| Summary: | A new algorithm is presented to compute nonrigid, possibly partial comparisons of shapes defined by unstructured triangulations of their surfaces. The algorithm takes as input a pair of surfaces with each surface given by a distinct and unrelated triangulation. Its goal is to define a possibly partial correspondence between the vertices of the two triangulations, with a cost associated with this correspondence that can serve as a measure of the similarity of the two shapes. To find this correspondence, the vertices in each triangulation are characterized by a signature vector of features. We tested both the LD-SIFT signatures, based on the concept of spin images, and the wave kernel signatures obtained by solving the Shrödinger equation on the triangulation. A cost matrix <i>C</i> is constructed such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></semantics></math></inline-formula> is the norm of the difference of the signature vectors of vertices <i>k</i> and <i>l</i>. The correspondence between the triangulations is then computed as the transport plan that solves the optimal transport or optimal partial transport problem between their sets of vertices. We use a statistical physics approach to solve these problems. The presentation of the proposed algorithm is complemented with examples that illustrate its effectiveness and manageable computing cost.
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