for all $i\neq j$ since the
configuration is assumed to be free of conincidences for
$t\in{\mathcal{T}^{\prime}}$. Hence, the above is a finite union of
$(d-1)$-dimensional hyperplanes in $\mathbb{R}^{d}$ intersected with the
sphere $S^{d-1}$. Since those intersections are $\mathcal{H}^{d-1}$-nullsets,
their union is as well. ∎

######  Lemma 54 (Ghost particle sets in projections).

Let ${\mathcal{T}^{\prime}}$ be a finite set of times. The ghost particles of
a particle configuration $S$ and of its projections $\mathrm{Rj}_{\theta}S$
satisfy the following relation.

  1. 1.

For all $\theta\in S^{d-1}$, as long as $\mathrm{Rj}_{\theta}$ is injective on
$S$, it holds that
$G(\mathrm{Rj}_{\theta}(S),{\mathcal{T}^{\prime}})\supset\mathrm{Rj}_{\theta}(G(S,{\mathcal{T}^{\prime}}))$.

  2. 2.

For $\mathcal{H}^{d-1}$-almost all $\theta\in S^{d-1}$ we even have
$G(\mathrm{Rj}_{\theta}(S),{\mathcal{T}^{\prime}})=\mathrm{Rj}_{\theta}(G(S,{\mathcal{T}^{\prime}}))$.

###### Proof.

To prove the first statement, take a ghost particle $({x},{v})\in
G(S,{\mathcal{T}^{\prime}})$ of the original configuration. By definition,
there exist pairwise distinct indices
$(i_{t})_{t\in{\mathcal{T}^{\prime}}}\subset\\{1,\dotsc,N\\}$ such that

$x_{i_{t}}+tv_{i_{t}}={x}+t{v}\quad\text{for all }t\in{\mathcal{T}^{\prime}}.$

Taking the inner product with $\theta$ on both sides of these equations, we
see that the projected ghost particle
$(\theta\cdot{x},\theta\cdot{v})=\mathrm{Rj}_{\theta}({x},{v})$ is a ghost
particle of the projected configuration $\\{(\theta\cdot x_{i},\theta\cdot
v_{i})\;|\;i=1,\dotsc,N\\}$. The assumption of injectivity is necessary to
ensure that the indices $(i_{t})_{t}$ still address distinct particles after
the projection.

To prove the second statement, we first show that the inclusion from the first
statement holds for $\mathcal{H}^{d-1}$-almost all $\theta\in S^{d-1}$.
Indeed, in order for $\mathrm{Rj}_{\theta}$ to not be injective on $S$, there
need to exist indices $1\leq i,j\leq N,i\neq j$, such that $(\theta\cdot
x_{i},\theta\cdot v_{i})=(\theta\cdot x_{j},\theta\cdot v_{j})$. We can thus
write

$\\{\theta\in S^{d-1}\;|\;\text{$\mathrm{Rj}_{\theta}$ is not injective on
$S$}\\}\subset\bigcup_{i\neq j}\\{\theta\in
S^{d-1}\;|\;\text{$\theta\cdot(x_{i}-x_{j})=0$ and
$\theta\cdot(v_{i}-v_{j})=0$}\\}.$

Since the particles in the original configuration are assumed to be distinct,
for $i\neq j$ we either have $x_{i}-x_{j}\neq 0$ or $v_{i}-v_{j}\neq 0$, which
implies that the sets in the union are intersections of $S^{d-1}$ with
hyperplanes and thus $\mathcal{H}^{d-1}$-nullsets. Thus $\mathrm{Rj}_{\theta}$
is injective on $S$ for $\mathcal{H}^{d-1}$-almost all $\theta$ as desired.

It remains to show the inclusion
$G(\mathrm{Rj}_{\theta}(S),{\mathcal{T}^{\prime}})\subset\mathrm{Rj}_{\theta}(G(S,{\mathcal{T}^{\prime}}))$
for $\mathcal{H}^{d-1}$-almost all $\theta\in S^{d-1}$. To this end, we define
the set

$M\coloneqq\\{\theta\in
S^{d-1}\;|\;\mathrm{Rj}_{\theta}(G(S,{\mathcal{T}^{\prime}}))\subsetneq
G(\mathrm{Rj}_{\theta}(S),{\mathcal{T}^{\prime}})\\}$

of projection directions for which the above inclusion is violated. We need to
show that this set has measure zero. Writing
${\mathcal{T}^{\prime}}=\\{t_{1},\dotsc,t_{r}\\}$, $M$ can be expressed as

$\displaystyle M$ $\displaystyle=\left\\{\theta\in
S^{d-1}\;\middle|\;\mathrm{Rj}_{\theta}(G(S,{\mathcal{T}^{\prime}}))\subsetneq\bigcup_{(i_{p})_{p}}\bigcap_{p=1}^{{r}}L_{i_{p},t_{p}}(\theta)\right\\}$
$\displaystyle\subset\bigcup_{(i_{p})_{p}}\left\\{\theta\in
S^{d-1}\;\middle|\;\bigcap_{p=1}^{{r}}L_{i_{p},t_{p}}(\theta)\setminus\mathrm{Rj}_{\theta}(G(S,{\mathcal{T}^{\prime}}))\neq\varnothing\right\\},$
(17)

where the union runs over all families of distinct indices $(i_{p})_{1\leq
p\leq{r}}\subset\\{1,\dotsc,N\\}$. Since the union is finite, it suffices to
show that every set in the union is a $\mathcal{H}^{d-1}$-nullset.

Let $(i_{p})_{1\leq p\leq{r}}$ be such a family of indices and fix a
$\theta\in S^{d-1}$. It is straightforward to see that the pairwise
intersections $L_{i_{p},t_{p}}\cap L_{i_{p+1},t_{p+1}}$ are singletons. We
write

$\\{(a_{p},b_{p})\\}=L_{i_{p},t_{p}}\cap L_{i_{p+1},t_{p+1}},\quad
p=1,\dotsc,{r}-1,$

and calculate

$\bigcap_{p=1}^{r}L_{i_{p},t_{p}}(\theta)=\bigcap_{p=1}^{{r}-1}L_{i_{p},t_{p}}(\theta)\cap
L_{i_{p+1},t_{p+1}}(\theta)\\\
=\bigcap_{p=1}^{{r}-1}\mathrm{Rj}_{\theta}(L_{i_{p},t_{p}}\cap
L_{i_{p+1},t_{p+1}})=\bigcap_{p=1}^{{r}-1}\mathrm{Rj}_{\theta}\\{(a_{p},b_{p})\\}.$
(18)

Now we can distinguish two cases:

  1. 1.

All points $(a_{p},b_{p})$ coincide, that is,

$\bigcap_{p=1}^{r}L_{i_{p},t_{p}}=\\{(a_{1},b_{1})\\}.$

In this case, $(a_{1},b_{1})\in G(S,{\mathcal{T}^{\prime}})$ is a ghost
particle of the full-dimensional configuration, and (18) implies

$\bigcap_{p=1}^{r}L_{i_{p},t_{p}}(\theta)\setminus\mathrm{Rj}_{\theta}(G(S,{\mathcal{T}^{\prime}}))=\varnothing.$

  2. 2.

There exists a $q\in\\{1,\dotsc,{r}-1\\}$ such that
$(a_{q},b_{q})\neq(a_{q+1},b_{q+1})$. Assume without loss of generality that
$a_{q}\neq a_{q+1}$. If the intersection in (18) is nonempty, we see from its
right-hand side that

$(\theta\cdot a_{q},\theta\cdot
b_{q})=\mathrm{Rj}_{\theta}(a_{q},b_{q})=\mathrm{Rj}_{\theta}(a_{q+1},b_{q+1})=(\theta\cdot
a_{q+1},\theta\cdot b_{q+1}).$

This can only hold if $\theta$ is contained in the hyperplane

$\\{\theta\in\mathbb{R}^{d}\;|\;\theta\cdot(a_{q}-a_{q+1})=0\\}.$

By these cases, every set in the union in (17) is either empty or contained in
a hyperplane intersected with $S^{d-1}$, which means that $M$ is contained in
a finite union of $\mathcal{H}^{d-1}$-nullsets. ∎

The following Lemma is an immediate consequence.

######  Lemma 55 (Ghost particles in projections).

Let ${\mathcal{T}^{\prime}}$ be a finite set of times and assume that the
particle configuration $S$ admits no ghost particle with respect to
${\mathcal{T}^{\prime}}$. Then for $\mathcal{H}^{d-1}$-almost all $\theta$ the
projected configuration $\mathrm{Rj}_{\theta}(S)$ does not admit any ghost
particle with respect to ${\mathcal{T}^{\prime}}$ either.

In fact, the arguments in this section even show that $\Theta^{\prime}$ from
Theorem 50 even differs from $\Theta$ at most by a set of finite
$(d-2)$-dimensional Hausdorff measure. Theorem 4 is now merely a combination
of Theorems 49, 50, 51, 53, 55 and 26.

###  3.3 Exact reconstruction with finitely many directions

It is clear intuitively that exact reconstruction becomes more difficult the
more particles need to be reconstructed. Thus, given $N$ particles, a natural
question is how large the set ${\mathcal{T}^{\prime}}$ of reconstructible
snapshots needs to be in order to exactly reconstruct the correct velocity
information in any $\gamma_{\theta}$. Likewise one can ask how big a set
$\Theta$ is necessary in order to be able to also exactly reconstruct all
snapshots (not just those that are reconstructible directly from the
measurements). To this end, we briefly generalize a theorem on the X-ray
transform due to Hajós and Rényi [22] to the setting of the Radon transform.
Several proofs can be found in the literature for the theorem by Hajós and
Rényi; a particularly simple one is given in [8] whose idea we also use here.

######  Proposition 56 (Determination of finite set from Radon transform).

Let $S\subset\mathbb{R}^{d}$ be finite, containing at most $N$ points, and let
$\Theta\subset S^{d-1}$ contain $(d-1)N+1$ directions such that any $d$ of
them are linearly independent. Then $S$ is uniquely determined by its Radon
transform $\mathrm{Rd}_{\theta}S$ for $\theta\in\Theta$.

###### Proof.

Let the set $S^{\prime}$ satisfy
$\mathrm{Rd}_{\theta}S=\mathrm{Rd}_{\theta}S^{\prime}$ for all
$\theta\in\Theta$. Assume there exists $x\in S^{\prime}\setminus S$, then for
each of the $(d-1)N+1$ directions $\theta\in\Theta$ there is a point in $S$
being projected under $\mathrm{Rd}_{\theta}$ to the same position as $x$.
Since there are at most $N$ points in $S$, there is at least one $y\in S$
which is projected onto the same position as $x$ for at least $d$ many
projections $\mathrm{Rd}_{\theta}$. As a point is uniquely determined by its
projections onto $d$ linearly independent coordinate lines, we must have
$y=x$, a contradiction. Knowing now that $S^{\prime}\subset S$ and thus that
$S^{\prime}$ contains no more than $N$ points, the same argument with the
roles of $S$ and $S^{\prime}$ swapped yields $S\subset S^{\prime}$. ∎

######  Corollary 57 (Determination of discrete measure from Radon transform).

Let $\nu\in\mathcal{M}_{+}(\mathbb{R}^{d})$ be a nonnegative Radon measure
with support in at most $N$ points, and let $\Theta\subset S^{d-1}$ contain
$(d-1)N+1$ directions such that any $d$ of them are linearly independent. Then
$\nu$ is uniquely determined among all nonnegative Radon measures by its Radon
transform $\mathrm{Rd}_{\theta}\nu$ for $\theta\in\Theta$.

###### Proof.

For a contradiction assume there is a second nonnegative Radon measure
$\nu^{\prime}$ satisfying
$\mathrm{Rd}_{\theta}\nu=\mathrm{Rd}_{\theta}\nu^{\prime}$ for all
$\theta\in\Theta$. Then in particular
$\mathrm{Rd}_{\theta}\operatorname{supp}\nu=\mathrm{Rd}_{\theta}\operatorname{supp}\nu^{\prime}$
for all $\theta\in\Theta$ so that necessarily
$\operatorname{supp}\nu=\operatorname{supp}\nu^{\prime}$. Now consider the
signed measure $\xi=\nu-\nu^{\prime}\in\mathcal{M}(\mathbb{R}^{d})$ and
decompose it into its positive and negative part $\xi_{+}$ and $\xi_{-}$. By
linearity of $\mathrm{Rd}_{\theta}$, we have
$\mathrm{Rd}_{\theta}\xi_{+}=\mathrm{Rd}_{\theta}\xi_{-}$ for all
$\theta\in\Theta$. Therefore, by Proposition 56 we have
$\operatorname{supp}\xi_{+}=\operatorname{supp}\xi_{-}$, which is impossible
unless $\xi_{+}=\xi_{-}=0$. ∎

An immediate consequence is the following result. It exploits the fact,
already used implicitly before, that the move operator $\mathrm{Mv}$ is
equivalent to the two-dimensional Radon transform in the sense

$\mathrm{Mv}_{t}\nu={[s\mapsto\sqrt{1+t^{2}}s]}_{\\#}(\mathrm{Rd}_{(1,t)/\sqrt{1+t^{2}}}\nu).$

######  Corollary 58 (Exact reconstruction).

Let the ground truth particle configuration $\lambda^{\dagger}$ contain at
most $N$ particles, let $\Theta\subset S^{d-1}$ contain $(d-1)N+1$ directions
such that any $d$ of them are linearly independent, and let
$\mathcal{H}_{\Theta}$ be the counting measure on $\Theta$. If there are $N+1$
times $t$ at which the static problem
$\mathrm{P}_{\text{stat}}^{t}(f^{\dagger}_{t})$ reconstructs exactly, then the
solution $(u,\gamma)$ to $\mathrm{P}_{0}(f^{\dagger})$ satisfies
$(u,\gamma)=(u^{\dagger},\gamma^{\dagger})$.

###### Proof.

Let ${\mathcal{T}^{\prime}}$ denote the $N+1$ times, then by Theorem 49 we
have $u_{t}=u^{\dagger}_{t}$ for all $t\in{\mathcal{T}^{\prime}}$. Now for any
$\theta\in\Theta$ we know
$\mathrm{Mv}_{t}\gamma_{\theta}=\mathrm{Rd}_{\theta}u_{t}=\mathrm{Rd}_{\theta}u^{\dagger}_{t}=\mathrm{Mv}_{t}\gamma^{\dagger}_{\theta}$
and thus
$\mathrm{Rd}_{(1,t)/\sqrt{1+t^{2}}}\gamma_{\theta}=\mathrm{Rd}_{(1,t)/\sqrt{1+t^{2}}}\gamma^{\dagger}_{\theta}$
for all $N+1$ times $t\in{\mathcal{T}^{\prime}}$. By 57 this uniquely
determines $\gamma_{\theta}=\gamma^{\dagger}_{\theta}$ for all
$\theta\in\Theta$. Now for any $t\in\Sigma$ we have
$\mathrm{Rd}_{\theta}u_{t}=\mathrm{Mv}_{t}\gamma_{\theta}=\mathrm{Mv}_{t}\gamma^{\dagger}_{\theta}=\mathrm{Rd}_{\theta}u^{\dagger}_{t}$
for all $(d-1)N+1$ many directions $\theta\in\Theta$, which again by 57
uniquely determines $u_{t}=u_{t}^{\dagger}$. ∎

The above estimates can be improved in various directions. We only considered
the worst case analysis: we even obtain exact reconstruction if the times
${\mathcal{T}^{\prime}}$ and the directions $\Theta$ are configured in the
worst possible case (we did so since the ${\mathcal{T}^{\prime}}$ is in
general prescribed by the given observations and cannot be chosen). If one
chooses the times ${\mathcal{T}^{\prime}}$ and directions $\Theta$ optimally,
exact reconstruction may even be achieved for more points than considered
above. To indicate directions of possible improvement we briefly summarize
results from the literature on the two-dimensional Radon transform:

  * •

If the directions $\Theta$ are not a subset of the sides of an affinely
deformed regular $N$-gon (or if the points do not form an affinely deformed
regular $N$-gon), then $N$ points are uniquely determined by $m$ projections
already if $m>N-\frac{\sqrt{N+9}-3}{2}$ [8, Prop. 3].

  * •

If the directions $\Theta$ are chosen optimally (not just satisfying the
previous condition), then $N$ points are uniquely determined by $m$
projections already if $N\leq\max\\{m_{0},2^{cm/\log m}\\}$ for two constants
$m_{0},c>0$ [19, Thm. 1.1]. This bound is quite sharp since for $N>6^{m/3}$
one can always find $N$ points that are not uniquely determined by $m$ fixed
projections [19, Thm. 1.2].

  * •

In higher dimensions $d$, the previous point still holds true (up to changed
constants) for the X-ray transform, while for $m$ arbitrary, non-optimal
projection directions spanning $\mathbb{R}^{d}$ one can asymptotically no
longer guarantee reconstruction of arbitrary $N$ points as soon as
$N>m^{d+1+\epsilon}$ for $\epsilon>0$ arbitrarily small [5, Cor. 4.1].

All above results are concerned with worst case point configurations, that is,
they ask for fixed projections such that all possible configurations of $N$
points can be uniquely reconstructed. In applications it is probably of higher
relevance whether _most_ point configurations can be reconstructed.

######  Proposition 59 (Determination of generic set from Radon transform).

Let $\Theta\subset S^{d-1}$ contain $d+1$ directions such that any $d$ of them
are linearly independent, and consider any probability distribution of $N$
points in $\mathbb{R}^{d}$ which is absolutely continuous with respect to the
$Nd$-dimensional Lebesgue measure (for instance Gaussian i.i.d. points). Then
almost surely a set $S$ of $N$ points is uniquely determined among all subsets
of $\mathbb{R}^{d}$ by its Radon transform $\mathrm{Rd}_{\theta}S$ for
$\theta\in\Theta$.

###### Proof.

Let the directions in $\Theta$ be numbered as
$\theta_{1},\ldots,\theta_{d+1}$. Consider a set
$S=\\{x_{1},\ldots,x_{N}\\}\subset\mathbb{R}^{d}$ and denote the hyperplanes
through those points normal to $\theta\in\Theta$ by
$H^{i}_{\theta}=\\{x\in\mathbb{R}^{d}\,|\,(x-x_{i})\cdot\theta=0\\}$. The set
$S$ is uniquely determined if the only mappings
$\iota:\Theta\to\\{1,\ldots,N\\}$ such that
$H^{\iota(\theta_{1})}_{\theta_{1}}\cap\ldots\cap
H^{\iota(\theta_{d+1})}_{\theta_{d+1}}\neq\varnothing$ are the constant
mappings $\iota\equiv i$ (in which case
$H^{\iota(\theta_{1})}_{\theta_{1}}\cap\ldots\cap
H^{\iota(\theta_{d+1})}_{\theta_{d+1}}=\\{x_{i}\\}$). Thus it remains to show
$H^{\iota(\theta_{1})}_{\theta_{1}}\cap\ldots\cap
H^{\iota(\theta_{d+1})}_{\theta_{d+1}}=\varnothing$ almost surely for all
nonconstant $\iota$. Now the condition
$H^{\iota(\theta_{1})}_{\theta_{1}}\cap\ldots\cap
H^{\iota(\theta_{d+1})}_{\theta_{d+1}}\neq\varnothing$ is equivalent to the
existence of a solution $x\in\mathbb{R}^{d}$ to the overdetermined system of
equations

$Ax=b_{\iota}\quad\text{for }A=(\theta_{1}\ \cdots\ \theta_{d+1})^{T},\quad
b_{\iota}=(\theta_{1}\cdot x_{\iota(\theta_{1})}\ \cdots\ \theta_{d+1}\cdot
x_{\iota(\theta_{d+1})})^{T}$

or equivalently (by the Fredholm alternative) to $b_{\iota}\cdot y=0$ for
$y\in S^{d}$ the vector (unique up to sign) spanning $\ker A^{T}$. Thus we
require that for any nonconstant $\iota$ the solutions
$(x_{1},\ldots,x_{N})\in\mathbb{R}^{Nd}$ to
$\sum_{i=1}^{d+1}y_{i}\theta_{i}^{T}x_{\iota(\theta_{i})}=b_{\iota}\cdot y=0$
form a set of dimension strictly smaller than $Nd$ and thus a Lebesgue-
nullset. However, the solution set can only have dimension $Nd$ if the
equation is trivial, which is never the case: The coefficient of
$x_{\iota(\theta_{1})}$ is the sum of at most $d$ terms of the form
$y_{i}\theta_{i}^{T}$. Since all $y_{i}$ are nonzero and any $d$ directions
$\theta_{i}$ are linearly independent, it is therefore nonzero. ∎

######  Corollary 60 (Determination of generic measures from Radon transform).

Let $\Theta\subset S^{d-1}$ contain $d+1$ directions such that any $d$ of them
are linearly independent, and consider any probability distribution of
discrete nonnegative measures $\sum_{i=1}^{n}a_{i}\delta_{x_{i}}$ such that
the induced distribution of their $N$ support points is absolutely continuous
with respect to the $Nd$-dimensional Lebesgue measure. Then almost surely such
a measure $\nu$ is uniquely determined among all nonnegative Radon measures by
its Radon transform $\mathrm{Rd}_{\theta}\nu$ for $\theta\in\Theta$.

###### Proof.

Let $\nu$ be a measure consisting of $N$ weighted Dirac measures, then its
support is almost surely uniquely determined by Proposition 59. Supposing that
its support is indeed uniquely determined, it remains to show that also its
weights are uniquely determined. Thus let $\nu^{\prime}$ be another admissible
measure with $\operatorname{supp}\nu^{\prime}=\operatorname{supp}\nu$ and
$\mathrm{Rd}_{\theta}\nu^{\prime}=\mathrm{Rd}_{\theta}\nu$ for all
$\theta\in\Theta$. Let $\xi_{+}$ and $\xi_{-}$ denote the positive and
negative part, respectively, of $\nu^{\prime}-\nu$, then
$\mathrm{Rd}_{\theta}\xi_{+}=\mathrm{Rd}_{\theta}\xi_{-}$ for all
$\theta\in\Theta$. Since
$\operatorname{supp}\xi_{+}\cup\operatorname{supp}\xi_{-}\subset\operatorname{supp}\nu$
also the supports of $\xi_{+}$ and $\xi_{-}$ are uniquely determined by their
Radon transforms, which therefore implies
$\operatorname{supp}\xi_{+}=\operatorname{supp}\xi_{-}$. This is only possible
if $\xi_{+}=\xi_{-}=0$. ∎

As a direct consequence we obtain generic exact reconstruction result even
with only $d+1$ directions in $\Theta$, covering in the result of Theorem 5.

######  Corollary 61 (Exact reconstruction).

Assume the ground truth $\lambda^{\dagger}$ to contain $N$ particles (some
potentially with zero mass) whose location vector in
$(\mathbb{R}^{d}\times\mathbb{R}^{d})^{N}$ is distributed according to a
probability distribution which is absolutely continuous with respect to the
$2Nd$-dimensional Lebesgue measure. Let $\Theta\subset S^{d-1}$ contain $d+1$
directions such that any $d$ of them are linearly independent, and let
$\mathcal{H}_{\Theta}$ be the counting measure on $\Theta$. If there are three
times $t$ at which the static problem
$\mathrm{P}_{\text{stat}}^{t}(f^{\dagger}_{t})$ reconstructs exactly, then
almost surely the solution $(u,\gamma)$ to $\mathrm{P}_{0}(f^{\dagger})$
satisfies $(u,\gamma)=(u^{\dagger},\gamma^{\dagger})$.

###### Proof.

Let ${\mathcal{T}^{\prime}}$ denote the three times, then by Theorem 49 we
have $u_{t}=u^{\dagger}_{t}$ for all $t\in{\mathcal{T}^{\prime}}$. Now for any
$\theta\in\Theta$ we know
$\mathrm{Mv}_{t}\gamma_{\theta}=\mathrm{Rd}_{\theta}u_{t}=\mathrm{Rd}_{\theta}u^{\dagger}_{t}=\mathrm{Mv}_{t}\gamma^{\dagger}_{\theta}$
and thus
$\mathrm{Rd}_{(1,t)/\sqrt{1+t^{2}}}\gamma_{\theta}=\mathrm{Rd}_{(1,t)/\sqrt{1+t^{2}}}\gamma^{\dagger}_{\theta}$
for all three times $t\in{\mathcal{T}^{\prime}}$. By 60 this almost surely
uniquely determines $\gamma_{\theta}=\gamma^{\dagger}_{\theta}$ for all
finitely many $\theta\in\Theta$. Now for any $t\in\Sigma$ we have
$\mathrm{Rd}_{\theta}u_{t}=\mathrm{Mv}_{t}\gamma_{\theta}=\mathrm{Mv}_{t}\gamma^{\dagger}_{\theta}=\mathrm{Rd}_{\theta}u^{\dagger}_{t}$
for all $d+1$ many directions $\theta\in\Theta$, which again by 60 almost
surely uniquely determines $u_{t}=u_{t}^{\dagger}$ for each fixed
$t\in\Sigma$. That the fixed $t\in\Sigma$ may even be replaced by all
$t\in\Sigma$ follows from reinspection of the proof of Proposition 59, which
can be extended to show that generically all $u_{t}$ (or rather their
supports) can simultaneously be uniquely reconstructed. ∎

######  Remark 62.

The results in this section were obtained independent of the ones in Section
3.1, in particular not relying on assumptions on ghost particles and
coincidences. Alternatively, it would have also been possible to investigate
the set $\Theta^{\prime}$ of Theorem 50 also for the case of finitely many
directions, and show that it is sufficiently large to ensure exact
reconstruction (almost surely). While this would have provided a common
framework for $\mathcal{H}_{\Theta}$ being either the Hausdorff or the
counting measure, a direct treatment of the later turned out to be more
concise and, for this reason, was preferred here.

##  4 Reconstruction from noisy data

While the previous Section dealt with noisefree observations $f^{\dagger}$ and
a reconstruction via $\mathrm{P}_{0}(f^{\dagger})$, in this section we will
consider noisy observations $f^{\delta}$ and a reconstruction via
$\mathrm{P}_{\alpha}(f^{\delta})$ with positive $\alpha$. We aim to derive
error estimates for our reconstruction. To this end we will first introduce a
measure to quantify the error in Section 4.1, then derive abstract error
estimates for general inverse problems of Radon measures in Section 4.2, and
finally apply these results to our models
$\Pi$-$\mathrm{P}_{\alpha}(f^{\delta})$ and $\mathrm{P}_{\alpha}(f^{\delta})$
in Sections 4.3 and 4.4.

###  4.1 Unbalanced optimal transport as error measure

In the setting of particle reconstruction, the positions and masses of the
reconstructed particles will slightly deviate from the ground truth due to
noise in the measurement. A quantification of this deviation thus has to
account for both sources of error simultaneously, the mass relocation and the
mass change. A clear separation of these two is not meaningful since a Dirac
mass in the wrong place can both be explained by an incorrect positioning of a
ground truth Dirac mass or simply by some added mass to the ground truth
(which just happened to be placed in that position). Therefore we will
quantify the reconstruction error in so-called unbalanced Wasserstein
divergences (variants of optimal transport distances that allow for mass
changes). The following unbalanced Wasserstein divergences turn out to be the
most natural in our setting.

######  Definition 63 (Unbalanced Wasserstein divergence).

Let $\nu_{1},\nu_{2}\in\mathcal{M}_{+}(\mathbb{R}^{d})$ be nonnegative Radon
measures on $\mathbb{R}^{d}$. The _Wasserstein- $p$ distance_ between
$\nu_{1}$ and $\nu_{2}$ is defined for $p\geq 1$ as

$W_{p}(\nu_{1},\nu_{2})=\inf\left\\{\left(\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\operatorname{dist}(x,y)^{p}\,{\mathrm{d}}\pi(x,y)\right)^{1/p}\,\middle|\,\pi\in\mathcal{M}_{+}(\mathbb{R}^{d}\times\mathbb{R}^{d}),\right.\\\
\left.\vphantom{\left(\int_{\mathbb{R}^{d}}\right)^{1/p}}{[(x,y)\mapsto
x]}_{\\#}\pi=\nu_{1},{[(x,y)\mapsto y]}_{\\#}\pi=\nu_{2}\right\\}.$

If $\nu_{1}$ and $\nu_{2}$ have unequal mass
$\nu_{1}(\mathbb{R}^{d})\neq\nu_{2}(\mathbb{R}^{d})$, the above infimum is
over the empty set and thus infinite. For fixed parameter $R>0$ we define the
_unbalanced Wasserstein- $p$ divergence_ between $\nu_{1}$ and $\nu_{2}$ as

$W_{p,R}^{p}(\nu_{1},\nu_{2})=\inf\left\\{W_{p}^{p}(\nu,\nu_{2})+\tfrac{1}{2}R^{p}\|\nu_{1}-\nu\|_{\mathcal{M}}\,\middle|\,\nu\in\mathcal{M}_{+}(\mathbb{R}^{d})\right\\}.$

######  Remark 64 (Interpretation of unbalanced Wasserstein divergence).

The Wasserstein distance is well-known to be a metric on the space of
nonnegative Radon measures with fixed mass, supported on a fixed compact
domain, and it metrizes weak-$\ast$ convergence on that space. The quantity
$W_{p}^{p}(\nu_{1},\nu_{2})$ measures the cost for transporting the material
from $\nu_{1}$ such that after the transport it is distributed according to
$\nu_{2}$. The unbalanced Wasserstein divergence
$W_{p,R}^{p}(\nu_{1},\nu_{2})$ thus measures the cost for first changing the
mass of $\nu_{1}$ to some intermediate measure $\nu$ and then transporting
that new mass distribution to $\nu_{2}$. Thus it can be used to quantify
weak-$\ast$ convergence of nonnegative measures of not necessarily the same
mass. The weight $R^{p}$ of the mass change cost governs up to which distance
a mass transport is less costly than removing the mass in the initial position
and regrowing it in the destination; this distance is exactly $R$.

######  Remark 65 (Symmetry of unbalanced Wasserstein divergence).

Obviously, an alternative definition would have been based on changing the
mass of $\nu_{2}$ or of both $\nu_{1}$ and $\nu_{2}$. It can readily be seen
that those are equivalent: Indeed, any good candidate intermediate measure
$\nu$ should be absolutely continuous with respect to and no larger than
$\nu_{1}+\nu_{2}$. Then it is easy to check that
$\tilde{\nu}=\nu_{2}+\nu_{1}-\nu\in\mathcal{M}_{+}(\mathbb{R}^{d})$ satisfies
$W_{p}^{p}(\nu,\nu_{2})=W_{p}^{p}(\nu_{1},\tilde{\nu})$ and
$\|\nu_{1}-\nu\|_{\mathcal{M}}=\|\nu_{2}-\tilde{\nu}\|_{\mathcal{M}}$.

The unbalanced Wasserstein divergence is nonnegative, and
$W_{p,R}^{p}(\nu_{1},\nu_{2})=0$ implies $\nu_{1}=\nu_{2}$. Furthermore, it is
symmetric by the previous remark. However, it does not satisfy the triangle
inequality (unless $p=1$). Therefore it is not a metric, but metric variants
can easily be defined.

A bound on the unbalanced Wasserstein divergence between a discrete measure
$\nu_{1}=\sum_{i=1}^{N}m_{i}\delta_{x_{i}}\in\mathcal{M}_{+}(\mathbb{R}^{d})$
and $\nu_{2}\in\mathcal{M}_{+}(\mathbb{R}^{d})$ is readily obtained from
estimates of the mass distribution within and outside balls around
$x_{1},\ldots,x_{N}$. Such estimates were first derived by Candès and
Fernandez-Granda to analyse spike recovery from noisy data, and in the
subsequent sections we will show how they can also be derived for our
dimension-reduced dynamic particle tracking setting.

######  Theorem 66 (Unbalanced Wasserstein divergence from mass estimates).

Let $p\geq 1$, $R>0$, and let
$\nu_{1}=\sum_{i=1}^{N}m_{i}\delta_{x_{i}}\in\mathcal{M}_{+}(\mathbb{R}^{d})$,
with the distance between the $x_{i}$ being at least $2R$, and
$\nu_{2}\in\mathcal{M}_{+}(\mathbb{R}^{d})$ satisfy

$\displaystyle|\nu_{2}|(\mathbb{R}^{d}\setminus
B_{R}(\\{x_{1},\ldots,x_{N}\\}))$ $\displaystyle\leq A,$
$\displaystyle\sum_{i=1}^{N}|(\nu_{1}-\nu_{2})(B_{R}(\\{x_{i}\\}))|$
$\displaystyle\leq B,$
$\displaystyle\sum_{i=1}^{N}\int_{B_{R}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{p}\>\mathrm{d}\nu_{2}(x)$
$\displaystyle\leq C_{p},$

where $B_{R}(S)$ denotes the open $R$-neighbourhood of the set
$S\subset\mathbb{R}^{d}$. Then we have

$\displaystyle W_{p,R}^{p}(\nu_{1},\nu_{2})$
$\displaystyle\leq\tfrac{1}{2}R^{p}(A+B)+C_{p},$ $\displaystyle
W_{q,R}^{q}(\nu_{1},\nu_{2})$
$\displaystyle\leq\tfrac{1}{2}R^{q}(A+B)+R^{q-p}C_{p}\text{ if }q>p,$
$\displaystyle W_{q,R}^{q}(\nu_{1},\nu_{2})$
$\displaystyle\leq\tfrac{1}{2}R^{q}(A+B)+\tfrac{p}{p-q}\left(\tfrac{p-q}{q}\right)^{q/p}C_{p}^{q/p}\|\nu_{2}\|_{\mathcal{M}}^{1-q/p}\text{
if }q<p.$

###### Proof.

In the definition of the unbalanced Wasserstein divergence simply take
$\nu=\nu_{2}{\mbox{\LARGE$\llcorner$}}(\mathbb{R}^{d}\setminus
B_{R}(\\{x_{1},\ldots,x_{N}\\}))+\sum_{i=1}^{N}\nu_{2}(B_{R}(\\{x_{i}\\}))\delta_{x_{i}}$,
then the first estimate immediately follows. The second one follows, using the
same $\nu$, from $\operatorname{dist}(x,x_{i})^{q}\leq
R^{q-p}\operatorname{dist}(x,x_{i})^{p}$ on $B_{R}(\\{x_{i}\\})$ for any
$q>p$. Finally, if $q<p$ we again pick the same $\nu$ and estimate

$\int_{B_{R}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{q}\>\mathrm{d}\nu_{2}(x)\leq\int_{B_{R}(\\{x_{i}\\})\setminus
B_{r}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{q}\>\mathrm{d}\nu_{2}(x)+\int_{B_{r}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{q}\>\mathrm{d}\nu_{2}(x)\\\
\leq r^{q-p}\int_{B_{R}(\\{x_{i}\\})\setminus
B_{r}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{p}\>\mathrm{d}\nu_{2}(x)+r^{q}\|\nu_{2}\|_{\mathcal{M}}\leq
r^{q-p}C_{p}+r^{q}\|\nu_{2}\|_{\mathcal{M}}$

for any $r>0$. Optimizing in $r$ yields
$r=(\frac{p-q}{q}\frac{C_{p}}{\|\nu_{2}\|_{\mathcal{M}}})^{1/p}$ and thus

$\int_{B_{R}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{q}\>\mathrm{d}\nu_{2}(x)\leq\frac{p}{p-q}\left(\frac{p-q}{q}\right)^{q/p}C_{p}^{q/p}\|\nu_{2}\|_{\mathcal{M}}^{1-q/p}$

so that the third estimate follows. ∎

###  4.2 General strategy for Bregman distance estimates

To derive noise-dependent convergence rates or error estimates for our
reconstruction in case of noisy data, we follow a classical strategy in
inverse problems that is based on duality. In this section we recapitulate the
corresponding procedure in an abstract setting. It reduces the derivation of
error estimates to the construction of appropriate dual variables. The
explicit construction of such dual variables for our specific setting will be
performed in the subsequent section.

The abstract setting is as follows. Let $X,Y$ be topological vector spaces,
$K:X\to Y$ a continuous linear observation operator, $G:X\to\mathbb{R}$ a
convex regularization energy, and $F_{f}:Y\to\mathbb{R}$ a nonnegative convex
fidelity term checking the fidelity of its argument to a measurement $f\in Y$.
We will assume $F_{f}(f)=0$ and $F_{f}>0$ elsewhere (otherwise the data
fidelity would not be able to discriminate between multiple, apparently
equally well fitting reconstructions). We will denote by $y^{\dagger}\in X$
the ground truth configuration with noise-free measurement
$f^{\dagger}=Ky^{\dagger}\in Y$, while the noisy measurement is denoted
$f^{\delta}$, the amount of noise $\delta$ being quantified by the condition

$F_{f^{\delta}}(f^{\dagger})\leq\delta.$

In our application, the spaces $X,Y$ will be given by products of spaces of
Radon measures, hence we assume $X$ and $Y$ to actually be dual spaces to
topological vector spaces (otherwise the roles of primal and dual spaces
simply swap in the following). For simplicity we denote the predual spaces by
$X^{*}$ and $Y^{*}$. We now consider the problem to reconstruct an
approximation to $y^{\dagger}$ by solving

$\min_{y^{\delta}\in X}E_{\alpha,f^{\delta}}(y^{\delta})\quad\text{for
}E_{\alpha,f^{\delta}}(y^{\delta})=G(y^{\delta})+\frac{1}{\alpha}F_{f^{\delta}}(Ky^{\delta}),$
(19)

where $\alpha,\delta>0$ corresponds to the noisy measurement case and
$\alpha=\delta=0$ to the noise-free setting (with the notational convention
$\tfrac{1}{0}F_{f}(\tilde{f})=0$ if $\tilde{f}=f$ and
$\tfrac{1}{0}F_{f}(\tilde{f})=\infty$ otherwise).

Generally, if $y^{\dagger}$ is the minimizer of a smooth energy $E$ and
$y^{\delta}$ an approximation, then the deviation of $y^{\delta}$ from
$y^{\dagger}$ can be estimated from the difference
$E(y^{\delta})-E(y^{\dagger})$ and lower bounds on the Hessian of $E$. In the
case of nonsmooth convex energies the role of Hessian-based distance estimates
is naturally replaced by Bregman distances for $E$, where the Bregman distance
of $a$ to $b$ with respect to a convex energy $E$ and a subgradient element
$w\in\partial E(b)$ is

$D^{E}_{w}(a,b)=E(a)-E(b)-\langle w,a-b\rangle\geq 0.$

However, since in our setting the energy difference
$E_{0,f^{\dagger}}(y^{\delta})-E_{0,f^{\dagger}}(y^{\dagger})$ is infinite if
$Ky^{\delta}\neq f^{\dagger}$, the resulting Bregman distance estimates would
not be very useful. Therefore, the fidelity term needs to be dualized first,
that is, for some fixed $w^{\dagger}\in Y^{*}$, instead of $E_{0,f^{\dagger}}$
we consider

$G(\cdot)+\langle
K^{*}w^{\dagger},\cdot\rangle-(\tfrac{1}{0}F_{f^{\dagger}})^{*}(w^{\dagger})$

(which by weak duality is never larger than $E_{0,f^{\dagger}}$). Above,
$K^{*}:Y^{*}\to X^{*}$ is the operator having $K$ as dual, and
$(\tfrac{1}{0}F_{f^{\dagger}})^{*}(w^{\dagger})=\sup_{y\in Y}\langle
y,w^{\dagger}\rangle-\tfrac{1}{0}F_{f^{\dagger}}(y)=\langle
f^{\dagger},w^{\dagger}\rangle$ represents the (predual) Legendre–Fenchel
conjugate. This then leads to the following estimates (as for instance derived
for quadratic $F_{f^{\dagger}}$ in [14, Thm. 2]).

######  Theorem 67 (Bregman distance estimate for noisy reconstruction).

Let $w^{\dagger}\in Y^{*}$ satisfy the source condition
$-K^{*}w^{\dagger}\in\partial G(y^{\dagger})$. Then a minimizer $y^{\delta}$
of (19) satisfies

$\displaystyle D^{G}_{-K^{*}w^{\dagger}}(y^{\delta},y^{\dagger})$
$\displaystyle\leq\left(3\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha w^{\dagger})\right)/(2\alpha),$
$\displaystyle F_{f^{\delta}}(Ky^{\delta})$
$\displaystyle\leq\left(3\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha w^{\dagger})\right),$
$\displaystyle\langle K^{*}w,y^{\delta}-y^{\dagger}\rangle$
$\displaystyle\leq\left(4\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(2\alpha w)+F_{f^{\delta}}^{*}(-2\alpha
w)\right)/(2\alpha)\quad\text{for all }w\in Y^{*}.$

###### Proof.

First observe that

$\left[G(y^{\delta})+\langle
K^{*}w^{\dagger},y^{\delta}\rangle-(\tfrac{1}{0}F_{f^{\dagger}})^{*}(w^{\dagger})\right]-\left[G(y^{\dagger})+\langle
K^{*}w^{\dagger},y^{\dagger}\rangle-(\tfrac{1}{0}F_{f^{\dagger}})^{*}(w^{\dagger})\right]\\\
=G(y^{\delta})-G(y^{\dagger})-\langle-K^{*}w^{\dagger},y^{\delta}-y^{\dagger}\rangle=D^{G}_{-K^{*}w^{\dagger}}(y^{\delta},y^{\dagger}).$

Now the energy difference on the left-hand side can be estimated from above
exploiting the optimality of $y^{\delta}$,

$G(y^{\delta})+\frac{1}{\alpha}F_{f^{\delta}}(Ky^{\delta})=E_{\alpha,f^{\delta}}(y^{\delta})\leq
E_{\alpha,f^{\delta}}(y^{\dagger})=G(y^{\dagger})+\frac{1}{\alpha}F_{f^{\delta}}(f^{\dagger})\leq
G(y^{\dagger})+\frac{\delta}{\alpha}.$

Furthermore, we use Fenchel’s inequality to obtain

$\langle K^{*}w,y^{\delta}-y^{\dagger}\rangle=\frac{\langle 2\alpha
w,Ky^{\delta}\rangle+\langle-2\alpha
w,Ky^{\dagger}\rangle}{2\alpha}\leq\frac{F_{f^{\delta}}(Ky^{\delta})+F_{f^{\delta}}^{*}(2\alpha
w)+F_{f^{\delta}}(Ky^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha w)}{2\alpha},$

which already proves the third statement in case the second holds. Using these
two inequalities we obtain

$\left[G(y^{\delta})+\langle
K^{*}w^{\dagger},y^{\delta}\rangle-(\tfrac{1}{0}F_{f^{\dagger}})^{*}(w^{\dagger})\right]-\left[G(y^{\dagger})+\langle
K^{*}w^{\dagger},y^{\dagger}\rangle-(\tfrac{1}{0}F_{f^{\dagger}})^{*}(w^{\dagger})\right]\\\
\leq\frac{\delta}{\alpha}-\frac{1}{\alpha}F_{f^{\delta}}(Ky^{\delta})+\langle
w^{\dagger},K(y^{\delta}-y^{\dagger})\rangle\leq\frac{1}{2\alpha}\left(3\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha
w^{\dagger})\right)-\frac{1}{2\alpha}F_{f^{\delta}}(Ky^{\delta}).$

Identifying the left-hand side with
$D^{G}_{-K^{*}w^{\dagger}}(y^{\delta},y^{\dagger})$ and adding
$\frac{1}{2\alpha}F_{f^{\delta}}(Ky^{\delta})$ on both sides, the first two
statements immediately follow. ∎

######  Remark 68 (Interpretation of source condition).

The source condition $-K^{*}w^{\dagger}\in\partial G(y^{\dagger})$ is one of
the two necessary and sufficient primal-dual optimality conditions for the
optimization problem; the other one is
$Ky^{\dagger}\in\partial(\tfrac{1}{0}F_{f^{\dagger}})^{*}(w^{\dagger})=\\{f^{\dagger}\\}$
and thus automatically satisfied for the ground truth.

######  Remark 69 (Rates from estimates).

Note that due to the strict convexity of $F_{f^{\delta}}$ in $f^{\delta}$, the
Legendre–Fenchel conjugate $F_{f^{\delta}}^{*}$ is differentiable in $0$ so
that $(F_{f^{\delta}}^{*}(2\alpha w)+F_{f^{\delta}}^{*}(-2\alpha w))/\alpha$
converges to zero as $\alpha\to 0$. Thus, by choosing $\alpha$ as the
minimizer of $(\delta+F_{f^{\delta}}^{*}(2\alpha
w)+F_{f^{\delta}}^{*}(-2\alpha w))/\alpha$, the right-hand sides in Theorem 67
become functions of $\delta$ that decrease to zero as $\delta\to 0$.

If the regularization contains the Radon norm of a measure, the above abstract
estimates of Bregman distances can be turned into unbalanced Wasserstein
estimates of how much $y^{\delta}$ deviates from $y^{\dagger}$. In essence,
this is the basic idea underlying the classical stability analysis of
superresolution or spike reconstruction as introduced by Candès and Fernandez-
Granda in [16].

######  Theorem 70 (Mass distribution from Bregman distance).

Assume $y^{\dagger}\in\mathcal{M}(\mathbb{R}^{d})$ to be a discrete measure
with support in $\\{x_{1},\ldots,x_{N}\\}\subset\mathbb{R}^{d}$, and let
$v^{\dagger}\in\partial\|\cdot\|_{\mathcal{M}}(y^{\dagger})\subset
C(\mathbb{R}^{d})$, that is, $|v^{\dagger}|\leq 1$ and
$v^{\dagger}(x_{i})=\operatorname{sgn}(y^{\dagger}(\\{x_{i}\\}))$. If
$v^{\dagger}$ satisfies

$|v^{\dagger}(x)|\leq
1-\kappa\min\\{R,\operatorname{dist}(x,\\{x_{1},\ldots,x_{N}\\})\\}^{2}$

for some $\kappa,R>0$ with $2R$ smaller than the minimum distance between the
$x_{i}$, then for any $y^{\delta}\in\mathcal{M}(\mathbb{R}^{d})$ we have

$\displaystyle|y^{\delta}|(\mathbb{R}^{d}\setminus
B_{R}(\\{x_{1},\ldots,x_{N}\\}))$ $\displaystyle\leq\frac{1}{\kappa
R^{2}}D^{\|\cdot\|_{\mathcal{M}}}_{v^{\dagger}}(y^{\delta},y^{\dagger}),$
$\displaystyle\sum_{i=1}^{N}\int_{B_{R}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{2}\>\mathrm{d}|y^{\delta}|(x)$
$\displaystyle\leq\frac{1}{\kappa}D^{\|\cdot\|_{\mathcal{M}}}_{v^{\dagger}}(y^{\delta},y^{\dagger}),\quad
i=1,\ldots,N.$

Furthermore, let $v\in\partial\|\cdot\|_{\mathcal{M}}(y)\subset
C(\mathbb{R}^{d})$ for
$y=\sum_{i=1}^{N}(y^{\delta}-y^{\dagger})(B_{R}(\\{x_{i}\\}))\delta_{x_{i}}$,
thus $v(x_{i})=\operatorname{sgn}(y(\\{x_{i}\\}))$ for $i=1,\ldots,N$. If for
some $\mu>0$ the function $v$ satisfies

$v(x_{i})v(x)\geq 1-\mu\operatorname{dist}(x,x_{i})^{2}\quad\text{for all
}x\in B_{R}(\\{x_{i}\\}),i=1,\ldots,N,$

then we additionally have

$\sum_{i=1}^{N}|(y^{\delta}-y^{\dagger})(B_{R}(\\{x_{i}\\}))|\leq\frac{1+\mu
R^{2}}{\kappa
R^{2}}D^{\|\cdot\|_{\mathcal{M}}}_{v^{\dagger}}(y^{\delta},y^{\dagger})+\langle
v,y^{\delta}-y^{\dagger}\rangle.$

###### Proof.

Abbreviate $B^{c}=\mathbb{R}^{d}\setminus B_{R}(\\{x_{1},\ldots,x_{N}\\})$ and
let $\frac{\mathrm{d}y^{\delta}}{\mathrm{d}|y^{\delta}|}$ denote the
Radon–Nikodym derivative of $y^{\delta}$ with respect to its total variation
measure, then the first two inequalities follow from

$\kappa
R^{2}|y^{\delta}|(B^{c})+\kappa\sum_{i=1}^{N}\int_{B_{R}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{2}\>\mathrm{d}|y^{\delta}|(x)\leq\int_{B^{c}}\frac{\mathrm{d}y^{\delta}}{\mathrm{d}|y^{\delta}|}-v^{\dagger}\>\mathrm{d}y^{\delta}(x)+\sum_{i=1}^{N}\int_{B_{R}(\\{x_{i}\\})}\frac{\mathrm{d}y^{\delta}}{\mathrm{d}|y^{\delta}|}-v^{\dagger}\>\mathrm{d}y^{\delta}(x)\\\
=\int_{\mathbb{R}^{d}}\frac{\mathrm{d}y^{\delta}}{\mathrm{d}|y^{\delta}|}-v^{\dagger}\>\mathrm{d}y^{\delta}(x)=\|y^{\delta}\|_{\mathcal{M}}-\langle
v^{\dagger},y^{\delta}\rangle=\|y^{\delta}\|_{\mathcal{M}}-\|y^{\dagger}\|_{\mathcal{M}}-\langle
v^{\dagger},y^{\delta}-y^{\dagger}\rangle=D^{\|\cdot\|_{\mathcal{M}}}_{v^{\dagger}}(y^{\delta},y^{\dagger}).$

The third inequality is obtained as follows. We have

$\sum_{i=1}^{N}|(y^{\delta}-y^{\dagger})(B_{R}(\\{x_{i}\\}))|=\int_{B_{R}(\\{x_{1},\ldots,x_{N}\\})}v\>\mathrm{d}(y^{\delta}-y^{\dagger})+\sum_{i=1}^{N}\int_{B_{R}(\\{x_{i}\\})}v(x_{i})-v\>\mathrm{d}(y^{\delta}-y^{\dagger}),$

where the first summand can be estimated as

$\int_{B_{R}(\\{x_{1},\ldots,x_{N}\\})}v\>\mathrm{d}(y^{\delta}-y^{\dagger})=\langle
v,y^{\delta}-y^{\dagger}\rangle-\int_{B^{c}}v\>\mathrm{d}(y^{\delta}-y^{\dagger})\\\
\leq\langle
v,y^{\delta}-y^{\dagger}\rangle+|y^{\delta}-y^{\dagger}|(B^{c})\leq\langle
v,y^{\delta}-y^{\dagger}\rangle+\frac{1}{\kappa
R^{2}}D^{\|\cdot\|_{\mathcal{M}}}_{v^{\dagger}}(y^{\delta},y^{\dagger}),$

while the second term is estimated from above via

$\sum_{i=1}^{N}\int_{B_{R}(\\{x_{i}\\})}v(x_{i})-v\>\mathrm{d}(y^{\delta}-y^{\dagger})\leq\sum_{i=1}^{N}\int_{B_{R}(\\{x_{i}\\})}\mu\operatorname{dist}(x,x_{i})^{2}\>\mathrm{d}|y^{\delta}|(x)\leq\frac{\mu}{\kappa}D^{\|\cdot\|_{\mathcal{M}}}_{v^{\dagger}}(y^{\delta},y^{\dagger}).\qed$

We finally combine Theorems 66, 70 and 67: Via Theorem 66 an unbalanced
Wasserstein error estimate can be expressed with the help of terms that can be
estimated by Bregman distances via Theorem 70, which in turn are estimated via
Theorem 67. In the case $G(y^{\delta})=\|y^{\delta}\|_{\mathcal{M}}$ the
combination is immediate, however, to allow for a little more generality
(which is required for the application to our setting) we shall assume

$G(y^{\delta})=\|y^{\delta}_{1}\|_{\mathcal{M}}+G_{2}(y^{\delta}_{2}),\quad\text{where
}y^{\delta}=(y^{\delta}_{1},y^{\delta}_{2}).$

######  Corollary 71 (Unbalanced Wasserstein estimate for noisy
reconstruction).

Consider the choice
$G(y^{\delta})=G(y^{\delta}_{1},y^{\delta}_{2})=\|y^{\delta}_{1}\|_{\mathcal{M}}+G_{2}(y^{\delta}_{2})$.
Let $y^{\dagger}_{1}\in\mathcal{M}_{+}(\mathbb{R}^{d})$ have support in
$\\{x_{1},\ldots,x_{N}\\}\subset\mathbb{R}^{d}$, let $y^{\delta}$ with
$y^{\delta}_{1}\in\mathcal{M}_{+}(\mathbb{R}^{d})$ be the approximation to
$y^{\dagger}$ obtained from (19), and let $w^{\dagger},w\in Y^{*}$ satisfy

$\displaystyle-K^{*}w^{\dagger}=-((K^{*}w^{\dagger})_{1},(K^{*}w^{\dagger})_{2})$
$\displaystyle\in\partial G(y^{\dagger}),$ $\displaystyle-(K^{*}w)_{1}$
$\displaystyle\in\partial\|\cdot\|_{\mathcal{M}}\left(\sum_{i=1}^{N}(y^{\delta}_{1}-y^{\dagger}_{1})(B_{R}(\\{x_{i}\\}))\delta_{x_{i}}\right),$
$\displaystyle|(K^{*}w^{\dagger})_{1}(x)|$ $\displaystyle\leq
1-\kappa\min\\{R,\operatorname{dist}(x,\\{x_{1},\ldots,x_{N}\\})\\}^{2},$
$\displaystyle(K^{*}w)_{1}(x_{i})(K^{*}w)_{1}(x)$ $\displaystyle\geq
1-\mu\operatorname{dist}(x,x_{i})^{2}\quad\text{for all }x\in
B_{R}(\\{x_{i}\\}),i=1,\ldots,N,$

for some $\mu,\kappa,R>0$ with $2R$ smaller than the minimum distance between
the $x_{i}$. Then we have

$\displaystyle|y^{\delta}_{1}|(\mathbb{R}^{d}\setminus
B_{R}(\\{x_{1},\ldots,x_{N}\\}))$ $\displaystyle\leq\frac{1}{2\alpha\kappa
R^{2}}\left(3\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha w^{\dagger})\right),$
$\displaystyle\sum_{i=1}^{N}\int_{B_{R}(\\{x_{i}\\})}\operatorname{dist}(x,x_{i})^{2}\>\mathrm{d}|y^{\delta}_{1}|(x)$
$\displaystyle\leq\frac{1}{2\alpha\kappa}\left(3\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha w^{\dagger})\right),$
$\displaystyle\sum_{i=1}^{N}|(y^{\delta}_{1}-y^{\dagger}_{1})(B_{R}(\\{x_{i}\\}))|$
$\displaystyle\leq\frac{1+\mu R^{2}}{2\alpha\kappa
R^{2}}\left(3\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha
w^{\dagger})\right)+\langle(K^{*}w)_{2},y^{\delta}_{2}-y^{\dagger}_{2}\rangle$
$\displaystyle\quad+\frac{1}{2\alpha}\left(4\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(2\alpha w)+F_{f^{\delta}}^{*}(-2\alpha
w)\right).$

In particular, by Theorem 66 this implies

$W_{2,R}^{2}(y^{\delta}_{1},y^{\dagger}_{1})\leq\tfrac{1}{2}R^{2}\langle(K^{*}w)_{2},y^{\delta}_{2}-y^{\dagger}_{2}\rangle\\\
+\frac{12+7\max\\{\kappa,\mu\\}R^{2}}{4\kappa\alpha}\left(\delta+F_{f^{\delta}}^{*}(2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(-2\alpha
w^{\dagger})+F_{f^{\delta}}^{*}(2\alpha w)+F_{f^{\delta}}^{*}(-2\alpha
w)\right).$

###### Proof.

This immediately follows from choosing $v^{\dagger}=(-K^{*}w^{\dagger})_{1}$,
$v=(-K^{*}w)_{1}$ in Theorem 70 and noting that
$D^{\|\cdot\|_{\mathcal{M}}}_{v^{\dagger}}(y^{\delta}_{1},y^{\dagger}_{1})\leq
D^{G}_{-K^{*}w^{\dagger}}(y^{\delta},y^{\dagger})$ as well as $\langle
v,y^{\delta}_{1}-y^{\dagger}_{1}\rangle=\langle-K^{*}w,y^{\delta}-y^{\dagger}\rangle+\langle(K^{*}w)_{2},y^{\delta}_{2}-y^{\dagger}_{2}\rangle$
so that Theorem 67 can be inserted. ∎

This way, the problem of proving the error estimate is reduced to the problem
of constructing $w^{\dagger}$ and $w$. Note for instance that for $G_{2}=0$
the condition $-(K^{*}w^{\dagger})_{2}\in\partial G_{2}(y^{\dagger}_{2})$
implies $(K^{*}w^{\dagger})_{2}=0$ so that the corresponding term in the
estimate of 71 vanishes.

######  Remark 72 (Adaptation for nonnegative measures and compact domains).

It is straightforward to check that all arguments stay the same when replacing
$\mathbb{R}^{d}$ with a compact domain $A$ and when taking the slightly
different regularizer choice

$G(y^{\delta})=G(y^{\delta}_{1},y^{\delta}_{2})=\|y^{\delta}_{1}\|_{+}+G_{2}(y^{\delta}_{2})\quad\text{with
}\|y^{\delta}_{1}\|_{+}=\begin{cases}\|y^{\delta}_{1}\|_{\mathcal{M}}&\text{if
}y^{\delta}_{1}\in\mathcal{M}_{+}(A),\\\ \infty&\text{else.}\end{cases}$

For this choice one obtains the estimates of 71 under the slightly weaker
conditions

$\displaystyle-K^{*}w^{\dagger}=-((K^{*}w^{\dagger})_{1},(K^{*}w^{\dagger})_{2})$
$\displaystyle\in\partial G(y^{\dagger}),$ $\displaystyle-(K^{*}w)_{1}$
$\displaystyle\in\partial\|\cdot\|_{\mathcal{M}}\left(\sum_{i=1}^{N}(y^{\delta}_{1}-y^{\dagger}_{1})(B_{R}(\\{x_{i}\\}))\delta_{x_{i}}\right),$
$\displaystyle-(K^{*}w^{\dagger})_{1}(x)$ $\displaystyle\leq
1-\kappa\min\\{R,\operatorname{dist}(x,\\{x_{1},\ldots,x_{N}\\})\\}^{2},$
$\displaystyle(K^{*}w)_{1}(x_{i})(K^{*}w)_{1}(x)$ $\displaystyle\geq
1-\mu\operatorname{dist}(x,x_{i})^{2}\quad\text{for all }x\in
B_{R}(\\{x_{i}\\}),i=1,\ldots,N.$

###  4.3 Construction of dual variables for $\Pi$-$\mathrm{P}_{\alpha}(f)$

In this and the next Section we prove error estimates for the particle
reconstruction from noisy observations. As in Section 3 we denote the ground
truth particle configuration by $\lambda^{\dagger}\in\mathcal{M}_{+}(\Lambda)$
with $\lambda^{\dagger}=\sum_{i=1}^{N}m_{i}\delta_{(x_{i},v_{i})}$, with
snapshots $u^{\dagger}_{t}=\mathrm{Mv}^{d}_{t}\lambda^{\dagger}$,
$t\in{\mathcal{T}}$, and with position-velocity projections
$\gamma^{\dagger}=\mathrm{Rj}\lambda^{\dagger}$ or
$\gamma^{\dagger}_{\theta}=\mathrm{Rj}_{\theta}\lambda^{\dagger}$,
$\theta\in\Theta$. The noisefree observations are denoted
$f^{\dagger}_{t}=\mathrm{Ob}_{t}\mathrm{Mv}^{d}_{t}\lambda^{\dagger}$, the
noisy ones $f^{\delta}_{t}$, $t\in{\mathcal{T}}$, with noise strength

$\delta=\tfrac{1}{2}\sum_{t\in{\mathcal{T}}}\|f_{t}^{\delta}-f_{t}^{\dagger}\|_{H}^{2}.$
(20)

The main result of these Sections is the below stated more quantitative
version of Theorem 6. Its error estimates depend on how close we are to a
coincidence or a ghost particle. This proximity to a degenerate configuration
is quantified via the notion of $\Delta$-coincidences and $\Delta$-ghost
particles that will be introduced in Definitions 78 and 81. Vanishing $\Delta$
indicates a degenerate configuration, and in this case the error estimates
become void (since the error is in those cases measured in the trivial
unbalanced Wasserstein divergence $W_{2,0}^{2}\equiv 0$). Finally note that
those snapshots that are not directly reconstructible from measurements will
instead be reconstructed from the position-velocity projections
$\gamma_{\theta}$ with $\theta$ from a certain set of _good directions_
$\Theta^{\prime}\subset\Theta$.

######  Theorem 73 (Reconstruction error estimates).

Let $(u,\mu)$ be the solution of $\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ or
$\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$, and let 3(1) hold for some
$\varnothing\neq{\mathcal{T}^{\prime}}\subset{\mathcal{T}}$. Pick some
$\Theta^{\prime}\subset\Theta$. Then there exist constants $R,C_{1}>0$
(depending on $\lambda^{\dagger}$, ${\mathcal{T}^{\prime}}$) and $C_{2}>0$
(depending on $\lambda^{\dagger}$, $\Theta^{\prime}$) such that

  1. 1.

we have $W_{2,R}^{2}(u_{t},u_{t}^{\dagger})\leq C_{1}\sqrt{\delta}$ for all
$t\in{\mathcal{T}^{\prime}}$,

  2. 2.

we have
$W_{2,R_{\theta}}^{2}(\gamma_{\theta},\gamma_{\theta}^{\dagger})\leq\frac{C_{1}}{R_{\theta}^{2}}\sqrt{\delta}$
for all $\theta\in\Theta$ ($\mathcal{H}_{\Theta}$-almost all $\theta\in\Theta$
in case of model $\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$), where
$R_{\theta}=\min\\{R,\Delta_{\theta}/3\\}/\sqrt{1+\max_{t\in{\mathcal{T}^{\prime}}}t^{2}}$
and $\Delta_{\theta}\geq 0$ is such that $\gamma_{\theta}^{\dagger}$ contains
no $\Delta_{\theta}$-coincidence or $\Delta_{\theta}$-ghost particle with
respect to ${\mathcal{T}^{\prime}}$ according to Definition 78,

  3. 3.

we have $W_{2,R_{t}}^{2}(u_{t},u_{t}^{\dagger})\leq
C_{1}C_{2}(t^{2}+\frac{1}{R_{t}^{4}})\sqrt{\delta}$ for all $t\in\Sigma$,
where $R_{t}=\min\\{R,\Delta_{t}/({3\sqrt{1+t^{2}}})\\}$ and
$\Delta_{t}\in[0,\inf_{\theta\in\Theta^{\prime}}\Delta_{\theta}]$ is such that
$u_{t}^{\dagger}$ contains no $\Delta_{t}$-coincidence or $\Delta_{t}$-ghost
particle with respect to $\Theta^{\prime}$ according to Definition 81.

Theorem 73 implies Theorem 6: Given ${\mathcal{T}^{\prime}}$, the minimum
$\Delta$ such that $\gamma^{\dagger}_{\theta}$ contains a $\Delta$-coincidence
or $\Delta$-ghost particle with respect to ${\mathcal{T}^{\prime}}$ is
continuous in $\theta$, thus it assumes a minimum $M$ at some
$\theta^{\prime}$ in the compact $\overline{\Theta}$. This minimum is positive
since $\gamma^{\dagger}_{\theta^{\prime}}$ contains no coincidence or ghost
particle. Hence, in Theorem 73(2) we may take $\Delta_{\theta}=\frac{M}{2}>0$
for all $\theta\in\overline{\Theta}$, which then yields Theorem 6(2). To
deduce Theorem 6(1) from Theorem 73(1) and (3) it suffices to show that
$t^{2}$ can be bounded from above and $\Delta_{t}$ away from zero, uniformly
for $t\in\overline{\Sigma}$. Boundedness of $t^{2}$ follows from compactness
of $\overline{\Sigma}$, so it remains to show uniform positivity of
$\Delta_{t}$. Picking $\Theta^{\prime}=\overline{\Theta}$, let $\Delta$ be
minimal such that $u^{\dagger}_{t}$ contains a $\Delta$-coincidence or
$\Delta$-ghost particle with respect to $\Theta^{\prime}$. As before, $\Delta$
depends continuously on $t$ and thus attains its minimum $N$ at some
$t^{\prime}$ in the compact $\overline{\Sigma}$. That the particles have
positive mutual distance in the projections
$(\mathrm{Rd}_{\theta}u^{\dagger}_{t^{\prime}})_{\theta\in\overline{\Theta}}$
and the support $\operatorname{supp}u^{\dagger}_{t^{\prime}}$ can be uniquely
reconstructed from
$(\operatorname{supp}(\mathrm{Rd}_{\theta}u^{\dagger}_{t^{\prime}}))_{\theta\in\overline{\Theta}}$
implies that $u^{\dagger}_{t^{\prime}}$ contains no $0$-coincidence or
$0$-ghost particle, hence $N>0$ and we may choose $\Delta_{t}=\frac{N}{2}>0$
for all $t\in\overline{\Sigma}$, as desired.

In this Section we prove the result for the (in this context simpler) model
$\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$; the proof will be extended to
the setting of $\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ in Section 4.4. We
will exploit that the optimization problem has multiple equivalent
formulations due to our constraint of nonnegative masses. Indeed, we may write
the regularizer as the norm of the snapshots or equivalently the norm of the
auxiliary variable $\gamma$. As seen in the previous section, the derivation
of error estimates requires the construction of appropriate dual variables,
which in our case is somewhat involved. In particular, to obtain error
estimates on the particle configurations at all time points outside
${\mathcal{T}^{\prime}}$, the error information needs to be passed from the
snapshots at times ${\mathcal{T}^{\prime}}$ to the variable $\gamma$ and then
back again, which is reflected in a stepwise construction of the corresponding
dual variables. In particular, we will reduce the derivation of error
estimates on $(u,\gamma)$ to the derivation of error estimates for the much
simpler static reconstruction problem of finding an approximation $u_{t}$ to
$u_{t}^{\dagger}$ from a noisy observation of
$\mathrm{Ob}_{t}u_{t}^{\dagger}$. This way the role of the observation
operator $\mathrm{Ob}_{t}$ is separated from the role of combining
measurements from multiple time points. An error analysis of the _static
problem_ is a classic superresolution problem and has for a particular choice
of $\mathrm{Ob}_{t}$ already been performed by Candès and Fernandez-Granda
[15] (even though they did not use our language of unbalanced Wasserstein
distances). Our contribution, in contrast, is to show how the error estimates
for the static problem transfer to error estimates of the _dynamic problem_.
Our analysis also applies to the reconstruction model (9) by Alberti et al.
[3], which forms the motivation of our work, but we will only perform it
explicitly for our novel reconstruction model $\Pi$-$\mathrm{P}_{\alpha}(f)$,
since this is the more complicated case and has the dimensionality advantage
over (9). If the static problem admits an error analysis we shall call the
problem reconstructible according to the following.

######  Definition 74 (Reconstructible measure).

We call a measure
$\nu^{\dagger}=\sum_{i=1}^{N}m_{i}\delta_{x_{i}}\in\mathcal{M}_{+}(\Omega)$
_reconstructible_ (from measurements via $\mathrm{Ob}_{t}$), if one can find
$\kappa,\mu,R>0$ ($R$ smaller than the minimum point separation in
$\nu^{\dagger}$) such that for any $N$-tuple $s\in\\{-1,1\\}^{N}$ and
associated measure
$\nu^{s}=\sum_{i=1}^{N}s_{i}\delta_{x_{i}}\in\mathcal{M}(\Omega)$ there exist
dual variables $v^{\dagger},v^{s}$ with
$-\mathrm{Ob}_{t}^{*}v^{\dagger}\in\partial\|\cdot\|_{\mathcal{M}}(\nu^{\dagger})$,
$-\mathrm{Ob}_{t}^{*}v^{s}\in\partial\|\cdot\|_{\mathcal{M}}(\nu^{s})$ and

$\displaystyle-\mathrm{Ob}_{t}^{*}v^{\dagger}(x)$ $\displaystyle\leq
1-\kappa\min\\{R,\operatorname{dist}(x,\\{x_{1},\ldots,x_{N}\\})\\}^{2}$
$\displaystyle\text{for all }x\in\Omega,$
$\displaystyle\mathrm{Ob}_{t}^{*}v^{s}(x_{i})\mathrm{Ob}_{t}^{*}v^{s}(x)$
$\displaystyle\geq 1-\mu\operatorname{dist}(x,x_{i})^{2}$
$\displaystyle\text{for all }x\in B_{R}(\\{x_{i}\\}),\,i=1,\ldots,N.$

We will say $\nu^{\dagger}$ is _$(\kappa,\mu,R)$ -reconstructible via
$v^{\dagger},v^{s}$_ to indicate the corresponding dual variables and values
of $\kappa$, $\mu$, and $R$.

Of course, the terminology comes from the fact that a reconstructible measure
can be reconstructed from its measurement.

######  Theorem 75 (Reconstructible measure).

If $\nu^{\dagger}$ is $(\kappa,\mu,R)$-reconstructible via $v^{\dagger},v^{s}$
and $f^{\delta}$ is an approximation to
$f^{\dagger}=\mathrm{Ob}_{t}\nu^{\dagger}$ with
$F_{f^{\delta}}(f^{\dagger})\leq\delta$, then the solution to

$\min_{\nu^{\delta}}\|\nu^{\delta}\|_{+}+\frac{1}{\alpha}F_{f^{\delta}}(\mathrm{Ob}_{t}\nu^{\delta})$

satisfies

$W_{2,R}^{2}(\nu^{\delta},\nu^{\dagger})\leq\tfrac{12\\!+\\!7\max\\{\kappa,\mu\\}R^{2}}{4\kappa\alpha}\left(\delta\\!+\\!F_{f^{\delta}}^{*}(2\alpha
v^{\dagger})\\!+\\!F_{f^{\delta}}^{*}(-2\alpha
v^{\dagger})\\!+\\!\\!\\!\max_{s\in\\{-1,1\\}^{N}}\\!\\!F_{f^{\delta}}^{*}(2\alpha
v^{s})\\!+\\!F_{f^{\delta}}^{*}(-2\alpha v^{s})\right).$

In particular, for $\delta=0$ the limit $\alpha\to 0$ implies
$\nu^{\delta}=\nu^{\dagger}$.

###### Proof.

This is just the special case of Remark 72 for $y^{\delta}_{1}=\nu^{\delta}$,
$K=\mathrm{Ob}_{t}$, $w^{\dagger}=v^{\dagger}$, $w=v^{s}$ with
$s_{i}=\operatorname{sgn}((\nu^{\delta}-\nu^{\dagger})(B_{R}(\\{x_{i}\\})))$,
$y^{\delta}_{2}$ and $G_{2}$ nonexistent. ∎

Depending on the measurement operator, simple criteria for reconstructibility
can be found. An example in one space dimension is proven in [15, Lem. 2.4 &
2.5] (higher-dimensional versions are also provided in that work, albeit
without detail).

######  Theorem 76 (Reconstructible measure for truncated Fourier series,
Candès & Fernandez-Granda 2013).

Let $\mathrm{Ob}_{t}$ be the truncated Fourier series on $\Omega=S^{1}$ which
only keeps coefficients up to a maximum absolute frequency $\Phi>0$. There
exist positive constants $c_{0},c_{1},c_{2},c_{3}>0$ such that, if the Dirac
locations in
$u^{\dagger}=\sum_{i=1}^{N}m_{i}\delta_{x_{i}}\in\mathcal{M}_{+}(\Omega)$ have
distance no smaller than $\Delta=c_{0}/\Phi$ to each other, then $u^{\dagger}$
is $(c_{1}\Phi^{2},c_{2}\Phi^{2},c_{3}/\Phi)$-reconstructible.

Before we begin the derivation of error estimates for
$\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$, let us introduce the
abbreviations

$\displaystyle\underline{\mathrm{Rd}}$
$\displaystyle:\mathcal{M}(\Omega)^{\Sigma}\to\mathcal{M}(\Xi)^{\Theta\times\Sigma},$
$\displaystyle\underline{\mathrm{Rd}}u$
$\displaystyle=(\mathrm{Rd}_{\theta}u_{t})_{\theta\in\Theta,t\in\Sigma},$
$\displaystyle\underline{\mathrm{Mv}^{1}}$
$\displaystyle:\mathcal{M}(\Gamma)^{\Theta}\to\mathcal{M}(\Xi)^{\Theta\times\Sigma},$
$\displaystyle\underline{\mathrm{Mv}^{1}}\gamma$
$\displaystyle=(\mathrm{Mv}^{1}_{t}\gamma_{\theta})_{\theta\in\Theta,t\in\Sigma},$
$\displaystyle\underline{\mathrm{Ob}}$
$\displaystyle:\mathcal{M}(\Omega)^{\Sigma}\to H^{|{\mathcal{T}}|},$
$\displaystyle\underline{\mathrm{Ob}}u$
$\displaystyle=(\mathrm{Ob}_{t}u_{t})_{t\in{\mathcal{T}}},$
$\displaystyle\iota_{+}$ $\displaystyle:\prod_{i\in
I}\mathcal{M}(A_{i})\to\\{0,\infty\\},$ $\displaystyle\iota_{+}(\nu)$
$\displaystyle=\begin{cases}0&\text{if }\nu_{i}\in\mathcal{M}_{+}(A_{i})\text{
for all }i\in I,\\\ \infty&\text{else,}\end{cases}$

where $I$ denotes an arbitrary index set and $A_{i}$ are Lipschitz domains.
Note that the above linear operators are all continuous with respect to the
product topology (a map $M:\prod_{i\in I}X_{i}\to\prod_{j\in J}Y_{j}$ between
product spaces with the product topology is continuous if and only if
$p_{j}\circ M$ is continuous for all projections $p_{j}:\prod_{j\in J}Y_{j}\to
Y_{j}$, and for all above operators $p_{j}\circ M$ is the composition of a
continuous projection $q_{i}:\prod_{i\in I}X_{i}\to X_{i}$ with a continuous
linear operator between Banach spaces). Furthermore, $\iota_{0}$ shall be the
convex indicator function of the set $\\{0\\}$. To translate the setting from
the previous section to $\Pi$-$\mathrm{P}_{\alpha}(f)$ we choose

$\displaystyle
y^{\dagger}\equiv(u^{\dagger},\gamma^{\dagger}),\,y^{\delta}\equiv(u,\gamma)\in\mathcal{M}(\Omega)^{\Sigma}\times\mathcal{M}(\Gamma)^{\Theta}\equiv
X,$ $\displaystyle
K\equiv\left(\begin{smallmatrix}\underline{\mathrm{Ob}}&0\\\
\underline{\mathrm{Rd}}&-\underline{\mathrm{Mv}^{1}}\end{smallmatrix}\right):X\to
H^{|{\mathcal{T}}|}\times\mathcal{M}(\Xi)^{\Theta\times\Sigma}\equiv Y,$
$\displaystyle
F_{f}(a,b)=\tfrac{1}{2}\sum_{t\in{\mathcal{T}}}\|a_{t}-f_{t}\|_{H}^{2}+\iota_{0}(b),\,G(u,\gamma)=\iota_{+}(u,\gamma)+\ast,$

where $\ast$ will either be $\|\gamma_{\theta}\|_{\mathcal{M}}$ or
$\|u_{t}\|_{\mathcal{M}}$ for particular choices of $\theta$ and $t$,
depending on what is most convenient for the desired estimate (by Remarks 20
and 28 and the constraint
$\underline{\mathrm{Rd}}u=\underline{\mathrm{Mv}^{1}}\gamma$, all these
choices are equivalent). The corresponding predual spaces are given by

$\displaystyle X^{*}$ $\displaystyle=\\{(\phi,\psi)\in
C(\Omega)^{\Sigma}\times C(\Gamma)^{\Theta}\,|\text{ only finitely many
}\phi_{t},\psi_{\theta}\text{ are nonzero}\\},$ $\displaystyle Y^{*}$
$\displaystyle=\\{(v,q)\in(H^{*})^{|{\mathcal{T}}|}\times
C(\Xi)^{\Theta\times\Sigma}\,|\text{ only finitely many }q_{\theta,t}\text{
are nonzero}\\},$

endowed with the final topology induced by the inclusions
$\phi_{t}\mapsto(\phi,\psi)\in X^{*}$ with $\phi_{\tilde{t}}=0$ for all
$\tilde{t}\neq t$, $\psi_{\theta}\mapsto(\phi,\psi)\in X^{*}$ with
$\psi_{\tilde{\theta}}=0$ for all $\tilde{\theta}\neq\theta$ and analogously
for $Y^{*}$ (it is straightforward to see that any continuous linear
functional on $X^{*}$ or $Y^{*}$ is an element of $X$ or $Y$). Note that above
we slightly changed our notation compared to the previous section in that $f$
here only denotes the first part of $Ky^{\dagger}$. This change has no
consequence for the validity of the results and arguments, hence we decided
not to complicate the notation of the previous section further by directly
including this from the start. This choice of fidelity has Legendre–Fenchel
conjugate
$F_{f}^{*}(w_{1},w_{2})=\sum_{t\in{\mathcal{T}}}\tfrac{1}{2}\|(w_{1})_{t}\|_{H^{*}}^{2}+\langle(w_{1})_{t},f_{t}\rangle$,
thus

$F_{f}^{*}(2\alpha w)+F_{f}^{*}(-2\alpha
w)=4\alpha^{2}\sum_{t\in{\mathcal{T}}}\|(w_{1})_{t}\|_{H^{*}}^{2}=:4\alpha^{2}|w_{1}|^{2}.$
(21)

For easier reference let us also recall the following subgradients,

$\displaystyle w\in\partial\|\cdot\|_{+}(\nu)\quad$
$\displaystyle\Leftrightarrow\quad w\leq 1\text{ everywhere, }w=1\text{ on
}\operatorname{supp}\nu,$ $\displaystyle w\in\partial\iota_{+}(\nu)\quad$
$\displaystyle\Leftrightarrow\quad w\leq 0\text{ everywhere, }w=0\text{ on
}\operatorname{supp}\nu.$

We now begin with estimating the reconstruction properties for those snapshots
that are reconstructible.

######  Theorem 77 (Error estimates for good snapshots).

Let $t\in{\mathcal{T}}$ be such that $u_{t}^{\dagger}$ is
$(\kappa,\mu,R)$-reconstructible via $v^{\dagger},v^{s}$ from measurements
with $\mathrm{Ob}_{t}$. Then the solution $(u,\gamma)$ to
$\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ with $\alpha=\sqrt{\delta}$
satisfies

$\displaystyle W_{2,R}^{2}(u_{t},u_{t}^{\dagger})$
$\displaystyle\leq\frac{12\\!+\\!7\max\\{\kappa,\mu\\}R^{2}}{4\kappa}\left(1+4|v^{\dagger}|^{2}+\max_{s\in\\{-1,1\\}^{N}}4|v^{s}|^{2}\right)\sqrt{\delta}.$

###### Proof.

We have $u_{t}^{\dagger}=\sum_{i=1}^{N}m_{i}\delta_{x_{i}+tv_{i}}$. Set
$s=(\operatorname{sgn}((u_{t}-u_{t}^{\dagger})(B_{R}(\\{x_{i}+tv_{i}\\}))))_{i=1,\ldots,N}$
and let $v^{\dagger}$ and $v^{s}$ be the dual variables from Definition 74
(with $\nu^{\dagger}=u_{t}^{\dagger}$). In order to apply Remark 72 we split
$y^{\delta}=(u,\gamma)$ into $y^{\delta}_{1}=u_{t}$ and
$y^{\delta}_{2}=((u_{\tilde{t}})_{\tilde{t}\neq t},\gamma)$ and consider the
regularizer

$G(u,\gamma)=\|u_{t}\|_{+}+\iota_{+}((u_{\tilde{t}})_{\tilde{t}\neq
t},\gamma).$

Then the choice $w^{\dagger}={\tilde{v}^{\dagger}\choose 0}$,
$w={\tilde{v}\choose 0}$ satisfies the conditions of Remark 72, where
$\tilde{v}^{\dagger}_{t}=v^{\dagger}$, $\tilde{v}_{t}=v^{s}$ and
$\tilde{v}^{\dagger}_{\tilde{t}}=\tilde{v}_{\tilde{t}}=0$ for all
$\tilde{t}\neq t$. Furthermore, $(K^{*}w)_{2}=0$ so that the result follows
from 71. ∎

Next, we use the just constructed dual variables to obtain dual variables for
error estimates on $\gamma$. Here we will also obtain an unbalanced
Wasserstein bound; its maximum transport radius $R$ will depend on how close
the configuration is to producing a coincidence or a ghost particle in the
following sense.

######  Definition 78 ($\Delta$-coincidence and $\Delta$-ghost particle).

Let
$\nu=\sum_{i=1}^{N}m_{i}\delta_{(x_{i},v_{i})}\in\mathcal{M}_{+}(\mathbb{R}^{n}\times\mathbb{R}^{n})$
represent a set of particles and ${\mathcal{T}^{\prime}}\subset\mathbb{R}$ a
set of times.

  1. 1.

Configuration $\nu$ contains a _$\Delta$ -coincidence_ with respect to
${\mathcal{T}^{\prime}}$ if two distinct particles $i\neq j$ come
$\Delta$-close at some time $t\in{\mathcal{T}^{\prime}}$, that is,
$\operatorname{dist}(x_{j}+tv_{j},x_{i}+tv_{i})\leq\Delta$ .

  2. 2.

Configuration $\nu$ contains a _$\Delta$ -ghost particle_ at
$(x,v)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ with respect to
${\mathcal{T}^{\prime}}$ if at each $t\in{\mathcal{T}^{\prime}}$ there is a
particle $i$ (not the same for all $t\in{\mathcal{T}^{\prime}}$) with
$\operatorname{dist}(x+tv,x_{i}+tv_{i})\leq\Delta$.

A $\Delta$-ghost particle is thus a nonexistent particle which would be
consistent with all snapshots if the finest measurement resolution at the
different time points were $\Delta$. Plain ghost particles are $0$-ghost
particles in this sense, plain coincidences are $0$-coincidences.

######  Theorem 79 (Error estimates for $\gamma$).

Let $\theta\in\Theta$ and assume there is
${\mathcal{T}^{\prime}}\subset{\mathcal{T}}$ such that $u_{t}^{\dagger}$ is
$(\hat{\kappa},\hat{\mu},\hat{R})$-reconstructible via
$\hat{v}_{t}^{\dagger},\hat{v}^{s}_{t}$ from measurements with
$\mathrm{Ob}_{t}$ for all $t\in{\mathcal{T}^{\prime}}$. Then the solution
$(u,\gamma)$ to $\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ satisfies

$\displaystyle W_{2,R}^{2}(\gamma_{\theta},\gamma_{\theta}^{\dagger})$
$\displaystyle\leq\frac{12+7\kappa
R^{2}}{4\kappa}\left(1+4\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}_{t}^{\dagger}|^{2}\right)\sqrt{\delta}+\frac{1}{2R^{2}}\max_{t\in{\mathcal{T}^{\prime}}}W_{2,R}^{2}(u_{t},u^{\dagger}_{t}).$

for $\kappa=\sigma\hat{\kappa}$ and
$R=\min\\{\hat{R},\Delta/3,\Delta^{\prime}\\}/\sqrt{1+\max_{t\in{\mathcal{T}^{\prime}}}t^{2}}$,
where $\sigma$ is the smallest eigenvalue of
$\frac{1}{\operatorname{card}{\mathcal{T}^{\prime}}}\sum_{t\in{\mathcal{T}^{\prime}}}{1\choose-t}\otimes{1\choose-t}$
and $\Delta,\Delta^{\prime}\geq 0$ are such that $\gamma_{\theta}^{\dagger}$
contains no $\Delta$-coincidence or $\Delta^{\prime}$-ghost particle with
respect to ${\mathcal{T}^{\prime}}$.

###### Proof.

This time split $y^{\delta}=(u,\gamma)$ into $y^{\delta}_{1}=\gamma_{\theta}$
and $y^{\delta}_{2}=(u,(\gamma_{\tilde{\theta}})_{\tilde{\theta}\neq\theta})$
and consider the regularizer

$G(u,\gamma)=\|\gamma_{\theta}\|_{+}+\iota_{+}(u,(\gamma_{\tilde{\theta}})_{\tilde{\theta}\neq\theta}).$

In order to apply Remark 72 we need to construct appropriate dual variables
$w^{\dagger}$ and $w$. To this end define
$v^{\dagger}\in(H^{*})^{|{\mathcal{T}}|}$ via
$v^{\dagger}_{t}=\hat{v}^{\dagger}_{t}/\operatorname{card}{{\mathcal{T}^{\prime}}}$
for all $t\in{\mathcal{T}^{\prime}}$ and $v^{\dagger}_{t}=0$ else. Furthermore
define $q^{\dagger}\in C(\Xi)^{\Theta\times\Sigma}$ as

$q^{\dagger}_{\theta,t}(x)=(1-\hat{\kappa}\min\\{\bar{R},\operatorname{dist}(x,\operatorname{supp}\mathrm{Rd}_{\theta}u_{t}^{\dagger})\\}^{2})/\operatorname{card}{{\mathcal{T}^{\prime}}}\quad\text{for
}t\in{\mathcal{T}^{\prime}}$

and $q^{\dagger}_{\tilde{\theta},\tilde{t}}=0$ for
$\tilde{t}\notin{\mathcal{T}^{\prime}}$ or $\tilde{\theta}\neq\theta$, where
$\bar{R}=\min\\{\hat{R},\Delta/3,\Delta^{\prime}\\}$. Due to
$\bar{R}\leq\Delta/3$, each $q_{\theta,t}^{\dagger}$ contains
$\operatorname{card}{\mathcal{T}^{\prime}}$ distinct parabolas of width
$2\bar{R}$. Also, let
$s_{i}=\operatorname{sgn}(\gamma_{\theta}-\gamma^{\dagger}_{\theta})(B_{R}(\\{(x_{i}\cdot\theta,v_{i}\cdot\theta)\\}))$,
$i=1,\ldots,N$, and define $q\in C(\Xi)^{\Theta\times\Sigma}$ as
$q_{\tilde{\theta},\tilde{t}}=0$ for $\tilde{t}\notin{\mathcal{T}^{\prime}}$
or $\tilde{\theta}\neq\theta$ and, for $t\in{\mathcal{T}^{\prime}}$, as

$q_{\theta,t}(x)=s_{i}/\operatorname{card}{{\mathcal{T}^{\prime}}}\quad\text{if
}x\in B_{\bar{R}}(\\{\theta\cdot(x_{i}+tv_{i})\\}),$

continuously extended onto the rest of $\Xi$ with absolute value no larger
than $1/\operatorname{card}{{\mathcal{T}^{\prime}}}$ (which again is possible
due to $\bar{R}\leq\Delta/3$). Now $w^{\dagger}={v^{\dagger}\choose
q^{\dagger}}$ and $w={0\choose q}$ satisfy the conditions of Remark 72 with
$\mu=0$. Indeed, due to $\bar{R}\leq\hat{R}$ we have
$-\mathrm{Rd}_{\theta}^{*}q^{\dagger}_{\theta,t}\leq\mathrm{Ob}_{t}^{*}v^{\dagger}_{t}$
for all $t\in{\mathcal{T}^{\prime}}$ by construction (with equality on the
support of $u_{t}^{\dagger}$) so that
$\underline{\mathrm{Ob}}^{*}v^{\dagger}+\underline{\mathrm{Rd}}^{*}q^{\dagger}\geq
0$, thus

$-(\underline{\mathrm{Ob}}^{*}v^{\dagger}+\underline{\mathrm{Rd}}^{*}q^{\dagger})\in\partial\iota_{+}.$
(22)

Furthermore, by construction, in the neighbourhood

$\mathcal{N}_{i}=\bigcap_{t\in{\mathcal{T}^{\prime}}}\textstyle\left\\{{x\choose
v}\in\mathbb{R}^{2}\,\middle|\,-\bar{R}\leq{1\choose t}\cdot\left[{x\choose
v}-{x_{i}\cdot\theta\choose v_{i}\cdot\theta}\right]\leq\bar{R}\right\\}$

of $(\theta\cdot x_{i},\theta\cdot v_{i})$ we have

$((\underline{\mathrm{Mv}^{1}})^{*}q^{\dagger})_{\theta}(x,v)=\sum_{t\in{\mathcal{T}^{\prime}}}\mathrm{Mv}^{1}_{t}q_{\theta,t}^{\dagger}(x,v)=1-\hat{\kappa}(x-\theta\cdot
x_{i},v-\theta\cdot v_{i})Q{x-\theta\cdot x_{i}\choose v-\theta\cdot v_{i}}$

for the positive definite matrix
$Q=\frac{1}{\operatorname{card}{\mathcal{T}^{\prime}}}\sum_{t\in{\mathcal{T}^{\prime}}}{1\choose-t}\otimes{1\choose-t}$
so that from $\kappa\leq\sigma\hat{\kappa}$ we obtain

$((\underline{\mathrm{Mv}^{1}})^{*}q^{\dagger})_{\theta}(x,v)\leq
1-\kappa\operatorname{dist}((x,v),\operatorname{supp}{\gamma_{\theta}^{\dagger}})^{2}$

in the neighbourhood $\bigcup_{i=1}^{N}\mathcal{N}_{i}$ of
$\operatorname{supp}\gamma_{\theta}^{\dagger}$. In particular,
$((\underline{\mathrm{Mv}^{1}})^{*}q^{\dagger})_{\theta}\in\partial\|\cdot\|_{+}(\gamma^{\dagger}_{\theta})$
(note that it is everywhere bounded by one). Furthermore, due to
$\bar{R}\leq\Delta^{\prime}$ there are no regions other than the
$\mathcal{N}_{i}$ in which
$((\underline{\mathrm{Mv}^{1}})^{*}q^{\dagger})_{\theta}=\sum_{t\in{\mathcal{T}^{\prime}}}\mathrm{Mv}^{1}_{t}q_{\theta,t}^{\dagger}$
can be greater than $1-\hat{\kappa}\bar{R}^{2}$. Since
$B_{R}(\\{x_{i}\\})\subset\mathcal{N}_{i}$ for the choice
$R=\bar{R}/\sqrt{1+\max_{t\in{\mathcal{T}^{\prime}}}t^{2}}$ we thus actually
have

$((\underline{\mathrm{Mv}^{1}})^{*}q^{\dagger})_{\theta}(x,v)\leq
1-\kappa\min\\{R,\operatorname{dist}((x,v),\operatorname{supp}{\gamma^{\dagger}_{\theta}})\\}^{2}.$
(23)

Summarizing, $w^{\dagger}$ satisfies the conditions of Remark 72. Finally, by
the same argument,

$\displaystyle\textstyle((\underline{\mathrm{Mv}^{1}})^{*}q)_{\theta}\in\partial\|\cdot\|_{\mathcal{M}}\left(\sum_{(x,v)\in\operatorname{supp}\gamma_{\theta}^{\dagger}}(\gamma_{\theta}-\gamma^{\dagger}_{\theta})(B_{R}(\\{(x,v)\\}))\delta_{(x,v)}\right)$
$\displaystyle\text{and
}((\underline{\mathrm{Mv}^{1}})^{*}q)_{\theta}(x,v)=s_{i}\quad\text{for
}(x,v)\in B_{R}(\\{(x_{i}\cdot\theta,v_{i}\cdot\theta)\\})$

so that also $w$ satisfies the conditions of Remark 72 and we obtain the
estimate from 71 with $\mu=0$.

To conclude, we note that by construction
$1=|\operatorname{card}{{\mathcal{T}^{\prime}}}(\underline{\mathrm{Rd}}^{*}q)_{t}(x)|$
for $x\in B_{\bar{R}}(\operatorname{supp}u_{t}^{\dagger})$ and any
$t\in{\mathcal{T}^{\prime}}$. Thus,

$\langle(K^{*}w)_{2},y^{\delta}_{2}-y^{\dagger}_{2}\rangle=\sum_{t\in{\mathcal{T}^{\prime}}}\langle(\underline{\mathrm{Rd}}^{*}q)_{t},u_{t}-u^{\dagger}_{t}\rangle\leq\frac{1}{\operatorname{card}{{\mathcal{T}^{\prime}}}}\sum_{t\in{\mathcal{T}^{\prime}}}\bigg{(}|u_{t}-u^{\dagger}_{t}|(\Omega\setminus
B_{R}(\operatorname{supp}u_{t}^{\dagger}))\\\
+\sum_{x\in\operatorname{supp}u_{t}^{\dagger}}|(u_{t}-u_{t}^{\dagger})(B_{R}(\\{x\\}))|\bigg{)}\leq\frac{1}{2R^{2}\operatorname{card}{{\mathcal{T}^{\prime}}}}\sum_{t\in{\mathcal{T}^{\prime}}}W_{2,R}^{2}(u_{t},u^{\dagger}_{t}),$

where we exploited that in the unbalanced Wasserstein divergence $W_{2,R}^{2}$
mass is never transported further than $R$. ∎

######  Remark 80 (Signed particles).

For the above error estimates it is essential that all particles have
nonnegative mass. We exploited this explicitly by the fact that each time we
can choose our regularizer to be the total variation of exactly that variable
for which we would like to derive an error estimate. Furthermore, we exploit
this in the fact that we only require
$\underline{\mathrm{Ob}}^{*}v^{\dagger}+\underline{\mathrm{Rd}}^{*}q^{\dagger}\geq
0$ in the above construction of dual variables. Had we considered signed
measures, then this condition would turn into
$\underline{\mathrm{Ob}}^{*}v^{\dagger}+\underline{\mathrm{Rd}}^{*}q^{\dagger}=0$.
However, the range of the predual observation operator
$\underline{\mathrm{Ob}}^{*}$ and the predual Radon transform
$\underline{\mathrm{Rd}}^{*}$ will typically have a very small intersection
which leaves too little freedom to construct appropriate dual variables. For
instance, if we want $q^{\dagger}_{\theta,t}$ to be nonzero only for one
$\theta$ and if $\mathrm{Ob}_{t}$ measures the Fourier coefficients at a
finite number of integer frequencies, then for irrational $\theta$ the only
possible $q^{\dagger}_{\theta,t}$ is a constant.

Finally, we derive an error estimate for all those snapshots that are not
reconstructible from $\mathrm{Ob}_{t}$, for instance because their particles
are too close together. Their reconstruction works nevertheless based solely
on those snapshots that _are_ reconstructible from $\mathrm{Ob}_{t}$. The
construction of corresponding dual variables is analogous to the construction
in Theorem 79, only the roles of $u$ and $\gamma$ as well as of
${\mathcal{T}^{\prime}}$ and $\Theta^{\prime}$ swap. This also means that we
require an analogue of $\Delta$-coincidences and $\Delta$-ghost particles.

######  Definition 81 ($\Delta$-coincidence and $\Delta$-ghost particle with
respect to directions).

Let $u=\sum_{i=1}^{N}m_{i}\delta_{x_{i}}\in\mathcal{M}_{+}(\mathbb{R}^{d})$
represent a set of particles and $\Theta^{\prime}\subset S^{d-1}$ a set of
directions.

  1. 1.

Configuration $u$ contains a _$\Delta$ -coincidence_ with respect to
$\Theta^{\prime}$ if two distinct particles $i\neq j$ seem $\Delta$-close in
some projection $\theta\in\Theta^{\prime}$, that is,
$\operatorname{dist}(\theta\cdot x_{j},\theta\cdot x_{i})\leq\Delta$ .

  2. 2.

Configuration $u$ contains a _$\Delta$ -ghost particle_ at
$x\in\mathbb{R}^{d}$ with respect to $\Theta^{\prime}$ if for each
$\theta\in\Theta^{\prime}$ there is a particle $i$ (not the same for all
$\theta\in\Theta^{\prime}$) with $\operatorname{dist}(\theta\cdot
x,\theta\cdot x_{i})\leq\Delta$.

######  Theorem 82 (Error estimates for bad snapshots).

Assume there is ${\mathcal{T}^{\prime}}\subset{\mathcal{T}}$ such that
$u_{t}^{\dagger}$ is $(\hat{\kappa},\hat{\mu},\hat{R})$-reconstructible via
$\hat{v}_{t}^{\dagger},\hat{v}^{s}_{t}$ from measurements with
$\mathrm{Ob}_{t}$ for all $t\in{\mathcal{T}^{\prime}}$. Let
$t\in\Sigma\setminus{\mathcal{T}^{\prime}}$ and pick any finite
$\Theta^{\prime}\subset\Theta$. Then the solution $(u,\gamma)$ to
$\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ satisfies

$\displaystyle W_{2,\bar{R}}^{2}(u_{t},u_{t}^{\dagger})$
$\displaystyle\leq\frac{12+7\bar{\kappa}\bar{R}^{2}}{4\bar{\kappa}}\left(1+4\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}_{t}^{\dagger}|^{2}\right)\sqrt{\delta}+\frac{1}{2\bar{R}^{2}}\max_{\theta\in\Theta^{\prime}}W_{2,\bar{R}}^{2}(\gamma_{\theta},\gamma_{\theta}^{\dagger})$

for $\bar{\kappa}=\sigma\kappa/(1+t^{2})$ and
$\bar{R}=\min\\{(1+t^{2})R,\Delta/(3\sqrt{1+t^{2}}),\Delta^{\prime}\\}$, where
$\sigma$ is the smallest eigenvalue of
$\frac{1}{\operatorname{card}\Theta^{\prime}}\sum_{\theta\in\Theta^{\prime}}\theta\otimes\theta$,
$\Delta,\Delta^{\prime}\geq 0$ are such that $u^{\dagger}_{t}$ contains no
$\Delta$-coincidence or $\Delta^{\prime}$-ghost particle with respect to
$\Theta^{\prime}$, and $\kappa,R\geq 0$ are the minimum values from the
statement of Theorem 79, applied to all $\theta\in\Theta^{\prime}$.

###### Proof.

This time we consider the splitting of $y^{\delta}=(u,\gamma)$ into
$y^{\delta}_{1}=u_{t}$ and $y^{\delta}_{2}=((u_{\tilde{t}})_{\tilde{t}\neq
t},\gamma)$ as well as the regularizer choice

$G(u,\gamma)=\|u_{t}\|_{+}+\iota_{+}((u_{\tilde{t}})_{\tilde{t}\neq
t},\gamma).$

For $\tilde{\kappa}$ to be determined later, define $q^{\dagger}\in
C(\Xi)^{\Theta\times\Sigma}$ as

$q^{\dagger}_{\theta,t}(x)=-(1-\tilde{\kappa}\min\\{\bar{R},\operatorname{dist}(x,\operatorname{supp}\mathrm{Mv}^{1}_{t}\gamma^{\dagger}_{\theta})\\}^{2})/\operatorname{card}{\Theta^{\prime}}\quad\text{for
}\theta\in\Theta^{\prime}$

and $q^{\dagger}_{\tilde{\theta},\tilde{t}}=0$ for $\tilde{t}\neq t$ or
$\tilde{\theta}\notin\Theta^{\prime}$. Due to $\bar{R}\leq\Delta/3$, each
$q_{\theta,t}^{\dagger}$ contains $\operatorname{card}{\mathcal{T}^{\prime}}$
distinct parabolas of width $2\bar{R}$. Also, let
$s_{i}=\operatorname{sgn}(u_{t}-u_{t}^{\dagger})(B_{\bar{R}}(\\{x_{i}+tv_{i}\\}))$,
$i=1,\ldots,N$, and define $q\in C(\Xi)^{\Theta\times\Sigma}$ as
$q_{\tilde{\theta},\tilde{t}}=0$ for $\tilde{t}\neq t$ or
$\tilde{\theta}\notin\Theta^{\prime}$ and, for $\theta\in\Theta^{\prime}$, as

$q_{\theta,t}(x)=-s_{i}/\operatorname{card}{\Theta^{\prime}}\quad\text{if
}x\in B_{\Delta/3}(\theta\cdot(x_{i}+tv_{i})),$

continuously extended onto the rest of $\Xi$ with absolute value no larger
than $1/\operatorname{card}{\Theta^{\prime}}$. Let further $w^{\theta}\in
Y^{*}$ for $\theta\in\Theta^{\prime}$ denote the first dual variable
constructed in the proof of Theorem 79 (there denoted $w^{\dagger}$), and
define

$w^{\dagger}={0\choose
q^{\dagger}}+\frac{1}{\operatorname{card}{\Theta^{\prime}}}\sum_{\theta\in\Theta^{\prime}}w^{\theta},\quad
w={0\choose q}.$

We now pick $\tilde{\kappa}$ small enough such that the conditions of Remark
72 are satisfied with $\mu=0$. First note that
$((0,(\underline{\mathrm{Mv}^{1}})^{*})w^{\dagger})_{\theta}$ is zero for all
$\theta\notin\Theta^{\prime}$. Now recall from (23) that for
$\theta\in\Theta^{\prime}$ we have
$(0,(\mathrm{Mv}^{1})^{*})w^{\theta}(x,v)\leq
1-\kappa\min\\{R,\operatorname{dist}((x,v),\operatorname{supp}\gamma_{\theta}^{\dagger})\\}^{2}$.
Thus, for the choice $\tilde{\kappa}=\kappa/(1+t^{2})$ and due to
$\bar{R}\leq(1+t^{2})R$ and $\bar{R}\leq\Delta/3$ we have
$(0,(\underline{\mathrm{Mv}^{1}})^{*})w^{\dagger}\leq 0$ everywhere with
equality on $\operatorname{supp}\gamma^{\dagger}_{\theta}$. Next recall from
(22) that $w^{\theta}$ satisfies
$(\underline{\mathrm{Ob}}^{*},\underline{\mathrm{Rd}}^{*})w^{\theta}\geq 0$
(with equality on the support of $u^{\dagger}$), thus by construction we have
$(-(\underline{\mathrm{Ob}}^{*},\underline{\mathrm{Rd}}^{*})w^{\dagger})_{\tilde{t}}\leq
0$ for $\tilde{t}\neq t$ with equality on
$\operatorname{supp}u_{\tilde{t}}^{\dagger}$. Furthermore,
$(-(\underline{\mathrm{Ob}}^{*},\underline{\mathrm{Rd}}^{*})w^{\dagger})_{t}=-(\underline{\mathrm{Rd}}^{*}q^{\dagger})_{t}\leq
1$ with equality on $\operatorname{supp}u_{t}$. Even more, by construction, in
the neighbourhood

$\mathcal{N}_{i}=\bigcap_{\theta\in\Theta^{\prime}}\textstyle\left\\{x\in\mathbb{R}^{d}\,\middle|\,-\bar{R}\leq\theta\cdot\left[x-(x_{i}+tv_{i})\right]\leq\bar{R}\right\\}$

of $x_{i}+tv_{i}$ we have

$-(\underline{\mathrm{Rd}}^{*}q^{\dagger})_{t}=-\sum_{\theta\in\Theta}\mathrm{Rd}_{\theta}^{*}q^{\dagger}_{\theta,t}=1-\tilde{\kappa}(x-x_{i}-tv_{i})^{T}Q(x-x_{i}-tv_{i})$

for the symmetric positive semi-definite
$Q=\frac{1}{\operatorname{card}\Theta^{\prime}}\sum_{\theta\in\Theta^{\prime}}\theta\otimes\theta$
so that from $\bar{\kappa}\leq\sigma\tilde{\kappa}$ we obtain

$-(\underline{\mathrm{Rd}}^{*}q^{\dagger})_{t}(x)\leq
1-\bar{\kappa}\min\\{\bar{R},\operatorname{dist}(x,\operatorname{supp}u_{t})\\}^{2}$

(here we exploited $\bar{R}\leq\Delta^{\prime}$). Summarizing, $w^{\dagger}$
fulfills the conditions from Remark 72. By the same argument,

$\displaystyle\textstyle-(\underline{\mathrm{Rd}}^{*}q)_{t}\in\partial\|\cdot\|_{\mathcal{M}}\left(\sum_{x\in\operatorname{supp}u_{t}^{\dagger}}(u_{t}-u_{t}^{\dagger})(B_{\bar{R}}(x))\delta_{x}\right)$
$\displaystyle\text{and }-(\underline{\mathrm{Rd}}^{*}q)_{t}(x)=s_{i}\text{
for }x\in B_{\bar{R}}(\\{x_{i}+tv_{i}\\})$

so that also $w$ satisfies the conditions of Remark 72 and we obtain the
estimate from 71 with $\mu=0$ (note that
$w^{\dagger}_{1}=v^{\dagger}=\frac{1}{\operatorname{card}{\mathcal{T}^{\prime}}}\hat{v}^{\dagger}$
using the notation from the proof of Theorem 79).

To conclude, we note that, due to $\bar{R}\leq\Delta/(3\sqrt{1+t^{2}})$, by
construction we have
$1=|\operatorname{card}{\Theta^{\prime}}((\underline{\mathrm{Mv}^{1}})^{*}q)_{\theta}(x,v)|$
for $(x,v)\in B_{\bar{R}}(\operatorname{supp}\gamma_{\theta}^{\dagger})$ and
any $\theta\in\Theta^{\prime}$. Thus,

$\langle(K^{*}w)_{2},y^{\delta}_{2}-y^{\dagger}_{2}\rangle=-\sum_{\theta\in\Theta^{\prime}}\langle(\mathrm{Mv}^{1})^{*}q_{\theta,t},\gamma_{\theta}-\gamma_{\theta}^{\dagger}\rangle\leq\frac{1}{\operatorname{card}{\Theta^{\prime}}}\sum_{\theta\in\Theta^{\prime}}\bigg{(}|\gamma_{\theta}-\gamma_{\theta}^{\dagger}|(\Gamma\setminus
B_{\bar{R}}(\operatorname{supp}\gamma_{\theta}^{\dagger}))\\\
+\sum_{(x,v)\in\operatorname{supp}\gamma_{\theta}^{\dagger}}|(\gamma_{\theta}-\gamma_{\theta}^{\dagger})(B_{\bar{R}}(\\{(x,v)\\}))|\bigg{)}\leq\frac{1}{2\bar{R}^{2}\operatorname{card}{\Theta^{\prime}}}\sum_{\theta\in\Theta^{\prime}}W_{2,\bar{R}}^{2}(\gamma_{\theta},\gamma_{\theta}^{\dagger}),$

where again we exploited that in the unbalanced Wasserstein divergence
$W_{2,\bar{R}}^{2}$ mass is never transported further than $\bar{R}$. ∎

Theorem 73 for $\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ now follows from
combining Theorems 77, 79 and 82.

###  4.4 Construction of dual variables for $\mathrm{P}_{\alpha}(f)$

We now modify the error analysis from the previous Section for reconstruction
via $\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ to an error analysis for
the model $\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$. Since in this model the
slices $\gamma_{\theta}$ are only defined for $\mathcal{H}_{\Theta}$-almost
every $\theta\in\Theta$, our error estimate will also just hold for
$\mathcal{H}_{\Theta}$-almost every slice. This time we employ the operators

$\displaystyle\overline{\underline{\mathrm{Rd}}}$
$\displaystyle:\mathcal{M}(\Omega)^{\Sigma}\to\mathcal{M}(\Theta\times\Xi)^{\Sigma},$
$\displaystyle\overline{\underline{\mathrm{Rd}}}u$
$\displaystyle=(\mathrm{Rd}u_{t})_{t\in\Sigma},$
$\displaystyle\overline{\underline{\mathrm{Mv}^{1}}}$
$\displaystyle:\mathcal{M}(\Theta\times\Gamma)\to\mathcal{M}(\Theta\times\Xi)^{\Sigma},$
$\displaystyle\overline{\underline{\mathrm{Mv}^{1}}}\gamma$
$\displaystyle=(\overline{\mathrm{Mv}^{1}_{t}}\gamma)_{t\in\Sigma},$

and choose

$\displaystyle
y^{\dagger}\equiv(u^{\dagger},\gamma^{\dagger}),\,y^{\delta}\equiv(u^{\delta},\gamma^{\delta})\in\mathcal{M}(\Omega)^{|{\mathcal{T}}|}\times\mathcal{M}(\Theta\times\Gamma)\equiv
X,$ $\displaystyle
K\equiv\left(\begin{smallmatrix}\underline{\mathrm{Ob}}&0\\\
\overline{\underline{\mathrm{Rd}}}&-\overline{\underline{\mathrm{Mv}^{1}}}\end{smallmatrix}\right):X\to
H^{|{\mathcal{T}}|}\times\mathcal{M}(\Theta\times\Xi)^{\Sigma}\equiv Y,$
$\displaystyle
F_{f}(a,b)=\tfrac{1}{2}\sum_{t\in{\mathcal{T}}}\|a_{t}-f_{t}\|_{H}^{2}+\iota_{0}(b),\,G(u,\gamma)=\iota_{+}(u,\gamma)+\ast,$

where $\ast$ will either be $\|u_{t}\|_{\mathcal{M}}$ for some
$t\in{\mathcal{T}}$ or a weighted total variation of $\gamma$,

$\|\gamma\|_{h}=\int_{\Theta\times\Gamma}h(\theta)\>\mathrm{d}|\gamma|(\theta,(x,v))=\int_{\Theta}\|\gamma_{\theta}\|_{\mathcal{M}}h(\theta)\>\mathrm{d}\mathcal{H}_{\Theta}$

for some fixed weight $h:\Theta\to[0,\infty)$ with
$\int_{\Theta}h\>\mathrm{d}\mathcal{H}_{\Theta}=1$ (the rightmost expression
is well-defined for all $\gamma$ admissible in $\mathrm{P}_{\alpha}(f)$ by
Lemma 39). As before, all these choices for $\ast$ are equivalent.

Essentially, the necessary changes to the analysis are the following, where we
now work backwards through the previous Sections.

  * •

For the estimates we needed to construct dual variables $w^{\dagger}$ (and
similarly $w$) such that $K^{*}w^{\dagger}\in\partial G(y^{\dagger})$. Since
in the previous Section the regularizer acted separately on each $u_{t}$ and
$\gamma_{\theta}$ for all $t\in{\mathcal{T}}$ and $\theta\in\Theta$, this
amounted to constructing continuous functions $q_{\theta,t}^{\dagger}$
separately for each $t,\theta$. Now, however, that our variable $\gamma$ stems
from $\mathcal{M}_{+}(\Theta\times\Gamma)$, these dual variables will have to
be continuous in $\theta$, thus we need new versions of Theorems 79 and 82
(while the construction from Theorem 77 stays unchanged apart from replacing
the space $\mathcal{M}_{+}(\Gamma)^{\Theta}$ by
$\mathcal{M}_{+}(\Theta\times\Gamma)$ everywhere). Roughly, the strategy is to
smooth $q_{\theta,t}^{\dagger}$ a little out in $\theta$.

  * •

So far we only had to estimate the mass distribution of approximations to
discrete measures, for which we used Bregman distances with respect to the
total variation. Now however, when we aim for estimating the mass distribution
in $\gamma$, the corresponding ground truth measure $\gamma^{\dagger}$ is not
discrete, but rather a measure concentrated on codimension-2 manifolds (each
$\theta$-slice $\gamma_{\theta}^{\dagger}$ is discrete, however, these are
only well-defined $\mathcal{H}_{\Theta}$-almost everywhere so that our earlier
estimates do not carry over). Hence we will require a new version of Theorem
70, where weighted averages of mass distributions of $\gamma_{\theta}$ are
estimated based on Bregman distances with respect to weighted total
variations.

  * •

An additional argument is required to pass from mass distribution estimates of
weighted averages of $\gamma_{\theta}$ to estimates for
$\mathcal{H}_{\Theta}$-almost all $\theta$.

We begin with a new version of Theorem 70, for simplicity directly adapted to
our setting of estimating the mass distribution in $\gamma$.

######  Theorem 83 (Weighted mass distribution from Bregman distance).

Let $h:\Theta\to[0,\infty)$ be continuous with
$\int_{\Theta}h\>\mathrm{d}\mathcal{H}_{\Theta}=1$. Assume
$\gamma^{\dagger}=\mathrm{Rj}\lambda^{\dagger}\in\mathcal{M}_{+}(\Theta\times\Gamma)$
for a discrete measure $\lambda^{\dagger}$ with particles
$(x_{1},v_{1}),\ldots,(x_{N},v_{N})$, and let
$v^{\dagger}\in\partial\|\cdot\|_{h}(\gamma^{\dagger})\subset
C(\Theta\times\Gamma)$. If $v^{\dagger}$ satisfies

$v^{\dagger}(\theta,(x,v))\leq
h(\theta)(1-\kappa\min\\{R,\operatorname{dist}((x,v),\operatorname{supp}\gamma_{\theta}^{\dagger})\\}^{2})$

for some $\kappa,R>0$ with $2R$ smaller than the minimum distance between the
points in the support of $\gamma_{\theta}^{\dagger}$ for
$\theta\in\operatorname{supp}h$, then for any
$\gamma\in\mathcal{M}_{+}(\Theta\times\Gamma)$ of the form
$\gamma=\mathcal{H}_{\Theta}{\otimes}_{\theta}\gamma_{\theta}$ we have

$\displaystyle\int_{\Theta}h(\theta)|\gamma_{\theta}|(\Gamma\setminus
B_{R}(\operatorname{supp}\gamma_{\theta}^{\dagger}))\>\mathrm{d}\mathcal{H}_{\Theta}$
$\displaystyle\leq\frac{1}{\kappa
R^{2}}D^{\|\cdot\|_{h}}_{v^{\dagger}}(\gamma,\gamma^{\dagger}),$
$\displaystyle\sum_{i=1}^{N}\int_{\Theta}h(\theta)\int_{B_{R}(\\{p_{i}^{\theta}\\})}\operatorname{dist}(p,p_{i}^{\theta})^{2}\>\mathrm{d}|\gamma_{\theta}|(p)\>\mathrm{d}\mathcal{H}_{\Theta}$
$\displaystyle\leq\frac{1}{\kappa}D^{\|\cdot\|_{h}}_{v^{\dagger}}(\gamma,\gamma^{\dagger}),$

where we abbreviated $p_{i}^{\theta}=(\theta\cdot x_{i},\theta\cdot v_{i})$.
Furthermore, let $v\in\partial\|\cdot\|_{h}(\bar{\gamma})\subset
C(\Theta\times\Gamma)$ for

$\bar{\gamma}=\sum_{i=1}^{N}s_{i}\mathcal{H}^{d-1}{\mbox{\LARGE$\llcorner$}}\\{p_{i}^{\theta}\,|\,\theta\in\Theta\\}\quad\text{
with
}s_{i}=\int_{\Theta}h(\theta)(\gamma_{\theta}-\gamma^{\dagger}_{\theta})(B_{R}(\\{p_{i}^{\theta}\\}))\>\mathrm{d}\mathcal{H}_{\Theta}.$

If $v$ satisfies

$s_{i}v(\theta,p)\geq
h(\theta)(1-\mu\operatorname{dist}(p,p_{i}^{\theta})^{2})\quad\text{for all
}p\in B_{R}(\\{p_{i}^{\theta}\\}),\,i=1,\ldots,N,$

for some $\mu>0$, then we additionally have

$\sum_{i=1}^{N}\left|\int_{\Theta}h(\theta)(\gamma_{\theta}-\gamma^{\dagger}_{\theta})(B_{R}(\\{p_{i}^{\theta}\\}))\>\mathrm{d}\mathcal{H}_{\Theta}\right|\leq\frac{1+\mu
R^{2}}{\kappa
R^{2}}D^{\|\cdot\|_{h}}_{v^{\dagger}}(\gamma,\gamma^{\dagger})+\langle
v,\gamma-\gamma^{\dagger}\rangle.$

###### Proof.

Abbreviate $B_{\theta}^{c}=\Gamma\setminus
B_{R}(\operatorname{supp}\gamma_{\theta}^{\dagger})$, then the first two
inequalities follow from

$\int_{\Theta}h(\theta)\kappa
R^{2}|\gamma_{\theta}|(B_{\theta}^{c})+h(\theta)\kappa\sum_{i=1}^{N}\int_{B_{R}(\\{p_{i}^{\theta}\\})}\operatorname{dist}(p,p_{i}^{\theta})^{2}\>\mathrm{d}|\gamma_{\theta}|p\>\mathrm{d}\mathcal{H}_{\Theta}(\theta)\\\
\leq\int_{\Theta}\int_{B_{\theta}^{c}}h(\theta)-v^{\dagger}(\theta,p)\>\mathrm{d}\gamma_{\theta}(p)+\sum_{i=1}^{N}\int_{B_{R}(\\{p_{i}^{\theta}\\})}h(\theta)-v^{\dagger}(\theta,p)\>\mathrm{d}\gamma_{\theta}(p)\>\mathrm{d}\mathcal{H}_{\Theta}(\theta)\\\
=\int_{\Theta\times\Gamma}(h(\theta)-v^{\dagger}(\theta,p))\>\mathrm{d}\gamma(\theta,p)=\|\gamma\|_{h}-\langle
v^{\dagger},\gamma\rangle=\|\gamma\|_{h}-\|\gamma^{\dagger}\|_{h}-\langle
v^{\dagger},\gamma-\gamma^{\dagger}\rangle=D^{\|\cdot\|_{h}}_{v^{\dagger}}(\gamma,\gamma^{\dagger}).$

The third inequality is obtained as follows. We have

$\sum_{i=1}^{N}\left|\int_{\Theta}h(\theta)(\gamma_{\theta}-\gamma^{\dagger}_{\theta})(B_{R}(\\{p_{i}^{\theta}\\}))\>\mathrm{d}\mathcal{H}_{\Theta}\right|\\\
=\int_{\\{(\theta,p)\,|\,\operatorname{dist}(p,\operatorname{supp}\gamma_{\theta}^{\dagger})<R\\}}v\>\mathrm{d}(\gamma-\gamma^{\dagger})+\sum_{i=1}^{N}\int_{\Theta}\int_{B_{R}(\\{p_{i}^{\theta}\\})}h(\theta)s_{i}-v\>\mathrm{d}(\gamma_{\theta}-\gamma^{\dagger}_{\theta})\>\mathrm{d}\mathcal{H}_{\Theta},$

where the first summand can be estimated as

$\int_{\\{(\theta,p)\,|\,\operatorname{dist}(p,\operatorname{supp}\gamma_{\theta}^{\dagger})<R\\}}v\>\mathrm{d}(\gamma-\gamma^{\dagger})=\langle
v,\gamma-\gamma^{\dagger}\rangle-\int_{\Theta}\int_{B_{\theta}^{c}}v\>\mathrm{d}(\gamma_{\theta}-\gamma_{\theta}^{\dagger})\>\mathrm{d}\mathcal{H}_{\Theta}(\theta)\\\
\leq\langle
v,\gamma-\gamma^{\dagger}\rangle+\int_{\Theta}h(\theta)|\gamma_{\theta}-\gamma_{\theta}^{\dagger}|(B_{\theta}^{c})\>\mathrm{d}\mathcal{H}_{\Theta}\leq\langle
v,\gamma-\gamma^{\dagger}\rangle+\frac{1}{\kappa
R^{2}}D^{\|\cdot\|_{h}}_{v^{\dagger}}(\gamma,\gamma^{\dagger}),$

while the second term satisfies

$\sum_{i=1}^{N}\int_{\Theta}\int_{B_{R}(\\{p_{i}^{\theta}\\})}h(\theta)s_{i}-v\>\mathrm{d}(\gamma_{\theta}-\gamma^{\dagger}_{\theta})\>\mathrm{d}\mathcal{H}_{\Theta}\\\
\leq\sum_{i=1}^{N}\int_{\Theta}h(\theta)\int_{B_{R}(\\{p_{i}^{\theta}\\})}\mu\operatorname{dist}(p,p_{i}^{\theta})^{2}\>\mathrm{d}|\gamma_{\theta}|(p)\>\mathrm{d}\mathcal{H}_{\Theta}(\theta)\leq\frac{\mu}{\kappa}D^{\|\cdot\|_{h}}_{v^{\dagger}}(\gamma,\gamma^{\dagger}).\mbox{\qed}$

The previous mass distribution estimate can now be used to obtain estimates on
$\gamma$.

######  Theorem 84 (Error estimates for
$\gamma\in\mathcal{M}_{+}(\Theta\times\Gamma)$).

Let $h:\Theta\to[0,\infty)$ be continuous with
$\int_{\Theta}h\>\mathrm{d}\mathcal{H}_{\Theta}=1$ and assume there is
${\mathcal{T}^{\prime}}\subset{\mathcal{T}}$ such that $u_{t}^{\dagger}$ is
$(\hat{\kappa},\hat{\mu},\hat{R})$-reconstructible via
$\hat{v}_{t}^{\dagger},\hat{v}_{t}^{s}$ from measurements with
$\mathrm{Ob}_{t}$ for all $t\in{\mathcal{T}^{\prime}}$. Then, abbreviating
$p_{i}^{\theta}=(\theta\cdot x_{i},\theta\cdot v_{i})$, the solution
$(u,\gamma)$ to $\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ satisfies

$\displaystyle\int_{\Theta}h(\theta)|\gamma_{\theta}|(\Gamma\setminus
B_{R}(\operatorname{supp}\gamma_{\theta}^{\dagger}))\>\mathrm{d}\mathcal{H}_{\Theta}$
$\displaystyle\leq\frac{1}{\kappa
R^{2}}\left(\frac{3}{2}+2\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}_{t}^{\dagger}|^{2}\right)\sqrt{\delta},$
$\displaystyle\sum_{i=1}^{N}\int_{\Theta}\\!\\!h(\theta)\\!\\!\int_{B_{R}(\\{p_{i}^{\theta}\\})}\\!\\!\\!\\!\operatorname{dist}(p,p_{i}^{\theta})^{2}\>\mathrm{d}|\gamma_{\theta}|(p)\>\mathrm{d}\mathcal{H}_{\Theta}$
$\displaystyle\leq\frac{1}{\kappa}\left(\frac{3}{2}+2\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}_{t}^{\dagger}|^{2}\right)\sqrt{\delta},$
$\displaystyle\sum_{i=1}^{N}\left|\int_{\Theta}h(\theta)(\gamma_{\theta}-\gamma^{\dagger}_{\theta})(B_{R}(\\{p_{i}^{\theta}\\}))\>\mathrm{d}\mathcal{H}_{\Theta}\right|$
$\displaystyle\leq\frac{1+\kappa R^{2}}{\kappa
R^{2}}\left(2\\!+\\!2\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}_{t}^{\dagger}|^{2}\right)\sqrt{\delta}\\!+\\!\frac{1}{2R^{2}}\max_{t\in{\mathcal{T}^{\prime}}}W_{2,R}^{2}(u_{t},u^{\dagger}_{t})$

for $\kappa=\sigma\hat{\kappa}$ and any
$R\leq\min\\{\hat{R},\Delta/3,\Delta^{\prime}\\}/\sqrt{1+\max_{t\in{\mathcal{T}^{\prime}}}t^{2}}$,
where $\sigma$ is the smallest eigenvalue of
$\frac{1}{\operatorname{card}{\mathcal{T}^{\prime}}}\sum_{t\in{\mathcal{T}^{\prime}}}{1\choose-t}\otimes{1\choose-t}$
and $\Delta,\Delta^{\prime}\geq 0$ are such that $\gamma_{\theta}^{\dagger}$
for any $\theta\in\operatorname{supp}h$ contains no $\Delta$-coincidence or
$\Delta^{\prime}$-ghost particle with respect to ${\mathcal{T}^{\prime}}$.

###### Proof.

We split $y^{\delta}=(u,\gamma)$ into $y^{\delta}_{1}=\gamma$ and
$y^{\delta}_{2}=u$ and consider the regularizer

$G(u,\gamma)=\|\gamma\|_{h}+\iota_{+}(\gamma)+\iota_{+}(u).$

As in the proof of Theorem 79 define $v^{\dagger}\in(H^{*})^{|{\mathcal{T}}|}$
via
$v^{\dagger}_{t}=-\hat{v}^{\dagger}_{t}/\operatorname{card}{{\mathcal{T}^{\prime}}}$
for all $t\in{\mathcal{T}^{\prime}}$ and $v^{\dagger}_{\tilde{t}}=0$ else. The
dual variables are now constructed as weighted averages (with weight $h$) of
the dual variables from Theorem 79. Indeed, we define $q^{\dagger}\in
C(\Theta\times\Gamma)^{\Sigma}$ as

$q^{\dagger}_{t}(\theta,x)=h(\theta)(1-\hat{\kappa}\min\\{\bar{R},\operatorname{dist}(x,\operatorname{supp}\mathrm{Rd}_{\theta}u_{t}^{\dagger})\\}^{2})/\operatorname{card}{{\mathcal{T}^{\prime}}}\quad\text{for
}t\in{\mathcal{T}^{\prime}}$

and $q^{\dagger}_{t}=0$ for $t\notin{\mathcal{T}^{\prime}}$, where
$\bar{R}=\min\\{\hat{R},\Delta/3,\Delta^{\prime}\\}$. Also, let
$s_{i}=\operatorname{sgn}\int_{\Theta}h(\theta)(\gamma_{\theta}-\gamma^{\dagger}_{\theta})(B_{R}(\\{p_{i}^{\theta}\\}))\>\mathrm{d}\mathcal{H}_{\Theta}$,
$i=1,\ldots,N$, where the slices $\gamma_{\theta}$ are defined via the
relation $\gamma=\mathcal{H}_{\Theta}{\otimes}_{\theta}\gamma_{\theta}$ (which
is possible by Lemma 39). With this $s_{i}$ we define $q\in
C(\Theta\times\Gamma)^{\Sigma}$ as $q_{t}=0$ for
$t\notin{\mathcal{T}^{\prime}}$ and, for $t\in{\mathcal{T}^{\prime}}$, as

$q_{t}(\theta,x)=h(\theta)s_{i}/\operatorname{card}{{\mathcal{T}^{\prime}}}\quad\text{if
}x\in B_{\bar{R}}(\theta\cdot(x_{i}+tv_{i})),$

continuously extended onto the rest of $\Theta\times\Gamma$ with absolute
value no larger than $h(\theta)/\operatorname{card}{{\mathcal{T}^{\prime}}}$.
Now analogously to the proof of Theorem 79 it follows that
$w^{\dagger}={v^{\dagger}\choose q^{\dagger}}$ and $w={0\choose q}$ satisfy

$\displaystyle-K^{*}w^{\dagger}=-((K^{*}w^{\dagger})_{1},(K^{*}w^{\dagger})_{2})$
$\displaystyle\in\partial G(y^{\dagger}),$ $\displaystyle-(K^{*}w)_{1}$
$\displaystyle\in\partial\|\cdot\|_{h}\left(\sum_{i=1}^{N}s_{i}\mathcal{H}^{d-1}{\mbox{\LARGE$\llcorner$}}\\{p_{i}^{\theta}\,|\,\theta\in\Theta\\}\right),$
$\displaystyle-(K^{*}w^{\dagger})_{1}(\theta,p)$ $\displaystyle\leq
h(\theta)(1-\kappa\min\\{R,\operatorname{dist}(p,\operatorname{supp}\gamma_{\theta}^{\dagger})\\}^{2}),$
$\displaystyle-(K^{*}w)_{1}(\theta,p)$
$\displaystyle=s_{i}h(\theta)\quad\text{for all }p\in
B_{R}(\\{p_{i}^{\theta}\\}).$

Thus, using (in this order) Theorem 83, Theorem 67 and (21) we conclude

$\int_{\Theta}h(\theta)|\gamma_{\theta}|(\Gamma\setminus
B_{R}(\operatorname{supp}\gamma_{\theta}^{\dagger}))\>\mathrm{d}\mathcal{H}_{\Theta}\leq\frac{1}{\kappa
R^{2}}D^{\|\cdot\|_{h}}_{-(K^{*}w^{\dagger})_{1}}(\gamma,\gamma^{\dagger})\leq\frac{1}{\kappa
R^{2}}D^{G}_{-K^{*}w^{\dagger}}(y^{\delta},y^{\dagger})\\\
{}^{\leq}\left(3\delta+F_{f^{\delta}}^{*}(2\sqrt{\delta}w^{\dagger})+F_{f^{\delta}}^{*}(-2\sqrt{\delta}w^{\dagger})\right)/(2\kappa
R^{2}\sqrt{\delta})=\frac{3+4|v^{\dagger}|^{2}}{2\kappa
R^{2}}\sqrt{\delta}\leq\frac{3+4\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}^{\dagger}_{t}|^{2}}{2\kappa
R^{2}}\sqrt{\delta},$

proving the first inequality of the statement. The second and third inequality
follow in exactly the same way (using $\mu=0$ in Theorem 83), where the third
needs one additional estimate for
$\langle-(K^{*}w)_{1},\gamma-\gamma^{\dagger}\rangle$: We employ
$\langle-(K^{*}w)_{1},\gamma-\gamma^{\dagger}\rangle=\langle-K^{*}w,(u,\gamma)-(u^{\dagger},\gamma^{\dagger})\rangle+\langle(K^{*}w)_{2},u-u^{\dagger}\rangle$,
where the first summand can be estimated via Theorem 67 and (21) as

$\langle-K^{*}w,(u,\gamma)-(u^{\dagger},\gamma^{\dagger})\rangle\leq
2(1+|v^{\dagger}|)\sqrt{\delta}\leq
2(1+\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}^{\dagger}_{t}|)\sqrt{\delta},$

and for the second summand we note that by construction
$1=|\operatorname{card}{{\mathcal{T}^{\prime}}}(\overline{\underline{\mathrm{Rd}}}^{*}q)_{t}(x)|$
for $x\in B_{\bar{R}}(\operatorname{supp}u_{t}^{\dagger})$ and any
$t\in{\mathcal{T}^{\prime}}$. Thus,

$\langle(K^{*}w)_{2},y^{\delta}_{2}-y^{\dagger}_{2}\rangle=\sum_{t\in{\mathcal{T}^{\prime}}}\langle(\overline{\underline{\mathrm{Rd}}}^{*}q)_{t},u_{t}-u^{\dagger}_{t}\rangle\leq\frac{1}{\operatorname{card}{{\mathcal{T}^{\prime}}}}\sum_{t\in{\mathcal{T}^{\prime}}}\bigg{(}|u_{t}-u^{\dagger}_{t}|(\Omega\setminus
B_{R}(\operatorname{supp}u_{t}^{\dagger}))\\\
+\sum_{x\in\operatorname{supp}u_{t}^{\dagger}}|(u_{t}-u_{t}^{\dagger})(B_{R}(x))|\bigg{)}\leq\frac{1}{2R^{2}\operatorname{card}{{\mathcal{T}^{\prime}}}}\sum_{t\in{\mathcal{T}^{\prime}}}W_{2,R}^{2}(u_{t},u^{\dagger}_{t}),$

where we exploited that in the unbalanced Wasserstein divergence $W_{2,R}^{2}$
mass is never transported further than $R$. ∎

This result can now be turned into an unbalanced Wasserstein estimate for the
slices of $\gamma$.

######  Theorem 85 (Error estimates for slices of
$\gamma\in\mathcal{M}_{+}(\Theta\times\Gamma)$).

The statement of Theorem 79 holds for $\mathcal{H}_{\Theta}$-almost every
$\theta$ if $\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ is replaced with
$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$.

###### Proof.

Based on Theorem 66, it suffices to show that one may replace
$h\mathcal{H}_{\Theta}$ in Theorem 84 with a Dirac measure $\delta_{\theta}$
for $\mathcal{H}_{\Theta}$-almost all $\theta$. We only show this for the
first inequality of Theorem 84 (the others follow analogously). Now let

$\displaystyle\Delta(\theta)$ $\displaystyle=\sup\\{\Delta\geq
0\,|\,\gamma_{\theta}^{\dagger}\text{ contains no $\Delta$-coincidence with
respect to }{\mathcal{T}^{\prime}}^{\prime}\\},$
$\displaystyle\Delta^{\prime}(\theta)$
$\displaystyle=\sup\\{\Delta^{\prime}\geq
0\,|\,\gamma_{\theta}^{\dagger}\text{ contains no $\Delta^{\prime}$-ghost
particle with respect to }{\mathcal{T}^{\prime}}^{\prime}\\}$

and note that both are continuous in $\theta$. Thus, so is
$R(\theta)=\min\\{\hat{R},\Delta(\theta)/3,\Delta^{\prime}(\theta)\\}/\sqrt{1+\max_{t\in{\mathcal{T}^{\prime}}}t^{2}}$.
We will show that for any nonnegative continuous function $\hat{R}(\theta)\leq
R(\theta)$ and any $\rho<1$ there is a piecewise constant function
$\bar{R}:S^{d-1}\to[0,\infty)$ with
$\rho^{2}\hat{R}(\theta)\leq\bar{R}(\theta)\leq\hat{R}(\theta)$ and such that
$|\gamma_{\theta}|(\Gamma\setminus
B_{\bar{R}(\theta)}(\operatorname{supp}\gamma_{\theta}^{\dagger}))\leq\frac{1}{\kappa\bar{R}(\theta)^{2}}(\frac{3}{2}+2\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}_{t}^{\dagger}|^{2})\sqrt{\delta}$
for $\mathcal{H}_{\Theta}$-almost every $\theta$, which then implies the
desired result due to the arbitrariness of $\rho$ and $\hat{R}$. To this end,
fix some $\rho\in(0,1)$ and let $\epsilon>0$ be arbitrary. Abbreviate

$S=\\{\theta\in\Theta\,|\,\hat{R}(\theta)=0\\},$

then it suffices to prove the inequality for $\mathcal{H}_{\Theta}$-almost
every $\theta$ in $\Theta\setminus S$ (since on $S$ there is nothing to show).
Due to the regularity of $\mathcal{H}_{\Theta}$ we can pick a compact
$C\subset\Theta\setminus S$ such that $\mathcal{H}_{\Theta}((\Theta\setminus
S)\setminus C)<\epsilon$. We will show the inequality for
$\mathcal{H}_{\Theta}$-almost every $\theta$ in $C$, then the result for
$\Theta\setminus S$ follows from the arbitrariness of $\epsilon$. For each
$\theta\in C$ define

$r(\theta)=\sup\\{r\in\mathbb{R}\,|\,\rho^{2}\hat{R}(\theta^{\prime})<\rho\hat{R}(\theta)<\hat{R}(\theta^{\prime})\text{
for all $\theta^{\prime}\in S^{d-1}$ with
}\operatorname{dist}(\theta,\theta^{\prime})<r\\},$

then $r(\theta)>0$ due to the continuity of $\hat{R}(\theta)$. Consider the
open $r(\hat{\theta})$-ball $B\subset S^{d-1}$ around $\hat{\theta}\in C$. It
suffices to prove the desired inequality for $\mathcal{H}_{\Theta}$-almost
every $\theta\in B$ (choosing $\bar{R}(\theta)=\rho\hat{R}(\hat{\theta})$),
since the compactum $C$ can be covered with finitely many of these balls and
the function $\bar{R}(\theta)$ can then simply be chosen constant on each ball
(minus the finitely many balls where it was already defined). Now on $B$, by
Theorem 84 we have

$\int_{\Theta}h(\theta)\left[|\gamma_{\theta}|(\Gamma\setminus
B_{R}(\operatorname{supp}\gamma_{\theta}^{\dagger}))-\frac{1}{\kappa
R^{2}}\left(\frac{3}{2}+2\max_{t\in{\mathcal{T}^{\prime}}}|\hat{v}_{t}^{\dagger}|^{2}\right)\sqrt{\delta}\right]\>\mathrm{d}\mathcal{H}_{\Theta}\leq
0$

for $R=\bar{R}(\theta)=\rho\hat{R}(\hat{\theta})$ and for any nonnegative
continuous $h:B\to[0,\infty)$ of compact support. Therefore the square-
bracketed term must be nonpositve for $\mathcal{H}_{\Theta}$-almost every
$\theta\in B$, which is what we needed to prove. ∎

The error estimate for all remaining snapshots finally stays the same for
model $\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ as for model
$\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$.

######  Theorem 86 (Error estimates for bad snapshots).

Theorem 82 still holds when replacing model
$\Pi$-$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ with
$\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$.

###### Proof.

The left- and right-hand side of the estimate depend continuously on
$\bar{R},\bar{\kappa}$ so that it suffices to prove the estimate for any
$\bar{\kappa},\bar{R}$ strictly smaller than their values from Theorem 82. As
in Theorem 82 we consider the splitting of $y^{\delta}=(u,\gamma)$ into
$y^{\delta}_{1}=u_{t}$ and $y^{\delta}_{2}=((u_{\tilde{t}})_{\tilde{t}\neq
t},\gamma)$ as well as the regularizer choice

$G(u,\gamma)=\|u_{t}\|_{+}+\iota_{+}((u_{\tilde{t}})_{\tilde{t}\neq
t},\gamma).$

We will also use almost exactly the same dual variables as in Theorem 82, only
we will smooth them out a little in $\theta$-direction. To this end, for a
finite set $\Theta^{\prime\prime}\subset S^{d-1}$ let

$\displaystyle\Delta(\Theta^{\prime\prime})$ $\displaystyle=\sup\\{\Delta\geq
0\,|\,u_{t}^{\dagger}\text{ contains no $\Delta$-coincidence with respect to
}\Theta^{\prime\prime}\\},$
$\displaystyle\Delta^{\prime}(\Theta^{\prime\prime})$
$\displaystyle=\sup\\{\Delta^{\prime}\geq 0\,|\,u_{t}^{\dagger}\text{ contains
no $\Delta^{\prime}$-ghost particle with respect to }\Theta^{\prime\prime}\\}$

and let $\sigma(\Theta^{\prime\prime})$ be the smallest eigenvalue of
$\frac{1}{\operatorname{card}\Theta^{\prime\prime}}\sum_{\theta\in\Theta^{\prime\prime}}\theta\otimes\theta$.
Furthermore, let $\kappa(\Theta^{\prime\prime})$ and
$R(\Theta^{\prime\prime})$ be the values of $\kappa$ and $R$ from the
statement of Theorem 82 for the choice
$\Theta^{\prime}=\Theta^{\prime\prime}$. Now let $\zeta>0$ be small enough and
pick $\tilde{\kappa},\tilde{R}$ such that the balls $B_{\zeta}(\\{\theta\\})$
are disjoint for all $\theta\in\Theta^{\prime}$ and such that
$\bar{\kappa}\leq\sigma(\Theta^{\prime\prime})\tilde{\kappa}$,
$\tilde{\kappa}\leq\kappa(\Theta^{\prime\prime})/(1+t^{2})$,
$\bar{R}\leq\min\\{(1+t^{2})R(\Theta^{\prime\prime}),\tilde{R}/\sqrt{1+t^{2}},\Delta^{\prime}(\Theta^{\prime\prime})\\}$
and $\tilde{R}\leq\Delta(\Theta^{\prime\prime})/3$ for any perturbation
$\Theta^{\prime\prime}$ of $\Theta^{\prime}$ that perturbs no
$\theta\in\Theta^{\prime}$ by more than distance $\zeta$ (this is possible
since we assume $\bar{\kappa}$ and $\bar{R}$ strictly smaller than their
values in Theorem 82 and since $\Delta(\Theta^{\prime\prime})$,
$\Delta^{\prime}(\Theta^{\prime\prime})$, $\sigma(\Theta^{\prime\prime})$,
$\kappa(\Theta^{\prime\prime})$ and $R(\Theta^{\prime\prime})$ depend
continuously on $\Theta^{\prime\prime}$). Furthermore, for
$\bar{\theta}\in\Theta^{\prime}$ let $h_{\bar{\theta}}:\Theta\to[0,\infty)$ be
continuous with $\operatorname{supp}h_{\bar{\theta}}\subset
B_{\zeta}(\\{\bar{\theta}\\})$ and
$\int_{\Theta}h_{\bar{\theta}}\mathcal{H}_{\Theta}=1$. Now define
$q^{\dagger}\in C(\Theta\times\Xi)^{\Sigma}$ as

$q^{\dagger}_{t}(\theta,x)=-h_{\bar{\theta}}(\theta)(1-\tilde{\kappa}\min\\{\bar{R},\operatorname{dist}(x,\operatorname{supp}\mathrm{Mv}^{1}_{t}\gamma^{\dagger}_{\theta})\\}^{2})/\operatorname{card}{\Theta^{\prime}}\quad\text{for
}\theta\in B_{\zeta}(\\{\bar{\theta}\\}),\bar{\theta}\in\Theta^{\prime}$

and $q^{\dagger}_{\tilde{t}}=0$ for $\tilde{t}\neq t$. Also, let
$s_{i}=\operatorname{sgn}(u_{t}-u_{t}^{\dagger})(B_{\bar{R}}(\\{x_{i}+tv_{i}\\}))$,
$i=1,\ldots,N$, and define $q\in C(\Theta\times\Xi)^{\Sigma}$ as
$q_{\tilde{t}}=0$ for $\tilde{t}\neq t$ and, otherwise,

$q_{t}(\theta,x)=-\sum_{\bar{\theta}\in\Theta^{\prime}}h_{\bar{\theta}}(\theta)s_{i}/\operatorname{card}{\Theta^{\prime}}\quad\text{if
}x\in B_{\tilde{R}}(\theta\cdot(x_{i}+tv_{i})),$

continuously extended onto the rest of $\Theta\times\Xi$ with absolute value
no larger than
$h_{\bar{\theta}}(\theta)/\operatorname{card}{\Theta^{\prime}}$. Finally, let
$w^{\theta}\in Y^{*}$ for $\theta\in\Theta$ denote the first dual variables
constructed in Theorem 79, and define

$w^{\dagger}={0\choose
q^{\dagger}}+\frac{1}{\operatorname{card}{\Theta^{\prime}}}\sum_{\bar{\theta}\in\Theta^{\prime}}\int_{\Theta}h_{\bar{\theta}}(\theta)w^{\theta}\>\mathrm{d}\mathcal{H}_{\Theta}(\theta),\quad
w={0\choose q}.$

The remainder of the proof (checking the properties of the constructed dual
variables and estimating one dual pairing) proceeds analogously to the proof
of Theorem 82. ∎

Theorem 73 for $\mathrm{P}_{\sqrt{\delta}}(f^{\delta})$ now follows from
combining Theorems 77, 85 and 86.

##  5 Discretization and numerical validation

Our goal is now to develop a numerical method in order to solve the dimension-
reduced problem $\mathrm{P}_{\alpha}(f)$ or $\Pi$-$\mathrm{P}_{\alpha}(f)$.
Such a method has to translate the dimension reduction from the continuous
formulation to a discrete formulation, which then is more efficient to solve
than comparable discretizations of the full-dimensional problem. A
straightforward discretization of $\mathrm{P}_{\alpha}(f)$ is achieved by
simply gridding the parameter space and replacing Radon measures on the
continuous parameter space by vectors representing the weights of a discrete
measure supported in (or around) the grid points.

In this section, we assume that the sets $\Sigma$ and $\Theta$ are finite.
Some choices of these parameter sets might be better suited than others, see
Section 3.3. To be able to compare different choices of $\Sigma$ for our
method without changing datasets or domains, we define a slightly less
restrictive domain in phase space: We require particles to stay in $\Omega$
only during the measurement times ${\mathcal{T}}$ instead of $\Sigma$, i. e.,
instead of $\Lambda$ as defined in (8) we use

$\tilde{\Lambda}\coloneqq\\{(x,v)\in\mathbb{R}^{d}\times\mathbb{R}^{d}\;|\;x+tv\in\Omega\text{
for all }t\in{\mathcal{T}}\\}.$

The code to reproduce the results of this paper will be made available at
https://github.com/alexschlueter/supermops.

###  5.1 Gridding for snapshots and position-velocity projections

The first challenge is that problem $\mathrm{P}_{\alpha}(f)$ contains multiple
Radon measures as variables that should be discretized consistently, namely
measures $\gamma_{\theta}$ for directions $\theta\in\Theta$ as well as
snapshots $u_{t}$ for times $t\in\Sigma$. For each of these variables we need
to generate a corresponding grid.

One option for $\gamma$ would be to use a single grid on $\Gamma$ in order to
discretize all $\gamma_{\theta}$ equally, independent of the direction
$\theta$. This however would result in wasted degrees of freedom, because the
$\gamma_{\theta}$ occupy different parts of $\Gamma$. For this reason we will
generate individual grids for each direction $\theta$. These should be
compatible with the (non-discretized) phase space domain $\tilde{\Lambda}$ in
the following sense: If $\lambda\in\mathcal{M}(\tilde{\Lambda})$ is a high-
dimensional phase space measure, the grid for $\gamma_{\theta}$ should cover
enough of $\Gamma$ such that the joint Radon transform of $\lambda$ along
direction $\theta$ can still be represented, thus at least
$\mathrm{Rj}_{\theta}(\tilde{\Lambda})\subset\Gamma$. Analogous considerations
hold for $u$ so that we also use individual grids for all $u_{t}$.

In the following we will consider

$\Omega=[0,1]^{d}$

and furthermore centre without loss of generality the times ${\mathcal{T}}$
around $0$ so that there exists $T>0$ with

$[\min{\mathcal{T}},\max{\mathcal{T}}]=[-T,T].$

Based on this we now compute the domains of all $\gamma_{\theta}$ and $u_{t}$
in order to generate tailored grids on these.

######  Proposition 87 (Domain of $\gamma_{\theta}$).

For a given $\theta\in S^{d-1}$ define
$s_{+}\coloneqq\sum_{j:\theta_{j}>0}\theta_{j}$ and
$s_{-}\coloneqq\sum_{j:\theta_{j}<0}\theta_{j}$. The set
$\mathrm{Rj}_{\theta}(\tilde{\Lambda})\subset\Gamma$ is a parallelogram in
$\mathbb{R}^{2}$ with vertices

$\begin{pmatrix}s_{-}\\\ 0\end{pmatrix},\begin{pmatrix}s_{+}\\\
0\end{pmatrix},\begin{pmatrix}\frac{s_{+}+s_{-}}{2}\\\
\frac{s_{+}-s_{-}}{2T}\end{pmatrix},\begin{pmatrix}\frac{s_{+}+s_{-}}{2}\\\
\frac{s_{-}-s_{+}}{2T}\end{pmatrix}.$

###### Proof.

Since we assume $\Omega=[0,1]^{d}$, we have

$\displaystyle(x,v)\in\tilde{\Lambda}$ $\displaystyle\iff\forall
j=1,\dotsc,d:0\leq x_{j}+Tv_{j}\leq 1\land 0\leq x_{j}-Tv_{j}\leq 1$
$\displaystyle\iff\forall
j=1,\dotsc,d:(x_{j},v_{j})\in\operatorname{conv}\left\\{\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}1\\\ 0\end{pmatrix},\begin{pmatrix}1/2\\\
1/(2T)\end{pmatrix},\begin{pmatrix}1/2\\\ -1/(2T)\end{pmatrix}\right\\}.$

Denote the convex hull above by $\Lambda_{1}\subset\mathbb{R}^{2}$. For a
given $\theta\in S^{d-1}$ we have, using the Minkowski sum of sets,

$\displaystyle\mathrm{Rj}_{\theta}(\tilde{\Lambda})$
$\displaystyle=\\{(\theta\cdot x,\theta\cdot
v)\;|\;(x,v)\in\tilde{\Lambda}\\}$
$\displaystyle=\bigg{\\{}(y,w)\in\mathbb{R}^{2}\,\bigg{|}\exists(x_{j},v_{j})\in\Lambda_{1},j=1,\dotsc,d:(y,w)=\sum_{j=1}^{d}\theta_{j}(x_{j},v_{j})\bigg{\\}}$
$\displaystyle=\theta_{1}\Lambda_{1}+\dotsb+\theta_{d}\Lambda_{1}$
$\displaystyle=\sum_{j:\theta_{j}>0}(\theta_{j}\Lambda_{1})+\sum_{j:\theta_{j}<0}(\theta_{j}\Lambda_{1}).$

By convexity of $\Lambda_{1}$, we can use the distributivity for Minkowski
sums after introducing a minus sign,

$\sum_{j:\theta_{j}>0}(\theta_{j}\Lambda_{1})+\sum_{j:\theta_{j}<0}(\theta_{j}\Lambda_{1})=\sum_{j:\theta_{j}>0}(\theta_{j}\Lambda_{1})+\sum_{j:\theta_{j}<0}((-\theta_{j})(-\Lambda_{1}))=s_{+}\Lambda_{1}+(-s_{-})(-\Lambda_{1}).$

Note that $-\Lambda_{1}=\Lambda_{1}-(1,0)$ so that

$\displaystyle
s_{+}\Lambda_{1}+(-s_{-})(-\Lambda_{1})=s_{+}\Lambda_{1}+(-s_{-})\Lambda_{1}+(s_{-},0)=(s_{+}-s_{-})\Lambda_{1}+(s_{-},0).$

Applying the scaling and translation in the last expression to the four points
inside the definition of $\Lambda_{1}$, we obtain the stated result. ∎

######  Proposition 88 (Domain of $u_{t}$).

Let $t\in\mathbb{R}$ be given.

  1. 1.

If $|t|\leq T$, we have $\mathrm{Mv}^{d}_{t}(\tilde{\Lambda})=[0,1]^{d}$.

  2. 2.

If $|t|>T$, we have
$\mathrm{Mv}^{d}_{t}(\tilde{\Lambda})=\left[\frac{1}{2}-\frac{|t|}{2T},\frac{1}{2}+\frac{|t|}{2T}\right]^{d}$.

###### Proof.

Write

$S\coloneqq\left\\{\begin{pmatrix}0\\\ 0\end{pmatrix},\begin{pmatrix}1\\\
0\end{pmatrix},\begin{pmatrix}1/2\\\ 1/(2T)\end{pmatrix},\begin{pmatrix}1/2\\\
-1/(2T)\end{pmatrix}\right\\}$

and define again $\Lambda_{1}=\operatorname{conv}S$. We have

$\mathrm{Mv}^{d}_{t}(\tilde{\Lambda})=\\{x+tv\;|\;(x,v)\in\tilde{\Lambda}\\}=\bigtimes_{j=1}^{d}\mathrm{Mv}^{1}_{t}(\Lambda_{1})$

and by linearity of $\mathrm{Mv}^{1}_{t}$ we know that

$\mathrm{Mv}^{1}_{t}(\Lambda_{1})=\mathrm{Mv}^{1}_{t}(\operatorname{conv}S)=\operatorname{conv}\mathrm{Mv}^{1}_{t}(S)=\operatorname{conv}\left\\{0,1,\frac{1}{2}+\frac{t}{2T},\frac{1}{2}-\frac{t}{2T}\right\\}\,,$

from which the claim follows easily. ∎

Now knowing the exact domains, we generate regular grids for each variable,
that is, grids $G_{u}(t)$ for each $t\in\Sigma$ and grids $G_{\gamma}(\theta)$
for each $\theta\in\Theta$. Examples can be seen in Fig. 1. The grids
$G_{\gamma}(\theta)$ are generated by applying an appropriate affine transform
to a standard rectangular grid as suggested by Proposition 87. We employ the
same number of $M$ grid cells along each dimension to obtain a more or less
isotropic resolution (in our numerical experiments, $M=100$ or $200$). Each
variable $u_{t}$, $t\in\Sigma$, and $\gamma_{\theta}$, $\theta\in\Theta$, is
now discretized as a piecewise constant function on the corresponding grid,
whose masses in the different grid cells (function value times cell volume)
are collected in the vectors $\vec{u}_{t}\in\mathbb{R}^{M^{d}}$, $t\in\Sigma$,
and $\vec{\gamma}_{\theta}\in\mathbb{R}^{M^{2}}$, $\theta\in\Theta$.

###  5.2 Discretized operators and optimization problem

Figure 1: Example grids and projections. The green line in projection
direction $\theta$ is subdivided into red bins corresponding to the grid
$G_{r}(\theta,t)$. The contribution of a grid cell to the projection onto a
red bin $[r_{i},r_{i+1}]$ is determined by its relative area of intersection
(visualized by opacity) with the strip orthogonal to $\theta$.

Next, the operators $\mathrm{Rd}_{\theta}$ and $\mathrm{Mv}_{t}^{1}$ need to
be discretized. We restrict our attention to the case $d=2$ for simplicity and
note that, for two spatial dimensions, these operators are identical up to a
rescaling: For every $t\in\mathbb{R}$, we have
$\mathrm{Mv}_{t}^{1}={[x\mapsto\sqrt{1+t^{2}}x]}_{\\#}\mathrm{Rd}_{\theta(t)}$,
where $\theta(t)\coloneqq(1,t)/\sqrt{1+t^{2}}$. Thus it will be sufficient to
only describe the implementation of the operator $\mathrm{Rd}_{\theta}$ in the
following.

For every direction $\theta\in\Theta$, we need to discretize the operation
$\mathrm{Rd}_{\theta}u_{t}$, where $u_{t}$ is represented by the vector
$\vec{u}_{t}$ of function values on the grid $G_{u}(t)$. The result of the
projection $\mathrm{Rd}_{\theta}u_{t}$ is a measure on $\Xi\subset\mathbb{R}$,
whose domain
$\mathrm{Rd}_{\theta}(\mathrm{Mv}^{d}_{t}(\tilde{\Lambda}))=\mathrm{Mv}^{1}_{t}(\mathrm{Rj}_{\theta}(\tilde{\Lambda}))\subset\Xi$
depends on $t$ and $\theta$ and is again discretized via a one-dimensional
equispaced grid $G_{r}(\theta,t)$. To have a comparable resolution as for the
variables $u_{t}$ and $\gamma_{\theta}$ we pick $G_{r}(\theta,t)$ to have the
same number $M$ of intervals as the other grids have along each direction.
Again, measures on $\Xi$ are discretized as piecewise constant functions on
that grid.

After discretization, the Radon transform $\mathrm{Rd}_{\theta}u_{t}$ will be
represented by $M_{\mathrm{Rd}}(t,\theta)\vec{u}_{t}$ for a matrix
$M_{\mathrm{Rd}}(t,\theta)\in\mathbb{R}^{M\times M^{2}}$. (In a completely
analogous manner the move operator $\mathrm{Mv}_{t}^{1}$ in
$\mathrm{Mv}_{t}^{1}\gamma_{\theta}$ will be represented by a matrix
$M_{\mathrm{Mv}}(\theta,t)\in\mathbb{R}^{M\times M^{2}}$). The entry of the
$i$-th row and $j$-th column in $M_{\mathrm{Rd}}(t,\theta)$ is given by the
relative area of intersection between the strip

$\\{x\in\mathbb{R}^{2}\;|\;r_{i}\leq\theta\cdot x\leq r_{i+1}\\}$

(where $[r_{i},r_{i+1}]$ represents the $i$-th grid cell of the grid
$G_{r}(\theta,t)$) and the $j$-th cell of the grid $G_{u}(t)$ (cf. Fig. 1).

Finally, we take $M_{\mathrm{Ob}}(t)$ to be appropriate matrices discretizing
the forward operators $\mathrm{Ob}_{t}$ on the grids $G_{u}(t)$ for all
$t\in{\mathcal{T}}$. In our experiments we consider as forward operator the
truncated discrete Fourier transform on $\Omega=[0,1]^{2}$ from 35. To compute
the entries of $M_{\mathrm{Ob}}(t)$ we simply apply $\mathrm{Ob}_{t}$ to Dirac
measures located at the centres of the grid cells in $G_{u}(t)$ for each time
$t\in{\mathcal{T}}$.

In principle one would now obtain a discretized version of
$\Pi$-$\mathrm{P}_{\alpha}(f)$ by replacing all variables and operators by
their discretized counterparts. However, the constraints
$\mathrm{Rd}_{\theta}u_{t}=\mathrm{Mv}_{t}^{1}\gamma_{\theta}$ should not be
enforced exactly in the discretized setting since this is likely to introduce
discretization artefacts. Therefore we replace them by a bound $\tau>0$ on the
violation so that the discretized problem reads

$\begin{split}\min_{\begin{subarray}{c}\vec{\gamma}_{\theta}\in\mathbb{R}^{M^{2}},\theta\in\Theta\\\
\vec{u}_{t}\in\mathbb{R}^{M^{d}},t\in\Sigma\end{subarray}}\,\sum_{\theta\in\Theta}\lVert\vec{\gamma}_{\theta}\rVert_{1}+\sum_{t\in\Sigma}\lVert\vec{u}_{t}\rVert_{1}+\frac{1}{2\alpha}\sum_{t\in{\mathcal{T}}}\lVert
M_{\mathrm{Ob}}(t)\vec{u}_{t}-f_{t}\rVert_{2}^{2}\\\ \quad\text{ such that
}\left(\sum_{\theta\in\Theta,t\in\Sigma}\lVert
M_{\mathrm{Mv}}(\theta,t)\vec{\gamma}_{\theta}-M_{\mathrm{Rd}}(t,\theta)\vec{u}_{t}\rVert_{2}^{2}\right)^{1/2}\leq\tau.\end{split}$
(24)

A coarse parameter search has shown a value of $\tau=0.001$ to be a good
choice, which we use in all our experiments. The resulting second-order cone
program is solved using the commercial MOSEK solver [7] called through the
Python library CVXPY [18, 1].

###  5.3 Application to truncated Fourier measurements

As a model super-resolution problem, we consider the case of truncated Fourier
measurements from 35 in $d=2$ dimensions, where we set the frequency cutoff to
$\Phi=2$. This setting in particular serves to benchmark our dimension
reduction against the full-dimensional dynamic reconstruction proposed by
Alberti et al.

#### Measurement times.

We are interested in evaluating the performance of the method for different
numbers of measurement times ${\mathcal{T}}$. For simplicity, we choose times
${\mathcal{T}}\subset[-1,1]$ centred around 0 as follows: Given a
$K\in\mathbb{N}$, choose
${\mathcal{T}}\coloneqq\\{k/K\;|\;k\in\\{-K,\dotsc,K\\}\\}$. The values of $K$
used in the following are $K=1,2,3$ resulting in 3, 5 and 7 times
respectively.

#### Datasets.

To compare different parameter choices for the algorithm and to benchmark it
against other methods, we generate three datasets $D_{3}$, $D_{5}$, $D_{7}$
consisting of 2000 particle configurations in $d=2$ dimensions, one dataset
for each choice of times ${\mathcal{T}}$. Each configuration within a dataset
consists of $4\leq N\leq 20$ particles with randomly chosen positions and
velocities $(x_{i},v_{i})\in\tilde{\Lambda}$ and weights $m_{i}$ drawn from
the uniform distribution on $[0.9,1.1]$.

We define the minimal dynamic separation of a particle configuration
$S=\\{(x_{i},v_{i})\;|\;i=1,\dotsc,N\\}$ as

$\Delta_{\mathrm{dyn}}(S)\coloneqq\min_{t\in{\mathcal{T}}}\min_{i\neq j}\lVert
x_{i}+tv_{i}-(x_{j}+tv_{j})\rVert.$

Since we are interested in testing the reconstruction methods for a range of
separation values, we want configurations with low and high separations to be
equally well represented in each dataset. To this end, we use a _rejection-
sampling_ method adapted from [4], which roughly works by drawing
configurations from the uniform distribution on $\tilde{\Lambda}$ and
accepting or rejecting the new configuration based on an acceptance
probability. For our datasets this probability is chosen as to make the
dynamic separation $\Delta_{\mathrm{dyn}}$ uniformly distributed on $[0,0.1]$.

#### Postprocessing and error measures.

Because static reconstruction will be one of the methods we compare against,
we only consider the reconstruction results for the centre time $t=0$, that
is, the reconstructed snapshot $u_{0}$. We use two different error measures to
evaluate the reconstruction results. The first is simply the unbalanced
Wasserstein divergence $W_{2,R}^{2}(u_{0}^{\dagger},u_{0})$ as defined in
Section 4.1, where $u_{0}^{\dagger}$ is the ground truth as before and $u_{0}$
is the measure obtained by placing a Dirac measure at each cell centre of
$G_{u}(0)$ with weights as given in $\vec{u}_{0}$. The radius is chosen as
$R=0.05$ to balance transport and mass creation inside the unit square. The
second error measure uses a clustering method: Since usually the ground truth
particle is not exactly representable on the discrete grid, the solution to
the discretized problem will typically contain a cluster of positive weights
near a true particle position. Therefore, in a post-processing step, we first
set all weights with values below a threshold $w_{\mathrm{min}}=0.1$ to zero.
Afterwards, we cluster neighbouring non-zero weights and create a list of
detected particles, where each cluster is assumed to be a detected particle
with its position equal to the cluster’s centre of mass. Now, a configuration
of particles is seen as correctly reconstructed if there exists a pairing
between true and reconstructed particles at time $t=0$ such that the paired
particles have distance less than $0.01$ and there remain no unpaired
particles.

#### Methods and parameters.

Our goal is to compare the following methods:

  * •

Static: We reconstruct $u_{0}$ by solving

$\min_{u\in\mathcal{M}_{+}(\Omega)}\lVert
u\rVert_{\mathcal{M}}+\frac{1}{2\alpha}\lVert\mathrm{Ob}_{0}u-f_{0}\rVert_{2}^{2}$

(the fidelity-term version of the static reconstruction problem (1)),
disregarding the data $f_{t}$ for $t\in{\mathcal{T}}\setminus\\{0\\}$. In
contrast to (24), the discretization of this problem is straightforward. In
practice, we reuse our implementation of the dimension-reduced problem (24),
setting $\Theta=\varnothing$.

  * •

ADCG: We use the implementation of Alberti et al. in [4], which applies the
Alternating Descent Conditional Gradient method (ADCG) from [12] to the full-
dimensional dynamic formulation

$\min_{\lambda\in\mathcal{M}_{+}(\tilde{\Lambda})}\|\lambda\|_{\mathcal{M}}\quad\text{such
that }\mathrm{Ob}_{t}\,\mathrm{Mv}^{d}_{t}\lambda=f_{t}\quad\text{for all
}t\in{\mathcal{T}}.$

In short, this method is an extension of the well-known Frank–Wolfe method and
works by alternating between adding new source points and differentiable
optimization of source position and weights.

  * •

Dimred. low: Our method with $|\Theta|=3$ and
$|\Sigma\setminus{\mathcal{T}}|=0$.

  * •