# A Physicist's View on Partial 3D Shape Matching
## 1 Introduction  
Our perception and analysis of the world around us have become mostly digital. Many scientific disciplines clearly benefit from this digital revolution. Scientists have created a wide variety of digital sensors that enable them to report on their research subjects at various time and length scales. This has led them to generate extensive quantitative and visual information, and the burden has now been placed into extracting knowledge from this information. In the case of 3D visual data, this amounts to identifying the shapes they contain. Computational geometry, computer vision, and computer graphics all face the challenge of developing effective algorithms to define, quantify, and compare those shapes. The advancements in machine learning and computational geometry have been extremely beneficial to those algorithms. This paper provides another source of evidence in support of enhancing such algorithms. We demonstrate that we can generate a potentially partial mapping between 3D shapes using statistical physics approaches. In our method, the cost of the correspondence acts as a gauge of the shapes' similarity. We demonstrate the efficacy of this strategy on both simulated and actual anatomical data. Methods that compare shapes fall under the general framework of morphometrics, the study of form, a concept that includes size and shape. While morphometrics is most often associated with biology and natural sciences, its techniques apply to any shape matching problems. Two types of such techniques can usually be identified, those based on global measures of the forms and those based on computing correspondences, or maps between the shapes. We review these two approaches briefly below. Traditionally, morphometrics rely on measurements of lengths, widths, areas, masses, ratios, and/or angles that are then compared to assess the similarities between shapes [1]. A significant drawback of such an approach is that the measurements are usually correlated and therefore include a significant amount of redundancy. Recent improvements within this approach include computation of geometric moments or Zernike moments over the shape [2; 3; 4; 5], or of spherical harmonics over its surface [6]. All those methods are based on global properties of the forms under study, thereby preventing partial matching between such forms. An alternate approach to shape comparison is to first build a correspondence between the shapes. Finding correspondences, or maps between shapes, is a common problem in geometric processing with a wide range of applications (for reviews, see, for example, [7; 8; 9; 10]). Here, we briefly discuss correspondence methods that are pertinent to our study. A promising method for such shape correspondence is to view the shapes as metric spaces. Two shapes are then equal if they are isometric. Otherwise, a map is built between the two shapes that is as isometric as possible, such that the difference between the two shapes is measured as the distance between the map and the isometry. For discrete shapes equipped with discrete metrics, the difference with an isometry is measured by computing the distortion in distances between pairs of points on the surface of the shapes, where the distance can be Euclidean or geometric. This idea has lead to the concepts of Gromov-Hausdorff and Gromov-Wasserstein distances between shapes (see, for example, [11; 12; 13; 14; 15; 16; 17] in the context of shape comparison). Unfortunately, computing such distance amounts to solving a quadratic assignment problem, which is NP-hard. Despite recent progress (see, for example, [18] for computing the Gromov-Wasserstein distance), efforts have focused on alternate approaches to finding shape correspondence. The first such approach proceeds by mapping the shapes into a common parameterizing metric domain, so that it is possible to directly compute a distance between points on different shapes. Methods in this category usually proceed in three steps.