  The alignment of shapes has been a crucial step in statistical shape
analysis, for example, in calculating mean shape, detecting locational
differences between two shape populations, and classification. Procrustes
alignment is the most commonly used method and state of the art. In this work,
we uncover that alignment might seriously affect the statistical analysis. For
example, alignment can induce false shape differences and lead to misleading
results and interpretations. We propose a novel hierarchical shape
parameterization based on local coordinate systems. The local parameterized
shapes are translation and rotation invariant. Thus, the inherent alignment
problems from the commonly used global coordinate system for shape
representation can be avoided using this parameterization. The new
parameterization is also superior for shape deformation and simulation. The
method's power is demonstrated on the hypothesis testing of simulated data as
well as the left hippocampi of patients with Parkinson's disease and controls.
