Elastic Shape Analysis of Planar Objects Using Tensor Field Representations

Abstract

Shape analysis of objects in images is a critical area of research, and several approaches, including those that utilize elastic Riemannian metrics, have been proposed. While elastic techniques for shape analysis of curves are pretty advanced, the corresponding results for higher-dimensional objects (surfaces and disks) are less developed. This paper studies shapes of solid planar objects that are embeddings of a compact domain—a unit square or a unit disk—in ${\mathbb {R}}^2$. Specifically, it introduces a mathematical representation of objects using tensor fields and uses a re-parametrization-invariant Riemannian metric on these tensor fields to analyze object shapes elastically. The essential contribution here is developing an efficient numerical technique to map tensor fields back to the object space, allowing one to approximate geodesic paths in these objects’ shape spaces. Finally, the paper extends this framework to reach landmark-driven registration and improve geodesic computations. The paper illustrates this framework using several simulated and natural objects.

Appendix: Geometry of \(\mathrm{{Sym}}^+(n)\)

For any $A\in \mathrm{{Sym}}^+(n)$ and $U, V\in T_A\mathrm{{Sym}}^+(n)$, consider the following Riemannian metric defined on $\mathrm{{Sym^+(n)}}$

$$\begin{aligned} \begin{aligned} \langle U, V\rangle _A&=\left( {\mathrm{{tr}}}\left( A^{-1}U_0A^{-1}V_0\right) \right. \\&\left. \quad +\kappa {\mathrm{{tr}}}\left( A^{-1}U\right) {\mathrm{{tr}}}\left( A^{-1}V\right) \right) \sqrt{\det A}, \end{aligned} \end{aligned}$$

(10)

where $\kappa >0$, $U_0 = U - \frac{1}{n}{\mathrm{{tr}}}(A^{-1}U)A$ and $V_0 = V - \frac{1}{n}{\mathrm{{tr}}}(A^{-1}V)A$ satisfying ${\mathrm{{tr}}}(A^{-1}U_0)=0$ and ${\mathrm{{tr}}}(A^{-1}V_0) = 0$. This Riemannian metric was first introduced in [12] and is termed the split Ebin metric. The geometric structure of $\mathrm{{Sym^+(n)}}$ endowed with the split Ebin metric was studied thoroughly in [12] and we borrow some results from it directly.

Theorem 3

Let $A\in \mathrm{{Sym}}^+(n)$ and $K\in T_A\mathrm{{Sym}}^+(n)$. Define $q = 1+\frac{{\mathrm{{tr}}}(A^{-1}K)}{4},$ $\theta = \frac{\sqrt{\kappa ^{-1}{\mathrm{{tr}}}\left( A^{-1}K_0A^{-1}K_0\right) }}{4}$, where $K_0 = K - \frac{1}{n}{\mathrm{{tr}}}(A^{-1}K)A$ satisfying ${\mathrm{{tr}}}(A^{-1}K_0)=0$. Then, the exponential map starting at A in the direction of K is given by

$$\begin{aligned} \begin{aligned}&\mathrm{{Exp}}_A(K) \\&= {\left\{ \begin{array}{ll} \left( q^2+\theta ^2\right) ^{\frac{2}{n}}A\exp \left( \frac{\arctan (\theta /q)}{\theta }A^{-1}K_0\right) \quad \text {if} \ K_0\ne 0,\\ q^{\frac{4}{n}}A \quad \text {if} \ K_0=0. \end{array}\right. } \end{aligned} \end{aligned}$$

(11)

Theorem 4

Let $A, B \in \mathrm{{Sym}}^+(n)$, we have K $=$ $A\log (A^{-1}B) \in \mathcal{{U}}$ $=$ $T_A\mathrm{{Sym}}^+(n)\backslash (-\infty ,-4/n]A$. Then the inverse of the exponential map is given by the following:

$$\begin{aligned} \begin{aligned}&\mathrm{{Exp}}^{-1}_A(B) = \\&{\left\{ \begin{array}{ll} \frac{4}{n}\left( \beta \cos \theta -1\right) A +\frac{1}{\theta }\beta \sin \theta K_0 \quad &{}\text {if} \ K_0\ne 0,\\ \frac{4}{n}\left( \beta -1\right) A \quad &{}\text {if} \ K_0=0, \end{array}\right. } \end{aligned} \end{aligned}$$

(12)

where $\beta =\exp \left( \frac{{\mathrm{{tr}}}(A^{-1}K)}{4}\right)$, $K_0$ and $\theta$ are define as Theorem 3.

Then, the geodesic on $\mathrm{{Sym}}^+(n)$ can be solved explicitly.

Theorem 5

Let $A, B \in \mathrm{{Sym}}^+(n)$, $u \triangleq \mathrm{{Exp}}^{-1}_A(B)$, the geodesic between A and B is given as

$$\begin{aligned} \phi (t) = \mathrm{{Exp}}_A(ut), t \in [0,1]. \end{aligned}$$

(13)

Let $\mathcal{{M}}$ be the space of positive semi-definite symmetric matrices. Then the metric completion of $\mathrm{{Sym}}^+(n)$ is given by ${\mathcal{{M}}}/\sim$, where $A\sim B$ if they are both degenerate.

When $\kappa = \frac{1}{n}$, we call the Riemannian metric in Eq. 10 as the standard Ebin metric. It is the usual metric on the space of all Riemannian metrics considered by [8, 10, 11]. [7] gives the geodesic distance between any two matrices in $\mathrm{{Sym}}^+(n)$ of the standard Ebin metric as follows.

Theorem 6

For $A, B\in \mathrm{{Sym}}^+(n)$. Let $K = A\log (A^{-1}B)$. Then the square of the geodesic distance for the metric (10) between A and B is given by

$$\begin{aligned} \begin{aligned} d(A, B)^2&= \frac{16}{n}\left( \sqrt{\det (A)} -2\root 4 \of {\det (A)}\root 4 \of {\det (B)}\cos \theta \right. \\&\left. \quad + \sqrt{\det (B)}\right) , \end{aligned} \end{aligned}$$

(14)

where $\theta = \min \left\{ \pi , \frac{\sqrt{n{\mathrm{{tr}}}(A^{-1}K_0A^{-1}K_0)}}{4}\right\}$, $K_0$ is defined as Theorem 3.

In this paper, we use the standard Ebin metric ($\kappa = \frac{1}{2}$) for computation.

Using these point wise geodesics, we can compute a geodesic path between any two elements of $\mathcal{{G}}$. For any $g_1, g_2 \in \mathcal{{G}}$, let $\varPhi (s,t) = \phi (t)$, where $\phi$ is a parameterized geodesic between $g_1(s)$ and $g_2(s)$ in ${\text {Sym}}^+(2)$, for all $s \in D$.