This dissertation studies statistical shape analysis of planar objects. The focus is on two different representations. The first one considers only the boundary of planar shapes, a comprehensive analysis framework including quantification, registration, statistical summary and modeling are illustrated in (A. Srivastava and E. Klassen, 2016). Here, we study the hypothesis testing problem with this boundary representation. The second representation considers both the boundary and the interior of the objects. The goal is to construct a shape analysis framework as in (A. Srivastava and E. Klassen, 2016). First, we apply the DISCO test, a nonparametric k-sample test proposed for Euclidean space, to the shape space of planar closed curves. There are two common frameworks for shape analysis of planar closed curves -- Kendall's shape analysis and elastic shape analysis. The former treats a planar closed curve as discrete points while the latter treats it as a continuous parametric function. The test statistic for the DISCO test is based on pairwise distances between curves. Thus, we deploy five shape metrics from the two shape spaces while considering the scales of curves. The specific problem that we are interested in is whether acute exercise (Run) will affect mitochondrial morphology as observed in the images of skeletal muscles of mice. In data collection, some other factors are also involved, including cell, animal, type of mitochondria (SS/IMF) and exercise (Sedentary/Run). These factors formulate a hierarchical data structure which makes it hard to test the factor exercise. We propose a compression method to rule out the effect of the inner factor and then test on the outer factor. After confirming the significance of factor cell and ignoring the significance of factor animal, the SS mitochondria and the IMF mitochondria are found to be significant for all the five shape metrics. However, when testing on factor exercise, the only significant case happens with scaled elastic metric for SS mitochondria. Second, we construct elastic shape analysis framework for planar shapes including the boundary as well as the interior. Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides comprehensive frameworks for simultaneous registration, deformation, and comparison of shapes, such as functions, curves and surfaces. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into Euclidean metrics, allowing for applications of standard algorithms and statistical tools. For analyzing shapes of embeddings of unit square/disk in ℝ², we introduced a tensor field of symmetric positive definite matrix as the mathematical representation. In addition to the desirable invariant properties, the main advantage of this representation is that it simplifies the calculation of the elastic metric, geodesics and registration. The computation of inversion from the tensor field to the embedding is complicated and we use a gradient descent method. The optimal reparametrization between two embeddings is found through the multiresolution algorithm (Hamid Laga, Qian Xie, Ian H Jermyn, and Anuj Srivastava, 2017). We demonstrate the proposed theory using a statistical analysis of fly wings and leaves.
