Statistical Analysis of Locally Parameterized Shapes

Abstract

In statistical shape analysis, the establishment of correspondence and defining shape representation are crucial steps for hypothesis testing to detect and explain local dissimilarities between two groups of objects. Most commonly used shape representations are based on object properties that are either extrinsic or noninvariant to rigid transformation. Shape analysis based on noninvariant properties is biased because the act of alignment is necessary, and shape analysis based on extrinsic properties could be misleading. Besides, a mathematical explanation of the type of dissimilarity, e.g., bending, twisting, stretching, etc., is desirable. This work proposes a novel hierarchical shape representation based on invariant and intrinsic properties to detect and explain locational dissimilarities by using local coordinate systems. The proposed shape representation is also superior for shape deformation and simulation. The power of the method is demonstrated on the hypothesis testing of simulated data as well as the left hippocampi of patients with Parkinson's disease and controls.

Fitting the Discrete Swept Skeletal Representation to Slabular Objects

“…Following Pizer et al (2022) and Taheri and Schulz (2022), in this work, we consider an SO as a slab with a swept skeletal structure such that each cross-section is a 2D GC, the length of the spine (i.e., the SO's center curve) is notably larger than the length of the skeleton of each cross-section. The intersection of the spine with each 2D GC is a point on and (approximately) at the middle of the 2D GC's skeleton.…”

“…The LP-dss-rep can be used for both shape analysis (i.e., after removing the size) or size-and-shape analysis (i.e., by preserving the size) (Dryden and Mardia, 2016). Since the LP-dss-rep is invariant to rigid transformation, for shape analysis we remove the scale by dividing the vectors' lengths by the size of the LP-dss-rep, which we call LP-size (Taheri and Schulz, 2022). The LP-size is the geometric mean of the vectors' length as…”

“…To make the analysis alignment independent and to capture the type of local dissimilarities (e.g., protrusion, elongation, etc. ), we adapt the idea of locally parameterized ds-rep (LP-ds-rep) suggested by Taheri and Schulz (2022) to introduce locally parameterized dss-rep (LP-dss-rep) by parameterizing the dss-rep based on a tree-like structure of its skeletal sheets equipped with local frames as discussed in Section 5.…”

Skeletons, Object Shape, Statistics

“…Nevertheless, as presented in Section Statistics on S-reps for Single and Multiple Objects, Taheri and Schulz (2021) and Liu et al (2021b) have shown serious advantages to classification and hypothesis testing when discretized features according to the fitted frame were used in the statistics.…”

“…However, research so far has shown little advantage to the alternative method of Euclideanization in various classification applications, so our standard technique is sphere-by-sphere Euclideanization. Schulz et al (2015) and Taheri and Schulz (2021) have used these features to do hypothesis testing between s-rep classes, studying which geometric object properties (feature tuples capturing a single geometric property, such as a spoke direction) differ significantly between the classes. Taheri showed in experiments comparing hippocampi between typical humans and those with Parkinson's disease that the fitted-frame based features produce superior detections of differences than those using global coordinates.…”