A continuous scale space of diffeomorphisms

TL;DR

The paper develops a continuous scale-space framework for diffeomorphic shape registration by introducing a nested family of RKHSs of scale-dependent vector fields, enabling flows that evolve across a scale interval $[s_1,s_2]$. It provides a detailed kernel-theoretic characterization of the composite scale-space space $\mathbb{W}$, along with practical numerical schemes (piecewise-constant and Fourier-based) to approximate the reproducing kernel $K_{\mathbb{W}}$ efficiently. A multiscale LDDMM formulation (MS-LDDMM) is proposed, with an optimal-control (PMP) approach that yields explicit expressions for the time-and-scale dependent control $v$, and existence results ensuring minimizers when the end-cost depends on finitely many base scales. Experimental simulations on 2D landmark matching illustrate how deformations decompose across scales and how choosing different base scales affects the interpolation between coarser and finer transformations, highlighting the added interpretability from the scale-resolved residuals.

Abstract

In this paper, we define and study a nested family of reproducing kernel Hilbert spaces of vector fields that is indexed by a range of scales, from which we construct a reproducing kernel Hilbert space of scale-dependent vector fields. We provide a characterization of the reproducing kernel of that space, with numerical approximations ensuring quick evaluations when this kernel does not have a closed form. We then introduce a multiscale version of the large deformation diffeomorphic metric mapping (LDDMM) problem and prove the existence of solutions. Finally, we provide numerical experiments performing landmark matching using multiscale LDDMM.

Figures (22)

• Figure 1: Schematic human: template and target.
• Figure 2: Schematic human: deformation at different scales. Base scales: $r_1$, $r_{20}$.
• Figure 3: Schematic human: First row: log Jacobian; Second row: log Jacobian of residuals $\rho^v_{r_i} = \psi^v_{r_{i}}\circ (\psi^v_{r_{i-1}})^{-1}$. Base scales: $r_1, r_{20}$.
• Figure 4: Schematic human: deformations. Base scale: $r_1$.
• Figure 5: Schematic human: deformations. Base scale: $r_{20}$.

Theorems & Definitions (20)

• Lemma 2.1
• Proposition 2.2
• Proposition 2.4
• proof
• Lemma 2.5
• Proposition 2.6
• Lemma 3.1
• proof
• Remark 3.2
• Theorem 3.3