Learning Near-Isometric Matching Models

Shape matching problems are typically described in terms of structural properties, encoding (for example) the ‘elasticity’ of certain joints in a rigid object. Unfortunately, encoding such constraints often results in a hard optimization problem (such as quadratic assignment), meaning that approximate solutions are typically employed; this presents a problem for many learning approaches that require exact inference schemes as a subroutine. We note that certain types of constraints, such as isometries, result in ‘easier’ optimization problems that can be solved with low tree-width graphical models. This allows us to apply learning in near-isometric matching scenarios, encoding rich first-order properties, as well as isometric and topological information. Introduction The well-known ‘quadratic assignment’ problem takes the form f̂ = argmin f :A→B ∑ a,b∈A Da,b,f(a),f(b), where f is an injective function. Many structural matching problems can be expressed in this form, for instance isometric matching can be written as f̂ = argmin f :S→T ∑ si,sj∈S ∣∣d(si, sj)− d(f(si), f(sj))∣∣, (1) where S and T represent shapes (so that each si defines the coordinates of a point), f is an injective function mapping point coordinates in S to point coordinates in T (for instance, mapping the ‘joints’ in a deformable model to pixels in an image), and d is the standard Euclidean distance function. In cases where a zero-cost solution exists to (eq. 1), we note that far more efficient solutions to this problem can be found: we need not consider all edges between pairs of points in S, but rather a subset of edges that define a ‘globally rigid’ graph. We have investigated several graphical models whose embeddings in the plane are globally rigid. Examples are shown in Figure 1 (an ‘embedding’ is shown at right). Some examples demonstrating the matching problem are shown in Figure 2. These models continue to maintain good performance even as the assumption of a zero-cost solution becomes substantially violated. Structured Matching Objectives Given a graph G with a globally rigid embedding, we can augment our potentials to encode structural constraints other than distance preservation. Generally, we write f̂ = argmin f :S→T ∑ i,j∈G 〈 Φi,j(si, sj , f(si), f(sj)), θ 〉 . Here Φ is a feature vector describing the mapping from (si, sj) to (f(si), f(sj)). One component of this vector would be |d(si, sj) − d(f(si), f(sj)) ∣∣ as in (eq. 1), though other features could include topological information (for example in OCR applications where connectedness is important), and first-order properties such as Shape Contexts or SIFT features; we can also create higher-order features Φi,j,k encoding angle and scale information. ∗The authors are with the Statistical Machine Learning Group at NICTA, and the Australian National University. Queries should be addressed to [email protected].