Abstract
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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References
Ahn, C.B., Kim, J.H., Cho, Z.H.: High-speed spiral-scan echo planar NMR imaging-I. IEEE Trans. Med. Imaging 5(1), 2 (1986)
Candes, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)
Candes, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)
Candes, E., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)
Cochran, D., Gelb, A., Wang, Y.: Edge detection from truncated Fourier data using spectral mollifiers, preprint, submitted to Advances in Computational Mathematics (2011)
Engelberg, S., Tadmor, E.: Recovery of edges from spectral data with noise—a new perspective. SIAM J. Numer. Anal. 46(5), 2620–2635 (2008)
Fessler, J.A., Sutton, B.P.: Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans. Signal Process. 51(2), 560–574 (2003)
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., 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 data based on the minmod limiter. J. Sci. Comput. 28(2–3), 279–306 (2006)
Gottlieb, D., Shu, C.W.: On the Gibbs phenomenon and its resolution. SIAM Rev. 39(4), 644–668 (1997)
Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 1.21 (2010)
Grant, M., Boyd, S., Ye, Y.: Disciplined convex programming. In: Nonconvex Optimization and its Applications, pp. 155–210. Springer, Berlin (2006)
Guo, W., Yin, W.: EdgeCS: edge guided compressive sensing reconstruction. Technical report, Rice CAAM Report TR10-02, 2010
Hoge, R.D., Kwan, R.K., Pike, G.B.: Density compensation functions for spiral MRI. Magn. Reson. Med. 38(1), 117–128 (1997)
Kadec, M.I.: The exact value of the Paley-Wiener constant. Sov. Math. Dokl. 5, 559–561 (1964)
O’Sullivan, J.D.: A fast sinc function griding algorithm for Fourier inversion in computer tomography. IEEE Trans. Med. Imaging 4(4), 200–207 (1985)
Pipe, J.G., Menon, P.: Sampling density compensation in MRI: Rationale and an iterative numerical solution. Magn. Reson. Med. 41(1), 179–186 (1999)
Sedarat, H., Nishimura, D.G.: On the optimality of the griding reconstruction algorithm. IEEE Trans. Med. Imaging 19(4), 306–317 (2000)
Shizgal, B., Jung, J.H.: Towards the resolution of the Gibbs phenomena. J. Comput. Appl. Math. 161(1), 41–65 (2003)
Stefan, W., Renaut, R.A., Gelb, A.: Improved total variation-type regularization using higher-order edge detectors. SIAM J. Imaging Sci. 3(2), 232–251 (2010)
Steidl, G.: A note on fast Fourier transforms for nonequispaced grids. Adv. Comput. Math. 9(3), 337–352 (1998)
Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999). Special issue on Interior Point Methods (CD supplement with software), http://sedumi.mcmaster.ca/
Tadmor, E.: Filters, mollifiers and the computation of the Gibbs phenomenon. Acta Numer. 16, 305–378 (2007)
Tadmor, E., Zou, J.: Three novel edge detection methods for incomplete and noisy spectral data. J. Fourier Anal. Appl. 14(5–6), 744–763 (2008)
Toh, K., Tütüncü, R., Todd, M.: SDPT3 4.0 (beta) (software package). Technical report, Department of Mathematics National University of Singapore, September 2006. http://www.math.nus.edu.sg/mattohkc/sdpt3.html
Viswanathan, A.: Imaging from Fourier spectral data: problems in discontinuity detection, non-harmonic Fourier reconstruction and point-spread function estimation. PhD thesis, Arizona State University (2010)
Viswanathan, A., Gelb, A., Cochran, D.: Iterative design of concentration factors for edge detection. J. Sci. Comput. (2011, to appear)
Wang, Y., Yin, W., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. SIAM J. Imaging Sci. 1(3), 248–272 (2008)
Young, R.M.: An Introduction to Nonharmonic Fourier Analysis. Academic Press, New York (2001)
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Stefan, W., Viswanathan, A., Gelb, A. et al. Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data. J Sci Comput 50, 536–556 (2012). https://doi.org/10.1007/s10915-011-9536-9
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DOI: https://doi.org/10.1007/s10915-011-9536-9