A new variational model for shape graph registration with partial matching constraints

TL;DR

The paper extends elastic shape analysis to handle shape graphs with partial matching and topological changes by introducing an inexact, varifold-based relaxation of the matching problem. It develops a general variational framework using higher-order Sobolev metrics on shape graphs, augmented with a weight function on components that is regularized by total variation to enable erasure or creation of parts in the match. Existence of minimizers is established for both unweighted graphs and weighted graphs under scale-invariant Sobolev metrics, and a detailed SFISTA-based optimization scheme is proposed to solve the discretized problem efficiently. The approach is demonstrated on partially observed and topology-varying data, with open-source code and discussion of practical parameter choices and potential extensions.

Abstract

This paper introduces a new extension of Riemannian elastic curve matching to a general class of geometric structures, which we call (weighted) shape graphs, that allows for shape registration with partial matching constraints and topological inconsistencies. Weighted shape graphs are the union of an arbitrary number of component curves in Euclidean space with potential connectivity constraints between some of their boundary points, together with a weight function defined on each component curve. The framework of higher order invariant Sobolev metrics is particularly well suited for constructing notions of distances and geodesics between unparametrized curves. The main difficulty in adapting this framework to the setting of shape graphs is the absence of topological consistency, which typically results in an inadequate search for an exact matching between two shape graphs. We overcome this hurdle by defining an inexact variational formulation of the matching problem between (weighted) shape graphs of any underlying topology, relying on the convenient measure representation given by varifolds to relax the exact matching constraint. We then prove the existence of minimizers to this variational problem when we choose Sobolev metrics of sufficient regularity and a total variation (TV) regularization on the weight function. We propose a numerical optimization approach which adapts the smoothed fast iterative shrinkage-thresholding (SFISTA) algorithm to deal with TV norm minimization and allows us to reduce the matching problem to solving a sequence of smooth unconstrained minimization problems. We finally illustrate the capabilities of our new model through several examples showcasing its ability to tackle partially observed and topologically varying data.

Figures (8)

• Figure 1: Parametrized shape graph (left) with associated adjacency matrix (right). The shape graph $c = \prod_{k=1}^K c^k$ has $K=4$ component curves, where $c^1$ is a closed curve (red), $c^2$ is an immersion with self intersection (blue), and $c^3$ and $c^4$ are open curves (yellow and green respectively).
• Figure 1: Geodesic between two shape graphs with the same topology: the source $c_0$ (left) and target $c_1$ (in red on the right). The target is overlayed on the transformed source $c(1)$ at $t=1$. The estimated geodesic distance is $\overline{\operatorname{dist}}(c_0,c_1)=0.83$.
• Figure 1: Matching incomplete leaves. Geodesics between Swedish leaves with partial matching constraints. The source (blue at $t=0$) and target (red at $t=1$) have distinct topologies, with the source being a closed curve, and the target being e.g., an open curve or having multiple connected components. The target is overlayed on the transformed source $c(1)$ on the right, and parts of the transformed source which get "erased", i.e. where the estimated weight function vanishes, are colored in progressively transparent shades of blue. (Top to bottom) The estimated geodesic distances are (i) $\overline{\operatorname{dist}}(c_0,c_1)= 1.29$, (ii) $\overline{\operatorname{dist}}(c_0,c_1)= 0.95$, (iii) $\overline{\operatorname{dist}}(c_0,c_1)= 0.51$, (iv) $\overline{\operatorname{dist}}(c_0,c_1)= 1.35$.
• Figure 2: Geodesic between source (blue at $t=0$) and target (red at $t=1$) shape graphs having different topologies. (Top row) We use the relaxed shape graph matching framework described in \ref{['eq:relaxed_shape_graph_match']}, which only allows for a geometric deformation of the source. The estimated geodesic distance is $\overline{\operatorname{dist}}(c_0,c_1)=1.44$, around 1.7 times higher than in Fig. \ref{['fig:match_2branches_fixed_weights']}. (Bottom row) Result obtained from the weighted shape graph matching framework described in \ref{['eq:relaxed_weighted_shape_graph_match']} that jointly estimates a deformation and weight changes on the source. Components of the source which get "erased" are colored in progressively transparent shades of blue. The estimated geodesic distance here is $\overline{\operatorname{dist}}(c_0,c_1)=0.77$, now fairly comparable to Fig. \ref{['fig:match_2branches_fixed_weights']}.
• Figure 2: 3D maize root systems. The source (blue at $t=0$) is a 3D maize root system with multiple lateral roots, and the target (red at $t=1$) is another maize root system with only 2 lateral roots. The extra branches of the transformed source which get "erased" are colored in progressively transparent shades of blue. The estimated geodesic distance here is $\overline{\operatorname{dist}}(c_0,c_1)=0.64$. We note that the left branch of the target matches with the particular branch on the transformed source shown above because matching with this specific branch requires the least amount of geometric deformation in ${\mathbb R}^3$ (and hence the least amount of energy) when compared to any of the 4 other branches. The algorithm then decides to erase these other 4 branches. \newlabelfig:match_3droots_source_weight0

Theorems & Definitions (20)

• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Proof 1
• Theorem 2.5
• Proof 2
• Remark 2.6
• Theorem 3.1
• Remark 3.2