neural network $\Phi$. To relate this estimate to $\mathsf{DWF}(f)$, we apply
wavefront set inpainting with a neural network $\Psi$ of U-Net architecture
[72], this is explained in detail in Subsection 5.5. Overall, this leads to an
operator $\mathcal{Q}$ that takes as an input $\boldsymbol{g}$ and the neural
networks $\Phi$ and $\Psi$ and outputs an estimate of $\boldsymbol{f}$ and
$\mathsf{DWF}(f)$. We therefore use the loss function

$\displaystyle\operatorname{\mathcal{L}}(\Phi,\Psi,\boldsymbol{g},\boldsymbol{f},\mathsf{DWF}(f))\operatorname{:=}C\operatorname{\ell}_{1}(\boldsymbol{f},\Phi(\boldsymbol{g}))+(1-C)\operatorname{\ell}_{2}(\mathsf{DWF}(f),\Psi(\mathcal{Q}(\boldsymbol{g},\Phi))),$
(5.1)

where $C\in(0,1]$ and $\operatorname{\ell}_{1},\operatorname{\ell}_{2}$ are
appropriate distance measures on discretised images and discretised wavefront
sets that will be discussed in Subsection 3.4 below. This strategy that
jointly trains a reconstruction and a task (wavefront set inpainting) falls in
the framework of _task-adapted reconstruction_ was introduced in [1]. Based on
the loss function of (5.1), we now train the neural networks $\Phi$ and $\Psi$
to minimise the objective

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\operatorname{\mathcal{L}}(\Phi,\Psi,\boldsymbol{g}_{i},\boldsymbol{f}_{i},\mathsf{DWF}(f_{i})).$
(5.2)

To find a minimiser, we use stochastic gradient descent-based optimisation
over the neural network’s weights as is standard in deep learning. Finally,
Subsection 5.6 explores in detail the joint reconstruction in the digital
setting.

###  5.2 Statistical learning framework

We would like to frame the problem of reconstruction and joint wavefront set
extraction as a statistical learning problem [77, 20]. For this, we first
introduce some relevant notions:

######  Definition 5.1 (Setting of statistical learning theory).

Let $X,Y$ be two sets. We call $X$ the _sample space_ , and $Y$ the _label
space_. Further, let $\mathcal{D}$ be a distribution on $X\times Y$, and let
$\ell\colon Y\times Y\to\mathbb{R}$ be a measurable function. We refer to
$\ell$ as _loss function_. Then the _risk_ $\mathrm{R}$ of a measurable
function $h:X\to Y$ is defined as

$\mathrm{R}(h)\operatorname{:=}\mathbb{E}_{(x,y)\sim\mathcal{D}}(\ell(h(x),y)).$

For $m\in\mathbb{N}$, a set of samples
$(x_{i},y_{i})_{i=1}^{m}\sim\mathcal{D}^{m}$ and for a hypothesis class
$\mathcal{H}$, we define the _empirical risk minimiser_ $h^{*}$ as

$h^{*}\operatorname{:=}\operatorname*{arg\min}_{h\in\mathcal{H}}\frac{1}{m}\sum_{i=1}^{m}\ell(h(x_{i}),y_{i})).$

Under certain assumptions on the complexity of the hypothesis set
$\mathcal{H}$, it can be shown that for sufficiently large $m$, the function
$h^{*}$ will also achieve a small risk [77, 20]. This means that the empirical
risk minimiser $h^{*}$ resolves the unknown relation between input and output
described by $\mathcal{D}$. Because of this, we treat the empirical risk
minimisation problem in this work as the central problem of interest.

Next we would like to phrase the problems of digital tomographic
reconstruction, of digital visible wavefront set extraction, of full digital
wavefront set extraction via inpainting, and of the joint reconstruction and
digital wavefront set extraction and inpainting in the framework of
statistical learning theory.

###  5.3 Digital tomographic reconstruction

A joint probability distribution on function spaces as described through the
statistical formulation of the tomographic reconstruction problem in
Definition 3.3 implies, via the discretisation procedure of Subsection 3.4, a
joint probability distribution $\mathcal{D}_{\mathrm{rec}}$ on discretised
image-sinogram data. In the discrete problem, the sample space is
$\mathbb{R}^{I_{s}}\eqqcolon X$ for $I_{s}\subset[m_{1}]\times[m_{2}]$, where
$[M]\operatorname{:=}\\{1,\dots,M\\}$, with the label space being defined by
$\mathbb{R}^{n_{1}\times n_{2}}\eqqcolon Y$. As loss function we choose
$\ell_{\mathrm{rec}}$ to be the squared Euclidean distance between two
elements in $Y$, i.e.,

$\displaystyle\ell_{\mathrm{rec}}(y_{1},y_{2})\operatorname{:=}\sum_{i,j=1}^{n_{1},n_{2}}|(y_{1})_{i,j}-(y_{2})_{i,j}|^{2}.$
(5.3)

Moreover, we choose as a hypothesis set
$\mathcal{H}_{\mathrm{rec}}\operatorname{:=}(h_{\vartheta})_{\vartheta\in\Theta_{\mathrm{res}}}$
the set of functions which perform the mapping $f_{0}\mapsto f_{I}$ according
to Algorithm 1 parametrised via the weights of the discrete two-dimensional
convolutional ResNets introduced in Definition 2.1.

###  5.4 Digital visible wavefront set extraction

Similarly to the previous Subsection 5.3, the discretisation procedure of
Subsection 3.4 implies a joint distribution between the digital wavefront set
of a measured sinogram and the digital visible wavefront set of the image,
precisely defined below:

######  Definition 5.2.

Let $I_{s}\subset[m_{1}]\times[m_{2}]$ be a digital grid in the sinogram
domain and $I_{d}\subset[180]$. The _digital visible wavefront set of the
sinogram data_ is given by
$X_{\mathrm{vis}}\operatorname{:=}\mathbb{R}^{I_{s}}\times\\{0,1\\}^{I_{d}}$,
where $I_{d}$ represents the set of _visible angles_. Moreover, let
$J_{s}\subset[n_{1}]\times[n_{2}]$ be a digital grid in the image domain and
$J_{d}\subset[180]$, determined through the canonical relation for the Radon
transform. The _digital visible wavefront set of the image data_ is given by
$Y_{\mathrm{vis}}\operatorname{:=}\mathbb{R}^{J_{s}}\times\\{0,1\\}^{J_{d}}$.

The underlying loss function over $Y_{\mathrm{vis}}$ is:

$\ell_{\mathrm{vis}}(y,y^{\prime})\operatorname{:=}-\sum_{(i_{1},i_{2})\in
J_{s}}\sum_{i_{3}\in
J_{d}}y_{i_{1},i_{2},i_{3}}\log\bigl{(}y_{i_{1},i_{2},i_{3}}^{\prime}\bigr{)}\quad\text{for
$y,y^{\prime}\in Y_{\mathrm{vis}}$.}$

As a hypothesis set we choose
$\mathcal{H}_{\mathrm{vis}}\operatorname{:=}(\hat{h}_{\vartheta})_{\vartheta\in\Theta_{\mathrm{res}}}$
as the set of maps resulting from the digital canonical relation described in
Subsection 4.3.6. These maps are parametrised via the weights of the discrete
two-dimensional convolutional ResNets introduced in Definition 2.1. The
important point to notice here is that _$\mathcal{H}_{\mathrm{vis}}$ and
$\mathcal{H}_{\mathrm{rec}}$ are parametrised by the same parameter set_.

###  5.5 Digital wavefront set extraction and inpainting

To reconstruct the full digital wavefront set of an image from its sinogram
data, we combine the visible wavefront set extraction of the previous
subsection with a wavefront set inpainting step which will be performed using
a U-Net. In this case, the sample set is
$X_{\mathrm{inp}}\operatorname{:=}\mathbb{R}^{I_{s}}\times\\{0,1\\}^{I_{d}}$,
where $I_{s}\subset[m_{1}]\times[m_{2}]$ and $I_{d}\subset[180]$, and label
space has consequently to be chosen as
$Y_{\mathrm{inp}}\operatorname{:=}\mathbb{R}^{n_{1}\times
n_{2}}\times\\{0,1\\}^{180}$. The loss function is then

$\ell_{\mathrm{inp}}(y,y^{\prime})\operatorname{:=}-\sum_{i_{1},i_{2}=1}^{n_{1},n_{2}}\sum_{i_{3}=1}^{180}y_{i_{1},i_{2},i_{3}}\log\bigl{(}y_{i_{1},i_{2},i_{3}}^{\prime}\bigr{)}\quad\text{
for }y,y^{\prime}\in Y_{\mathrm{inp}}.$

As a hypothesis set $\mathcal{H}_{\mathrm{inp}}$, we use the set of maps
resulting from the digital canonical relation described in Subsection 4.3.6
composed with a U-Net. These maps are parametrised via the weights of the
discrete two-dimensional convolutional ResNets introduced in Definition 2.1 as
well as by the weights of the U-Net. Because of this, we can write
$\mathcal{H}_{\mathrm{inp}}\operatorname{:=}(\boldsymbol{h}_{\vartheta,\theta})_{\vartheta\in\Theta_{\mathrm{res}},\theta\in\Theta_{U}}$.

###  5.6 Joint reconstruction and digital wavefront set extraction and
inpainting

We will now describe a joint approach in which the reconstruction problem as
well as the wavefront set reconstruction and inpainting problem are solved
simultaneously. This task certainly requires that the hypotheses classes
underlying the risk minimisation problem are, at least partially, parametrised
by the same parameter set.

This joint approach leads to the sample space being chosen as
$X_{\mathrm{joint}}\operatorname{:=}\mathbb{R}^{I_{s}}\times\mathbb{R}^{I_{s}}\times\\{0,1\\}^{I_{d}}$
for $I_{s}\subset[m_{1}]\times[m_{2}]$ and $I_{d}\subset[180]$, and the label
space being defined as
$Y_{\mathrm{inp}}\operatorname{:=}\mathbb{R}^{n_{1}\times
n_{2}}\times(\mathbb{R}^{n_{1}\times n_{2}}\times\\{0,1\\}^{180})$. The loss
function needs then to be chosen $\ell_{\mathrm{joint}}$ as a weighted average
of $\ell_{\mathrm{rec}}$ and $\ell_{\mathrm{inp}}$ in the following sense: For
a $\lambda\in(0,1]$ we set, for $g_{1},g_{2}\in\mathbb{R}^{n_{1}\times
n_{2}},(h_{1},h_{2})\in\mathbb{R}^{n_{1}\times n_{2}}\times\\{0,1\\}^{180}$

$\displaystyle\ell_{\mathrm{joint}}((g_{1},h_{1}),(g_{2},h_{2}))\operatorname{:=}\lambda\ell_{\mathrm{rec}}(g_{1},g_{2})+(1-\lambda)\ell_{\mathrm{inp}}(h_{1},h_{2}).$
(5.4)

The task-adapted reconstruction architecture involving the joint loss function
(5.4) is depicted in Figure 4.

Finally, the hypothesis set for the joint approach is given by
$\mathcal{H}_{\mathrm{joint}}\operatorname{:=}((h_{\vartheta},\boldsymbol{h}_{\vartheta,\theta}))_{\vartheta\in\Theta_{\mathrm{res}},\theta\in\Theta_{U}}$.

##  6 Numerical results

We now provide numerical results for the joint reconstruction and wavefront
set extraction algorithm of Section 5. We first present the set-up, followed
by a description of our results, and finally an interpretation of the outcome.

###  6.1 Set-up

The set-up follows our viewpoint taken in Subsection 5.6, namely regarding the
joint reconstruction problem from an empirical risk minimisation standpoint.
Therefore, as a training set we use an artificial data set comprised of
piecewise smooth functions, where the singularity curves are given by random
splines of degree at most four and the smooth regions are polynomials of
degree at most two. We call this data set the _random cartoon-functions data
set_. Some examples of functions in this data set were shown in Figure 2.
Notice that for the images in this data set the digital wavefront set is
known. We then study two types of partial data, namely limited-angle data,
where we assume that a wedge of $40^{\circ}$ is missing (a total angle
interval of $80^{\circ}$), and sparse-view, where we only measure $40$ angles.
This is followed by evaluating the trained task-adapted architecture (see
Figure 4) on the OASIS dataset [53] formed by real brain scans (see
https://www.oasis-brains.org/).

The results are then compared with classical approaches for the limited-angle
and sparse-view tomography, including filtered back-projection [50], Tikhonov
[82], and total variation [88]. In addition, we have also performed varitional
regularisation using as regulariser the L2 and L1-norm of the shearlet
coefficients of the reconstruction [37, 52]. We named these methods
_Shearlet-L2 sparse_ and _Shearlet-L1 sparse_. The deep-learning based
benchmarks, namely the Learned Primal-Dual [2] and the Phantom-Net
architectures [13], were similarly trained using the random cartoon functions
data set. The code to reproduce our experiments can be found in
http://shearlab.math.lmu.de/applications.

###  6.2 Results

We now present some exemplary results for our algorithm in the case of
limited-angle tomography ($80^{\circ}$ wedge) and sparse-view ($40$ measured
angles). The visual results are depicted in Figures 5 and 6. The comparison
with the benchmark approaches from Subsection 6.1 is shown in Table
LABEL:table:T3.

In addition, we show in Figures 7 and 8 reconstruction results of the limited-
angle and sparse-view tomography for the benchmarks already mentioned in
Subsection 6.1. In all cases, the joint approach outperforms all classical
approaches. Table LABEL:table:T3 presents the performance measure of the
benchmarks in terms of the average self similarity (SSIM) and peak signal-to-
noise ratio (PSNR).

Ground truth

Reconstruction using LPD without the wavefront set prior
(PSNR 24.90)

Reconstruction using the joint approach introduced in this work
(PSNR 30.20)

Visible wavefront set of the ground truth extracted by DeNSE

Inpainted wavefront set using U-Net without simultaneous reconstruction

Reconstructed wavefront set using our joint approach

Figure 5: Realistic data set results on the joint CT reconstruction and WFset
inpainting for limited-angle case, missing wedge = $40^{\circ}$.

Ground truth

Reconstruction using LPD without the wavefront set prior
(PSNR 26.91)

Reconstruction using the joint approach introduced in this work
(PSNR 30.22)

Visible wavefront set of the ground truth extracted by DenSE

Inpainted wavefront set using U-Net without simultaneous reconstruction

Reconstructed wavefront set using our joint approach

Figure 6: Realistic dataset results on the joint CT reconstruction and WFset
inpainting for sparse-view case, number of angles = 40.

###  6.3 Interpretation

In Table LABEL:table:T3, we see that the joint approach significantly
outperforms all competing approaches. The same observation can be made by
observing Figures 8 and 7 that demonstrate a strong improvement in the
reconstruction accuracy in comparison with the other methods and in particular
to the Learned Primal-Dual architecture. In this context, it is noteworthy
that the performance gap in this application is much more pronounced than in
the sparse-view application. Since the Learned Primal-Dual corresponds to one
half of the pipeline of the joint approach, we observe that the additional
wavefront set information is especially helpful in the limited-angle case.
Since the gaps in the visible wavefront set are significantly smaller in the
sparse-view set-up, we expect that, in this set-up, considerably more
information on the wavefront set is already included in the observed data,
whereas in the limited-angle set-up more prior knowledge needs to be invoked.
It appears as if this necessary prior knowledge was very successfully
identified in the training phase of the joint approach.

In Figures 5 and 6 we performed a type of ablation study by looking at the
prowess of the inpainting algorithm alone. It transpires from those figures
that the inpainting is less successful if it is not coupled with the
reconstruction pipeline. This clearly shows that the strength of the joint
approach does not solely lie in wavefront set identification and inpainting
but in the interplay of inpainting and reconstruction jointly.

We wish to remark that _our algorithm is not trained on real-world images_ but
instead on artificial phantoms from the random cartoon-functions data set.
This shows that the joint prior that incorporates physical information is not
based on memorisation. From the standpoint of applications, this could be a
very desirable feature, as it prevents overfitting on specific biases in real-
world data sets.

In Figure 6, we see that the learned primal dual, as well as the joint
approach, appear to produce reconstructions that exhibit almost no texture-
like areas but instead are piecewise smooth. This effect could potentially be
an effect of the underlying data set. In this context, it may be worthwhile to
expand the training data set by artificial images with more texture like
features.

## Acknowledgements

GK acknowledges support from DFG-SFB/TR 109 Grant C09 and DFG-SPP 2298 Grant
KU 1446/32-1. HA acknowledges support from DFG-SFB/TR 109 Grant C09.

Ground truth

Filtered Back-projection
(PSNR 14.40)

Tikhonov regularisation
(PSNR 21.97)

Total variation regularisation
(PSNR 22.98)

Shearlet-L2 sparse regularisation
(PSNR 22.15)

Shearlet-L1 sparse regularisation
(PSNR 22.35)

PhantomNet
(PSNR 26.02)

Learned primal-dual
(PSNR 25.02)

Joint approach
(PSNR 29.95)

Figure 7: Realistic dataset results for general benchmarks for limited-angle
CT, wedge = $40^{\circ}$.

Ground truth

Filtered back-projection
(PSNR 18.95)

Tikhonov regularisation
(PSNR 25.05)

Total variation regularisation
(PSNR 26.32)

Shearlet-L2 sparse regularisation
(PSNR 24.22)

Shearlet-L1 sparse regularisation
(PSNR 25.31)

Learned Primal-Dual
(PSNR 26.91)

Joint approach
(PSNR 30.22)

Figure 8: Realistic data set results for general benchmarks for sparse-view CT
using $40$ angles only.

##  Appendix A Notions and results from distribution theory

Below, we collect some essential notions and results from distribution theory
and the analysis of the wavefront set. These were used extensively in
Subsection 4.3.4.

######  Definition A.1 (Essential support [73]).

Let $\Omega\subset\mathbb{R}^{n}$ be a domain. The _essential support_ of
$f\in\mathcal{S}^{\prime}(\Omega)$ is the defined as the set

$\operatorname*{ess\\!\sup}(f)\operatorname{:=}\mathbb{R}^{n}\setminus\bigcup_{U\in\mathfrak{U}}U,$

where $\mathfrak{U}\operatorname{:=}\bigl{\\{}U\subset\Omega:U\text{ is open
and }f\bigl{|}_{U}=0\bigr{\\}}$ with $f\bigl{|}_{U}$ denoting the restriction
of $f$ to $U\subset\Omega$.

Note that if $f\in\mathcal{C}(\Omega)$, then
$\operatorname*{ess\\!\sup}(f)=\operatorname{supp}(f)$, i.e., the essential
support coincides with the usual support. We also introduce the notion of
positive and negative supports of a distribution.

######  Definition A.2 (Positive support, [73]).

Let $\Omega\subset\mathbb{R}^{n}$ be a domain. The _positive support_ of
$f\in\mathcal{S}^{\prime}(\Omega)$ is the defined as the set

$\operatorname{supp}_{+}(f)\operatorname{:=}\overline{\bigcup_{U\in\mathfrak{U}}U},$

where $\mathfrak{U}\operatorname{:=}\bigl{\\{}U\subset\Omega:f(\phi)>0\text{
for all
}\phi\in\mathcal{S}(\Omega)\setminus\\{0\\},\operatorname{supp}\phi\subset
U,\phi\geq 0\bigr{\\}}$. Finally, the _negative support_ of $f$ is defined as
$\operatorname{supp}_{-}(f)\operatorname{:=}(\overline{\operatorname{supp}_{+}(f)})^{c}$.

From the definition, we see that

$\operatorname{supp}_{+}(f)=\operatorname{supp}\bigl{(}|f|+f\bigr{)}=\overline{\\{x\in\Omega:f(x)>0\bigr{\\}}}\quad\text{whenever
$f\in\mathcal{C}^{\infty}(\Omega)$.}$

We now turn our attention to defining the product of two distributions.

######  Definition A.3 (Product of distributions, [11, Definition 2]).

Let $\Omega\subset\mathbb{R}^{n}$ be a fixed domain and consider the
distributions $u,v\in\mathcal{S}^{\prime}(\Omega)$. We say that
$w\in\mathcal{S}^{\prime}(\Omega)$ is the _product of $u$ and $v$_ if an only
if, for each $x\in\mathbb{R}^{n}$, there exists a test function
$\psi\in\mathcal{S}(\Omega)$, with $\psi=1$ on a neighbourhood of $x$, so that

$\widehat{\psi^{2}w}(\xi)=\bigl{(}\widehat{\psi u}\ast\widehat{\psi
v}\bigr{)}(\xi)=\int_{\mathbb{R}^{n}}\widehat{\psi u}(\xi)\,\widehat{\psi
v}(\xi-\eta)\,d\eta$ (A.1)

converges absolutely for each $\xi\in\mathbb{R}^{n}$. In addition, under this
conditions, we can define $w$ by (A.1).

######  Theorem A.4 (Product theorem/Hörmander condition, [11, Section 3.2]).

Let $\Omega\subset\mathbb{R}^{n}$ and $u,v\in\mathcal{S}^{\prime}(\Omega)$.
Assume there are no points $(x,\xi)\in\operatorname{WF}(u)$ such that
$(x,-\xi)\in\operatorname{WF}(v)$. Then, $w$ defined as in Definition A.3 is a
well-defined unique distribution that is the product $uv$. Moreover, in this
case we have

$\operatorname{WF}(uv)\subset S_{+}\cup S_{u}\cup S_{v},$

where

$\displaystyle S_{+}$
$\displaystyle\operatorname{:=}\Bigl{\\{}\bigl{(}x,(\xi+\omega)/\|\xi+\omega\|_{2}\bigr{)}:(x,\xi)\in\operatorname{WF}(u)\text{
and }(x,\omega)\in\operatorname{WF}(v)\Bigr{\\}},$ $\displaystyle S_{u}$
$\displaystyle\operatorname{:=}\bigl{\\{}(x,\xi):(x,\xi)\in\operatorname{WF}(u)\text{
and }x\in\operatorname{supp}(v)\bigr{\\}},$ $\displaystyle S_{v}$
$\displaystyle\operatorname{:=}\bigl{\\{}(x,\omega):(x,\omega)\in\operatorname{WF}(v)\text{
and }x\in\operatorname{supp}(u)\bigr{\\}}.$

######  Remark A.5.

If $u,v\in L^{2}_{\mathrm{loc}}(\mathbb{R})$ and we define $uv(x)=u(x)v(x)$
almost everywhere, then the multiplication of $u$ and $v$ defined in (A.1)
coincides with $uv$ almost everywhere. This holds even if there exist
$(x,\lambda)\in\operatorname{WF}(u)$ such that
$(x,-\lambda)\in\operatorname{WF}(v)$. To see this, let $x\in\mathbb{R}^{2}$
and $\psi$ be as in Definition A.3. Then

$\widehat{\psi^{2}(uv)}(\xi)=\int_{\mathbb{R}^{2}}\widehat{\psi
u}(\xi)\widehat{\psi v}(\nu-\xi)d\xi$

holds in an $L^{2}$ sense. Moreover, by Plancherel’s identity, we have that
$\widehat{\psi u},\widehat{\psi v}\in L^{2}(\mathbb{R}^{2})$, which yields
with the Cauchy-Schwarz identity, that

$\int_{\mathbb{R}^{2}}\bigl{|}\widehat{\psi u}(\xi)\widehat{\psi
v}(\nu-\xi)\bigr{|}d\xi\leq\|\widehat{\psi u}\|_{2}\,\bigl{\|}\widehat{\psi
v}(\nu-\cdot)\bigr{\|}_{2}=\|\widehat{\psi u}\|_{2}\,\|\widehat{\psi
v}\|_{2}<\infty.$

This yields absolute convergence in (A.1).

######  Definition A.6.

The $L^{2}$-support of $f\in\mathcal{S}^{\prime}(\Omega)$ is defined as the
largest open set on $\Omega$ where $f$ is given by an $L^{2}$-function:

$\operatorname{supp}_{L^{2}}(h)\operatorname{:=}\bigcup\Bigl{\\{}U\subset\Omega\text{
open }:f\bigl{|}_{U}\in\mathcal{S}^{\prime}(U)\Bigr{\\}}.$

Thus, if $x\in\operatorname{supp}_{L^{2}}(h)$ then there is an open set $x\in
U\subset\Omega$ and $f_{U}\in L^{2}(U)$ such that
$f(\phi)=\int_{U}f_{U}(x)\phi(x)dx$ for all $\phi\in\mathcal{S}(U)$.

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