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Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data

Stefan, W. and Viswanathan, A. and Gelb, A. and Renaut, R. (2012) Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data. Journal of Scientific Computing, 50 (3). pp. 536-556. ISSN 0885-7474.

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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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Additional Information:© 2011 Springer Science+Business Media, LLC. Received: 4 August 2010; Revised: 31 August 2011; Accepted: 1 September 2011; Published online: 22 September 2011. The work of A. Gelb and A. Viswanathan was supported in part by NSF grants DMS 0652833 and DMS 0608844. The work of R. Renaut was supported in part by the NSF grant DMS 0903737. The work of W. Stefan at Rice University was supported in part by NSF Grant DMS-07-48839 and the U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-09-1-0383. We would also like to thank Dr. Wotao Yin at Rice University for many fruitful discussions.
Funding AgencyGrant Number
NSFDMS 0652833
NSFDMS 0608844
NSFDMS 0903737
Army Research LaboratoryUNSPECIFIED
Army Research Office (ARO)W911NF-09-1-0383
Subject Keywords:Edge detection; Concentration factors; Compressive sensing; Partial Fourier data; Higher order methods; Sparseness
Issue or Number:3
Record Number:CaltechAUTHORS:20120502-113023843
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:31269
Deposited By: Tony Diaz
Deposited On:02 May 2012 19:52
Last Modified:03 Oct 2019 03:50

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