Edge detection from truncated Fourier data using spectral mollifiers
- PDF / 754,071 Bytes
- 26 Pages / 439.37 x 666.142 pts Page_size
- 94 Downloads / 210 Views
Edge detection from truncated Fourier data using spectral mollifiers Doug Cochran · Anne Gelb · Yang Wang
Received: 17 September 2011 / Accepted: 6 November 2011 / Published online: 17 December 2011 © Springer Science+Business Media, LLC 2011
Abstract Edge detection from a finite number of Fourier coefficients is challenging as it requires extracting local information from global data. The problem is exacerbated when the input data is noisy since accurate high frequency information is critical for detecting edges. The noise furthermore increases oscillations in the Fourier reconstruction of piecewise smooth functions, especially near the discontinuities. The edge detection method in Gelb and Tadmor (Appl Comput Harmon Anal 7:101–135, 1999, SIAM J Numer Anal 38(4):1389–1408, 2000) introduced the idea of “concentration kernels” as a way of converging to the singular support of a piecewise smooth function. The kernels used there, however, and subsequent modifications to reduce the impact of noise, were generally oscillatory, and as a result oscillations were always prevalent in the neighborhoods of the jump discontinuities. This paper revisits concentration kernels, but insists on uniform convergence to the “sharp
Communicated by Yuesheng Xu. A. Gelb and D. Cochran were supported in part by NSF-DMS-FRG award 0652833. Y. Wang was supported in part by NSF-DMS awards 0813750 and 1043034. D. Cochran Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287, USA e-mail: [email protected] A. Gelb School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA e-mail: [email protected] Y. Wang (B) Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA e-mail: [email protected]
738
D. Cochran et al.
peaks” of the function, that is, the edge detection method converges to zero away from the jumps without introducing new oscillations near them. We show that this is achievable via an admissible class of spectral mollifiers. Our method furthermore suppresses the oscillations caused by added noise. Keywords Edge detection · Truncated Fourier data · Gibbs phenomenon · Poisson summation formula · Spectral mollifier Mathematics Subject Classifications (2010) 42A10 · 42A50 · 65T10
1 Introduction Detecting edges in piecewise smooth functions and images represents a fundamental problem for a variety of applications, including signal processing and numerical partial differential equation simulation. Algorithms using given pixel data, such as those developed by Canny [2] and Sobel [12] are generally based on constructing the numerical gradient of the image. Point values corresponding to large gradients constitute the edges of the image. In a similar fashion, the celebrated ENO scheme measures the gradients of neighboring pixel values to avoid crossing over jump discontinuities, or shocks, in solving partial differential equations that admit shocks as part of their solutions [10]. The situation is somewhat different, however, when the input data is gi
Data Loading...
