# Gauge Invariance of Equilibrium Statistical Mechanics

Johanna Müller  Theoretische Physik II, Physikalisches Institut, Universität
Bayreuth, D-95447 Bayreuth, Germany  Sophie Hermann  Theoretische Physik II,
Physikalisches Institut, Universität Bayreuth, D-95447 Bayreuth, Germany
Florian Sammüller  Theoretische Physik II, Physikalisches Institut,
Universität Bayreuth, D-95447 Bayreuth, Germany  Matthias Schmidt
Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95447
Bayreuth, Germany<EMAIL_ADDRESS>
(27 June 2024)

###### Abstract

We identify a recently proposed shifting operation on classical phase space as
a gauge transformation for statistical mechanical microstates. The
infinitesimal generators of the continuous gauge group form a non-commutative
Lie algebra, which induces exact sum rules when thermally averaged. Gauge
invariance with respect to finite shifting is demonstrated via Monte Carlo
simulation in the transformed phase space which generates identical
equilibrium averages. Our results point towards a deeper basis of statistical
mechanics than previously known and they offer avenues for systematic
construction of exact identities and of sampling algorithms.

One of the arguably most important applications of Noether’s theorem of
invariant variations [1, 2, 3, 4] is the systematic treatment of local gauge
invariances within the fundamental physical field theories for the
electromagnetic, weak, and strong interactions. Using the corresponding
continuous gauge groups $\rm U(1)$, $\rm SU(2)$, and $\rm SU(3)$ as
fundamental building blocks for theory construction is one of the most
successful strategies in modern physics. The nature of the physical mechanisms
that underlie the symmetry do however not feature explicitly in Noether’s
theorem. The theorem rather constitutes a power tool to obtain exact
equations, usually in the form of global or local conservation laws, from an
underlying continuous symmetry that needs to have been identified within (or
input into) a variational formulation of the considered physics.

The roots of statistical mechanics are older than the modern gauge field
theories. Nevertheless, Noether’s theorem has been applied only relatively
recently in various different productive ways to the physics of equilibrium
and nonequilibrium many-body systems [5, 6, 7, 8, 9, 10, 11, 12]. The role
that exact sum rules [13, 14, 15, 16, 17, 18] play in statistical mechanics is
akin to that of conservation laws in dynamical theories, in that they allow to
constrain and rationalize the nature of the physics, without in general
determining the full solution of the problem at hand.

In a range of recent investigations Noether’s theorem has been applied to a
specific shifting operation on phase space [19, 20, 21, 22, 23, 24, 25], where
instead of the more usual conservation laws both well-known and new
statistical mechanical sum rules were obtained systematically. Thereby
Noether’s concept of invariance against continuous transformation is applied
to statistical mechanical functionals, such as the partition sum. While
similarities with global spatial translational invariance, as generates linear
momentum conservation, were discussed [19, 20], neither the physical nature
nor the mathematical structure of the general phase space shifting
transformation [22, 23, 24, 25, 26, 27] have been unravelled.

Here we identify the phase space shifting transformation [19, 20, 21, 22, 23,
24, 25] as a local gauge symmetry transformation that is inherent to the
statistical mechanics of particle-based systems. Realizing the defining
feature of a gauge transformation, the application of the local shifting has
no effect on any physical observables. Despite the shifting being geometric,
the transformation is non-commutative, even when displacing only
infinitesimally. A non-commutative Lie algebra of generators characterizes
infinitesimal transformations. Corresponding exact sum rules follow for
thermal averages. Finite transformations retain the gauge invariance as we
demonstrate via Monte Carlo computer simulations.

The shifting operation put forward in Refs. [19, 20, 22, 21, 23, 24, 25]
affects the positions ${\mathbf{r}}_{i}$ and momenta ${\bf p}_{i}$ of each
particle $i=1,\ldots,N$ via the following transformation:

$\displaystyle{\mathbf{r}}_{i}$
$\displaystyle\to{\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}({\mathbf{r}}_{i})=\tilde{\mathbf{r}}_{i},$
(1) $\displaystyle{\bf p}_{i}$
$\displaystyle\to[{\mathbbm{1}}+\nabla_{i}{\boldsymbol{\epsilon}}({\mathbf{r}}_{i})]^{-1}\cdot{\bf
p}_{i}=\tilde{\bf p}_{i},$ (2)

where the $d$-dimensional vector field
${\boldsymbol{\epsilon}}({\mathbf{r}}_{i})$ is such that Eq. (1) is a
diffeomorphism, i.e., together with its inverse is bijective and smooth; $d$
is the spatial dimensionality and the tilde indicates the new phase space
variables. In Eq. (2) the symbol ${\mathbbm{1}}$ denotes the $d\times d$ unit
matrix, $\nabla_{i}$ is the derivative with respect to ${\mathbf{r}}_{i}$, and
the superscript $-1$ denotes matrix inversion. The transformation is canonical
in the sense of classical mechanics [28] and hence the differential phase
space volume element is preserved, $d{\mathbf{r}}_{i}d{\bf
p}_{i}=d\tilde{\mathbf{r}}_{i}d\tilde{\bf p}_{i}$. This property is
fundamental for thermal averages to arise as invariant under the application
of Eqs. (1) and (2).

To be specific, we consider the statistical mechanics of Hamiltonians $H$ with
the standard form

$\displaystyle H$ $\displaystyle=\sum_{i}\frac{{\bf
p}_{i}^{2}}{2m}+u({\mathbf{r}}^{N})+\sum_{i}V_{\rm ext}({\mathbf{r}}_{i}),$
(3)

where the sums run over all $N$ particle indices $i$, $m$ denotes the particle
mass, $u({\mathbf{r}}^{N})$ is the interparticle interaction potential, and
$V_{\rm ext}({\mathbf{r}})$ is an external one-body potential. We use the
shorthand notation ${\mathbf{r}}^{N}={\mathbf{r}}_{1},\ldots,{\mathbf{r}}_{N}$
and ${\bf p}^{N}={\bf p}_{1},\ldots,{\bf p}_{N}$ to indicate the phase space
variables of all particles. The statistical mechanics is based on the grand
ensemble with chemical potential $\mu$ and temperature $T$. The grand
partition sum is $\Xi={\rm Tr}\,{\rm e}^{-\beta(H-\mu N)}$, where the
classical trace is defined as ${\rm
Tr}\,\cdot=\sum_{N=0}^{\infty}(N!h^{dN})^{-1}\int d{\mathbf{r}}^{N}d{\bf
p}^{N}\cdot$, with $\int d{\mathbf{r}}^{N}d{\bf p}^{N}$ denoting the phase
space integral over the position and momentum coordinates of all $N$
particles, $\beta=1/(k_{B}T)$, and $k_{B}$ denoting the Boltzmann constant.
The grand potential is $\Omega=-k_{B}T\ln\Xi$ and thermal averages are
obtained as $\langle\cdot\rangle={\rm Tr}\,\cdot{\rm e}^{-\beta(H-\mu
N)}/\Xi$.

We here introduce operator methods to capture the essence of the phase space
shifting (1) and (2). Specifically we define the following, at each position
${\mathbf{r}}$ localized, phase space shifting operators:

$\displaystyle\boldsymbol{\sigma}({\mathbf{r}})$
$\displaystyle=\sum_{i}\big{[}\delta({\mathbf{r}}-{\mathbf{r}}_{i})\nabla_{i}+{\bf
p}_{i}\nabla\delta({\mathbf{r}}-{\mathbf{r}}_{i})\cdot\nabla_{{\bf
p}_{i}}\big{]},$ (4)

where $\delta(\cdot)$ denotes the Dirac distribution in $d$ dimensions,
$\nabla$ indicates the derivative with respect to position ${\mathbf{r}}$,
$\nabla_{{\bf p}_{i}}$ is the momentum derivative with respect to ${\bf
p}_{i}$, and we recall that $\nabla_{i}$ is the derivative with respect to
${\mathbf{r}}_{i}$. The shifting operators (4) possess two key properties.
First $\boldsymbol{\sigma}({\mathbf{r}})$ is anti-self-adjoint on phase space:

$\displaystyle\boldsymbol{\sigma}^{\dagger}({\mathbf{r}})=-\boldsymbol{\sigma}({\mathbf{r}}).$
(5)

The adjoint operator is indicated by the dagger and it has the standard
definition: $\int d{\mathbf{r}}^{N}d{\bf
p}^{N}f\boldsymbol{\sigma}({\mathbf{r}})g=\int d{\mathbf{r}}^{N}d{\bf
p}^{N}g\boldsymbol{\sigma}^{\dagger}({\mathbf{r}})f$ for arbitrary phase space
functions $f({\mathbf{r}}^{N},{\bf p}^{N})$ and $g({\mathbf{r}}^{N},{\bf
p}^{N})$. Equation (5) is readily proven via phase space integration by parts
and the product rule (for $f$ and $g$ being well-behaved).

Secondly, the consecutive action of two shifting operators that are
respectively localized at positions ${\mathbf{r}}$ and ${\mathbf{r}}^{\prime}$
satisfies the commutator relation:

$\displaystyle[\boldsymbol{\sigma}({\mathbf{r}}),\boldsymbol{\sigma}({\mathbf{r}}^{\prime})]$
$\displaystyle=\boldsymbol{\sigma}({\mathbf{r}}^{\prime})[\nabla\delta({\mathbf{r}}-{\mathbf{r}}^{\prime})]+[\nabla\delta({\mathbf{r}}-{\mathbf{r}}^{\prime})]\boldsymbol{\sigma}({\mathbf{r}}).$
(6)

We have used the standard definition of commutators of vectors:
$[\boldsymbol{\sigma}({\mathbf{r}}),\boldsymbol{\sigma}({\mathbf{r}}^{\prime})]=\boldsymbol{\sigma}({\mathbf{r}})\boldsymbol{\sigma}({\mathbf{r}}^{\prime})-\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})^{\sf
T}$, where the superscript $\sf T$ denotes matrix transposition, such that the
(Cartesian) $ab$-component is
$[\sigma_{a}({\mathbf{r}}),\sigma_{b}({\mathbf{r}}^{\prime})]=\sigma_{a}({\mathbf{r}})\sigma_{b}({\mathbf{r}}^{\prime})-\sigma_{b}({\mathbf{r}}^{\prime})\sigma_{a}({\mathbf{r}})$.
Equation (6) follows from explicit calculation via applying the sequence of
two shifting operators (4) and simplifying. It is also straightforward to show
that the commutator (6) is anti-self-adjoint:
$[\boldsymbol{\sigma}({\mathbf{r}}),\boldsymbol{\sigma}({\mathbf{r}}^{\prime})]^{\dagger}=-[\boldsymbol{\sigma}({\mathbf{r}}),\boldsymbol{\sigma}({\mathbf{r}}^{\prime})]$,
as is a general property of the commutator of two anti-self-adjoint operators.
Furthermore the commutator (6) is anti-symmetric:
$[\boldsymbol{\sigma}({\mathbf{r}}),\boldsymbol{\sigma}({\mathbf{r}}^{\prime})]=-[\boldsymbol{\sigma}({\mathbf{r}}^{\prime}),\boldsymbol{\sigma}({\mathbf{r}})]^{\sf
T}$, and it satisfies the Jacobi identity:
$[\sigma_{a}({\mathbf{r}}),[\sigma_{b}({\mathbf{r}}^{\prime}),\sigma_{c}({\mathbf{r}}^{\prime\prime})]]+[\sigma_{b}({\mathbf{r}}^{\prime}),[\sigma_{c}({\mathbf{r}}^{\prime\prime}),\sigma_{a}({\mathbf{r}})]]+[\sigma_{c}({\mathbf{r}}^{\prime\prime}),[\sigma_{a}({\mathbf{r}}),\sigma_{b}({\mathbf{r}}^{\prime})]]=0$,
as can be be proven by explicit calculation.

The above set of distinctive properties of $\boldsymbol{\sigma}({\mathbf{r}})$
is closely connected to a Lie algebra structure of infinitesimal phase space
shifting, as we lay out in the following. That the operators (6) represent
infinitesimal versions of the phase space shifting according to (1) and (2)
can be seen by multiplying with a given shifting field
${\boldsymbol{\epsilon}}({\mathbf{r}})$ and integrating over ${\mathbf{r}}$ to
generate an operator $\Sigma[{\boldsymbol{\epsilon}}]=\int
d{\mathbf{r}}{\boldsymbol{\epsilon}}({\mathbf{r}})\cdot\boldsymbol{\sigma}({\mathbf{r}})$
that shifts according to the given form of
${\boldsymbol{\epsilon}}({\mathbf{r}})$. Using
$\boldsymbol{\sigma}({\mathbf{r}})$ in the form (4) and integrating gives

$\displaystyle\Sigma[{\boldsymbol{\epsilon}}]=\sum_{i}\big{\\{}{\boldsymbol{\epsilon}}({\mathbf{r}}_{i})\cdot\nabla_{i}-[\nabla_{i}{\boldsymbol{\epsilon}}({\mathbf{r}}_{i})]:{\bf
p}_{i}\nabla_{{\bf p}_{i}}\big{\\}}.$ (7)

The colon in Eq. (7) indicates a double tensor contraction (trace of the
product of the two matrices) and the phase space shifting operator
$\Sigma[{\boldsymbol{\epsilon}}]$ depends functionally on the shifting field
${\boldsymbol{\epsilon}}({\mathbf{r}})$ as is indicated by the brackets.

By construction a given phase space function $f({\mathbf{r}}^{N},{\bf p}^{N})$
is transported from ${\mathbf{r}}^{N},{\bf p}^{N}$ to
$\tilde{\mathbf{r}}^{N},\tilde{\bf p}^{N}$ to first order in
${\boldsymbol{\epsilon}}({\mathbf{r}}_{i})$ and
$\nabla_{i}{\boldsymbol{\epsilon}}({\mathbf{r}}_{i})$ by applying Eq. (7)
according to:

$\displaystyle f(\tilde{\mathbf{r}}^{N},\tilde{\bf p}^{N})$
$\displaystyle=f({\mathbf{r}}^{N},{\bf
p}^{N})+\Sigma[{\boldsymbol{\epsilon}}]f({\mathbf{r}}^{N},{\bf p}^{N}).$ (8)

Here the phase space variables with and without tilde are related by the
transformation (1) and (2).

Applying two consecutive shifts with respective vector fields
${\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i})$ and
${\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})$ constitutes repeated
application of the shifting operator according to
$\Sigma[{\boldsymbol{\epsilon}}_{2}]\Sigma[{\boldsymbol{\epsilon}}_{1}]$. As
thereby $\Sigma[{\boldsymbol{\epsilon}}_{2}]$ acts on phase space points that
have already been displaced by $\Sigma[{\boldsymbol{\epsilon}}_{1}]$, the
order of consecutive shifting is relevant. The non-commutative nature of the
displacements is illustrated in Fig. 1.

Figure 1: Illustration of the non-commutative nature of phase space shifting.
Given are two different shifting vector fields
${\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i})$ (solid arrows) and
${\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})$ (dashed arrows). Starting from
${\mathbf{r}}_{i}$, the first shift yields the intermediate position
${\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i})$. At this
point the second shifting field
${\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i}))$
is used to give
${\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i})+{\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i}))$
as the final position (two green arrows). Applying the opposite order of
shifting and starting again at ${\mathbf{r}}_{i}$ gives a different
intermediate point
${\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})$. Evaluating
${\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i}))$
at this position yields
${\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})+{\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i}+{\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i}))$
as the final location (two blue arrows). The two final destinations mismatch
in general (red dotted arrow) while the differential phase space volume
element is conserved (boxes).

The commutator of two shifting operators $\Sigma[{\boldsymbol{\epsilon}}_{1}]$
and $\Sigma[{\boldsymbol{\epsilon}}_{2}]$ quantifies the degree of their non-
commutativity. Via explicit calculation on the basis of Eq. (7) one obtains
straightforwardly the identity

$\displaystyle[\Sigma[{\boldsymbol{\epsilon}}_{1}],\Sigma[{\boldsymbol{\epsilon}}_{2}]]$
$\displaystyle=\Sigma[{\boldsymbol{\epsilon}}_{\Delta}].$ (9)

The difference shifting vector field
${\boldsymbol{\epsilon}}_{\Delta}({\mathbf{r}}_{i})$ is thereby obtained from
the given forms of ${\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i})$ and
${\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})$ via

$\displaystyle{\boldsymbol{\epsilon}}_{\Delta}({\mathbf{r}}_{i})$
$\displaystyle={\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i})\cdot[\nabla_{i}{\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})]-{\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})\cdot[\nabla_{i}{\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i})].$
(10)

The right hand side of Eq. (10) constitutes the standard form [29] of the Lie
bracket
$[{\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i}),{\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})]_{\rm
L}$ of the two vector fields ${\boldsymbol{\epsilon}}_{1}({\mathbf{r}}_{i})$
and ${\boldsymbol{\epsilon}}_{2}({\mathbf{r}}_{i})$. Via replacing the
functional argument ${\boldsymbol{\epsilon}}_{\Delta}({\mathbf{r}}_{i})$ on
the right hand side of Eq. (9) by the Lie bracket, we can hence alternatively
express Eq. (9) compactly as
$[\Sigma[{\boldsymbol{\epsilon}}_{1}],\Sigma[{\boldsymbol{\epsilon}}_{2}]]=\Sigma[[{\boldsymbol{\epsilon}}_{1},{\boldsymbol{\epsilon}}_{2}]_{\rm
L}]$.

The relationship (9) constitutes a non-commutative Lie algebra of
infinitesimal generators $\Sigma[{\boldsymbol{\epsilon}}]$ due to three
properties: i) anti-symmetry, ii) bilinearity, as respectively follows from
the definition of the commutator and from linearity of the differential
operator (7), and iii) the Jacobi identity:
$[\Sigma_{1},[\Sigma_{2},\Sigma_{3}]]+[\Sigma_{2},[\Sigma_{3},\Sigma_{1}]]+[\Sigma_{3},[\Sigma_{1},\Sigma_{2}]]=0$,
as can be verified by explicit calculation on the basis of Eqs. (7)–(10); we
have used the shorthand notation
$\Sigma_{1}=\Sigma[{\boldsymbol{\epsilon}}_{1}]$,
$\Sigma_{2}=\Sigma[{\boldsymbol{\epsilon}}_{2}]$, and
$\Sigma_{3}=\Sigma[{\boldsymbol{\epsilon}}_{3}]$.

The variational strategy of Refs. [22, 23, 24, 25] is based on eliminating
explicit occurrences of the shifting fields. This is inline with their present
status as mere gauge functions. Within the functional calculus methods [22,
23, 24, 25], one takes appropriate functional derivatives,
$\delta/\delta{\boldsymbol{\epsilon}}({\mathbf{r}})$, of relevant thermal
averages and then sets the shifting field to zero,
${\boldsymbol{\epsilon}}({\mathbf{r}})=0$. Applying the concept to the present
operator formalism leads to differentiating the shifting operator
$\Sigma[{\boldsymbol{\epsilon}}]$ functionally with respect to
${\boldsymbol{\epsilon}}({\mathbf{r}})$. Calculating
$\delta\Sigma[{\boldsymbol{\epsilon}}]/\delta{\boldsymbol{\epsilon}}({\mathbf{r}})=\boldsymbol{\sigma}({\mathbf{r}})$
on the basis of Eq. (7) is straightforward and reproduces
$\boldsymbol{\sigma}({\mathbf{r}})$ as defined via Eq. (4). The functional
derivative creates spatial localization via the Dirac distribution, as it
emerges from the chain rule and the identity
$\delta{\boldsymbol{\epsilon}}({\mathbf{r}}_{i})/\delta{\boldsymbol{\epsilon}}({\mathbf{r}})=\delta({\mathbf{r}}-{\mathbf{r}}_{i}){\mathbbm{1}}$.
Due to the linearity of Eq. (7) in ${\boldsymbol{\epsilon}}({\mathbf{r}})$ the
dependence on ${\boldsymbol{\epsilon}}({\mathbf{r}})$ has disappeared in Eq.
(4).

We can identify the effect of applying $\boldsymbol{\sigma}({\mathbf{r}})$ to
a generic phase space function $f({\mathbf{r}}^{N},{\bf p}^{N})$ to be
equivalent to the following functional derivatives of the considered function
in the transformed variables:

$\displaystyle\boldsymbol{\sigma}({\mathbf{r}})f({\mathbf{r}}^{N},{\bf
p}^{N})$ $\displaystyle=\frac{\delta f(\tilde{\mathbf{r}}^{N},\tilde{\bf
p}^{N})}{\delta{\boldsymbol{\epsilon}}({\mathbf{r}})}\Big{|}_{{\boldsymbol{\epsilon}}=0},$
(11)
$\displaystyle\boldsymbol{\sigma}({\mathbf{r}})\boldsymbol{\sigma}({\mathbf{r}}^{\prime})f({\mathbf{r}}^{N},{\bf
p}^{N})$ $\displaystyle=\frac{\delta^{2}f(\tilde{\mathbf{r}}^{N},\tilde{\bf
p}^{N})}{\delta{\boldsymbol{\epsilon}}({\mathbf{r}})\delta{\boldsymbol{\epsilon}}({\mathbf{r}}^{\prime})}\Big{|}_{{\boldsymbol{\epsilon}}=0}$
$\displaystyle\quad+[\nabla\delta({\mathbf{r}}-{\mathbf{r}}^{\prime})]\boldsymbol{\sigma}({\mathbf{r}})f({\mathbf{r}}^{N},{\bf
p}^{N}).$ (12)

We recall that the tilde indicates the transformed phase space variables (1)
and (2). Equation (11) follows from differentiating Eq. (8) and the analogous
second-order version Eq. (12) follows iteratively.

As an initial demonstration of the prowess of the localized shifting operator
(4) we apply it to the Hamiltonian with the result

$\displaystyle-\boldsymbol{\sigma}({\mathbf{r}})H$ $\displaystyle=\hat{\bf
F}({\mathbf{r}}),$ (13)

where $\hat{\bf F}({\mathbf{r}})$ is the (total) force density phase space
function, which consists of kinetic, interparticle, and external contributions
according to [30]:

$\displaystyle\hat{\bf F}({\mathbf{r}})$
$\displaystyle=\nabla\cdot\hat{\boldsymbol{\tau}}({\mathbf{r}})+\hat{\bf
F}_{\rm int}({\mathbf{r}})-\hat{\rho}({\mathbf{r}})\nabla V_{\rm
ext}({\mathbf{r}}).$ (14)

The three terms on the right hand side represent kinetic stress density
$\hat{\boldsymbol{\tau}}({\mathbf{r}})=-\sum_{i}{\bf p}_{i}{\bf
p}_{i}\delta({\mathbf{r}}-{\mathbf{r}}_{i})/m$, interparticle force density
$\hat{\bf F}_{\rm
int}({\mathbf{r}})=-\sum_{i}\delta({\mathbf{r}}-{\mathbf{r}}_{i})\nabla_{i}u({\mathbf{r}}^{N})$
and particle density
$\hat{\rho}({\mathbf{r}})=\sum_{i}\delta({\mathbf{r}}-{\mathbf{r}}_{i})$ as
phase space functions.

As a prerequisite for applying the localized shifting operator approach
$\boldsymbol{\sigma}({\mathbf{r}})$ to the thermal physics, we consider its
effect on the Boltzmann factor:

$\displaystyle\boldsymbol{\sigma}({\mathbf{r}}){\rm e}^{-\beta H}$
$\displaystyle=\beta\hat{\bf F}({\mathbf{r}}){\rm e}^{-\beta H}.$ (15)

The result (15) follows from applying the phase space derivatives in
$\boldsymbol{\sigma}({\mathbf{r}})$ as given in Eq. (4) to the exponential,
using the chain rule, and then generating $\hat{\bf F}({\mathbf{r}})$ via Eq.
(13). Applying $\boldsymbol{\sigma}({\mathbf{r}})$ to the entire grand
ensemble probability distribution ${\rm e}^{-\beta(H-\mu N)}/\Xi$ gives no
additional terms, as the partition sum $\Xi$ is not a phase space function and
is hence unaffected by the action of $\boldsymbol{\sigma}({\mathbf{r}})$ and
the presence of the chemical potential contribution ${\rm e}^{\beta\mu N}$
also has no adverse effect. Hence in full analogy to Eq. (15),
$\boldsymbol{\sigma}({\mathbf{r}}){\rm e}^{-\beta(H-\mu N)}/\Xi=\beta\hat{\bf
F}({\mathbf{r}}){\rm e}^{-\beta(H-\mu N)}/\Xi$.

We are now ready to apply the operator algebra to the thermal physics. We
first demonstrate how prior results follow from the framework and hence start
with the thermal average $\langle\boldsymbol{\sigma}({\mathbf{r}})\rangle={\rm
Tr}\,\boldsymbol{\sigma}({\mathbf{r}}){\rm e}^{-\beta(H-\mu
N)}/\Xi=\langle\beta\hat{\bf F}({\mathbf{r}})\rangle$, as is readily obtained
from Eq. (15). On the other hand the anti-self-adjoint property (5) allows,
upon inserting a factor 1 before the shifting operator, to conclude
$\langle\boldsymbol{\sigma}({\mathbf{r}})\rangle=\langle
1\boldsymbol{\sigma}({\mathbf{r}})\rangle=\langle[\boldsymbol{\sigma}^{\dagger}({\mathbf{r}})1]\rangle=-\langle[\boldsymbol{\sigma}({\mathbf{r}})1]\rangle=-\langle
0\rangle=0$. Hence overall $\langle\hat{\bf F}({\mathbf{r}})\rangle=0$, which
is the exact equilibrium one-body force balance [31, 32, 13, 30, 22, 26, 27].

Higher-order identities follow with similar ease. Consider the two-point case,
where
$\langle\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})\rangle=0$,
which follows as above from the adjoint
$\boldsymbol{\sigma}^{\dagger}({\mathbf{r}}^{\prime})1=0$. Consecutively
applying two operators yields in a first step
$\langle\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})\rangle=\langle\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\beta\hat{\bf
F}({\mathbf{r}})\rangle$ according to Eq. (15). In the second step applying
$\boldsymbol{\sigma}({\mathbf{r}}^{\prime})$, using the product rule, and
bearing in mind that the overall result vanishes, one obtains the sum rule
$\beta\langle{\bf F}({\mathbf{r}}^{\prime}){\bf
F}({\mathbf{r}})\rangle=-\langle[\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\hat{\bf
F}({\mathbf{r}})]\rangle$, as previously identified in Refs. [23, 24]. The
term on the right hand side is the mean negative force gradient or
equivalently the mean Hessian of the Hamiltonian,
$-\langle[\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\hat{\bf
F}({\mathbf{r}})]\rangle=\langle[\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})H]\rangle=\langle\delta^{2}H(\tilde{\mathbf{r}}^{N},\tilde{\bf
p}^{N})/[\delta{\boldsymbol{\epsilon}}({\mathbf{r}})\delta{\boldsymbol{\epsilon}}({\mathbf{r}}^{\prime})]|_{{\boldsymbol{\epsilon}}=0}\rangle$,
as re-written first via Eq. (13) and then via the functional derivative
identity Eq. (12) and using that $\langle\hat{\bf F}({\mathbf{r}})\rangle=0$.

The involved force-force and force-gradient correlation functions were shown
to be highly useful measures for the spatial two-body structure of a wide
variety of soft matter systems [23, 24]. Similarly, by introducing a further
physical observable $\hat{A}({\mathbf{r}}^{N},{\bf p}^{N})$ of interest [33,
25], we consider $\langle\boldsymbol{\sigma}({\mathbf{r}})\hat{A}\rangle=0$,
which follows again from using the adjoint (5). Applying the shifting operator
to the functions on its right gives $\langle\beta\hat{\bf
F}({\mathbf{r}})\hat{A}\rangle+\langle[\boldsymbol{\sigma}({\mathbf{r}})\hat{A}]\rangle=0$,
which is the recent general hyperforce sum rule by Robitschko et al. [25].

Figure 2: Monte Carlo results for one-dimensional hard rods confined between
two hard walls with separation distance $L=10\sigma$. Shown are results in the
unshifted (left column) and shifted systems (right column). The shifting field
$\epsilon$ (top panels) displaces both coordinates and momenta. Despite the
different sampling and Markov chain the scaled density profile $\rho(x)\sigma$
(middle panels) and one-body phase space distribution function
$f(x,p)\sigma\sqrt{mk_{B}T}$ (bottom panels) remain numerically identical.

We re-write the second term as the thermal average ${\bf
S}_{A}({\mathbf{r}})=\langle\hat{\bf S}_{A}({\mathbf{r}})\rangle$ where we
have defined the hyperforce phase space function [25] as $\hat{\bf
S}_{A}({\mathbf{r}})=[\boldsymbol{\sigma}({\mathbf{r}})\hat{A}]$ (which is
explicitly $\hat{\bf
S}_{A}({\mathbf{r}})=\sum_{i}\delta({\mathbf{r}}-{\mathbf{r}}_{i})\nabla_{i}\hat{A}$
in case $\hat{A}$ is independent of momenta). The one-body hyperforce sum rule
[25] can then be expressed in the compact form

$\displaystyle\langle\beta\hat{\bf F}({\mathbf{r}})\hat{A}\rangle+{\bf
S}_{A}({\mathbf{r}})$ $\displaystyle=0.$ (16)

Repeated application of the localized shifting can be realized via similar
steps as described above. Sketching a typical case for two shifts, we have
$0=\langle[\boldsymbol{\sigma}^{\dagger}({\mathbf{r}}^{\prime})1]\hat{A}\boldsymbol{\sigma}({\mathbf{r}})\rangle=\langle\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\hat{A}\boldsymbol{\sigma}({\mathbf{r}})\rangle=\langle[\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\hat{A}]\boldsymbol{\sigma}({\mathbf{r}})\rangle+\langle\hat{A}\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})\rangle=\langle[\boldsymbol{\sigma}^{\dagger}({\mathbf{r}})\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\hat{A}]^{\sf
T}\rangle+\langle\hat{A}\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})\rangle$.
The very last term can be made more explicit as
$\langle\hat{A}\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})\rangle=\beta^{2}\langle\hat{A}\hat{\bf
F}({\mathbf{r}}^{\prime})\hat{\bf
F}({\mathbf{r}})\rangle-\beta\langle\hat{A}[\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})H]\rangle$,
which gives upon using Eq. (5) and re-arranging the overall result
$\beta^{2}\langle\hat{A}\hat{\bf F}({\mathbf{r}}^{\prime})\hat{\bf
F}({\mathbf{r}})\rangle=\beta\langle\hat{A}[\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\boldsymbol{\sigma}({\mathbf{r}})H]\rangle+\langle[\boldsymbol{\sigma}({\mathbf{r}})\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\hat{A}]^{\sf
T}\rangle$, which is the two-body hyperforce sum rule of Ref. [25] (the
shifting field in Ref. [25] is set to zero after each individual functional
derivative is taken).

That these exact correlation identities emerge with relatively little effort
from the present operator formalism points to its relevance. Besides technical
efficacy the formalism however allows to reveal the rich additional structure
that is generated by the Lie algebra (9). As a demonstration we multiply Eq.
(6) from the left by $\hat{A}({\mathbf{r}}^{N},{\bf p}^{N})$ and then build
the thermal average. Writing out the resulting sandwich structure of the
integral gives for the first term on the left hand side
$\langle\hat{A}\boldsymbol{\sigma}({\mathbf{r}})\boldsymbol{\sigma}({\mathbf{r}}^{\prime})\rangle=-\langle[\boldsymbol{\sigma}({\mathbf{r}})\hat{A}]\beta{\bf
F}({\mathbf{r}}^{\prime})\rangle=-\langle\hat{\bf
S}_{A}({\mathbf{r}})\beta\hat{\bf F}({\mathbf{r}}^{\prime})\rangle$. Similar
treatment of the second terms yields the following exact Lie sum rule:

$\displaystyle\langle\hat{\bf S}_{A}({\mathbf{r}})\beta\hat{\bf
F}({\mathbf{r}}^{\prime})\rangle-\langle\beta\hat{\bf F}({\mathbf{r}})\hat{\bf
S}_{A}({\mathbf{r}}^{\prime})\rangle$ $\displaystyle\qquad={\bf
S}_{A}({\mathbf{r}}^{\prime})[\nabla\delta({\mathbf{r}}-{\mathbf{r}}^{\prime})]+[\nabla\delta({\mathbf{r}}-{\mathbf{r}}^{\prime})]{\bf
S}_{A}({\mathbf{r}}).$ (17)

The right hand side of Eq. (17) follows from the average of the product of
$\hat{A}$ with the right hand side of Eq. (6) and noting that
$\langle\hat{A}\boldsymbol{\sigma}({\mathbf{r}})\rangle=\langle[-\boldsymbol{\sigma}({\mathbf{r}})\hat{A}]\rangle=-{\bf
S}_{A}({\mathbf{r}})$. The right hand side of Eq. (17) can alternatively be
re-written via Eq. (16).

The significance of the sum rule (17) is that it imprints the structure of the
Lie operator algebra (6) onto measurable spatial correlation functions. There
are two immediate consequences. First, for the case
${\mathbf{r}}\neq{\mathbf{r}}^{\prime}$, the right hand side of Eq. (17)
vanishes and the following nontrivial exchange symmetry emerges:

$\displaystyle\langle\hat{\bf S}_{A}({\mathbf{r}})\hat{\bf
F}({\mathbf{r}}^{\prime})\rangle$ $\displaystyle=\langle\hat{\bf
F}({\mathbf{r}})\hat{\bf S}_{A}({\mathbf{r}}^{\prime})\rangle.$ (18)

Equation (18) implies the invariance of the correlation against exchange of
the force and hyperforce densities at two distinct positions ${\mathbf{r}}$
and ${\mathbf{r}}^{\prime}$.

Secondly, the singular (“self”) contribution that occurs for
${\mathbf{r}}={\mathbf{r}}^{\prime}$ in Eq. (17) does not generate any new
one-body correlation functions. Rather, besides the gradient of the delta
distribution, the one-body hyperforce correlation function ${\bf
S}_{A}({\mathbf{r}})$, as it appears in the one-body hyperforce sum rule (16),
re-emerges. Hence the present example demonstrates both that the Lie algebra
(6) i) systematically interrelates the different $n$-body levels of
correlation identities and ii) that, despite possible re-writings of
equivalent expressions, the set of relevant correlation functions that is
associated with a given observable $\hat{A}$ is closed.

We have thus far considered the infinitesimal structure of phase space
shifting. In standard treatments using the exponential map one generates a Lie
group of finite transformations from a given Lie algebra [29]. Here we use an
alternative route to demonstrate directly the invariance of the thermal
physics under the general transformation (1) and (2). Via particle-based Monte
Carlo simulations we demonstrate the physical reality of the gauge invariance
by considering finite shifting which we perform numerically. We continue to
work with the full phase space variables (as are also relevant for equilibrium
Molecular Dynamics) and hence resolve both position and momentum.

As a representative example we choose the iconic one-dimensional hard rod
system, for which analytical solutions exist [34, 35]. To induce spatial
inhomogeneity we consider confinement between two hard walls. The hard core
nature of this test situation poses a stringent test for the gauge invariance
as the finite particle shifting will in general lead to overlapping particle
configurations and consequentially to a differing sequence of microstates in
the Monte Carlo Markov chain. The one-dimensional shifting field is chosen as
either $\epsilon(x_{i})=0$, which reproduces the original system and
constitutes our reference, or $\epsilon(x_{i})=0.5\sin(4\pi x_{i}/L)$, where
$L=10\sigma$ is the separation distance between the two hard walls, $\sigma$
is the particle diameter, and $x_{i}$ is the one-dimensional position
coordinate of particle $i$. According to Eqs. (1) and (2) the transformed
variables are $\tilde{x}_{i}=x_{i}+\epsilon(x_{i})$ and
$\tilde{p}_{i}=[1+\partial\epsilon(x_{i})/\partial x_{i}]^{-1}p_{i}$. The
construction of Monte Carlo trial moves is identical in the shifted and
unshifted sytems, in that $x_{i}$ and $p_{i}$ are displaced uniformly within a
maximal cutoff. In the unshifted system the new trial state enters the
Boltzmann factor to accept or reject the new state according to the Metropolis
criterion [36], as is the standard procedure. In the shifted system, the
Boltzmann factor is evaluated on the basis of the shifted variables
$\tilde{x}_{i}$ and $\tilde{p}_{i}$, both before and after the trial move. As
a representative observable, we show in Fig. 2 results for the density profile
$\rho(x)$ as obtained from histograms of particle positions $\tilde{x}_{i}$
from separate runs without and with shifting. We also show the position- and
momentum-resolved one-body phase space density $f(x,p)$, where $p$ denotes the
momentum variable. Consistent with the theoretical structure of the particle
gauge invariance, the results in the shifted system are identical to those in
the original system with $p$ following the correct Maxwellian. We have
ascertained that the same behaviour holds when replacing the hard core wall
and interparticle potentials by soft potentials, using the Lennard-Jones form
as representative. As expected the invariance holds already canonically, i.e.,
with fixed $N$.

In conclusion we have shown that statistical mechanical microstates carry an
intrinsic ambiguity with respect to the gauge shifting transformation (1) and
(2). The Lie algebra (6) for infinitesimal generators (4) is imprinted in
measurable physical correlation functions. Numerical implementation of the
finite shifting gives additional freedom for particle-based simulation
techniques and one can envisage rich cross fertilization with force sampling
schemes [37, 38, 39, 40, 41, 42] and the mapped averaging framework [43, 44,
45, 42]. Recent progress in microscopy-based measurement of locally resolved
forces in colloidal systems [46] offers exciting potential for carrying out
corresponding experimental work.

###### Acknowledgements.

We thank Thomas Kriecherbauer for useful discussions.

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CharacterMixer enables interpolation between two 3D models that have different
surface mesh topologies and rig skeletons. It preserves a posable rig
throughout interpolation. In the top row, we show an example of interpolating
source to target with a fixed pose. CharacterMixer constructs three types of
bones to handle different skeleton topologies: 1-to-1 matched bones are
Constrained, 1-to-many matches are Loose (tail), and 1-to-void matches are
Virtual (shoulder-mounted cannons). The bottom row shows the same
interpolation, but where the character’s pose changes continuously during a
run cycle (posed rigs are omitted for a clear visual of the geometry).

# CharacterMixer: Rig-Aware Interpolation of 3D Characters

X. Zhan10000-0003-1375-180X R. Fu10000-0002-0115-0831 D.
Ritchie10000-0002-8253-0069
1Brown University, USA

###### Abstract

We present CharacterMixer, a system for blending two rigged 3D characters with
different mesh and skeleton topologies while maintaining a rig throughout
interpolation. CharacterMixer also enables interpolation during motion for
such characters, a novel feature. Interpolation is an important shape editing
operation, but prior methods have limitations when applied to rigged
characters: they either ignore the rig (making interpolated characters no
longer posable) or use a fixed rig and mesh topology. To handle different mesh
topologies, CharacterMixer uses a signed distance field (SDF) representation
of character shapes, with one SDF per bone. To handle different skeleton
topologies, it computes a hierarchical correspondence between source and
target character skeletons and interpolates the SDFs of corresponding bones.
This correspondence also allows the creation of a single “unified skeleton”
for posing and animating interpolated characters. We show that CharacterMixer
produces qualitatively better interpolation results than two state-of-the-art
methods while preserving a rig throughout interpolation. Project page:
https://seanxzhan.github.io/projects/CharacterMixer.

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[500]Computing methodologies Shape analysis [500]Computing methodologies Shape
modeling

††volume: 43††issue: 2

##  1 Introduction

Interpolation is a fundamental operation in 3D shape modeling and editing.
Producing smooth blends between shapes can be used to create animations
[The23], to “fill in gaps” between shapes in a collection [MCA∗22], or to
create new hybrid shapes [ALX∗14]. One of the most common types of 3D shape is
a _3D character_ : an articulated body that is animated in some film, game, or
other 3D graphics experience. Interpolation between 3D characters can be used
for pose matching [ENK∗21] or for creating a range of blended characters from
a smaller set of hand-modeled ones (e.g. for creating crowds of background
characters) [CG 23].

When the shapes to be interpolated are 3D characters, the _rigs_ , or the
articulated skeletons that allow characters to be animated, for those
characters should be taken into account, which complicates the problem. In
practice, most systems which can interpolate between rigged characters are
based on parametric models which can produce variations of the character’s
body shape but always keep the same surface mesh and rig topologies, limiting
the range of characters that can be interpolated [LMR∗15]. Methods exist for
interpolating between different 3D shapes, but when applied to rigged
characters, they ignore the rigs, leading to intermediate shapes that are no
longer rigged and thus not directly posable [ENK∗21].

In this paper, we present CharacterMixer, the first system for interpolating
between two rigged characters with different mesh and skeleton topologies,
such that a rig is preserved. With the preserved rig, not only does
CharacterMixer allow interpolated characters to be posable, but it can also
generate animation sequences where interpolation happens at the same time
(Fig. CharacterMixer: Rig-Aware Interpolation of 3D Characters). Handling
different mesh and skeleton topologies is crucial for interpolation tasks
involving characters not created by the same artist, or when animators do not
control characters’ sources, such as uploaded assets in the online gaming
community. To handle characters with different mesh topologies, CharacterMixer
uses signed distance field (SDF) representations of the source and target
geometries. To make the system rig-aware, a character is represented as a
union of SDFs, one per each bone of the rig. Mesh-based methods such as
NeuroMorph [ENK∗21] are unable to interpolate the identities of characters;
they deform the source mesh to match the shape of the target, keeping the
source topology unchanged. In contrast, the SDF representation allows our
method to interpolate geometry and produce intermediate characters of
different identities (Fig. CharacterMixer: Rig-Aware Interpolation of 3D
Characters). To interpolate between two rigged characters with different
skeletal topologies, CharacterMixer computes a hierarchical correspondence
between two skeletons. This correspondence allows it to create a single
“unified skeleton” whose pose can drive the pose of both the source and target
characters. Given the unified skeleton, CharacterMixer interpolates between
the two characters’ geometries by linear interpolation of the SDFs of
corresponding bones.

We evaluate CharacterMixer by comparing to a state-of-the-art optimal
transport approach for shape interpolation [SdGP∗15] and a mesh-based data-
driven method for shape correspondence and interpolation [ENK∗21], showing
that CharacterMixer generates intermediate shapes with higher visual fidelity
while also maintaining a posable rig. In summary, our contributions are:

  * •

A method for computing hierarchical correspondence between two skeletons and
producing unified intermediate skeletons

  * •

A technique for posing and animating interpolated characters using the unified
skeletons

  * •

An interpolation approach for blending between two characters’ geometries
while preserving a rig

##  2 Related Works

Shape Interpolation. There is a significant body of prior work on shape
interpolation and blending. One family of work uses optimal transport,
treating the source and target shapes as probability distributions and finding
a transformation of the source to the target that moves as little probability
mass as possible [SdGP∗15, JCG20, MDZ∗21]. Another work interpolates the
interiors of shapes in an as-rigid-as-possible manner, restricting local
volumes to be least-distorting [ACOL00]. There are also data-driven
approaches, interpolating from a source shape to a target shape by finding a
path through a large collection of related shapes [AS21, GCLX16, GLHH13] or
using the structures of manufactured shapes [YML∗22, GYW∗19]. Most recently,
several works train neural networks to produce deformations from a source to
target shape [ENK∗21, YAK∗20, JHTG20]. Deep generative models can also be
viewed as interpolators, as their latent spaces allow interpolation between
shapes in the generator’s output domain [ADMG17, LLHF21, YHH∗19, CZ19,
ZLWT22]. These methods are all oblivious to character rigs and thus
intermediate interpolated shapes would not be posable, making it impossible to
interpolate throughout an animation.

To the best of our knowledge, no prior work focuses on rig-aware character
interpolation. Parametric body models, such as SMPL [LMR∗15], support
interpolation between body shapes with the same rig; these shapes all have the
same mesh and skeleton topology. Our method supports interpolation between
characters with different mesh and skeleton topologies.

Automated Character Rigging. Our system interpolates between 3D characters
such that the intermediate characters are still animatable. One could instead
use a rig-oblivious shape interpolation method and then attempt to
automatically compute a rig for the new intermediate shape. Several automated
rigging methods exist: some are restricted to characters created via a
specialized sketch-based modeling interface [BJD∗12, DSC∗20], whereas others
can take arbitrary shapes as input and produce a skeleton [XZKS19],
potentially with skinning weights [XZK∗20]. These methods can sometimes fail
to predict usable rigs, and they would produce different rigs for each step in
an interpolation sequence. Our method produces a single rig that can animate
all intermediate characters over an interpolation sequence. Alternatively, one
may opt not to use a skeleton to pose an intermediate character. This would
require posing one of the two input characters and using techniques such as
[LMHM18, HRE∗08, VCH∗21, ZYD∗21] to transfer the pose to the other character
before blending the two characters. However, this offers no direct rigging
control over the intermediate character.

Tree Correspondence. CharacterMixer computes a hierarchical correspondence
between two rig skeletons. This is related to prior work on computing
hierarchical correspondences between 3D shapes [ZYL∗17]. One work in this
space uses these correspondences to “interpolate” between shapes [ALX∗14],
though they focus on manufactured objects and produce transitions that involve
discrete structural switches; we instead focus on continuous blends.

Part-based character blending. One way to produce transitions or blends
between two 3D shapes is by gradually swapping their constituent parts.
Modeling-by-assembly could be used to do this, albeit with considerable user
interaction [FKS∗04, KJS07]. Some prior work can produce such transitions
automatically [JTRS12]. There also exists a system which can mix and match
parts from rigged character models such that the resulting chimera is also
rigged [NPC∗22]. Our goal is to produce a qualitatively different kind of
interpolation between characters: a continuous morph from one to the other.

Figure 1:  The overall pipeline of CharacterMixer. Given a source and target
character (represented as surface meshes + skeletal rigs), CharacterMixer uses
rig skinning weights to segment each character’s geometry into a set of parts.
It also computes a correspondence between the source and target skeletons,
which it uses to create a single unified skeleton given a time step. This
unified skeleton is used to guide interpolation between the geometries of
corresponding parts. Given a posed unified skeleton, CharacterMixer transfers
the pose to source and target characters and interpolates the deformed
geometries of the two posed characters. CharacterMixer enables interpolation
during animation, where poses change (red arrow) concurrently with
interpolation time steps (purple arrow) as shown in the bottom right.

##  3 Approach

Since prior shape interpolation methods [SdGP∗15, ENK∗21] ignore the
underlying structure of shapes, the intermediate results cannot be
manipulated. When the shapes are 3D characters, the interpolated characters
are not posable. CharacterMixer resolves this issue by rig-aware
interpolation; it maintains a rig throughout the interpolation process. Users
can pose an intermediate rig to animate an interpolated character.
Furthermore, while the geometry of the unified rig changes with varying
interpolation time steps, its topology remains the same. Thus, characters can
be interpolated for an animation sequence of a single unified skeleton, such
that intermediate characters’ poses and identities vary at the same time.

Fig. 1 illustrates CharacterMixer’s pipeline. Given a pair of source and
target characters in rest-pose and their rigs, CharacterMixer produces
animatable interpolated characters. CharacterMixer first finds hierarchical
correspondences between the skeletons of the two characters using recursively
defined cost functions (Section 4). It then creates a _unified skeleton_ using
the corresponding pairs (Section 5). The unified skeleton serves as a proxy
that guides geometry interpolation between topologically-different characters
and supports user interaction. Users may animate the unified skeleton, and
CharacterMixer transfers the poses to source and target skeletons by
propagating bone transformations (Section 6). If there is no pose input, the
source and target characters remain in their rest poses. Lastly,
CharacterMixer generates geometry for each bone in the unified skeleton by
interpolating between the corresponding character part SDFs (Section 7).

##  4 Skeleton Correspondence

Character interpolation should preserve semantics of body parts: legs should
be interpolated with legs, and arms should be interpolated with arms, etc.
This calls for a method to find corresponding body parts. While existing
methods find surface correspondence using functional maps [OBCS∗12] or neural
networks [ETLC20, ENK∗21], their interpolation ignores rigs such that
intermediate results are not posable. In contrast, we seek to maintain a rig
throughout the interpolation process. Thus, CharacterMixer finds hierarchical
correspondence between input _skeletons_. As part of preprocessing,
CharacterMixer segments rest-pose characters’ meshes into surface patches
representing body parts using skinning weights, where we assign each vertex to
the bone that has the highest skinning weight for that vertex. If two bones
are corresponded, their segmented body parts are also corresponded. We then
convert segmented surface patches of characters in rest-pose to SDFs, which is
further discussed in Section 7.

The process of finding bone correspondences is identified as the "Skeleton
Correspondence" module in Figure. 1. A bone within a skeleton hierarchy is
defined as $(\mathbf{h}_{i},l_{i},\mathbf{b}_{i},M(x_{i},y_{i},z_{i}))$, where
$\mathbf{h}_{i}$ is the world position of the head of the bone, $l_{i}$ is the
bone length, $\mathbf{b}_{i}$ is the tightest axis-aligned bounding box around
the part surface geometry in bone local space, and $M$ transforms from bone
local space to world space given by the bone’s $x,y,z$ axes. Note that the
y-axis of bone local space is along the direction of the bone from head to
tail, and the x, z axes are computed such that the rotation matrix from the
y-axis satisfies the damped track constraint [Ble23]. $\mathbf{b}_{i}$’s
y-axis is aligned along the bone’s y-axis. The need for $\mathbf{b}_{i}$ is
explained in Section 7.1. We define source skeleton as bones
$\mathbb{S}=\\{s_{i}\\}$ and target skeleton as $\mathbb{D}=\\{d_{j}\\}$.

CharacterMixer first produces initial bone correspondences, using Xu et al.
LayoutBlending’s hierarchical correspondence algorithm with our custom
heuristics suitable for 3D bone matching. The algorithm outputs 1-to-1 and
1-to-void pairs, where one bone can match with another bone or none. Having
only these two types of pairs is undesirable for 3D skeleton correspondence,
as some matching body parts may have different numbers of bones.
CharacterMixer addresses this issue by grouping as many 1-to-void
correspondences as possible into _1-to-many correspondences_ , where one bone
is matched to multiple bones. In this way, semantically matching body parts
with different numbers of bones can be corresponded correctly. To illustrate,
in Fig. 2, the heads of source and target shapes are correctly corresponded
after grouping.

###  4.1 Producing Initial Skeleton Correspondences

Figure 2:  Our pipeline for computing correspondences between skeletons and
constructing an unified skeleton. Left: All the bones from both source and
target skeletons. Middle: The initial correspondence phase which takes the
source skeleton (white) and the target skeleton (gray) and produces 1-to-1 and
1-to-void matches, where void is denoted by $\emptyset$. Right: Grouping 1-to-
void pairs to to create 1-to-many correspondences.

CharacterMixer finds bone correspondences between two input skeletons to
establish part correspondences, and it produces an intermediate character by
interpolating geometries of two corresponding parts. We adapt Xu et al.
LayoutBlending’s hierarchical correspondence algorithm to find 1-to-1 and
1-to-void bone mappings between two input hierarchies. Note that a bone in the
skeleton hierarchy can be either a leaf bone or a branch bone. There are five
correspondence cases: leaf-to-leaf, leaf-to-void, branch-to-void, branch-to-
leaf, and branch-to-branch. Xu et al.’s algorithm defines cost functions for
leaf-to-leaf and leaf-to-void correspondences, and the costs for the latter
three cases are recursively computed from the first two. A matrix encoding the
cost of matching any source bone to any target bone is then constructed, and
the Hungarian algorithm is used to solve for optimal 1-to-1 and 1-to-void
correspondences [Kuh55]. The cost function heuristics used in Xu et al.’s
algorithm are designed for 2D layouts, so we have developed custom heuristics
suitable for 3D skeletons. In the supplemental material, we enumerate our
heuristics and perform an ablation study that validates them.

###  4.2 Post-Processing Initial Skeleton Correspondences

Source and target characters may have semantically corresponding body parts
with different numbers of bones. Fig. 2 shows a source character having a head
with one bone (5) and a target character having a head with four bones (1, 6,
11, 16). When interpolating, since the head bones have 1-to-void
correspondence, the source head would shrink and the target head would grow
instead of interpolation. This is undesirable, so we post-process the initial
pairings by introducing 1-to-many correspondences. A source set and a target
set of 1-to-void correspondences can be grouped into 1-to-many correspondence
if the lowest common ancestor of the source nodes is in 1-to-1 correspondence
with the lowest common ancestor of the target nodes. In Fig. 2, the four
target bones (1, 6, 11, 16) are grouped together to correspond to the one
source bone (5) because all bones satisfy 1-to-void correspondence and the
source ancestor (0) is 1-to-1 corresponded with the target ancestor (0). In
the supplemental material, we provide detailed pseudocode for an algorithm
that eliminates as many 1-to-void correspondences as possible in favor of
1-to-1 and 1-to-many mappings. We define correspondence as $P=(r_{s},r_{d})$,
where $P$ has five cases: $(s,d),(S,d),(s,D),(s,\emptyset),(\emptyset,d)$,
$S\subset\mathbb{S},D\subset\mathbb{D},s\in\mathbb{S},d\in\mathbb{D}$.

##  5 Unified Skeleton Generation

We seek to preserve a rig throughout the interpolation process such that the
intermediate characters are posable. Given 1-to-1, 1-to-many, 1-to-void
skeleton correspondences (Section 4), CharacterMixer generates a _unified
skeleton_ at a time step $t$ (Fig. 3). This process is identified as
"UniSkelGen" in Fig. 1. The geometries of unified skeletons vary depending on
$t$, but their topologies remain the same. Thus, animators can interact with a
single unified skeleton and specify interpolation steps for each frame to
generate an animation sequence where motion and interpolation happen
simultaneously (Figs. CharacterMixer: Rig-Aware Interpolation of 3D
Characters, 8). A unified bone inherits the properties of a regular skeleton
bone, and it additionally references source and target bones, a source bone
and void, or a target bone and void. Define reference $R=(r_{s},r_{d})$, where
$R$ is one of
$(s,d),(s,\emptyset),(\emptyset,d),s\in\mathbb{S},d\in\mathbb{D}$. A unified
bone also carries $t$, denoting the interpolation time step. Thus, a unified
bone is defined as $(\mathbf{h}_{k},l_{k},\mathbf{b}_{k},R_{k},t)$,
$k\in\mathbb{K}$, where $\mathbb{K}$ is the set of all bones in an unified
skeleton. In this section, we will first discuss the three types of unified
bones then explain how to construct a unified bone. We define three types of
unified bones displayed in Fig. 3:

  * •

Constrained. Reference $R=(s,d)$, where $s,d$ satisfy 1-to-1 correspondence
$P=(s,d)$.

  * •

Loose. Reference $R=(s,d)$, where $s,d$ are associated via 1-to-many
correspondence $P=(S,d)$ or $(s,D)$, $s\in S,d\in D$.

  * •

Virtual. Reference $R=(s,\emptyset)$ or $(\emptyset,d)$, where $s,d$ satisfy
1-to-void correspondence $P=(s,\emptyset)$ or $(\emptyset,d)$.

Figure 3:  Generating unified bones at t = 0.5. The blue claw is a constrained
bone made from 1-to-1 correspondence. Each yellow head bone is a loose bone
made from 1-to-many correspondence. The red tail is a virtual bone made from
1-to-void correspondence.

Given a 1-to-1 pair $P=(s,d)$, CharacterMixer constructs a constrained unified
bone (Fig. 3, Constrained). Since we have 1-to-1 mapping, it’s straightforward
to make an intermediate bone; we linearly interpolate each attribute of $s,d$.
$\mathbf{h}_{k}=\text{lerp}(\mathbf{h}_{s},\mathbf{h}_{d},t)$,
$l_{k}=\text{lerp}(l_{s},l_{d},t)$,
$\mathbf{b}_{k}=\text{lerp}_{\text{box}}(\mathbf{b}_{s},\mathbf{b}_{d},t)$,
and $M_{k}=\text{slerp}(M_{s},M_{d},t)$, where $\text{lerp}_{\text{box}}$
linearly interpolates the eight corners of two bounding boxes. The constrained
unified bone references $s$ and $d$.

Given a 1-to-many correspondence pair $P=(S,d)$, CharacterMixer constructs
$|S|$ number of loose unified bones (Fig. 3, Loose). CharacterMixer first
splits up $d$ into $|S|$ parts whose lengths are proportional to
$l_{s_{i}},s_{i}\in S$. $\mathbf{b}_{k_{i}}$ is generated similarly by
splitting $\mathbf{b}_{d}$ into parts proportional to the geometries of
$\mathbf{b}_{s_{i}}$ (Section 7.2). We have now converted the problem into
1-to-1 interpolation. Each $k_{i}$ can then be constructed in the same way as
a constrained bone. For example, in Fig. 3, $k_{2}$ is linearly interpolated
between $s_{2}$ and $d^{\prime}_{2}$. The loose unified bone $k_{i}$
references $s_{i}$ and $d$, where $s_{i}\in S$.

Given a 1-to-void correspondence pair $P=(\emptyset,d)$, CharacterMixer
constructs a virtual unified bone (Fig. 3, Virtual). CharacterMixer uses
_bounding box mapping_ to map $\mathbf{h}_{d}$ to the source character, where
the details of bounding box mapping are explained in Section 7.1. Note that
the center of the bounding box sits at the head of the root bone, and the axes
of the box are aligned with those of the root bone. Then, CharacterMixer
linearly interpolates $\mathbf{h}_{d}$ and the projected point to acquire
$\mathbf{h}_{k}$. The virtual unified bone $k$’s length and bounding box is
computed by $l_{k}=\text{lerp}(0,l_{d},t)$, and
$\mathbf{b}_{k}=\text{lerp}_{\text{box}}(\mathbf{b}_{0},\mathbf{b}_{d},t)$,
where $\mathbf{b}_{0}$ denotes a bounding box whose height, width, and length
are all 0. The virtual unified bone references $\emptyset$ and $d$. The
supplemental material contains more detailed pseudocode for the unified
skeleton construction process.

##  6 Pose Transferring

Users can interact with the unified skeletons to pose or animate interpolated
characters. CharacterMixer also produces animation sequences where
interpolation happens simultaneously as shown in Fig. CharacterMixer: Rig-
Aware Interpolation of 3D Characters. Given animation input for a unified
skeleton at $t$, CharacterMixer transfers poses to source and target
characters’ skeletons and blends their geometries to generate a posed or
animated interpolated character at interpolation time $t$. Note that by
construction, even though the geometry of the unified skeleton varies
depending on time step $t$, its topology remains the same throughout the
interpolation. Thus, users can animate any unified skeleton and also vary the
interpolate time step such that characters are interpolated during an
animation sequence. In this section, we will discuss how pose is transferred
from a unified skeleton to source and destination skeletons. This process is
identified as "Pose Xfer." in Fig. 1.

CharacterMixer transfers the unified skeleton’s pose to source and target
skeletons by propagating bone transformations. We will explain how
CharacterMixer transfers poses from constrained, loose, and virtual unified
bones.

A constrained unified bone $k$ references one source bone $s$ and one target
bone $d$, where $s$ and $d$ have 1-to-1 correspondence. In this case, the
source and target nodes should transform in the same way as the unified bone.
Given user input joint angles for the unified bone, CharacterMixer computes a
local rotation matrix $Rot_{k}$ and sets $Rot_{s}=Rot_{d}=Rot_{k}$.

Loose unified bones are created from 1-to-many correspondence $P=(S,d)$, and
each unified bone $k_{i}$ references $R_{i}=(s_{i},d),s_{i}\in S$. When users
pose $k_{i}$, the behavior of $s_{i}$ should be the same;
${Rot}_{s_{i}}={Rot}_{k_{i}}$. To rotate bone $d$, one way would be to compute
the vector from the head of the $S$ linkage to its tail and rotate bone $d$ to
align with that vector. However, this would ignore the rotation around the
axis along bone $d$. Thus, CharacterMixer averages the joint angles of
$\\{k_{i}\\}$ to construct rotation matrix $Rot_{d}$.

A virtual unified bone $k$ references $R=(s,\emptyset)$ or $R=(\emptyset,d)$.
If $R=(s,\emptyset)$, rotating $k$ only affects $s$ and has no impact on the
target skeleton. Thus, CharacterMixer sets $Rot_{s}=Rot_{k}$ or
$Rot_{d}=Rot_{k}$.

##  7 Rig-Aware Geometry Interpolation

We have discussed how CharacterMixer computes skeleton correspondence and
constructs a unified skeleton. It is established that bone correspondence
implies part correspondence (Section 4), and each unified bone refers to a
source and/or a target bone in either 1-to-1, 1-to-many, or 1-to-void
correspondence (Section 5). The unified skeleton allows for rig-aware
character interpolation; a rig is maintained throughout interpolation, and
CharacterMixer builds geometries around the unified skeleton. Given user pose
or animation input for a unified skeleton, CharacterMixer transfers the pose
to source and target characters’ skeletons and interpolates their posed
geometries. If there is no pose input, CharacterMixer interpolates their rest-
pose geometries. Note that although the geometries of the unified skeletons
vary depending on the input interpolation time step $t$, their topologies
remain the same. Thus, users can pose a single unified skeleton generated at
an arbitrary $t$ and pass different interpolation steps to CharacterMixer’s
rig-aware geometry interpolation algorithm for each frame of animation.
CharacterMixer then produces an animation sequence where interpolation and
motion happen at the same time (Fig. CharacterMixer: Rig-Aware Interpolation
of 3D Characters bottom row). In Fig. 1, a user animates the unified skeleton
generated at $t=0.5$ (center) and passes different interpolation steps for
each animated frame to CharacterMixer and produces a run cycle sequence where
the source character is interpolated to the target character.

Figure 4:  Interpolating part geometries for a constrained unified bone. With
part-based SDFs, interpolating between two characters amounts to taking a
query point (cyan dot) within the bounding box of a unified skeleton bone in
the bone’s local space, finding which point in the source and target boxes
that it maps to, looking up SDF values at those points, and linearly
interpolating between them.

Since interpolation of source and target geometries should preserve part-level
semantics (e.g. legs should be interpolated with legs), CharacterMixer employs
part-based representation of characters, and CharacterMixer uses each unified
bone’s reference information $R$ to identify and interpolate source and target
parts. We choose signed distance fields (SDFs) to represent characters’ body
parts, as SDFs allow easy interpolation between geometries with different mesh
topologies. Furthermore, CharacterMixer’s geometry interpolation is aware of
the rig state, which allows for interpolating character geometries during
animation,; CharacterMixer keeps track of bone local frames and interpolates
parts in bone local spaces where SDFs are defined. To obtain an SDF for a
segmented body part, CharacterMixer converts the part to voxels on a $128^{3}$
regular grid then to SDFs using distance transform. When interpolating part
SDFs, CharacterMixer uses _bounding box mapping_ to ensure that the
interpolated geometry preserves characteristics from both parts. Fig. 4 shows
how CharacterMixer acquires SDF values for a blended part by mapping a query
point to the source and target spaces then interpolating. A query point is
evaluated by each unified bone $k\in\mathbb{K}$ to produce SDF value $v_{k}$;
CharacterMixer defines the final SDF value for that point as
$V_{k}=\min\\{v_{k}\\}$, which can be interpreted as a union operation. In the
following subsections, we explain bounding box mapping, discuss shape
interpolation for different types of unified bones, and show how to query
deformed SDFs given character poses.

###  7.1 Bounding Box Mapping

When interpolating between corresponding parts, the intermediate result should
preserve characteristics from both parts. CharacterMixer achieves this by re-
localizing query points with bounding box centers and scaling them when
transforming to source and target spaces. Fig. 5 shows two methods for part
interpolation. If we map a query point $p$ in the unified bone space to source
and target spaces relative to its position to the bone heads, interpolating a
hexagon and an ellipse results in an unexpected shape. In contrast, bounding
box mapping outputs a rounded hexagon. CharacterMixer first constructs
bounding boxes $B_{s},B_{d}$ around source and target part mesh geometries and
interpolate to obtain an intermediate bounding box
$B_{k}=\text{lerp}_{\text{box}}(\mathbf{b}_{s},\mathbf{b}_{d},t)$ for a
constrained or loose unified bone $k$. Then CharacterMixer finds $p$’s
location in the intermediate bounding box before scaling it to source and
target bounding box spaces. Let $\mathbf{b}^{c}$ denotes the center of
$\mathbf{b}$, and $\mathbf{b}^{x},\mathbf{b}^{y},\mathbf{b}^{z}$ denotes the
x, y, z axes lengths of $\mathbf{b}$. Point $p$ in the source bounding box
space is given by
$(p-\mathbf{b}^{c}_{k})\cdot(\frac{\mathbf{b}^{x}_{s}}{\mathbf{b}^{x}_{k}},\frac{\mathbf{b}^{y}_{s}}{\mathbf{b}^{y}_{k}},\frac{\mathbf{b}^{z}_{s}}{\mathbf{b}^{z}_{k}})$.
Note that a query point may land outside of the interpolated bounding box, but
the transformation still applies as bounding box geometries provide scaling.

###  7.2 Interpolation with Unified Bones

CharacterMixer uses a unified bone $k$’s reference information $R$ to
interpolate between source and destination part geometries.

It’s simple to compute geometry for a constrained unified bone $k$ which
references one source bone $s$ and one target bone $d$. Fig. 4 shows how to
obtain an interpolated SDF value given a query point $p$ in world space.
CharacterMixer first transforms $p$ into the interpolated bounding box
$\mathbf{b}_{k}=\text{lerp}_{\text{box}}(\mathbf{b}_{s},\mathbf{b}_{d},t)$.
Using bounding box mapping, CharacterMixer finds the position of $p$ in
$\mathbf{b}_{s}$, $\mathbf{b}_{d}$. SDF values of source and target parts are
defined in bone local space, so CharacterMixer transforms $p$ from bounding
box space to bone local space to query $v_{s},v_{d}$, where the point is
translated by $\mathbf{b}^{c}$, the bounding box center. The final SDF value
is given by $v_{k}=\text{lerp}(v_{s},v_{d},t)$.

When source and target bones are in 1-to-many correspondence $P=(S,d)$,
CharacterMixer constructs $|S|$ number of loose unified bones $k_{i}$, each
referencing $R_{i}=(s_{i},d),s_{i}\in S$, where each $s_{i}$ has bounding box
${\mathbf{b}_{s}}_{i}$. Similar to how CharacterMixer proportionally splits up
bone $d$ when generating $k_{i}$, CharacterMixer splits $d$’s bounding box
$\mathbf{b}_{d}$ into sub-boxes $\mathbf{b}^{\prime}_{d_{i}}$ whose y-axis
lengths are proportional to each of $\mathbf{b}_{s_{i}}$. In this way, an
interpolated bounding box can be generated for each loose unified bone
$k_{i}$, where
$\mathbf{b}_{k_{i}}=\text{lerp}_{\text{box}}(\mathbf{b}_{s_{i}},\mathbf{b}^{\prime}_{d_{i}},t)$.
CharacterMixer then proceeds to interpolate geometries as described in the
previous paragraph.

A virtual unified bone $k$ references $(s,\emptyset)$ or $(\emptyset,d)$.
Interpolation for virtual bones works similarly to constrained bones. When
either the source or destination bounding box is $\mathbf{b}_{0}$,
CharacterMixer doesn’t attempt to transform point $p$ into $\mathbf{b}_{0}$
but sets $v_{s}=0$ or $v_{d}=0$ then interpolates for $v_{k}$.

Figure 5:  Comparison between interpolation using bounding box mapping and
bone-relative mapping with 2D SDF visualization, where white outlines
geometries and grey denotes bones. The interpolated geometry (“interp") is
generated by mapping a point (cyan dot) in the unified bone’s space to source
and target spaces and then interpolating the queried SDFs. Interpolation with
bounding box mapping preserves characteristics of input geometries while the
other method does not.

###  7.3 Querying Deformed SDFs

Given a posed unified skeleton and an interpolation time step, CharacterMixer
queries deformed SDFs of source and target characters to generate an
interpolated character. After transferring the unified skeleton’s pose to
source and target skeletons, CharacterMixer poses the two input characters
using their respective skinning weights. CharacterMixer then segments the
characters into parts using the procedure described in the beginning of
Section 7, converts them to SDFs, and uses rig-aware interpolation (Section
7.2) to generate geometry for the posed unified skeleton.

Figure 6:  Qualitative comparisons between CharacterMixer, ConvWasser
[SdGP∗15], and NeuroMorph [ENK∗21] (“$\text{NMorph}^{*}$"). The
“$\text{NMorph}^{*}$" rows are produced using the method in Section 8.
CharacterMixer smoothly interpolates characters while the others do not.
ConvWasser is agnostic to correspondences, and NeuroMorph produces many
artifacts (arms at steps 3, 4 of top left; head and paws at steps 3, 4 of
bottom left; legs at steps 2, 3, 4 of top right; arms and head at steps 2, 3,
4, 5 of bottom right). In the supplemental document, we show two sets of
NeuroMorph interpolations generated from both directions for each pair. See
supplementary video for animated comparisons.

##  8 Results and Evaluation

In this section, we present results of interpolating a variety of characters
with CharacterMixer. We compare CharacterMixer with ConvWasser [SdGP∗15], a
rig-oblivious method for shape interpolation with optimal transport, and
NeuroMorph [ENK∗21], a learning-based approach that produces surface
correspondence and interpolation given two input meshes. We trained each
character pair with NeuroMorph for 5000 epochs to obtain results. For our
experiments, we used 36 character pairs from the RigNet dataset [XZK∗20],
sourced from Models Resource [VG 19], a publicly-available dataset of rigged
video game characters. Experiments were run on 6-core Intel i7 machine with
32GB RAM and a NVIDIA GTX 1080Ti GPU. We strongly encourage readers to watch
our supplementary video for animated results.

Figure 7:  Qualitatively comparing our correspondences with NeuroMorph. We
visualize CharacterMixer’s skeleton correspondence by segmenting meshes using
skinning weights and assigning different colors to surface patches associated
with corresponding bones. NeuroMorph colors denote vertex-level
correspondence.

Fig. 7 compares CharacterMixer and NeuroMorph correspondences. Leveraging
characters’ rigs, CharacterMixer produces more fine-grained correspondences
and correctly identifies corresponding body parts. For instance, in Fig. 7
column 3, we match the two heads of the character pair, while NeuroMorph
matches the shell of the first character to the head of the second character.
See the supplemental material for more of our correspondences.

Fig. 6 compares how well CharacterMixer interpolates rest-pose characters
compared to ConvWasser and $\text{NeuroMorph}^{*}$. NeuroMorph changes the
pose of one shape to match the other while leaving its identity unchanged. We
produced NeuroMorph’s interpolation for both directions from source $s$ to
target $d$ and $d$ to $s$ to obtain ${s_{t=0},...s_{t=1}}$ and
${d_{t=0},...d_{t=1}}$. Intermediate results, labeled $\text{NeuroMorph}^{*}$,
are computed by $\text{lerp}(\text{SDF}(s_{t}),\text{SDF}(d_{1-t}),t)$ where
$\text{SDF}(\cdot)$ converts a mesh to SDF. Our approach produces higher-
quality and semantic-preserving interpolations. For example, in the bottom
right of Fig. 6, our approach smoothly interpolates the heads, while
ConvWasser ignores correspondence and $\text{NeuroMorph}^{*}$ has many
artifacts around the intermediate characters’ head and arms. Furthermore, our
method allows intermediate characters to be posed, while ConvWasser and
NeuroMorph do not.

Fig. 8 shows interpolation during animation. For each sequence, an artist has
animated the unified skeleton generated at $t=0.5$ (Section 5) and specified
interpolation time steps for each frame. CharacterMixer then transfers the
poses to source and target characters (Section 6) and performs rig-aware
interpolation (Section 7). Since CharacterMixer maintains a rig with the same
topology throughout interpolation, it allows for interpolation while the
intermediate characters perform a motion sequence.

In terms of timing, CharacterMixer produces a posed interpolated character in
83 seconds on average. We experimented with another approach to generate
geometry for a posed intermediate character that is 17% faster with negligible
cost in quality (see the supplemental material).

Figure 8:  Interpolation during animation. For each character pair, the top
row shows correspondence and posed unified skeletons, and the row below shows
a motion sequence where character poses and interpolation steps vary at the
same time. During interpolation, CharacterMixer preserves a “unified skeleton"
whose geometry varies depending on time steps and topology remains the same.
Thus, animators can interact with a single unified skeleton and specify
interpolation steps for each frame to achieve interpolation during animation.
Pairs 1, 4 include 1-to-1, 1-to-void, and 1-to-many correspondences, and pairs
2, 3, 5, 6 include 1-to-1 and 1-to-void correspondences. Please watch our
supplementary video for animated sequences.

##  9 Conclusion

We presented CharacterMixer, a method to interpolate between 3D characters
with different mesh and rig topologies, such that users can pose the
intermediate interpolated characters. It also enables interpolation during
animation. To the best of our knowledge, CharacterMixer is the first system
that tackles this novel challenge. CharacterMixer achieves this goal by
maintaining a unified rig throughout interpolation, where the unified rig is
built from skeleton correspondences between two input rigs of potentially
different topologies. CharacterMixer is agnostic to mesh topologies as it
represents characters as a union of sign distance fields, one per each bone of
the character’s rig. We showed how to perform rig-aware interpolation of
characters and pose any intermediate interpolated character. Our experiments
show that CharacterMixer produces higher-quality character interpolations than
rig-oblivious shape interpolation methods [SdGP∗15, ENK∗21].

CharacterMixer is not without its limitations. Similar to Xu et al. [XLY∗22]’s
work, our skeleton correspondence algorithm can sometimes produce incorrect
correspondences that may not satisfy users. In this case, an interactive
system built on CharacterMixer could simply allow users to manually correct
the automatically-produced correspondences. Moreover, CharacterMixer can
struggle to produce a good correspondence between characters with drastically
different skeletons—indeed, in some cases, a meaningful correspondence might
not exist. In the supplemental material, we provide some correspondence
failure cases. Although we have shown that NeuroMorph [ENK∗21], a mesh-based
method, creates severe artifacts and does not support interpolation during
animation, CharacterMixer’s SDF-based representation has a lack of control
over output surfaces such as loss of original mesh topology and texture. In
terms of posing intermediate results by propagating poses to source and target
characters, the current pose transferring method works well when the local
frames of matching bones align well. It is an interesting future direction to
implement more dedicated pose retargeting to refine the intermediate
characters’ motion. Nevertheless, such dedicated techniques only need to deal
with local motion retargeting as we have simplified the problem to three
categories – constrained, loose, and virtual. Additionally, more work is
needed to optimize CharacterMixer for real-time use. To pose an interpolated
character, 70% of reconstruction time is spent on converting part-level mesh
geometries to SDFs with voxelization and distance transform. The runtime can
be improved by employing faster methods to convert from mesh to SDFs, such as
fast winding numbers [BDS∗18]. Neural SDFs [DNJ20] may further increase the
speed of interpolation.

##  10 Acknowledgements

We would like to thank Ivery Chen and Healey Koch for animating interpolation
sequences, and Arman Maesumi for providing radial basis function interpolator
code for our fast interpolation by advection method (Supplemental Section 3).

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