Metrics, Quantization and Registration in Varifold Spaces

Abstract

This paper is concerned with the theory and applications of varifolds to the representation, approximation and diffeomorphic registration of shapes. One of its purpose is to synthesize and extend several prior works which, so far, have made use of this framework mainly in the context of submanifold comparison and matching. In this work, we instead consider deformation models acting on general varifold spaces, which allow to formulate and tackle diffeomorphic registration problems for a much wider class of geometric objects and lead to a more versatile algorithmic pipeline. We study in detail the construction of kernel metrics on varifold spaces and the resulting topological properties of those metrics and then propose a mathematical model for diffeomorphic registration of varifolds under a specific group action which we formulate in the framework of optimal control theory. A second important part of the paper focuses on the discrete aspects. Specifically, we address the problem of optimal finite approximations (quantization) for those metrics and show a $\varGamma$-convergence property for the corresponding registration functionals. Finally, we develop numerical pipelines for quantization and registration before showing a few preliminary results for one- and two-dimensional varifolds.

Appendix

Proof of Theorem 3

We first prove that ${\mathcal {H}}^d(X\bigtriangleup Y) =0$. Let us denote by $W^{pos}$ and $W^G$ the RKHS associated with kernels $k^{pos}$ and $k^{G}$, respectively. Suppose that X and Y are rectifiable sets as above such that $\left\| \mu _X-\mu _Y \right\| _{W^*}=0$ and ${\mathcal {H}}^d(X\bigtriangleup Y) >0$. Without loss of generality, we may assume that ${\mathcal {H}}^d(X \setminus Y) >0$. From Lusin’s theorem, there exists a subset U of X such that $T|_{U}$ is continuous and ${\mathcal {H}}^d(X\setminus U) < {\mathcal {H}}^d(X\setminus Y)$. Let us denote by $E := U \cap (X \setminus Y)$, we see that ${\mathcal {H}}^d(E)>0$. Since for ${\mathcal {H}}^d \ a.e. \ x\in E$,

$$\begin{aligned} \limsup \limits _{r \rightarrow 0} \frac{{\mathcal {H}}^d(B_r(x)\cap E )}{\frac{\pi ^{\frac{d}{2}}}{\varGamma (\frac{d}{2}+1)}r^d} \ge \frac{1}{2^d}, \end{aligned}$$

(cf [24]), there exists $x_0 \in E, \ {\mathcal {H}}^d(B_r(x_0) \cap E)>0$ for any $r >0$.

Let $g:{\widetilde{G}}^n_d \rightarrow {\mathbb {R}}$ be defined by $g(\cdot ) = k^G(T(x_0),\cdot )$. Since $x \longmapsto g(T(x))$ is continuous on E and $g(T(x_0))>0$, there exists $r_0>0$ such that $\forall \ x \in B_{r_0}(x_0) \cap E, \ g(T(x))>0$. Let $A \doteq B_{r_0}(x_0) \cap E$ and $h(x) := {\mathbf {1}}_A(x)$, then ${\mathcal {H}}^d(A)>0$ and $g(T(x))>0, \ \forall \ x \in A$. Using the density of $C_c({\mathbb {R}}^n)$ in together with the fact that $k^{pos}$ is $C_0$-universal, there exist $\{f_j\}_{j=1}^{\infty } \subset C_c({\mathbb {R}}^n)$ and $\{h_j\}_{j=1}^{\infty } \subset W^{pos}$ such that $\lim \limits _{j \rightarrow \infty } f_j =h$ in and $\Vert f_j-h_j \Vert _{\infty }< \frac{1}{j}$. Now, since $h_j \otimes g \in W$ and $\mu _X = \mu _Y$ in $W^*$, we have

$$\begin{aligned} 0= & {} (\mu _X-\mu _Y)(h_j \otimes g) = \int _{X} h_j(x) g(T(x)) d {\mathcal {H}}^d(x) \\&- \int _{Y} h_j(y) g(S(y)) \text {d} {\mathcal {H}}^d(y) \rightarrow \int _A g(T(x)) \text {d} {\mathcal {H}}^d(x) >0, \end{aligned}$$

which is a contradiction. Hence, we have ${\mathcal {H}}^d(X\bigtriangleup Y) = 0$

Next, we show that $T(x) = S(x)$ ${\mathcal {H}}^d$-a.e.. Let $F := \{x \in X | T(x) = -S(x)\}$ and assume that ${\mathcal {H}}^d(F)>0$. From Lusin’s theorem, there exists subset $F' \subset F$ such that $T|_{F'}$ is continuous and ${\mathcal {H}}^d(F')>0$. Using the upper density argument as above, we can find $z_0 \in F'$ such that ${\mathcal {H}}^d(B_r(z_0) \cap F')>0$ for all $r>0$. Since the map $x \mapsto \langle T(x),T(z_0) \rangle$ restricted to $F'$ is continuous, there exists a $\delta _0>0$ satisfying:

$$\begin{aligned} \langle T(x),T(z_0) \rangle >0, \ \forall x \in B_{\delta _0}(z_0) \cap F'. \end{aligned}$$

Define $B:= B_{\delta _0}(z_0) \cap F'$, $\eta (\cdot ):= \gamma (\langle \cdot ,T(z_0) \rangle )$ and $u(x) := \eta (T(x)) - \eta (S(x))$. Observe that, from the assumption $\gamma (t) \ne \gamma (-t), \ \forall t \in [-1,1]$,

$$\begin{aligned} u(x) = \eta (T(x)) - \eta (-T(x)) \ne 0, \ \forall x \in F'. \end{aligned}$$

From this, we may assume that $u(x)>0, \ \forall x \in F'$. Let $\{f_j'\}_j$ and $\{h_j'\}_j$ be sequences in $C_c({\mathbb {R}}^n)$ and $W_{pos}$ such that $f_j'$ converges to ${\mathbf {1}}_B$ in and $\Vert f_j'-h_j'\Vert _{\infty } < 1/j$. We obtain

$$\begin{aligned} 0 = (\mu _X - \mu _Y| h_j' \otimes \eta ) = \int _X h_j'(x) u(x) d{\mathcal {H}}^d(x) \rightarrow \int _{B} u(x) d{\mathcal {H}}^d(x) > 0, \end{aligned}$$

which is impossible. $\square$

Proof of Theorem 8

Thanks to the first term in E, any minimizing sequence of E is bounded in $L^2([0,1],V)$. Let $\{v_j\}$ be a subsequence of such minimizing sequence which converges weakly to some ${\bar{v}}$ in $L^2([0,1],V)$. Using the results of [53] Chapter 7.2, we know that

$$\begin{aligned} \lim _{j \rightarrow \infty } \Vert (\varphi _1^{v_j} - \varphi _1^{{\bar{v}}})|_K \Vert _{1,\infty } = 0. \end{aligned}$$

Furthermore, for any $\omega \in W$, we have

$$\begin{aligned}&\left| \left( (\varphi _1^{v_j})_{\#} \mu _0 - (\varphi _1^{{\bar{v}}})_{\#} \mu _0 | \omega \right) \right| \\&\quad =\bigg | \int _{K} J_S \varphi _1^{v_j}(x) \omega (\varphi _1^{v_j}(x),d_x \varphi _1^{v_j} \cdot S) - J_S\varphi _1^{{\bar{v}}}(x) \omega (\varphi _1^{{\bar{v}}}(x),d_x \varphi _1^{{\bar{v}}} \cdot S) \text {d} \mu _0 \bigg | \\&\quad \le \int _{K} |J_S \varphi _1^{v_j}(x)| \left| \omega (\varphi _1^{v_j}(x),d_x \varphi _1^{v_j} \cdot S) - \omega (\varphi _1^{{\bar{v}}}(x),d_x \varphi _1^{{\bar{v}}} \cdot S) \right| \text {d} \mu _0 \\&\qquad + \int _{K} \left| J_S \varphi _1^{v_j}(x) - J_S\varphi _1^{{\bar{v}}}(x) \right| \left| \omega (\varphi _1^{{\bar{v}}}(x),d_x \varphi _1^{{\bar{v}}} \cdot S) \right| \text {d} \mu _0 \\ \end{aligned}$$

Now, using the embedding $W \hookrightarrow C_0^1({\mathbb {R}}^n \times {\widetilde{G}}^n_d)$

$$\begin{aligned}&\left| \left( (\varphi _1^{v_j})_{\#} \mu _0 - (\varphi _1^{{\bar{v}}})_{\#} \mu _0 | \omega \right) \right| \\&\quad \le \left( \int _{K} |J_S \varphi _1^{v_j}(x)| \text {d} \mu _0 \right) \Vert \omega \Vert _{1,\infty } \Vert (\varphi _1^{v^N} - \varphi _1^{{\bar{v}}})|_K \Vert _{1,\infty } + C \Vert (\varphi _1^{v_j} - \varphi _1^{{\bar{v}}})|_K \Vert _{1,\infty } \\&\quad \le C' \Vert (\varphi _1^{v_j} - \varphi _1^{{\bar{v}}})|_K \Vert _{1,\infty }. \end{aligned}$$

Taking supremum over all $\omega \in W$ with $\Vert \omega \Vert _W \le 1$, we obtain that

$$\begin{aligned} \Vert (\varphi _1^{v_j})_{\#} \mu _0 - (\varphi _1^{{\bar{v}}})_{\#} \mu _0 \Vert _{W^*} \le C' \Vert (\varphi _1^{v_j} - \varphi _1^{{\bar{v}}})|_K \Vert _{1,\infty } \rightarrow 0 \end{aligned}$$

as $j \rightarrow \infty$. Combining this with lower semicontinuity of $v \mapsto \Vert v\Vert _{L^2([0,1],V)}^2$, we finally obtain that

$$\begin{aligned} E({\bar{v}}) \le \liminf _{j \rightarrow \infty } E(v_j) \end{aligned}$$

and hence ${\bar{v}}$ is a global minimizer. $\square$

Proof of Proposition 9

Recall that for all $\phi \in \text {Diff}({\mathbb {R}}^n)$, $g(\phi ) = \lambda \Vert \phi _{\#} \mu _0 - \mu _{tar} \Vert _{W^*}^2$ which we may rewrite as

$$\begin{aligned} g(\phi ) = \lambda (\phi _{\#} \mu _0 | K_W(\phi _{\#} \mu _0 -2\mu _{tar})) + \lambda \Vert \mu _{tar}\Vert _{W^*}^2. \end{aligned}$$

Thus, the variation with respect to $\phi$ in the Banach space ${\mathcal {B}}$ writes

$$\begin{aligned} \partial _{\phi } g(\phi ) = \partial _{\phi } (\phi _{\#} \mu _0 | \omega _0) \end{aligned}$$

where $\omega _0 \doteq 2\lambda K_W(\phi _{\#} \mu _0 -\mu _{tar}) \in W$. Moreover,

$$\begin{aligned} (\phi _{\#} \mu _0 | \omega _0) = \int _{{\mathbb {R}}^n \times {\widetilde{G}}_d^n} \omega _0(\phi (x),d_x \phi \cdot T) J_T \phi (x) \text {d} \mu _0(x,T). \end{aligned}$$

Taking the variation with respect to $\phi$ along any $u \in C_0^1({\mathbb {R}}^n,{\mathbb {R}}^n)$, we obtain:

$$\begin{aligned} (\partial _{\phi } g(\phi ) | u) =&\int _{{\mathbb {R}}^n \times {\widetilde{G}}_d^n} \partial _x \omega _0(\phi (x),d_x \phi \cdot T) \cdot u(x) J_T \phi (x) \text {d} \mu _0(x,T) \nonumber \\&+\int _{{\mathbb {R}}^n \times {\widetilde{G}}_d^n} \partial _T \omega _0(\phi (x),d_x \phi \cdot T) \cdot (d_x u |_T) J_T \phi (x) d \mu _0(x,T) \nonumber \\&+\int _{{\mathbb {R}}^n \times {\widetilde{G}}_d^n} \omega _0(\phi (x),d_x \phi \cdot T). \text {div}_T u(x). J_T \phi (x) \text {d} \mu _0(x,T) \end{aligned}$$

(37)

where the last term follows from the differentiation of Gram determinant matrices, while the notation $\partial _T$ in the second term is a shortcut notation for differentiation on the Grassmannian which we do not explicit further here, we, however, refer to the similar computations done in [18] and to the developments in Sect. 6 for more details. For the first term, we can rely on the Young measure decomposition $\mu _0 = |\mu _0| \otimes \nu _x$ introduced at the end of Sect. 2.1 which gives:

$$\begin{aligned} (1)= & {} \int _{{\mathbb {R}}^n} {\tilde{\alpha }}(\phi ,x) \cdot u(x) \ d |\mu _0|(x), \ \ \text {where} \ \ {\tilde{\alpha }}(\phi ,x) \\= & {} \int _{{\widetilde{G}}_d^n} \partial _x \omega _0(\phi (x),d_x \phi \cdot T) J_T \phi (x) \text {d}\nu _x(T). \end{aligned}$$

We can also rewrite the third term as:

$$\begin{aligned} (3)= & {} \int _{{\mathbb {R}}^n \times {\widetilde{G}}_d^n} {\tilde{\gamma }}(\phi ,x,T) \ \text {div}_T u(x) \ \text {d} \mu _0(x,T), \ \ \text {with} \ \ {\tilde{\gamma }}(\phi ,x,T)\\= & {} \omega _0(\phi (x),d_x \phi \cdot T) \ J_T \phi (x). \end{aligned}$$

As for the second term in (37), for each (x, T) the integrand involves a linear combination (depending on $\phi$) of the partial derivatives of u along the subspace T, i.e., of the elements of the matrix $d_x u|_T \in {\mathbb {R}}^{n\times d}$. Thus, without attempting to specify this term explicitly, we can in general write it as ${\tilde{\beta }}(\phi ,x,T) d_x u|_T$ where ${\tilde{B}}$ is a continuous map from ${\mathcal {B}} \times {\mathbb {R}}^n \times {\tilde{G}}_d^n$ into ${\mathcal {L}}({\mathbb {R}}^{n\times d},{\mathbb {R}})$ giving us

$$\begin{aligned} (2) = \int _{{\mathbb {R}}^n \times {\widetilde{G}}_d^n} {\tilde{B}}(\phi ,x,T) d_x u|_T \ \text {d} \mu _0(x,T). \end{aligned}$$

The result of the theorem then follows by setting $\alpha (x) \doteq {\tilde{\alpha }}(\varphi _1^v,x)$, $\beta (x,T) \doteq {\tilde{\beta }}(\varphi _1^v,x,T)$ and $\gamma (x,T)={\tilde{\gamma }}(\varphi _1^v,x,T)$. $\square$

Proof of Proposition 14

We can treat the case of each particle i separately and thus, without loss of generality, we may directly assume that $N=1$. We write $q(t)=(x(t),u^{(1)}(t),\cdots ,u^{(d)}(t))$, $p(t)=(p^x(t),p^{u_1}(t),\ldots ,p^{u_d}(t))$ for the state and costate variables along an optimal trajectory and

$$\begin{aligned} U \doteq \text {Span}\{u^{(1)}(1),\cdots ,u^{(d)}(1)\}. \end{aligned}$$

Consider the group of linear transformations, $G \doteq \mathrm{SL} (U) \oplus \mathrm{GL} (U^{\perp })$, i.e., for any $\mathrm {g} \in G$,

where $x_{U}$ and $x_{U^{\perp }}$ are the orthogonal projections of x on U and $U^{\perp }$, with and $\mathrm {g}_{\perp } \in \mathrm{GL}(U^{\perp })$. The Lie algebra of G is ${\mathfrak {g}} = {\mathfrak {sl}}(U) \times {\mathcal {L}}(U^\perp )$ and ${\mathfrak {sl}}(U)$ is the set of all zero trace linear transformations of U. Now, consider the action of G on ${\mathbb {R}}^{(d+1)n}$ defined as:

$$\begin{aligned} \mathrm {g} \cdot q := (q_0,\mathrm {g}(q_1),\cdots ,\mathrm {g}(q_d)). \end{aligned}$$

for any $q=(q_0,\ldots ,q_{d}) \in {\mathbb {R}}^{(d+1)n}$. We see that $\mu ^{\mathrm {g}\cdot q(1)} = \mu ^{q(1)}$ for all $\mathrm {g} \in G$ and therefore $g(\mathrm {g}\cdot q(1)) = g(q(1))$.

Now, if we let $\{\mathrm {g}_t\}$ be a smooth curve in G that satisfies $\mathrm {g}_0 = id$ and $\frac{d}{\text {d}\tau }|_{\tau =0} \mathrm {g}_\tau = h \in {\mathfrak {g}}$, differentiating the equality $g(\mathrm {g}_\tau \cdot q(1)) = g(q(1))$ shows that for any $h \in {\mathfrak {g}}$, we have

$$\begin{aligned} 0 = (p(1)|h \cdot q(1)) = \sum _{k=1}^{d} \langle p^{u_k}(1) , h(u^{(k)}(1)) \rangle \end{aligned}$$

Since $h \in {\mathfrak {g}}$, we must have that $h|_{U}$ is a zero trace linear map. For any $1 \le i < j \le d$, we may choose h such that $h(u^{(i)}(1)) = -h(u^{(j)}(1))$ and $h(u^{(k)}(1)) = 0 , \ \forall k \notin \{i,j\}$, which leads to $\langle u^{(i)}(1), p^{u_i}(1) \rangle = \langle u^{(j)}(1), p^{u_j}(1) \rangle$. Consequently,

$$\begin{aligned} \langle u^{(1)}(1), p^{u_1}(1) \rangle = \cdots = \langle u^{(d)}(1), p^{u_d}(1) \rangle =\alpha \end{aligned}$$

for some constant $\alpha$. In addition, for any $i \ne j$, we can also choose h such that $h(u^{(i)}(1)) = u^{(j)}(1)$ and $h(u^{(k)}(1))= 0 , \ \forall k \notin \{i,j\}$, which gives $\langle u^{(i)}(1),p^{u_j}(1) \rangle = 0$. It results that $D(1) = \alpha .I_{d \times d}$.

Finally, since D(t) is constant by Lemma 2, we obtain that

$$\begin{aligned} D(t) = \left( \begin{array}{ccc} \alpha &{} &{} 0 \\ &{} \ddots &{} \\ 0 &{} &{} \alpha \end{array} \right) , \end{aligned}$$

for all $t \in [0,1]$. $\square$