Shapes From Pixels

Continuous-domain visual signals are usually captured as discrete (digital) images. This operation is not invertible in general, in the sense that the continuous-domain signal cannot be exactly reconstructed based on the discrete image, unless it satisfies certain constraints (\emph{e.g.}, bandlimitedness). In this paper, we study the problem of recovering shape images with smooth boundaries from a set of samples. Thus, the reconstructed image is constrained to regenerate the same samples (consistency), as well as forming a shape (bilevel) image. We initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We also introduce a requirement (called reducibility) that guarantees equivalence between the two problems. We illustrate that the reducibility property effectively sets a requirement on the minimum sampling density. One can draw analogy between the reducibility property and the so-called restricted isometry property (RIP) in compressed sensing which establishes the equivalence of the $\ell_0$ minimization with the relaxed $\ell_1$ minimization. We also evaluate the performance of the relaxed alternative in various numerical experiments.Comment: 13 pages, 14 figure

GEL Criteria for Moment Condition Models

GEL methods which generalize and extend previous contributions are defined and analysed for moment condition models specified in terms of weakly dependent data. These procedures offer alternative one-step estimators and tests that are asymptotically equivalent to their efficient two-step GMM counterparts. The basis for GEL estimation is via a smoothed version of the moment indicators using kernel function weights which incorporate a bandwidth parameter. Examples for the choice of bandwidth parameter and kernel function are provided. Efficient moment estimators based on implied probabilities derived from the GEL method are proposed, a special case of which is estimation of the stationary distribution of the data. The paper also presents a unified set of test statistics for over-identifying moment restrictions and combinations of parametric and moment restriction hypotheses.GMM, Generalized Empirical Likelihood, Efficient Moment Estimation,