Fundamental Limits on Parametric Shape Estimation Performance

This paper considers the problem of extraction of shape information from noise corrupted images acquired from a resolution limited imaging instrument, a problem that is closely related to shape estimation and segmentation. The problem is formulated as estimation of coefficients in a basis expansion of the boundary of 2D and 3D star–shaped objects. We derive an expression for the Fisher information matrix and the Cramèr– Rao (CR) bound under a polar shape descriptor model, homogeneous object intensity, Gaussian point spread function, and additive white Gaussian noise. We analyze boundary estimation performance for both finite and infinite dimensional sets of shape basis functions. We show that circles and spheres are the easiest to estimate in the sense that they minimize the CR bound over the class of star shaped 2D and 3D objects, respectively. We show that irregularly shaped objects with sharp corners are worst case shapes in the sense that they maximize the CR bound. Finally we discuss the CR bound sensitivity with respect to variation of the center point position for the star–shaped object. In particular, the object centroid is not in general the optimum center location.