Recovery of Edges from Spectral Data with Noise—A New Perspective

Engelberg, Shlomo; Tadmor, Eitan

doi:10.1137/070689899

Cited by 32 publications

(41 citation statements)

References 7 publications

(14 reference statements)

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“…3 are far off: minmod-based detection was used here for comparison purposes, demonstrating the effect noise. Concentration factors which are adapted to noisy data can be found in the recent work [9].…”

Section: Numerical Results For Sift Edge Detectionmentioning

confidence: 99%

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Tadmor

Zou

2008

J Fourier Anal Appl

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We propose three novel methods for recovering edges in piecewise smooth functions from their possibly incomplete and noisy spectral information. The proposed methods utilize three different approaches: #1. The randomly-based sparse Inverse Fast Fourier Transform (sIFT); #2. The Total Variation-based (TV) compressed sensing; and #3. The modified zero crossing. The different approaches share a common feature: edges are identified through separation of scales. To this end, we advocate here the use of concentration kernels (Tadmor, Acta Numer. 16:305-378, 2007), to convert the global spectral data into an approximate jump function which is localized in the immediate neighborhoods of the edges. Building on these concentration kernels, we show that the sIFT method, the TV-based compressed sensing and the zero crossing yield effective edge detectors, where finitely many jump discontinuities are accurately recovered. One-and two-dimensional numerical results are presented.

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Section: Numerical Results For Sift Edge Detectionmentioning

confidence: 99%

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Tadmor

Zou

2008

J Fourier Anal Appl

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“…Here we employ edge detection techniques based on concentration factors for noisy data [19], which works on Fourier coefficients of the signal. The Fourier coefficients are found from either the noisy signal or through a transformation on the DCT coefficients of the signal.…”

Section: Denoisingmentioning

confidence: 99%

Inverse polynomial reconstruction method in DCT domain

Dadkhahi

Gotchev

Егиазарян

2012

EURASIP J. Adv. Signal Process.

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The discrete cosine transform (DCT) offers superior energy compaction properties for a large class of functions and has been employed as a standard tool in many signal and image processing applications. However, it suffers from spurious behavior in the vicinity of edge discontinuities in piecewise smooth signals. To leverage the sparse representation provided by the DCT, in this article, we derive a framework for the inverse polynomial reconstruction in the DCT expansion. It yields the expansion of a piecewise smooth signal in terms of polynomial coefficients, obtained from the DCT representation of the same signal. Taking advantage of this framework, we show that it is feasible to recover piecewise smooth signals from a relatively small number of DCT coefficients with high accuracy. Furthermore, automatic methods based on minimum description length principle and cross-validation are devised to select the polynomial orders, as a requirement of the inverse polynomial reconstruction method in practical applications. The developed framework can considerably enhance the performance of the DCT in sparse representation of piecewise smooth signals. Numerical results show that denoising and image approximation algorithms based on the proposed framework indicate significant improvements over wavelet counterparts for this class of signals.

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“…(Low order CFs may not properly resolve the function variation in smooth regions, causing the misclassification of edges there.) Nonlinear filtering is able to remove or suppress some of the oscillations, [11,25,8,6]. Unfortunately these techniques become less effective when the Fourier data is at all corrupted (e.g.…”

Section: Introductionmentioning

confidence: 99%

Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data

et al. 2011

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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.

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Recovery of Edges from Spectral Data with Noise—A New Perspective

Cited by 32 publications

References 7 publications

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Three Novel Edge Detection Methods for Incomplete and Noisy Spectral Data

Inverse polynomial reconstruction method in DCT domain

Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data

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