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 ob- ject shape from sparse noisy samples of its Fourier trans- form. 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 ap- proximation 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 dis- crepancy between the true and estimated object. Numer- ical results demonstrate the effectiveness of the proposed approach. 1. PROBLEM STATEMENT Various applications, including magnetic resonance imag- ing, 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 recon- struction based on Fourier inversion can lead to severe arti- facts or even useless images. In this work, we propose in- stead 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 (1) This work was supported by a grant from DARPA under Contract F49620-98-1-0498, administered by AFOSR. is a deterministic functional of the unknown boundary . The observations (2) are noisy samples of at fixed and known spatial frequencies , corrupted by a white complex Gaussian noise sequence with variance . The problem is to estimate the unknown boundary from the noisy observations. 2. COMPLEXITY-REGULARIZED ESTIMATOR Shape estimation from a finite amount of data is in gen- eral an ill-posed problem. To obtain a regularized estima- tor, we apply a penalized-likelihood approach. A variety of known contour estimation methods used in image pro- cessing and computer vision (see [2–4] and references there in) are based on penalized-likelihood techniques. The major limitation of many of these methods is their non-adaptiveness. In this work, we use Rissanen’s MDL criterion [6]. It is one of fully automated estimation principles which are broadly applied in signal, image, and contour estimation [3–5]. Since both functions composing are periodic on , the boundary estimate can be represented using a Fourier series. The Fourier coefficients that describe a two dimen- sional boundary, are called Fourier descriptors (FD) (cf. [7] and references there in), and the parametrization can be used to describe arbitrarily located general-shaped objects. Let denote a set of vector functions of the form (3) where to guarantee uniqueness of the parametriza- tion (4) [7]. Denote by the combined column vector taking values in and by the full rank matrix of dimension containing the