A statistical model for contours in images

In this paper, we describe a statistical model for the gradient vector field of the gray level in images validated by different experiments. Moreover, we present a global constrained Markov model for contours in images that uses this statistical model for the likelihood. Our model is amenable to an iterative conditional estimation (ICE) procedure for the estimation of the parameters; our model also allows segmentation by means of the simulated annealing (SA) algorithm, the iterated conditional modes (ICM) algorithm, or the modes of posterior marginals (MPM) Monte Carlo (MC) algorithm. This yields an original unsupervised statistical method for edge-detection, with three variants. The estimation and the segmentation procedures have been tested on a total of 160 images. Those tests indicate that the model and its estimation are valid for applications that require an energy term based on the log-likelihood ratio. Besides edge-detection, our model can be used for semiautomatic extraction of contours, localization of shapes, non-photo-realistic rendering; more generally, it might be useful in various problems that require a statistical likelihood for contours.

[1]  Sean Dougherty,et al.  Edge detector evaluation using empirical ROC curves , 1999, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149).

[2]  Donald Geman,et al.  Boundary Detection by Constrained Optimization , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[3]  Harry Shum,et al.  Image segmentation by data driven Markov chain Monte Carlo , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[4]  Wojciech Pieczynski,et al.  Estimation of generalized mixture in the case of correlated sensors , 2000, IEEE Trans. Image Process..

[5]  Demetri Terzopoulos,et al.  Snakes: Active contour models , 2004, International Journal of Computer Vision.

[6]  Jitendra Malik,et al.  A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[7]  Max Mignotte,et al.  Unsupervised texture segmentation using a statistical wavelet-based hierarchical multidata model , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[8]  Van Nostrand,et al.  Error Bounds for Convolutional Codes and an Asymptotically Optimum Decoding Algorithm , 1967 .

[9]  Patrick Pérez,et al.  Sonar image segmentation using an unsupervised hierarchical MRF model , 2000, IEEE Trans. Image Process..

[10]  A. Cohen,et al.  Maximum Likelihood Estimation in the Weibull Distribution Based On Complete and On Censored Samples , 1965 .

[11]  J. Besag.  On the Statistical Analysis of Dirty Pictures , 1986 .

[12]  Didier Demigny,et al.  On optimal linear filtering for edge detection , 2002, IEEE Trans. Image Process..

[13]  John F. Canny,et al.  A Computational Approach to Edge Detection , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[14]  Wojciech Pieczynski,et al.  Adaptive Mixture Estimation and Unsupervised Local Bayesian Image Segmentation , 1995, CVGIP Graph. Model. Image Process..

[15]  Abdelwaheb Marzouki,et al.  Estimation of generalized mixtures and its application in image segmentation , 1997, IEEE Trans. Image Process..

[16]  Haluk Derin,et al.  Modeling and Segmentation of Noisy and Textured Images Using Gibbs Random Fields , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  William A. Barrett,et al.  Interactive Segmentation with Intelligent Scissors , 1998, Graph. Model. Image Process..

[18]  Rachid Deriche,et al.  Using Canny's criteria to derive a recursively implemented optimal edge detector , 1987, International Journal of Computer Vision.

[19]  Wojciech Pieczynski,et al.  Estimation of Generalized Multisensor Hidden Markov Chains and Unsupervised Image Segmentation , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[21]  Jun Shen,et al.  An optimal linear operator for step edge detection , 1992, CVGIP Graph. Model. Image Process..

[22]  Anil K. Jain,et al.  Object Matching Using Deformable Templates , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[23]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[24]  Max Mignotte,et al.  Unsupervised statistical sketching for non-photorealistic rendering models , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[25]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[26]  Alan L. Yuille,et al.  Statistical Edge Detection: Learning and Evaluating Edge Cues , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[27]  Wojciech Pieczynski,et al.  Parameter Estimation in Hidden Fuzzy Markov Random Fields and Image Segmentation , 1997, CVGIP Graph. Model. Image Process..

[28]  Stephen P. Banks,et al.  Signal Processing, Image Processing and Pattern Recognition , 1991 .

[29]  Tomaso Poggio,et al.  Probabilistic Solution of Ill-Posed Problems in Computational Vision , 1987 .

[30]  L. Younes.  Parametric Inference for imperfectly observed Gibbsian fields , 1989 .

[31]  G. Celeux,et al.  Asymptotic properties of a stochastic EM algorithm for estimating mixing proportions , 1993 .

[32]  François Destrempes.  Unsupervised Localization of Shapes Using Statistical Models , 2002 .

[33]  Alain Hillion,et al.  Estimation of fuzzy Gaussian mixture and unsupervised statistical image segmentation , 1997, IEEE Trans. Image Process..

[34]  Donald Geman,et al.  An Active Testing Model for Tracking Roads in Satellite Images , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[35]  François Destrempes.  Unsupervised Detection and Semi-Automatic Extraction of Contours Using a Statistical Model and Dynamic Programming , 2002 .

[36]  Patrick Pérez,et al.  JetStream: probabilistic contour extraction with particles , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.