Nonlinear Energy Minimization for
Surface Parameterizations

Mei-Heng Yueh

Department of Mathematics
National Taiwan Normal University

Outline

Introduction
Overview of State-of-the-Arts
Conformal and Authalic Parameterizations
Applications in Computer Graphics
Volume-Preserving Parameterizations
Application to Brain Tumor Segmentation

Introduction

Simplicial Surfaces

A simplicial surface is a piecewise linear manifold composed of simplices.


simplicial surface	simplicial 3-manifold
composed of triangles	composed of tetrahedrons

Parameterization Problem

The parameterization problem aims to find a bijective map $f$ that maps

a given simplicial manifold	to	a domain of a specified shape

	$\xrightarrow{~~~~~f~~~~~}$
$\mathcal{M}$		$\mathcal{D}$

while minimizing a specified distortion.

Coordinate Charts Induced by Parameterizations

Surface Parameterizations

A coordinate system on the surface is induced by the parameterization.

A desired parameterization usually preserves the local shape well.

Conformal mappings are good options!

Conformal Mappings

Uniformization theorem guarantees the existence of conformal maps.
The computation of conformal mappings has been extensively studied.

Overview of State-of-the-Arts

Disk Conformal Parameterizations

Double Covering ^{[HuGH14][ChLu18]}
Quasi-Conformal Approach ^[ChLu15]
Conformal Energy Minimization ^{[YuLW17][KuLY21]}
Constructive Algorithm ^[LiYu22]

[HuGH14] Huang, Gu, Huang, Lin, Lin and Yau, Geom Imag Comput, 1(2):223–258, 2014.
[ChLu15] Choi and Lui, J Sci Comput, 65(3):1065–1090, 2015.
[YuLW17] Yueh, Lin, Wu and Yau, J Sci Comput, 73(1):203–227, 2017.
[ChLu18] Choi and Lui, Adv Comput Math, 44:87–114, 2018.
[KuLY21] Kuo, Lin, Yueh and Yau, SIAM J Imaging Sci, 14(4):1790–1815, 2021.
[LiYu22] Liao and Yueh, J Sci Comput, 92(2), 2022.

Free-Boundary Conformal Parameterizations

Angle-Based Flattening ^{[ShSt01][ShLM05]}
Least Squares Conformal Map ^[LePR02]
Spectral Conformal Parameterization ^{[MuTA08][HuGL14]}
Boundary First Flattening ^[SaCr17]

[ShSt01] Sheffer and Sturler, Eng with Comput, 17(3):326–337, 2001.
[LePR02] Lévy, Petitjean, Ray and Maillot, ACM Trans Graph, 21(3):362–371, 2002.
[ShLM05] Sheffer, Lévy, Mogilnitsky and Bogomyakov, ACM Trans Graph, 24(2):311–330, 2005.
[MuTA08] Mullen, Tong, Alliez and Desbrun, SGP ’08, 27(5):1487–1494, 2008.
[HuGL14] Huang, Gu, Lin and Yau, J Sci Comput, 61(3):558–583, 2014.
[SaCr17] Sawhney and Crane, ACM Trans Graph, 37(1):5:1–5:14, 2017.

Spherical Conformal Parameterizations

Harmonic Map ^{[AnHT99][GuWC04][HuGH14]}
Quasi-Conformal Approach ^[ChLL15]
Conformal Energy Minimization ^[YuLL19]

[AnHT99] Angenent, Haker, Tannenbaum and Kikinis, IEEE Trans Med Imaging, 18(8):700–711, 1999.
[GuWC04] Gu, Wang, Chan, Thompson and Yau, IEEE Trans Med Imaging, 23(8):949–958, 2004.
[HuGH14] Huang, Gu, Huang, Lin, Lin and Yau, Geom Imag Comput, 1(2):223–258, 2014.
[ChLL15] Choi, Lam and Lui, SIAM J Imaging Sci, 8(1):67–94, 2015.
[YuLL19] Yueh, Li, Lin and Yau, SIAM J Imaging Sci, 12(2):1071–1098, 2019.

Authalic (Area-Preserving) Parameterizations

Stretch-Minimizing ^[YoBS04]
Lie Advection ^[ZoHG11]
Optimal Mass Transport ^[ZhSG13]
Stretch Energy Minimization ^[YuLW19]

[YoBS04] Yoshizawa, Belyaev and Seidel, Proceedings Shape Modeling Applications, 200-208, 2004.
[ZoHG11] Zou, Hu, Gu and Hua, IEEE T Vis Comput Gr, 17(12):2005-2014, 2011.
[ZhSG13] Zhao, Su, Gu, Kaufman, Sun, Gao and Luo, IEEE T Vis Comput Gr, 19(12):2838-2847, 2013.
[YuLW19] Yueh, Lin, Wu and Yau, J Sci Comput, 78(3): 1353–1386, 2019.

Volume-Preserving Parameterizations

Stretch-Minimizing ^[JiQZ15]
Lie Advection ^[ZoHG11]
Optimal Mass Transport ^[SuCL17]
Volumetric Stretch Energy Minimization ^{[YuLL19][YuLL20]}

[JiQZ15] Jin, Qian, Zhao, Chang, Tong and Zhang, J Comput Sci Technol, 30:553–564, 2015.
[SuCL17] Su, Chen, Lei, Zhang, Qian and Gu, Comput Aided Design, 82:42-56, 2017.
[YuLL19] Yueh, Li, Lin and Yau, SIAM J Imaging Sci, 12(2):1071–1098, 2019.
[YuLL20] Yueh, Li, Lin and Yau, SIAM J Imaging Sci, 13(3):1536–1564, 2020.

Energy Minimization-Based Methods

Minimize nonlinear energy functionals of the form $$ E(f) = \frac{1}{2}\mathbf{v}(f)^\top L(f) \, \mathbf{v}(f) $$ subject to a boundary shape constraint $f(\partial\mathcal{M}) = \partial\mathcal{D}$, where $\mathbf{v}(f)$ and $L(f)$ are vector and matrix dependent on $f$, respectively.

Main Targets


Surface	Disk Conformal	Disk Area-Preserving	Square Conformal	Square Area-Preserving

Prove the minimizer is the desired parameterization.
Develop efficient algorithm for the energy minimization.
Prove the convergence of the proposed algorithms.

Simplicial Surfaces and Maps

Simplicial Surface

A simplicial surface $\mathcal{M}$ is composed of \begin{align*} \text{vertices } \mathcal{V}(\mathcal{M}) &= \left\{ {v}_1, \ldots, {v}_n \right\} \subset \mathbb{R}^3,\\ \text{edges } \mathcal{E}(\mathcal{M}) &= \left\{ [ {v}_i, {v}_j ] \right\},\\ \text{faces } \mathcal{F}(\mathcal{M}) &= \left\{ [ {v}_i, {v}_j, {v}_k ] \right\}, \end{align*} $$ [ {v}_1, \ldots, v_d ] := \left\{ \sum_{k=1}^d \alpha_k v_k \,\left|\, \alpha_k>0, \,\sum_{k=1}^d \alpha_k = 1 \right.\right\}. $$

Simplicial Map

A simplicial map $f$ is a piecewise affine map with $f(v_k)=\mathbf{f}_k\in\mathbb{R}^2$.

$\xrightarrow{~~~~~f:\mathcal{M}\to\mathbb{R}^2~~~~~}$

The map $f$ is represented as a matrix $$ \mathbf{f} = \begin{bmatrix} f(v_1) \\ \vdots \\ f(v_n) \end{bmatrix} = \begin{bmatrix} \mathbf{f}_1^\top \\ \vdots \\ \mathbf{f}_n^\top \end{bmatrix} = \begin{bmatrix} \mathbf{f}^1_1 & \mathbf{f}^2_1 \\ \vdots & \vdots \\ \mathbf{f}^1_n & \mathbf{f}^2_n \end{bmatrix} = \begin{bmatrix} \mathbf{f}^1 & \mathbf{f}^2 \end{bmatrix}\in\mathbb{R}^{n\times 2}, $$ or a vector $({\mathbf{f}^1}^\top, {\mathbf{f}^2}^\top)^\top \in\mathbb{R}^{2n}$.

Simplicial Map

On the triangle $\sigma = [{v}_i, {v}_j, {v}_k]$, \begin{align} f|_{\sigma}({\color{yellow}v}) &= \frac{|[{\color{yellow}v}, {v}_j, {v}_k]|}{|\sigma|} \,{f}(v_i)\\ &+ \frac{|[{v}_i, {\color{yellow}v}, {v}_k]|}{|\sigma|} \,{f}(v_j) \\ &+ \frac{|[{v}_i, {v}_j, {\color{yellow}v}]|}{|\sigma|} \,{f}(v_k), \end{align} $|\sigma|$ denotes the area of $\sigma$.

Barycentric coordinates of ${\color{yellow}v}$:

$\left(\frac{|[{\color{yellow}v}, {v}_j, {v}_k]|}{|\sigma|}, \frac{|[{v}_i, {\color{yellow}v}, {v}_k]|}{|\sigma|}, \frac{|[{v}_i, {v}_j, {\color{yellow}v}]|}{|\sigma|}\right)$

Simplicial 3-Complex

A simplicial $3$-complex $\mathcal{M}$ is composed of \begin{align*} \text{vertices } \mathcal{V}(\mathcal{M}) &= \left\{ {v}_1, \ldots, {v}_n \right\} \subset \mathbb{R}^3,\\ \text{edges } \mathcal{E}(\mathcal{M}) &= \left\{ [ {v}_i, {v}_j ] \right\},\\ \text{faces } \mathcal{F}(\mathcal{M}) &= \left\{ [ {v}_i, {v}_j, {v}_k ] \right\},\\ \text{tetrahedrons } \mathcal{T}(\mathcal{M}) &= \left\{ [ {v}_i, {v}_j, {v}_k, {v}_\ell ] \right\}, \end{align*} where $$ [ {v}_1, \ldots, v_d ] := \left\{ \sum_{k=1}^d \alpha_k v_k \,\left|\, \alpha_k>0, \,\sum_{k=1}^d \alpha_k = 1 \right.\right\}. $$

Simplicial Map

A simplicial map $f$ is a piecewise affine map with $f(v_k)=\mathbf{f}_k\in\mathbb{R}^3$.

$\xrightarrow{~~~~~f:\mathcal{M}\to\mathbb{R}^3~~~~~}$

The map $f$ is represented as a matrix $$ \mathbf{f} = \begin{bmatrix} f(v_1) \\ \vdots \\ f(v_n) \end{bmatrix} = \begin{bmatrix} \mathbf{f}_1^\top \\ \vdots \\ \mathbf{f}_n^\top \end{bmatrix} = \begin{bmatrix} \mathbf{f}^1_1 & \mathbf{f}^2_1 & \mathbf{f}^3_1 \\ \vdots & \vdots & \vdots \\ \mathbf{f}^1_n & \mathbf{f}^2_n & \mathbf{f}^3_n \end{bmatrix} = \begin{bmatrix} \mathbf{f}^1 & \mathbf{f}^2 & \mathbf{f}^3 \end{bmatrix}\in\mathbb{R}^{n\times 3}, $$ or a vector $({\mathbf{f}^1}^\top, {\mathbf{f}^2}^\top, {\mathbf{f}^3}^\top)^\top \in\mathbb{R}^{3n}$.

Conformal Parameterizations

Conformal Energy Minimization (CEM) ^[KuLY21]

The CEM aims to find a disk conformal map.

$\xrightarrow{~~~~~f~~~~~}$

[KuLY21] Kuo, Lin, Yueh and Yau, SIAM J Imaging Sci, 14(4):1790–1815, 2021.

Conformal Energy Functional

The conformal energy of a smooth map $f$ is defined as \[ \mathcal{E}_C(f) = \mathcal{E}_D(f) - \mathcal{A}(f), \] where $\mathcal{E}_D$ is the Dirichlet energy defined as \[ \mathcal{E}_D\left({f}\right) = \frac{1}{2}\int_\mathcal{M} \|\nabla {f}\|^2 \,\mathrm{d} \sigma_\mathcal{M}. \]

Lower Bound of Dirichlet Energy

Theorem ^[Hutc91]

The Dirichlet energy satisfies $\mathcal{E}_D(f) \geq \mathcal{A}(f).$
The equality holds if and only if $f$ is conformal.

\begin{align*} \mathcal{E}_D\left({f}\right) &= \frac{1}{2}\int_\mathcal{M} \|\nabla {f}\|^2 \,\mathrm{d} \sigma_\mathcal{M} = \frac{1}{2}\int_\mathcal{M} ( \|{f}_u\|^2 + \|{f}_v\|^2 ) \,\mathrm{d} u \,\mathrm{d} v\\ &\geq \int_\mathcal{M} \|{f}_u\| \|{f}_v\| \,\mathrm{d} u \,\mathrm{d} v \geq \int_\mathcal{M} \|{f}_u \times {f}_v\| \,\mathrm{d} u \,\mathrm{d} v = \mathcal{A}(f). \end{align*} The equality holds if and only if $ \|{f}_u\| = \|{f}_v\| ~\text{ and }~ {f}_u \perp {f}_v. $

Corollary ^[Hutc91]

$\mathcal{E}_C(f) \equiv \mathcal{E}_D(f) - \mathcal{A}(f) = 0$ $\iff$ $f$ is conformal.

[Hutc91] Hutchinson, Proc Cent Math Appl, 26:140–161, 1991.

Discrete Dirichlet Energy

The discrete Dirichlet energy functional \begin{align*} {E}_D(f) &\equiv \frac{1}{2}\sum_{\tau\in\mathcal{F}(\mathcal{M})} \left\|(\nabla f)|_\tau \right\|^2 |\tau| \\ &= \frac{1}{2}\sum_{\tau\in\mathcal{F}(\mathcal{M})} \sum_{i\neq j}\frac{\cot\theta_{i,j}}{2}\|f(v_j)-f(v_i)\|^2. \end{align*}

Discrete Dirichlet Energy

The discrete Dirichlet energy can be represented as \[ {E}_{D}(f) = \frac{1}{2} \, \mathrm{trace}\left(\mathbf{f}^{\top} {L}_D \mathbf{f}\right) = \frac{1}{2} \,({\mathbf{f}^1}^\top, {\mathbf{f}^2}^\top) \begin{bmatrix} {L}_D \\ & {L}_D \end{bmatrix} \begin{bmatrix} \mathbf{f}^1\\ \mathbf{f}^2 \end{bmatrix}, \] where $L_D$ is the discrete Laplace–Beltrami operator with $$ [{L}_D]_{i,j} = \begin{cases} -\frac{1}{2}\left(\cot\theta_{i,j}+\cot\theta_{j,i}\right) &\mbox{if $[{v}_i,{v}_j]\not\subset\partial\mathcal{M}$,}\\ -\frac{1}{2}\cot\theta_{i,j} &\mbox{if $[{v}_i,{v}_j]\subset\partial\mathcal{M}$,}\\ -\sum_{k\neq i} [{L}_D]_{i,k} &\mbox{if $i = j$,} \\ 0 &\mbox{otherwise.} \end{cases} $$

Image Area

The area of the image $f(\mathcal{M})$ can be written as \begin{equation*}%\label{eqn:Area} \mathcal{A}(f) = \frac{1}{2} \sum_{[{v}_i,{v}_j] \in \partial \mathcal{M}} ({f}^1_{i} {f}^2_{j} - {f}^1_{j} {f}^2_{i}) = \frac{1}{2} \, ({\mathbf{f}^1}^\top, {\mathbf{f}^2}^\top) \begin{bmatrix} \mathbf{0} & {S}^{\top}\\ {S} & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{f}^1\\ \mathbf{f}^2 \end{bmatrix}, \end{equation*} with ${S}^{\top} = -{S}$.

Discrete Conformal Energy \begin{equation*} {E}_{C}({f}) = {E}_{D}({f}) - \mathcal{A}({f}) = \frac{1}{2} \, ({\mathbf{f}^1}^\top, {\mathbf{f}^2}^\top) \begin{bmatrix} {L}_D & {S}\\ -{S} & {L}_D \end{bmatrix} \begin{bmatrix} \mathbf{f}^1\\ \mathbf{f}^2 \end{bmatrix}. \end{equation*}

Disk Image Area

Let $n$ be the number of boundary vertices. The area of the polygon with vertices located on $\mathbb{S}^1$ can be represented in polar coordinates as \begin{align*} \mathcal{A}({f}) &=\frac{1}{2} \underbrace{\left[\begin{array}{c}\cos \theta_1\\\cos \theta_2\\\vdots\\\cos \theta_n\end{array}\right]^{\top}}_{\mathbf{c}^{\top}} \underbrace{\left[\begin{array}{cccc}0&1&&-1\\ -1&\ddots&\ddots&\\ &\ddots&\ddots&1\\ 1& & -1&0\end{array}\right]}_{D} \underbrace{\left[\begin{array}{c}\sin \theta_1\\\sin \theta_2\\\vdots\\\sin \theta_n\end{array}\right]}_{\mathbf{s}} \equiv \frac{1}{2}\mathbf{c}^\top D \mathbf{s}. \end{align*}

Discrete Conformal Energy

The discrete conformal energy can be reformulated as \begin{align*} E_C(\boldsymbol{\theta})=\frac{1}{2}\mathbf{c}^{\top}K\mathbf{c} + \frac{1}{2}\mathbf{s}^{\top}K\mathbf{s} - \frac{1}{2}\mathbf{c}^{\top}D\mathbf{s}, \end{align*} where $ K=[{L}_D]_{\mathtt{B},\mathtt{B}}-[{L}_D]_{\mathtt{I},\mathtt{B}}^{\top}[{L}_D]_{\mathtt{I},\mathtt{I}}^{-1}[{L}_D]_{\mathtt{I},\mathtt{B}}\in \mathbb{R}^{n\times n}, $ $$ \mathtt{B} = \{ b \mid v_b \in \partial\mathcal{M} \} ~\text{ and }~ \mathtt{I} = \{ i \mid v_i \in \mathcal{M}\backslash\partial\mathcal{M} \}. $$

Discrete CEM Problem

The discrete CEM problem is formulated as \begin{align*} \min\{E_C(\boldsymbol{\theta})\ |\ 0=\theta_1 < \cdots < \theta_n < 2\pi\}. \end{align*}

Theorem ^[KuLY21]

Under some mild conditions, the discrete CEM problem has a nontrivial local minimizer.

[KuLY21] Kuo, Lin, Yueh and Yau, SIAM J Imaging Sci, 14(4):1790–1815, 2021.

Gradient Descent for CEM

The gradient of $E_C$ can be written as \begin{align*} \nabla E_C(\boldsymbol{\theta}) &= {\rm diag}(\mathbf{c})K\mathbf{s} - {\rm diag}(\mathbf{s})K\mathbf{c} + \frac{1}{2}\left({\rm diag}(\mathbf{c})D\mathbf{c}+{\rm diag}(\mathbf{s})D\mathbf{s}\right). \end{align*} The critical point $\boldsymbol{\theta}$ satisfies $\nabla E_C(\boldsymbol{\theta})=\mathbf{0}$.

Theorem ^[KuLY21]

The gradient method for the CEM converges globally.

[KuLY21] Kuo, Lin, Yueh and Yau, SIAM J Imaging Sci, 14(4):1790–1815, 2021.

Numerical Results

Comparison of Angular Distortions

Comparison of Time Costs

Authalic Parameterizations

Stretch Energy Minimization (SEM) ^[YuLW19]

The SEM aims to find a disk authalic (area-preserving) map.

$\xrightarrow{~~~~~f~~~~~}$

[YuLW19] Yueh, Lin, Wu and Yau, J Sci Comput, 78(3): 1353–1386, 2019.

Stretch Energy Functional ^[YuLW19]

The stretch energy functional is defined as $$ {E}_S(f) = \frac{1}{2} \text{trace}\left(\mathbf{f}^{\top} {L_S(f)} \, \mathbf{f} \right) = \frac{1}{2} \,({\mathbf{f}^1}^\top, {\mathbf{f}^2}^\top) \begin{bmatrix} {L}_S(f) \\ & {L}_S(f) \end{bmatrix} \begin{bmatrix} \mathbf{f}^1\\ \mathbf{f}^2 \end{bmatrix}, $$ where $L_S(f)$ is the stretched Laplacian matrix with $$ [{L}_S(f)]_{i,j} = \begin{cases} -\frac{1}{2}\left(\frac{\cot\theta_{i,j}(f)}{\color{yellow}{\frac{|[v_i,v_j,v_k]|}{|f([v_i,v_j,v_k])|}}}+\frac{\cot\theta_{j,i}(f)}{\color{yellow}{\frac{|[v_j,v_i,v_\ell]|}{|f([v_j,v_i,v_\ell])|}}}\right) &\mbox{if $[{v}_i,{v}_j]\in\mathcal{F}(\mathcal{M})$,}\\ -\sum_{k\neq i} [{L}_S(f)]_{i,k} &\mbox{if $i = j$,} \\ 0 &\mbox{otherwise.} \end{cases} $$

[YuLW19] Yueh, Lin, Wu and Yau, J Sci Comput, 78(3): 1353–1386, 2019.

Lower Bound of Stretch Energy

Theorem ^[Yueh22]

The stretch energy satisfies ${E}_S(f) \geq \mathcal{A}(f).$
The equality holds if and only if $f$ is authalic.

\begin{align*} E_S(f) &= \frac{1}{2} \sum_{s=1}^2 \sum_{\tau=[v_i,v_j,v_k]\in\mathcal{F}(\mathcal{M})} \begin{bmatrix} f_i^s & f_j^s & f_k^s \end{bmatrix} [L_S(f)]_{\tau} \begin{bmatrix} f_i^s \\ f_j^s \\ f_k^s \end{bmatrix} \\ &= \sum_{\tau\in\mathcal{F}(\mathcal{M})} \frac{|f(\tau)|^2}{|\tau|} \geq \sum_{\tau\in\mathcal{F}(\mathcal{M})} |f(\tau)| = \mathcal{A}(f). \end{align*} The equality holds if and only if $|f(\tau)| = |\tau|$, for every $\tau\in\mathcal{F}(\mathcal{M})$.

Corollary ^[Yueh22]

${E}_A(f) \equiv {E}_S(f) - \mathcal{A}(f) = 0$ $\iff$ $f$ is authalic.

[Yueh22] Yueh, arXiv:2205.14414, 2022.

Gradient Formula of Stretch Energy

Proposition ^[Yueh22]

The gradient of $E_S$ can be formulated as \begin{align*} \nabla_{\mathbf{f}^s} E_S(f) &= 2 L_S(f) \, \mathbf{f}^s, ~\text{for $s=1,2$.} \end{align*}

$$ [\nabla_{\mathbf{f}^s} E_S(f)]_\ell = [L_S(f) \, \mathbf{f}^s]_\ell + \underbrace{\frac{1}{2} \sum_{t=1}^2 \sum_{i=1}^n \sum_{j=1}^n f_i^t f_j^t \frac{\partial}{\partial f_\ell^s}[L_S(f)]_{i,j}}_{(*)}. $$ It suffices to check $(*) = [L_S(f) \, \mathbf{f}^s]_\ell$.

Fixed Point Iteration for SEM

The critical point $f$ satisfies $L_S(f) \, \mathbf{f}^s =\mathbf{0}$, for $s=1,2$. Under a given boundary constraint $\mathbf{f}^s_\mathtt{B} = \mathbf{g}^s$, the critical point $f$ satisfies $$ [L_S(f)]_{\mathtt{I},\mathtt{I}} \mathbf{f}^s_{\mathtt{I}} = -[L_S(f)]_{\mathtt{I},\mathtt{B}} \mathbf{g}^s. $$

Let \begin{equation*} \varphi^s(f) = \begin{bmatrix} -[L_S(f)]_{\mathtt{I},\mathtt{I}}^{-1} [L_S(f)]_{\mathtt{I},\mathtt{B}} \mathbf{g}^s \\ \mathbf{g}^s \end{bmatrix}, ~\text{for $s=1,2$}. \end{equation*}

Algorithm ^[YuLW19]

$ {\mathbf{f}^s}^{(n+1)} = \varphi^s(f^{(n)}), ~ \text{for $s=1,2$}. $

[YuLW19] Yueh, Lin, Wu and Yau, J Sci Comput, 78(3): 1353–1386, 2019.

Numerical Results

Comparison of Area Distortions

Comparison of Time Costs

Applications in Computer Graphics

Image Editing

Surface Texture Mapping

Texture mapping aims to map the colors from an image to the surface.

Texture Mapping

The texture is mapped to the surface by a parameterization.

Animation with Texture

The motion can be constructed by the cubic spline homotopy.

Surface Alignment and Fusion ^[YuHL20]

[YuHL20] Yueh, Huang, Li, Lin and Yau, Elec Trans Num Anal, 53:383–405, 2020.

Fused Model

Volume-Preserving Parameterizations

Volumetric Stretch Energy Minimization (VSEM) ^[YuLL19]

The VSEM aims to find a spherical volume-preserving map.

$\xrightarrow{~~~~~f~~~~~}$

[YuLL19] Yueh, Li, Lin and Yau, SIAM J Imaging Sci, 12(2):1071–1098, 2019.

Volumetric Stretch Energy Functional ^[YuLL19]

The volumetric stretch energy functional is defined as $$ {E}_{\mathbb{S}}({f}) = \frac{1}{2}\mathrm{trace}\left(\mathbf{f}^\top L_{\mathbb{S}}(f) \, \mathbf{f}\right) = \frac{1}{2} \,({\mathbf{f}^1}^\top, {\mathbf{f}^2}^\top, {\mathbf{f}^3}^\top) (I_3 \otimes L_{\mathbb{S}}(f)) \begin{bmatrix} \mathbf{f}^1\\ \mathbf{f}^2\\ \mathbf{f}^3 \end{bmatrix}, $$ where $L_{\mathbb{S}}(f)$ is the stretched volumetric Laplacian matrix with

$\displaystyle [L_{\mathbb{S}}(f)]_{i,j} = \begin{cases} -\frac{1}{6} \sum_{k, \ell} \frac{\left|[\mathbf{f}_{k}, \mathbf{f}_{\ell}]\right| \cot\theta_{i,j}^{k,\ell}({f})}{\color{yellow}{\frac{|[v_i, v_j, v_k, v_\ell]|}{|f([v_i, v_j, v_k, v_\ell])|}}} &\mbox{if $[{v}_i,{v}_j]\in\mathcal{E}(\mathcal{M})$,}\\ -\sum_{\ell\neq i} [L_{\mathbb{S}(f)}]_{i,\ell} &\mbox{if $j = i$,}\\ 0 &\mbox{otherwise.} \end{cases} $

[YuLL19] Yueh, Li, Lin and Yau, SIAM J Imaging Sci, 12(2):1071–1098, 2019.

Lower Bound of Volumetric Stretch Energy

Theorem ^[HuLL22]

The volumetric stretch energy satisfies $${E}_\mathbb{S}(f) \geq \frac{3}{2}|f(\mathcal{M})|.$$ The equality holds if and only if $f$ is volume-preserving.

\begin{align*} E_\mathbb{S}(f) &= \frac{1}{2} \sum_{s=1}^3 \sum_{\tau=[v_i,v_j,v_k,v_\ell]\in\mathcal{T}(\mathcal{M})} \begin{bmatrix} f_i^s & f_j^s & f_k^s & f_\ell^s \end{bmatrix} [L_\mathbb{S}(f)]_{\tau} \begin{bmatrix} f_i^s \\ f_j^s \\ f_k^s \\ f_\ell^s \end{bmatrix} \\ &= \sum_{\tau\in\mathcal{T}(\mathcal{M})} \frac{3|f(\tau)|^2}{2|\tau|} \geq \frac{3}{2}\sum_{\tau\in\mathcal{T}(\mathcal{M})} |f(\tau)| = \frac{3}{2}|f(\mathcal{M})|. \end{align*} The equality holds if and only if $|f(\tau)| = |\tau|$, for every $\tau\in\mathcal{T}(\mathcal{M})$.

[HuLL22] Huang, Liao, Lin, Yueh and Yau, arXiv:2210.09654, 2022.

Gradient Formula of Volumetric Stretch Energy

Proposition ^[HuLL22]

The gradient of $E_\mathbb{S}$ can be formulated as \begin{align*} \nabla_{\mathbf{f}^s} E_\mathbb{S}(f) &= 3 L_\mathbb{S}(f) \, \mathbf{f}^s, ~\text{for $s=1,2,3$.} \end{align*}

$$ [\nabla_{\mathbf{f}^s} E_\mathbb{S}(f)]_\ell = [L_\mathbb{S}(f) \, \mathbf{f}^s]_\ell + \underbrace{\frac{1}{2} \sum_{t=1}^3 \sum_{i=1}^n \sum_{j=1}^n f_i^t f_j^t \frac{\partial}{\partial f_\ell^s}[L_\mathbb{S}(f)]_{i,j}}_{(*)}. $$ It suffices to check $(*) = 2 [L_\mathbb{S}(f) \, \mathbf{f}^s]_\ell$.

Fixed Point Iteration for VSEM

The critical point $f$ satisfies $L_\mathbb{S}(f) \, \mathbf{f}^s =\mathbf{0}$, for $s=1,2,3$. Under a given boundary constraint $\mathbf{f}^s_\mathtt{B} = \mathbf{g}^s$, the critical point $f$ satisfies $$ [L_\mathbb{S}(f)]_{\mathtt{I},\mathtt{I}} \mathbf{f}^s_{\mathtt{I}} = -[L_\mathbb{S}(f)]_{\mathtt{I},\mathtt{B}} \mathbf{g}^s. $$

Let \begin{equation} \varphi^s(f) = \begin{bmatrix} -[L_\mathbb{S}(f)]_{\mathtt{I},\mathtt{I}}^{-1} [L_\mathbb{S}(f)]_{\mathtt{I},\mathtt{B}} \mathbf{g}^s \\ \mathbf{g}^s \end{bmatrix}, ~\text{for $s=1,2,3$}. \end{equation}

Algorithm ^[YuLL19]

$ {\mathbf{f}^s}^{(n+1)} = \varphi^s(f^{(n)}), ~ \text{for $s=1,2,3$}. $

[YuLL19] Yueh, Li, Lin and Yau, SIAM J Imaging Sci, 12(2):1071–1098, 2019.

Application of VSEM to Optimal Mass Transportation

Optimal Mass Transportation (OMT) Problem

The OMT problem aims to find a measure-preserving map $f:\mathcal{M}\to \mathcal{N}$ that minimizes $$ \mathcal{C}(f) = \int_\mathcal{M} \| x - f(x)\|^2 \,\mathrm{d}\mu. $$

$\xrightarrow{~~~~~f~~~~~}$

$(\mathcal{M},\mu)$ $(\mathcal{N},\nu)$

Discrete OMT Problem

The discrete OMT problem aims to find a measure-preserving map $f:\mathcal{M}\to\mathcal{N}$ that minimizes $$ \mathcal{C}_V(f) = \sum_{v\in\mathcal{V}(\mathcal{M})} \|v - f(v)\|^2 \,\mu_V(v), ~ \text{where $\mu_V(v)=\frac{1}{4}\sum_{\tau\supset v} \mu(\tau)$.} $$

$\xrightarrow{~~~~~f~~~~~}$