Abstract
Edge detection is an essential task in image processing. In some applications, such as Magnetic Resonance Imaging, the information about an image is available only through its frequency (Fourier) data. In this case, edge detection is particularly challenging, as it requires extracting local information from global data. The problem is exacerbated when the data are noisy. This paper proposes a new edge detection algorithm which combines the concentration edge detection method (Gelb and Tadmor in Appl. Comput. Harmon. Anal. 7:101–135, 1999) with statistical hypothesis testing. The result is a method that achieves a high probability of detection while maintaining a low probability of false detection.
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References
Archibald, R., Gelb, A.: Reducing the effects of noise in image reconstruction. J. Sci. Comput. 17(1–4), 167–180 (2002)
Archibald, R., Gelb, A.: A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue boundary integrity. IEEE Trans. Med. Imaging 21(4), 305–319 (2002)
Archibald, R., Gelb, A., Yoon, J.: Polynomial fitting for edge detection in irregularly sampled signals and images. SIAM J. Numer. Anal. 43(1), 259–279 (2005)
Benjamini, Y., Yekutieli, D.: The control of the false discovery rate in multiple testing under dependency. Ann. Stat. 29(4), 1165–1188 (2001)
Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986)
Engelberg, S., Tadmor, E.: Recovery of edges from spectral data with noise—a new perspective. SIAM J. Numer. Anal. 46(5), 2620–2635 (2008)
Gelb, A., Cates, D.: Detection of edges in spectral data III—refinement of the concentration method. J. Sci. Comput. 36(1), 1–43 (2008)
Gelb, A., Cates, D.: Segmentation of images from Fourier spectral data, communications in computational. Physics 5, 326–349 (2009)
Gelb, A., Tadmor, E.: Detection of edges in spectral data. Appl. Comput. Harmon. Anal. 7, 101–135 (1999)
Gelb, A., Tadmor, E.: Detection of edges in spectral data II—nonlinear enhancement. SIAM J. Numer. Anal. 38(4), 1389–1408 (2000)
Gelb, A., Tadmor, E.: Adaptive edge detectors for piecewise smooth dara based on the minmod limiter. J. Sci. Comput. 28(2–3), 279–306 (2006)
Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)
Jung, J.-H., Gottlieb, S., Kim, S.O.: Iterative adaptive rbf methods for detection of edges in two dimensional functions. Appl. Numer. Math. 61, 77–91 (2011)
Kay, S.M.: Fundamentals of Statistical Signal Processing—Detection Theory. Prentice Hall, Englewood Cliffs (1993)
Sobel, I.: An isotropic 3 image gradient operator. In: Freeman, H. (ed.) Machine Vision for Three-Dimensional Scenes. Academic Press, Boston (1990)
Tadmor, E.: Filters, mollifiers and the computation of the Gibbs phenomenon. Acta Numer. 16, 305–378 (2007)
Tadmor, E., Zou, J.: Novel edge detection methods for incomplete and noisy spectral data. J. Fourier Anal. Appl. 14(5), 744–763 (2008)
Viswanathan, A., Cochran, D., Gelb, A., Cates, D.: Detection of signal discontinuities from noisy Fourier data. In: Signals, Systems and Computers, Forty-Second Asilomar Conference on Signals, Systems, and Computers, pp. 1705–1708 (2008)
Ziou, D., Tabbone, S.: Edge detection techniques—an overview. Pattern Recognit. Image Anal. 8(4), 537–559 (1998)
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Petersen, A., Gelb, A. & Eubank, R. Hypothesis Testing for Fourier Based Edge Detection Methods. J Sci Comput 51, 608–630 (2012). https://doi.org/10.1007/s10915-011-9523-1
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DOI: https://doi.org/10.1007/s10915-011-9523-1