# American Institute of Mathematical Sciences

November  2013, 7(4): 1331-1366. doi: 10.3934/ipi.2013.7.1331

## Edge-preserving reconstruction with contour-line smoothing and non-quadratic data-fidelity

 1 CREATIS (CNRS research unit UMR5220 and INSERM research unit U1044), INSA-Lyon, 69621 Villeurbanne Cedex, France, France 2 College of Life Science and Technology, Shanghai Jiao Tong University, Shanghai 200240, China

Received  March 2012 Revised  March 2013 Published  November 2013

The standard approach for image reconstruction is to stabilize the problem by including an edge-preserving roughness penalty in addition to faithfulness to the data. However, this methodology produces noisy object boundaries and creates a staircase effect. State-of-the-art methods to correct these undesirable effects either have weak convergence guarantees or are limited to specific situations; furthermore, most of them use a quadratic data-fidelity term. In this paper, we propose a simple alternative regularization model to improve contour regularity and to reduce the staircase effect-our model incorporates the smoothness of the edge field in an implicit way by adding a simple penalty term defined in the wavelet domain. We also derive an efficient half-quadratic algorithm to solve the resulting optimization problem, including the case when the data-fidelity term is not quadratic and the cost function is not convex. Our approach either extends or supplements existing methods and offers strong convergence guarantees. Numerical experiments show that it outperforms first-order total variation regularization as well as state-of-the-art second-order regularization techniques.
Citation: Marc C. Robini, Yuemin Zhu, Jianhua Luo. Edge-preserving reconstruction with contour-line smoothing and non-quadratic data-fidelity. Inverse Problems & Imaging, 2013, 7 (4) : 1331-1366. doi: 10.3934/ipi.2013.7.1331
##### References:
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Leahy, A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,, IEEE Trans. Med. Imag., 8 (1989), 194. doi: 10.1109/42.24868. Google Scholar [27] T. Hou, S. Wang and H. Qin, Image deconvolution with multi-stage convex relaxation and its perceptual evaluation,, IEEE Trans. Image Process., 20 (2011), 1057. doi: 10.1109/TIP.2011.2150236. Google Scholar [28] Y. Hu and M. Jacob, Higher degree total variation (HDTV) regularization for image recovery,, IEEE Trans. Image Process., 21 (2012), 2559. doi: 10.1109/TIP.2012.2183143. Google Scholar [29] J. Idier, Convex half-quadratic criteria and interacting auxiliary variables for image restoration,, IEEE Trans. Image Process., 10 (2001), 1001. doi: 10.1109/83.931094. Google Scholar [30] K. Ito, B. Jin and T. Takeuchi, A regularization parameter for nonsmooth Tikhonov regularization,, SIAM J. Sci. Comput., 33 (2011), 1415. doi: 10.1137/100790756. Google Scholar [31] P. Kohli, A. Shekhovtsov, C. Rother, V. Kolmogorov and P. Torr, On partial optimality in multi-label MRFs,, in Proc. $25^{th}$ Int. Conf. Machine Learning, (2008), 480. doi: 10.1145/1390156.1390217. Google Scholar [32] S. Lefkimmiatis, A. Bourquard and M. Unser, Hessian-based norm regularization for image restoration with biomedical applications,, IEEE Trans. Image Process., 21 (2012), 983. doi: 10.1109/TIP.2011.2168232. Google Scholar [33] S. Li, On discontinuity-adaptive smoothness priors in computer vision,, IEEE Trans. Pattern Anal. Machine Intell., 17 (1995), 576. doi: 10.1109/34.387504. Google Scholar [34] Y.-R. Li, L. Shen, D.-Q. Dai and B. Suter, Framelet algorithms for de-blurring images corrupted by impulse plus Gaussian noise,, IEEE Trans. Image Process., 20 (2011), 1822. doi: 10.1109/TIP.2010.2103950. Google Scholar [35] M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional,, Int. J. Comput. Vis., 66 (2006), 5. doi: 10.1007/s11263-005-3219-7. Google Scholar [36] J. Ma and G. Plonka, The curvelet transform,, IEEE Signal Processing Mag., 27 (2010), 118. doi: 10.1109/MSP.2009.935453. Google Scholar [37] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation,, IEEE Trans. Pattern Anal. Machine Intell., 11 (1989), 674. doi: 10.1109/34.192463. Google Scholar [38] J. Marroquin, S. Mitter and T. Poggio, Probabilistic solution of ill-posed problems in computational vision,, J. Amer. Statist. Assoc., 82 (1987), 76. doi: 10.1080/01621459.1987.10478393. Google Scholar [39] R. Meyer, Sufficient conditions for the convergence of monotonic mathematical programming algorithms,, J. Comput. System Sci., 12 (1976), 108. doi: 10.1016/S0022-0000(76)80021-9. Google Scholar [40] P. Moulin and J. Liu, Analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors,, IEEE Trans. Inform. Theory, 45 (1999), 909. doi: 10.1109/18.761332. Google Scholar [41] M. Nikolova, A variational approach to remove outliers and impulse noise,, J. Math. Imaging Vis., 20 (2004), 99. doi: 10.1023/B:JMIV.0000011920.58935.9c. Google Scholar [42] M. Nikolova, Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares,, Multiscale Model. Simul., 4 (2005), 960. doi: 10.1137/040619582. Google Scholar [43] M. Nikolova, M. Ng and C.-P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction,, IEEE Trans. Image Process., 19 (2010), 3073. doi: 10.1109/TIP.2010.2052275. Google Scholar [44] S. Ramani, T. Blu and M. Unser, Monte-Carlo SURE: a black-box optimization of regularization parameters for general denoising algorithms,, IEEE Trans. Image Process., 17 (2008), 1540. doi: 10.1109/TIP.2008.2001404. Google Scholar [45] M. C. Robini and I. Magnin, Optimization by stochastic continuation,, SIAM J. Imaging Sci., 3 (2010), 1096. doi: 10.1137/090756181. Google Scholar [46] M. C. Robini, T. Rastello and I. Magnin, Simulated annealing, acceleration techniques and image restoration,, IEEE Trans. Image Process., 8 (1999), 1374. doi: 10.1109/83.791963. Google Scholar [47] P. Rodríguez and B. Wohlberg, Efficient minimization method for a generalized total variation functional,, IEEE Trans. Image Process., 18 (2009), 322. doi: 10.1109/TIP.2008.2008420. Google Scholar [48] C. Rother, V. Kolmogorov, V. Lempitsky and M. Szummer, Optimizing binary MRFs via extended roof duality,, in Proc. IEEE Conf. Comp. Vis. Patt. Recognition, (2007), 1. doi: 10.1109/CVPR.2007.383203. Google Scholar [49] X.-C. Tai, J. Hahn and G. Chung, A fast algorithm for Euler's elastica model using augmented Lagrangian method,, SIAM J. Imaging Sci., 4 (2011), 313. doi: 10.1137/100803730. Google Scholar [50] M. Unser, Ten good reasons for using spline wavelets,, in Proc. SPIE, (3169), 422. doi: 10.1117/12.292801. Google Scholar [51] L. Wang, T.-T. Wong, P. Heng and J. Cheng, Template-matching approach to edge detection of volume data,, in Proc. Int. Workshop on Medical Imaging and Augmented Reality, (2001), 286. Google Scholar [52] T. Zhang, Analysis of multi-stage convex relaxation for sparse regularization,, J. Mach. Learn. Res., 11 (2010), 1081. Google Scholar [53] W. Zhu and T. Chan, Image denoising using mean curvature of image surface,, SIAM J. Imaging Sci., 5 (2012), 1. doi: 10.1137/110822268. Google Scholar [54] X. Zhu and P. Milanfar, Automatic parameter selection for denoising algorithms using a no-reference measure of image content,, IEEE Trans. Image Process., 19 (2010), 3116. doi: 10.1109/TIP.2010.2052820. Google Scholar

show all references

##### References:
 [1] E. Bae, J. Shi and X.-C. Tai, Graph cuts for curvature based image denoising,, IEEE Trans. Image Process., 20 (2011), 1199. doi: 10.1109/TIP.2010.2090533. Google Scholar [2] M. Banham and A. Katsaggelos, Digital image restoration,, IEEE Signal Processing Mag., 14 (1997), 24. doi: 10.1109/79.581363. Google Scholar [3] L. Bedini, L. Benvenuti, E. Salerno and A. Tonazzini, A mixed-annealing algorithm for edge preserving image reconstruction using a limited number of projections,, Signal Process., 32 (1993), 397. doi: 10.1016/0165-1684(93)90009-Y. Google Scholar [4] M. Belge, M. Kilmer and E. Miller, Wavelet domain image restoration with adaptive edge-preserving regularization,, IEEE Trans. Image Process., 9 (2000), 597. doi: 10.1109/83.841937. Google Scholar [5] M. Belge, M. Kilmer and E. Miller, Efficient determination of multiple regularization parameters in a generalized L-curve framework,, Inverse Problems, 18 (2002), 1161. doi: 10.1088/0266-5611/18/4/314. Google Scholar [6] L. Blanc-Féraud, S. Teboul, G. Aubert and M. Barlaud, Nonlinear regularization using constrained edges in image reconstruction,, in Proc. IEEE Int. Conf. Image Processing, 2 (1996), 449. doi: 10.1109/ICIP.1996.560882. Google Scholar [7] J.-F. Cai, R. Chan and M. Nikolova, Fast two-phase image deblurring under impulse noise,, J. Math. Imaging Vis., 36 (2010), 46. doi: 10.1007/s10851-009-0169-7. Google Scholar [8] R. Chan, Y. Dong and M. Hintermüller, An efficient two-phase $L^1$-TV method for restoring blurred images with impulse noise,, IEEE Trans. Image Process., 19 (2010), 1731. doi: 10.1109/TIP.2010.2045148. Google Scholar [9] T. Chan, S. Kang and J. Shen, Euler's elastica and curvature-based inpainting,, SIAM J. Appl. Math., 63 (2002), 564. doi: 10.1137/S0036139901390088. Google Scholar [10] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration,, SIAM J. Sci. Comput., 22 (2000), 503. doi: 10.1137/S1064827598344169. Google Scholar [11] P. Charbonnier, L. Blanc-Féraud, G. Aubert and M. Barlaud, Deterministic edge-preserving regularization in computed imaging,, IEEE Trans. Image Process., 6 (1997), 298. doi: 10.1109/83.551699. Google Scholar [12] H. Choi and R. Baraniuk, Wavelet statistical models and Besov spaces,, in Proc. SPIE, (3813), 489. doi: 10.1007/978-0-387-21579-2_2. Google Scholar [13] E. Chouzenoux, J. Idier and S. Moussaoui, A majorize-minimize strategy for subspace optimization applied to image restoration,, IEEE Trans. Image Process, 20 (2011), 1517. doi: 10.1109/TIP.2010.2103083. Google Scholar [14] A. Cohen, I. Daubechies and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets,, Comm. Pure Appl. Math., 45 (1992), 485. doi: 10.1002/cpa.3160450502. Google Scholar [15] I. Daubechies, M. Defrise and C. D. Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,, Comm. Pure Appl. Math., 57 (2004), 1413. doi: 10.1002/cpa.20042. Google Scholar [16] A. H. Delaney and Y. Bresler, Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography,, IEEE Trans. Image Process, 7 (1998), 204. doi: 10.1109/83.660997. Google Scholar [17] G. Demoment, Image reconstruction and restoration: Overview of common estimation structures and problems,, IEEE Trans. Acoust. Speech Signal Process, 37 (1989), 2024. doi: 10.1109/29.45551. Google Scholar [18] R. DeVore, Nonlinear approximation,, Acta Numer., 7 (1998), 51. doi: 10.1017/S0962492900002816. Google Scholar [19] D. Dobson and F. Santosa, Recovery of blocky images from noisy and blurred data,, SIAM J. Appl. Math., 56 (1996), 1181. doi: 10.1137/S003613999427560X. Google Scholar [20] Y. Dong, M. Hintermüller and M. Neri, An efficient primal-dual method for $L^1$-TV image restoration,, SIAM J. Imaging Sci., 2 (2009), 1168. doi: 10.1137/090758490. Google Scholar [21] H. Farid and E. Simoncelli, Differentiation of discrete multidimensional signals,, IEEE Trans. Image Process., 13 (2004), 496. doi: 10.1109/TIP.2004.823819. Google Scholar [22] D. Geman and G. Reynolds, Constrained restoration and the recovery of discontinuities,, IEEE Trans. Pattern Anal. Machine Intell., 14 (1992), 367. doi: 10.1109/34.120331. Google Scholar [23] S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,, IEEE Trans. Pattern Anal. Machine Intell., 6 (1984), 721. doi: 10.1109/TPAMI.1984.4767596. Google Scholar [24] S. Geman and D. McClure, Bayesian image analysis: an application to single photon emission tomography,, in Proc. Stat. Comput. Section: Annual meeting of the Amer. Stat. Assoc., (1985), 12. Google Scholar [25] P. Green, Bayesian reconstructions from emission tomography data using a modified EM algorithm,, IEEE Trans. Med. Imag., 9 (1990), 84. doi: 10.1109/42.52985. Google Scholar [26] T. Hebert and R. Leahy, A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,, IEEE Trans. Med. Imag., 8 (1989), 194. doi: 10.1109/42.24868. Google Scholar [27] T. Hou, S. Wang and H. Qin, Image deconvolution with multi-stage convex relaxation and its perceptual evaluation,, IEEE Trans. Image Process., 20 (2011), 1057. doi: 10.1109/TIP.2011.2150236. Google Scholar [28] Y. Hu and M. Jacob, Higher degree total variation (HDTV) regularization for image recovery,, IEEE Trans. Image Process., 21 (2012), 2559. doi: 10.1109/TIP.2012.2183143. Google Scholar [29] J. Idier, Convex half-quadratic criteria and interacting auxiliary variables for image restoration,, IEEE Trans. Image Process., 10 (2001), 1001. doi: 10.1109/83.931094. Google Scholar [30] K. Ito, B. Jin and T. Takeuchi, A regularization parameter for nonsmooth Tikhonov regularization,, SIAM J. Sci. Comput., 33 (2011), 1415. doi: 10.1137/100790756. Google Scholar [31] P. Kohli, A. Shekhovtsov, C. Rother, V. Kolmogorov and P. Torr, On partial optimality in multi-label MRFs,, in Proc. $25^{th}$ Int. Conf. Machine Learning, (2008), 480. doi: 10.1145/1390156.1390217. Google Scholar [32] S. Lefkimmiatis, A. Bourquard and M. Unser, Hessian-based norm regularization for image restoration with biomedical applications,, IEEE Trans. Image Process., 21 (2012), 983. doi: 10.1109/TIP.2011.2168232. Google Scholar [33] S. Li, On discontinuity-adaptive smoothness priors in computer vision,, IEEE Trans. Pattern Anal. Machine Intell., 17 (1995), 576. doi: 10.1109/34.387504. Google Scholar [34] Y.-R. Li, L. Shen, D.-Q. Dai and B. Suter, Framelet algorithms for de-blurring images corrupted by impulse plus Gaussian noise,, IEEE Trans. Image Process., 20 (2011), 1822. doi: 10.1109/TIP.2010.2103950. Google Scholar [35] M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional,, Int. J. Comput. Vis., 66 (2006), 5. doi: 10.1007/s11263-005-3219-7. Google Scholar [36] J. Ma and G. Plonka, The curvelet transform,, IEEE Signal Processing Mag., 27 (2010), 118. doi: 10.1109/MSP.2009.935453. Google Scholar [37] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation,, IEEE Trans. Pattern Anal. Machine Intell., 11 (1989), 674. doi: 10.1109/34.192463. Google Scholar [38] J. Marroquin, S. Mitter and T. Poggio, Probabilistic solution of ill-posed problems in computational vision,, J. Amer. Statist. Assoc., 82 (1987), 76. doi: 10.1080/01621459.1987.10478393. Google Scholar [39] R. Meyer, Sufficient conditions for the convergence of monotonic mathematical programming algorithms,, J. Comput. System Sci., 12 (1976), 108. doi: 10.1016/S0022-0000(76)80021-9. Google Scholar [40] P. Moulin and J. Liu, Analysis of multiresolution image denoising schemes using generalized gaussian and complexity priors,, IEEE Trans. Inform. Theory, 45 (1999), 909. doi: 10.1109/18.761332. Google Scholar [41] M. Nikolova, A variational approach to remove outliers and impulse noise,, J. Math. Imaging Vis., 20 (2004), 99. doi: 10.1023/B:JMIV.0000011920.58935.9c. Google Scholar [42] M. Nikolova, Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares,, Multiscale Model. Simul., 4 (2005), 960. doi: 10.1137/040619582. Google Scholar [43] M. Nikolova, M. Ng and C.-P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction,, IEEE Trans. Image Process., 19 (2010), 3073. doi: 10.1109/TIP.2010.2052275. Google Scholar [44] S. Ramani, T. Blu and M. Unser, Monte-Carlo SURE: a black-box optimization of regularization parameters for general denoising algorithms,, IEEE Trans. Image Process., 17 (2008), 1540. doi: 10.1109/TIP.2008.2001404. Google Scholar [45] M. C. Robini and I. Magnin, Optimization by stochastic continuation,, SIAM J. Imaging Sci., 3 (2010), 1096. doi: 10.1137/090756181. Google Scholar [46] M. C. Robini, T. 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