Complexity regularized shape estimation from noisy Fourier data

Natalia A. Schmid, Yoram Bresler, and Pierre Moulin University of Illinois at Urbana-Champaign Coordinated Science Laboratory, 1308 West Main, Urbana, IL 61801 nschmid ifp.uiuc.edu, ybresler uiuc.edu, moulin ifp.uiuc.edu ABSTRACT We consider the estimation of an unknown arbitrary 2D object shape from sparse noisy samples of its Fourier transform. The estimate of the closed boundary curve is parametrized by normalized Fourier descriptors (FDs). We use Rissanen’s MDL criterion to regularize this ill-posed non-linear inverse problem and determine an optimum tradeoff between approximation and estimation errors by picking an optimum order for the FD parametrization. The performance of the proposed estimator is quantified in terms of the area discrepancy between the true and estimated object. Numerical results demonstrate the effectiveness of the proposed approach. 1. PROBLEM STATEMENT Various applications, including magnetic resonance imaging, tomographic reconstruction, and synthetic aperture radar (SAR) involve estimation of an object shape from noisy sparse Fourier data. A similar Fourier formulation applies to linearizations of nonlinear inverse scattering problems using Born or physical optics approximations [1]. Depending on the amount of Fourier data available, conventional reconstruction based on Fourier inversion can lead to severe artifacts or even useless images. In this work, we propose instead a method based on statistical inference to reconstruct the shape of the object. Suppose that an object of interest with unknown support is located somewhere in a two-dimensional scene with a finite support The scene is described by two known continuous intensity functions for and for respectively. The closed boundary of the unknown shape can be represented as a vector function with and real periodic functions. The continuous Fourier transform of the scene