Globally Convergent Edge-preserving Reconstruction with Contour-line Smoothing

The standard approach to 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. The existing attempts to favor the formation of smooth contour lines take the edge field explicitly into account; they either are computationally expensive or produce disappointing results. In this paper, we propose to incorporate the smoothness of the edge field in an implicit way by means of an additional 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 non-quadratic and the cost function is nonconvex. Numerical experiments show that our technique preserves edge sharpness while smoothing contour lines; it produces visually pleasing reconstructions which are quantitatively better than those obtained without wavelet-domain constraints.

• Marc C. Robini
• Pierre-Jean Viverge
• Yuemin Zhu
• Jianhua Luo

[1] J. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of illposed
problems in computational vision," J. Amer. Statist. Assoc.,vol. 82, no. 397, pp. 76-89, 1987.
[2] G. Demoment, "Image reconstruction and restoration: overview of common
estimation structures and problems," IEEE Trans. Acoust. Speech
Signal Process., vol. 37, no. 12, pp. 2024-2036, 1989.
[3] P. Charbonnier, L. Blanc-F'eraud, G. Aubert, and M. Barlaud, "Deterministic
edge-preserving regularization in computed imaging," IEEE Trans.
Image Process., vol. 6, no. 2, pp. 298-311, 1997.
[4] M. Nikolova, "Analysis of the recovery of edges in images and signals
by minimizing nonconvex regularized least-squares," Multiscale Model.
Simul., vol. 4, no. 3, pp. 960-991, 2005.
[5] S. Geman and D. Geman, "Stochastic relaxation, Gibbs distributions, and
the Bayesian restoration of images," IEEE Trans. Pattern Anal. Machine
Intell., vol. 6, no. 6, pp. 721-741, 1984.
[6] L. Bedini, L. Benvenuti, E. Salerno, and A. Tonazzini, "A mixedannealing
algorithm for edge preserving image reconstruction using a
limited number of projections," Signal Process., vol. 32, no. 3, pp. 397-
408, 1993.
[7] L. Blanc-F'eraud, S. Teboul, G. Aubert, and M. Barlaud, "Nonlinear
regularization using constrained edges in image reconstruction," in Proc.
IEEE Int. Conf. Image Processing, vol. 2, (Lausanne, Switzerland),
pp. 449-452, Sept. 1996.
[8] J. Idier, "Convex half-quadratic criteria and interacting auxiliary variables
for image restoration," IEEE Trans. Image Process., vol. 10, no. 7,
pp. 1001-1009, 2001.
[9] D. Dobson and F. Santosa, "Recovery of blocky images from noisy and
blurred data," SIAM J. Appl. Math., vol. 56, no. 4, pp. 1181-1198, 1996.
[10] T. Chang, A. Marquina, and P. Mulet, "High-order total variation-based
image restoration," SIAM J. Sci. Comput., vol. 22, no. 2, pp. 503-516,
2000.
[11] D. Geman and G. Reynolds, "Constrained restoration and the recovery
of discontinuities," IEEE Trans. Pattern Anal. Machine Intell., vol. 14,
no. 3, pp. 367-383, 1992.
[12] A. H. Delaney and Y. Bresler, "Globally convergent edge-preserving
regularized reconstruction: an application to limited-angle tomography,"
IEEE Trans. Image Process., vol. 7, no. 2, pp. 204-221, 1998.
[13] M. J. Black and A. Rangarajan, "On the unification of line processes,
outlier rejection, and robust statistics with applications in early vision,"
Int. J. Comput. Vis., vol. 19, no. 1, pp. 57-91, 1996.