On Synergy Between Statistical Shape Analysis and Functional Data Analysis

• Activity Number: 378
• Type: Topic Contributed
• Date/Time: Tuesday, August 5, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistics in Imaging
• Author(s): Wei Wu*+ and Anuj Srivastava
• Companies: Florida State University and Florida State University
• Keywords: statistical shape analysis ; functional data analysis ; phase-amplitude separation ; Hilbert spaces ; nonlinear time warping

Abstract

The problem of statistical shape analysis (SSA) of objects has traditionally been formulated as an analysis of landmarks (registered points) modulo certain similarity transformations (rotation, translation, and scale). More recently SSA techniques have been extended to include shapes of continuous objects -- parameterized curves, surfaces, and their temporal evolutions -- by treating them as elements of Hilbert spaces. The branch of statistics on functional data -- functional data analysis (FDA) -- also deals with generating inferences on certain Hilbert spaces and shares some common issues and solutions with SSA. Specifically, the problem of phase-amplitude separation in FDA involves alignment of peaks and valleys of given functions using nonlinear time warpings. This relates to optimal registration of points in domains of the given functions using diffeomorphism maps. This exact problem has been termed as the registration in SSA. An elegant solution to this problem comes from use of a family of square-root transforms of the original functions or objects, along with the standard L2 norm. I will describe this framework using examples from FDA and SSA of curves and surfaces.