This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Close-Form Solution and Parameter Selection for Convex Minimization-Based Edge-Preserving Smoothing
September 1998 (vol. 20 no. 9)
pp. 916-932

Abstract—This work presents a new approach for the analysis of convex minimization-based edge-preserving image smoothing and the parameter selection therein. The global solution, that is, the response of a convex smoothing model to the ideal step edge, is derived in close-form. By analyzing the close-form solution, insights are drawn into how the optimal solution responds to edges in the data and how the parameter values affect resultant edges in the solution. Based on this, a scheme is proposed for selecting parameters to achieve desirable responses at edges. The theoretic results are substantiated by experiments.

[1] S. Geman and D. Geman, "Stochastic Relaxation, Gibbs Distribution and the Bayesian Restoration of Images," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 6, no. 6, pp. 721-741, Nov. 1984.
[2] J.L. Marroquin, Probabilistic Solution of Inverse Problems, PhD thesis, Massachusetts Institute of Tech nology, 1985.
[3] D. Mumford and J. Shah, "Boundary Detection by Minimizing Functionals: I," Proc. IEEE Computer Society Conf. Computer Vision and Pattern Recognition, pp. 22-26,San Francisco, June 1985.
[4] A. Blake and A. Zisserman, Visual Reconstruction.Cambridge, Mass.: MIT Press, 1987.
[5] S.Z. Li, Markov Random Field Modeling in Computer Vision.New York: Springer-Verlag, 1995.
[6] D. Shulman and J.Y. Herve, "Regularization of Discontinuous Flow Fields," Proc. Workshop Visual Motion, pp. 81-86, 1989.
[7] P.J. Green, Bayesian Reconstruction from Emission Tomography Data Using a Modified Em Algorithm IEEE Trans. Medical Imaging, vol. 9, pp. 84-93, 1990.
[8] K. Lange, "Convergence of EM Image Reconstruction Algorithm With Gibbs Smoothing," IEEE Trans. Medical Imaging, vol. 9, pp. 439-446, Dec. 1990.
[9] C. Bouman and K. Sauer, "A Generalized Gaussian Image Model for Edge Preserving MAP Estimation," IEEE Trans. Image Processing, vol. 2, pp. 296-310, July 1993.
[10] R.L. Stevenson, B.E. Schmitz, and E.J. Delp, "Discontinuity Preserving Regularization of Inverse Visual Problems," IEEE Trans. Systems, Man, and Cybernetics, vol. 24, pp. 455-469, Mar. 1994.
[11] S.Z. Li, Y.H. Huang, and J. Fu, "Convex MRF Potential Functions," Proc. IEEE Int'l Conf. Image Processing, vol. 2, pp. 296-299,Washington, D.C.,23-26 Oct. 1995.
[12] S.G. Nadabar and A.K. Jain, "Parameter Estimation in Markov Random Field Contextual Models Using Geometric Models of Objects," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 3, pp. 326-329, Mar. 1996.
[13] H. Jeong and C.I. Kim, "Adaptive Determination of Filter Scales for Edge Detection," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, pp. 579-585, 1992.
[14] P. Huber, Robust Statistics.New York: John Wiley, 1981.
[15] R. Stevenson and E. Delp, "Fitting Curves With Discontinuities," Proc. Int'l Workshop Robust Computer Vision, pp. 127-136,Seattle, Wash.,1-3 Oct. 1990.
[16] R.R. Schultz and R.L. Stevenson, "A Bayesian Approach to Image Expansion for Improved Definition," IEEE Trans. Image Processing, vol. 3, 1994.
[17] P. Perona and J. Malik, "Scale-Space and Edge Detection Using Anisotropic Diffusion," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629639, July 1990.
[18] S.Z. Li, "On Discontinuity-Adaptive Smoothness Priors in Computer Vision," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 6, pp. 576-586, June 1995.
[19] I. Pitas and A.N. Venetsanopoulos, "Edge Detectors Based on Nonlinear Filters," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, pp. 538-550, 1986.
[20] M. Petrou and J. Kittler, "Optimal Edge Detection for Ramp Edges," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, pp. 483-490, 1991.

Index Terms:
Convexity, edge preservation, energy minimization, image smoothing, Markov random field (MRF), maximum a posteriori (MAP), parameter selection, regularization.
Citation:
Stan Z. Li, "Close-Form Solution and Parameter Selection for Convex Minimization-Based Edge-Preserving Smoothing," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, no. 9, pp. 916-932, Sept. 1998, doi:10.1109/34.713359
Usage of this product signifies your acceptance of the Terms of Use.