On Inversion-Free Mapping and Distortion Minimization

Abstract

This paper addresses a general problem of computing inversion-free maps between continuous and discrete domains that induce minimal geometric distortions. We will refer to this problem as optimal mapping problem. Finding a good solution to the optimal mapping problem is a key part in many applications in geometry processing and computer vision, including: parameterization of surfaces and volumetric domains, shape matching and shape analysis. The first goal of this paper is to provide a self-contained exposition of the optimal mapping problem and to highlight the interrelationship of various aspects of the problem. This includes a formal definition of the problem and of the related unitarily invariant geometric measures, which we call distortions. The second goal is to identify novel properties of distortion measures and to explain how these properties can be used in practice. Our major contributions are: (i) formalization and juxtaposition of key concepts of the optimal mapping problem, which so far have not been formalized in a unified manner; (ii) providing a detailed survey of existing methods for optimal mapping, including exposition of recent optimization algorithms and methods for finding injective mappings between meshes; (iii) providing novel theoretical findings on practical aspects of geometric distortions, including the multi-resolution invariance of geometric energies and the characterization of convex distortion measures. In particular, we introduce a new family of convex distortion measures, and prove that, on meshes, most of the existing distortion energies are non-convex functions of vertex coordinates.

Appendix A Computational Aspects

We are now in the position to deliver some detail on the implementation. We explicitly write down in full matrix form the main expressions that arise in Sect.  4.

1.1 Appendix A.1 Jacobian of Simplicial Map

First, assume a general scenario (45) in which dimensions m and d of the embedding spaces may differ from the simplex dimension.

Let $(v_1, \ldots , v_{n+1})$ be a oriented n-simplex $s \in {{\,\mathrm{\mathcal {S}}\,}}$. To simplify our notations, we denote by $v_i$ an element of the vertex set (vertex index) and by $\varvec{v_i}$ we denote the source coordinates of that vertex. That is, if $\varvec{y}$ is the source coordinate vector, then $\varvec{v_i}\triangleq \varvec{y}_{v_i} \in \mathbb {R}^m$. In particular, by ${\varvec{v_i}}_j$ we denote the j-th coordinate of the point $\varvec{y}_{v_i}$. We employ similar notations for the target coordinates $\varvec{x}$, i.e., $f_s(\varvec{v}_i) = f_s(v_i)=\varvec{x}_{v_i}$, and $f(v_i)_j$ denotes the j-th coordinate of the point $\varvec{x}_{v_i}\in \mathbb {R}^d$.

Assuming that vertex coordinates are column vector, one can then represent simplex s as a matrix $M_s \in \mathbb {R}^{m \times (n+1)}$

$$\begin{aligned} M_s \triangleq \begin{pmatrix} {\varvec{v_1}}_1 &{} {\varvec{v_2}}_1 &{} \cdots &{} {\varvec{v_{\varvec{(n+1)}}}}_1\\ {\varvec{v_1}}_2 &{} {\varvec{v_2}}_2 &{} \cdots &{} {\varvec{v_{\varvec{(n+1)}}}}_2\\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ {\varvec{v_1}}_m &{} {\varvec{v_2}}_m &{} \vdots &{} {\varvec{v_{\varvec{(n+1)}}}}_m\\ \end{pmatrix}, \end{aligned}$$

where the order of vertices (columns) reflects simplex orientation. Then, the image of s under $f_s$ is represented by

$$\begin{aligned} M_{s'} \triangleq \begin{pmatrix} {f_s(v_1)}_1 &{} {f_s(v_2)}_1 &{} \cdots &{} {f_s(v_{n+1})}_1\\ {f_s(v_1)}_2 &{} {f_s(v_2)}_2 &{} \cdots &{} {f_s(v_{n+1})}_2\\ \vdots &{} \vdots &{} \cdots &{} \vdots \\ {f_s(v_1)}_d &{} {f_s(v_2)}_d &{} \cdots &{} {f_s(v_{n+1})}_d\\ \end{pmatrix}. \end{aligned}$$

(92)

Using hat functions (48), the expression for the map becomes

$$\begin{aligned} f_s({{\,\mathrm{\varvec{r}}\,}}) = M_{s'} \varvec{h}_s({{\,\mathrm{\varvec{r}}\,}}),\,\, {{\,\mathrm{\varvec{r}}\,}}\in {{\,\mathrm{conv}\,}}(s),\,\, \end{aligned}$$

where

$$\begin{aligned} \varvec{h}_s({{\,\mathrm{\varvec{r}}\,}})\triangleq ~\begin{bmatrix}h_{v_1}({{\,\mathrm{\varvec{r}}\,}})&h_{v_2}({{\,\mathrm{\varvec{r}}\,}})&\cdots&h_{v_{n+1}}({{\,\mathrm{\varvec{r}}\,}})\end{bmatrix}^{\top }~\in ~\mathbb {R}^{(n+1)\times 1}\,. \end{aligned}$$

(93)

Let $\nabla h_{v_i}$ be the gradient of $h_{v_i}$, computed with respect to ${{\,\mathrm{\varvec{r}}\,}}\in {{\,\mathrm{conv}\,}}(s)$, then $\nabla h_{v_i}$ is constant in ${{\,\mathrm{conv}\,}}(s)$ and according to (51)

$$\begin{aligned} dh_{v_i} = - \dfrac{\varvec{\upeta }_j}{n {{\,\mathrm{Vol}\,}}(s)} \in \mathbb {R}^{m\times 1} \,, \end{aligned}$$

where $\varvec{\upeta }_i$ is a normal vector defined in (51). Consequently, the Jacobian matrix of $\varvec{h}_s({{\,\mathrm{\varvec{r}}\,}})$, denoted by $d\varvec{h}_s$, is given by

$$\begin{aligned} d\varvec{h}_s = \begin{bmatrix}dh_{v_1}(\varvec{v})&dh_{v_2}(\varvec{v})&\cdots&dh_{v_{n+1}}(\varvec{v})\end{bmatrix} ^{\top }\in \mathbb {R}^{(n+1) \times m}\,, \end{aligned}$$

(94)

and the Jacobian of ${f}_s$ is

$$\begin{aligned} d{f}_s = M_{s'}\,d\varvec{h}_s,~~ d{f}_s\in \mathbb {R}^{d \times m}\,. \end{aligned}$$

(95)

Note that $\text {rank}\big (df_s\big ) \le n$. Therefore, $df_s$ has at most n non-zero singular values $\sigma _1(df_s),\dots ,\sigma _n(df_s)$. These singular values can be used for computing n-dimensional distortions of $df_s$. Equivalently, we can repeat the above computations for the canonical representation $\widetilde{f}_s$ of the map for obtaining the canonical form of the Jacobian, $d \widetilde{f}_s\in \mathbb {R}^{n\times n}$. Jacobian $df_s$ and its canonical form have the same n singular values,

$$\begin{aligned} \sigma _1(d \widetilde{f}_s)=\sigma _1(df_s),\dots , \sigma _n(d \widetilde{f}_s)=\sigma _n(df_s)\,, \end{aligned}$$

so that ${{\,\mathrm{\mathcal {D}}\,}}\big (\widetilde{f}_s\big )={{\,\mathrm{\mathcal {D}}\,}}\big (\sigma _1(df_s),\dots , \sigma _n(df_s)\big )$ for any distortion ${{\,\mathrm{\mathcal {D}}\,}}$.

1.2 Appendix A.2 Distortion Energy Gradient

Assume the scenario (43) of equal dimensions, $m=n=d$. Denote by $J\in \mathbb {R}^{n\times n}$ the Jacobian matrix $df_s$ of a simplicial map f on a simplex s, and denote by $U \mathrm {diag} (\sigma ) V^{\top }$ the SVD of J. According to [89] and [40] the derivation of a distortion ${{\,\mathrm{\mathcal {D}}\,}}$, computed with respect to J, is given by

$$\begin{aligned} \nabla _J {{\,\mathrm{\mathcal {D}}\,}}(J) = U \mathrm {diag}\big ( \nabla _{\sigma } {{\,\mathrm{\mathcal {D}}\,}}(\sigma )\big ) V^\top \,. \end{aligned}$$

(96)

If $M_{s'}\in \mathbb {R}^{n\times (n+1)}$ is the matrix (92) of target coordinates of s, then the derivation of $E_{{{\,\mathrm{\mathcal {D}}\,}}}$ with respect to $M_{s'}$ can be written as:

$$\begin{aligned} \nabla _{M_{s'}} E_{{{\,\mathrm{\mathcal {D}}\,}}}(M_{s'}) = {{\,\mathrm{Vol}\,}}(s) d\varvec{h} \nabla _{J_s} {{\,\mathrm{\mathcal {D}}\,}}(J_s)\in \mathbb {R}^{(n+1)\times n} \,, \end{aligned}$$

(97)

where $J_s$ denotes the Jacobian $df_s[\varvec{x}]$ and $d\varvec{h}$ is defined according to (94). The gradient $\nabla {E}_{\varvec{x}}(\varvec{x})\in \mathbb {R}^{n |{{\,\mathrm{\mathcal {V}}\,}}|}$ of E is computed with respect to the column vector $\varvec{x}$ by laying elements of (97) into their positions in the column vector,

$$\begin{aligned} \nabla _{M_s'} E(M_s') \mapsto \nabla E_{\varvec{x}}(M_s')\in \mathbb {R}^{n|{{\,\mathrm{\mathcal {V}}\,}}|}\,, \end{aligned}$$

and then $\nabla E_{\varvec{x}}(M_s')$ are summed for every simplex $s\in {{\,\mathrm{\mathcal {S}}\,}}$.

Note that (96) is also valid for the signed SVD representation of distortion measures, mentioned in Remark 2.1. See the supplemental material for a more general computation of $\nabla _{\varvec{x}} E$.