Scale Invariant Metrics of Volumetric Datasets

Abstract

Abstract. Nature reveals itself in similar structures of different scales. A child and an adult share similar organs yet dramatically differ in size. Comparing the two is a challenging task to a computerized approach as scale and shape are coupled. Recently, it was shown that a local measure based on the Gaussian curvature can be used to normalize the local metric of a surface and then to extract global features and distances. In this paper we consider higher dimensions; specifically, we construct a scale invariant metric for volumetric domains which can be used in analysis of medical datasets such as computed tomography (CT) and magnetic resonance imaging (MRI).Key words. scale invariant, differential geometry, Laplace-Beltrami, shape analysis AMS subject classifications. 53B21, 58D17DOI. 10.1137/1409876751. Introduction. A key idea in shape or image analysis is based on the design of invariants. Structures or images may look different on one regime, but given a different metric the true relationship is revealed. The study of metric invariants in computer vision was initiated more than 20 years ago in [8,9], where geometry of planar curves under projective transformations was examined. Scale and affine transformations attracted the main focus [28], where global and local invariants [11] were used in the process. Other approaches, for example, based on scale-space signatures [10] and semidifferential invariants [24], were found useful and built the foundations for future schemes.Images, on the other hand, require a different approach for designing invariants. Due to their high dimensionality, constructing invariants in images still remains a challenge in modern computer vision tasks. Lowe's milestone research on scale invariant feature transform (SIFT) [23] revolutionized the way we capture meaningful information in images. This was the beginning of an era where features became more sophisticated and were able to cleverly compensate for different deformations. Let us just mention SURF [2] and affine-SIFT (ASIFT) [25] as such predecessors. One more milestone we must mention was adopting the equi-affine metric into computer vision society [33], which was further used for important low-level vision tasks such as denoising.Nonrigid shapes became popular in recent years modeling natural behavior. Faces, for example, can be considered as bendable shapes where different mimics still preserve the intrinsic (geodesic) distances between pairwise points [4]. While such distances can be evaluated in linear time [19,38], their usage in comparison or alignment algorithms is not trivial. Shapes