Exploring Texture Ensembles by EÆcient Markov Chain Monte Carlo

This article presents a mathematical de nition of texture { the Julesz ensemble (h), which is the set of all images (de ned on Z) that share identical statistics h. Then texture modeling is posed as an inverse problem: given a set of images sampled from an unknown Julesz ensemble (h ), we search for the statistics h which de ne the ensemble. A Julesz ensemble (h) has an associated probability distribution q(I;h), which is uniform over the images in the ensemble and has zero probability outside. In a companion paper[32], q(I;h) is shown to be the limit distribution of the FRAME (Filter, Random Field, And Minimax Entropy) model[35] as the image lattice ! Z. This conclusion establishes the intrinsic link between the scienti c de nition of texture on Z and the mathematical models of texture on nite lattices. It brings two advantages to computer vision. 1). The engineering practice of synthesizing texture images by matching statistics has been put on a mathematical foundation. 2). We are released from the burden of learning the expensive FRAME model in feature pursuit, model selection and texture synthesis. In this paper, an eÆcient Markov chain Monte Carlo algorithm is proposed for sampling Julesz ensembles. The algorithm generates random texture images by moving along the directions of lter coeÆcients and thus extends the traditional single site Gibbs sampler. This paper also compares four popular statistical measures in the literature, namely, moments, recti ed functions, marginal histograms and joint histograms of linear lter responses in terms of their descriptive abilities. Our experiments suggest that a small number of bins in marginal histograms are suÆcient for capturing a variety of texture patterns. We illustrate our theory and algorithm by successfully synthesizing a number of natural textures.