No abstract

We propose a new edge detection method that is effective on multivariate irregular data in any domain. The method is based on a local polynomial annihilation technique and can be characterized by its convergence to zero for any value away from discontinuities. The method is numerically cost efficient and entirely independent of any specific shape or complexity of boundaries. Application of the minmod function to the edge detection method of various orders ensures a high rate of convergence away from the discontinuities while reducing the inherent oscillations near the discontinuities. It further enables distinction of jump discontinuities from steep gradients, even in instances where only sparse nonuniform data is available. These results are successfully demonstrated in both one and two dimensions.

Abstract. We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f ](x) := f (x+) − f (x−) = 0. Our approach is based on two main aspects-localization using appropriate concentration kernels and separation of scales by nonlinear enhancement.To detect such edges, one employs concentration kernels, Kǫ(·), depending on the small scale ǫ. It is shown that odd kernels, properly scaled, and admissible (in the sense of having small, thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form K σ N (t) = σ(k/N ) sin kt to detect edges from the first 1/ǫ = N spectral modes of piecewise smooth f 's. Here we improve in generality and simplicity over our previous study in [A. Gelb and E. Tadmor, Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135]. Both periodic and nonperiodic spectral projections are considered. We identify, in particular, a new family of exponential factors, σ exp (·), with superior localization properties.The other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges, where Kǫ * f (x) ∼ [f ](x) = 0, and the smooth regions where Kǫ * f = O(ǫ) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.

Fourier samples are collected in a variety of applications including magnetic resonance imaging and synthetic aperture radar. The data are typically under-sampled and noisy. In recent years, l 1 regularization has received considerable attention in designing image reconstruction algorithms from under-sampled and noisy Fourier data. The underlying image is assumed to have some sparsity features, that is, some measurable features of the image have sparse representation. The reconstruction algorithm is typically designed to solve a convex optimization problem, which consists of a fidelity term penalized by one or more l 1 regularization terms. The Split Bregman Algorithm provides a fast explicit solution for the case when TV is used for the l 1 regularization terms. Due to its numerical efficiency, it has been widely adopted for a variety of applications. A well known drawback in using TV as an l 1 regularization term is that the reconstructed image will tend to default to a piecewise constant image. This issue has been addressed in several ways. Recently, the polynomial annihilation edge detection method was used to generate a higher order sparsifying transform, and was coined the "polynomial annihilation (PA) transform." This paper adapts the Split Bregman Algorithm for the case when the PA transform is used as the l 1 regularization term. In so doing, we achieve a more accurate image reconstruction method from under-sampled and noisy Fourier data. Our new method compares favorably to the TV Split Bregman Algorithm, as well as to the popular TGV combined with shearlet approach.

We are concerned with the detection of edges-the location and amplitudes of jump discontinuities of piecewise smooth data realized in terms of its discrete grid values. We discuss the interplay between two approaches. One approach, realized in the physical space, is based on local differences and is typically limited to low-order of accuracy. An alternative approach developed in our previous work [Gelb and Tadmor, Appl. Comp. Harmonic Anal., 7, 101-135 (1999)] and realized in the dual Fourier space, is based on concentration factors; with a proper choice of concentration factors one can achieve higherorders-in fact in [Gelb and Tadmor, SIAM J. Numer. Anal., 38, 1389-1408(2001] we constructed exponentially accurate edge detectors. Since the stencil of these highly-accurate detectors is global, an outside threshold parameter is required to avoid oscillations in the immediate neighborhood of discontinuities. In this paper we introduce an adaptive edge detection procedure based on a cross-breading between the local and global detectors. This is achieved by using the minmod limiter to suppress spurious oscillations near discontinuities while retaining high-order accuracy away from the jumps. The resulting method provides a family of robust, parameter-free edge-detectors for piecewise smooth data. We conclude with a series of one-and two-dimensional simulations.

The classical Gibbs phenomenon exhibited by global Fourier projections and interpolants can be resolved in smooth regions by reprojecting in a truncated Gegenbauer series, achieving high resolution recovery of the function up to the point of discontinuity. Unfortunately, due to the poor conditioning of the Gegenbauer polynomials, the method suffers both from numerical round-off error and the Runge phenomenon. In some cases the method fails to converge. Following the work in [D. Gottlieb, C.W. Shu, Atti Conv. Lincei 147 (1998) 39-48], a more general framework for reprojection methods is introduced here. From this insight we propose an additional requirement on the reprojection basis which ameliorates the limitations of the Gegenbauer reconstruction. The new robust Gibbs complementary basis yields a reliable exponentially accurate resolution of the Gibbs phenomenon up to the discontinuities.  2005 Elsevier Inc. All rights reserved.MSC: 41A25; 42A10; 42A16; 42A20; 42C25; 65B10; 65T40

Gibbs ringing is a well known artifact that effects reconstruction of images having discontinuities. This is a problem in the reconstruction of magnetic resonance imaging (MRI) data due to the many different tissues normally present in each scan. The Gibbs ringing artifact manifests itself at the boundaries of the tissues, making it difficult to determine the structure of the brain tissue. The Gegenbauer reconstruction method has been shown to effectively eliminate the effects of Gibbs ringing in other applications. This paper presents the application of the Gegenbauer reconstruction method to neuro-imaging.

No abstract