No abstract

We present a new method for estimating the edges in a piecewise smooth function from blurred and noisy Fourier data. The proposed method is constructed by combining the so called concentration factor edge detection method, which uses a finite number of Fourier coefficients to approximate the jump function of a piecewise smooth function, with compressed sensing ideas. Due to the global nature of the concentration factor method, Gibbs oscillations feature prominently near the jump discontinuities. This can cause the misidentification of edges when simple thresholding techniques are used. In fact, the true jump function is sparse, i.e. zero almost everywhere with non-zero values only at the edge locations. Hence we adopt an idea from compressed sensing and propose a method that uses a regularized deconvolution to remove the artifacts. Our new method is fast, in the sense that it only needs the solution of a single l 1 minimization. Numerical examples demonstrate the accuracy and robustness of the method in the presence of noise and blur.

The concentration method of edge detection was developed to compute the locations and values of jump discontinuities in a piecewise-analytic function from its first few Fourier series coefficients. The accuracy and characteristic features of the resulting jump approximation depend on Fourier space "filter" factors known as concentration factors. In this paper, we provide a flexible, iterative framework for the design of these factors. Previously devised concentration factors are shown to be the solutions of specific problem formulations within this new framework. We also provide sample formulations of the procedure applicable to the design of concentration factors for data with missing spectral bands. Several illustrative examples are used to demonstrate the capabilities of the method.

Abstract-The concentration method for identifying the locations, magnitudes, and signs of jump discontinuities in analog signals from truncated Fourier series is well established in mathematical literature. Its performance in the presence of noise on the Fourier data has only recently started to receive attention, however. This paper examines the performance of the concentration method in the presence of noise from a detection-theoretic point of view. In particular, receiver operating characteristics for the elemental problem of detecting a unit step discontinuity are developed for this method. Additionally, the problem of optimally combining data obtained from multiple concentration factors is addressed.

No abstract