Detection of Edges in Spectral Data II. Nonlinear Enhancement

Abstract

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.

Noise reduction, smoothing and time interval segmentation of noisy signals using an energy optimisation method

“…Let us first assume that our sampled data are characterised by one single discontinuity defined by (15) and (16), as depicted in Fig. 2 gðn þ 1Þ À gðnÞ ¼ M .…”

“…Image restoration, also known as the 'inverse problem', was developed by Rudin et al [7] as an optimisation method based on the concept of total variation. In another development within computer vision, a method known as the 'snake' algorithm was introduced for object segmentation in images by Kass et al [14,15]. The objective of this spectral method is then to recover the segmented signals from artefacts introduced by the Gibbs phenomenon, by using the Gegenbauer reconstruction algorithm [16 -18], whereas in our case, noise reduction of the segmented signals is the main aim of the proposed method.…”

“…A novel method is therefore introduced in this paper to model signals as piecewise continuous functions for noise reduction and smoothing purposes based on the energy optimisation. A similar modelling approach for signals is proposed in [14 -18] in which discontinuities are initially detected using (pseudo-) spectral methods based on Fourier and Legendre series [14,15]. The objective of this spectral method is then to recover the segmented signals from artefacts introduced by the Gibbs phenomenon, by using the Gegenbauer reconstruction algorithm [16 -18], whereas in our case, noise reduction of the segmented signals is the main aim of the proposed method.…”

Edge Detection Free Postprocessing for Pseudospectral Approximations

“…The most powerful methods need to know the exact location of all discontinuities (or edges). Methods to locate edges in spectral data are developed in references [5,6,7]. Computational efficiency in higher dimensions is also an issue with some of the postprocessing methods.…”

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

“…(3) The improved zero crossing algorithm, e.g., [15]. In this improved approach, we combine compressed sensing with zero crossing technique, inspired by the investigation of concentration kernels (1.1c) in [11,12].…”