482 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 21, NO. 6, JUNE 1999 A Markov Pixon Information Approach for Low-Level Image Description Xavier Descombes and Frithjof Kruggel Abstract—The problem of extracting information from an image which corresponds to early stage processing in vision is addressed. We propose a new approach (the MPI approach) which simultaneously provides a restored image, a segmented image and a map which reflects the local scale for representing the information. Embedded in a Bayesian framework, this approach is based on an information prior, a pixon model and two Markovian priors. This model based approach is oriented to detect and analyze small parabolic patches in a noisy environment. The number of clusters and their parameters are not required for the segmentation process. The MPI approach is applied to the analysis of Statistical Parametric Maps obtained from fMRI experiments. Index Terms—Information, Pixon, Markov Random Fields, image restoration, fMRI analysis. ——————————F—————————— 1 INTRODUCTION CRUCIAL step before interpreting a scene from an im- age consists in extracting the underlying information. This step is referred to as the early stage in vision and usu- ally denoted as low-level processing. The main areas in this domain concern image segmentation, image restoration and primitive extraction. The aim of this process is to obtain a synthetic description of the image. The first level of infor- mation concerns the detection and the localization of the objects. The second level is a description of the shape of these objects. To achieve this description we have some noisy and/or blurred data and some contextual a priori information. These issues refer to well-known areas of im- age processing. The extraction of objects is first performed by reducing the number of gray levels to simplify the de- scription of the scene and is referred to as classification or clustering. Adding a priori knowledge leads to segmenta- tion. The object shapes description refers to the problem of image restoration. In this paper, we propose a new approach to solve these three problems simultaneously. We consider a stochastic model embedded in a Bayesian framework. We describe the scene from data (original image) by three maps referred to as the restored image, the segmented image and the pixon map. The pixon concept has been introduced by Piña and Puetter in [1] to restore astronomy images, and further been refined in [2], [3]. The term pixon refers to pixel information. Piña and Puetter model the restored image by the local convolution of a pseudo-image, whose entropy is maxi- mized, by the pixon map. The size of the pixels is given by the resolution of the data. However, to locally describe an image the required resolution is not spatially homogeneous. The background and regions do not require a fine resolu- tion whereas the description of edges and fine objects re- quires to use all the information given by the data resolu- tion. The pixons are geometric features of varying size. The size of a pixon defines locally the scale of the underlying information. Herein, we define the pseudo-image as the segmentation of the scene. In the classical pixon approach, the aim of the entropy term is to regularize the pseudo- image. In our context, the entropy is used to reduce the number of gray levels in the description (i.e. to define the classes of the segmentation). Therefore, we minimize the entropy of the segmented image histogram. The regulari- zation is performed by adding a Markovian prior. The seg- mented image provides a rough analysis of the scene using a piecewise constant description. By convolving the seg- mented image with the pixon map, we obtain a fine de- scription of edges and fine structures. Herein, we consider the restoration of small parabolic patches in a noisy envi- ronment. Therefore, the pixon basis consists of a set of pa- rabolas. We propose to estimate the pixon map and the segmented image by modeling them with Markov random fields (MRFs). The construction of the model involves three steps. We first address the classification problem to obtain the seg- mented image. However, we do not assume to know the number of classes in the segmented image. This is a major advantage with respect to other algorithms reported in the literature. We introduce an information prior on the seg- mented image (also referred to as the entropy prior) to per- form the classification by extracting a reduced number of gray levels (the labels) describing the scene. The most sig- nificant gray levels are selected by the information prior and represent the different classes. The pixels are classified by minimizing a distance with respect to these classes in the same procedure. The information concept was introduced by Shannon in 1948 [4] to quantify the information in a 0162-8828/99/$10.00 © 1999 IEEE ²²²²²²²²²²²²²²²²  X. Descombes was with Max Planck Institute of Cognitive Neuroscience, Image Processing Group, 22-26 Inselstrasse, 04103, Leipzig, Germany. He is now with INRIA, 2004, route des Lucioles BP 93, 06902 Sophia Antipo- lis Cedex, France.  E-mail: xavier.descombes @sophia.inria.fr.  F. Kruggel is with Max Planck Institute of Cognitive Neuroscience, Image Processing Group, 22-26 Inselstrasse, 04103, Leipzig, Germany.  E-mail: kruggel @cns.mpg.de. Manuscript received 5 Sept. 1997; revised 15 Sept. 1998. Recommended for accep- tance by K. Mardia. For information on obtaining reprints of this article, please send e-mail to: tpami@computer.org, and reference IEEECS Log Number 107575. A