A Parametric Texture Model based onJoint Statistics of Complex Wavelet

We present a universal statistical model for texture images in the context of an over-complete complex wavelet transform. The model is parameterized by a set of statistics computed on pairs of coeecients corresponding to basis functions at adjacent spatial locations , orientations, and scales. We develop an eecient algorithm for synthesizing random images subject to these constraints, by iteratively projecting onto the set of images satisfying each constraint, and we use this to test the perceptual validity of the model. In particular, we demonstrate the necessity of subgroups of the parameter set by showing examples of texture synthesis that fail when those parameters are removed from the set. We also demonstrate the power of our model by successfully synthesizing examples drawn from a diverse collection of artiicial and natural textures. Vision is the process of extracting information from the images that enter the eye. The space of all such images is vast, and yet only a small fraction of these are likely to be encountered in a natural setting 38, 23, 17, 53]. Nevertheless, it has proven diicult to characterize this set of \natural" images, using either deterministic or statistical models. The class of images that we commonly call \visual texture" seems most amenable to statistical modeling. Loosely speaking, texture images contain repeated elements, often subject to some randomization in their location, size, color, orientation, etc. Julesz pioneered the statistical characterization of textures by hypothesizing that the Nth-order joint empirical densities of image pixels (for some unspeciied N), could be used to partition textures into classes that are preattentively indistinguishable to a human observer 34]. This work established the description of texture using homogeneous (stationary) random elds, the goal of determining a minimal set of statistical measurements for characterization, and the validation of texture models through human perceptual comparisons. Julesz et. al. later proposed that pairwise (N = 2) statistics were suucient 36], but then disproved this conjecture by producing example pairs of textures with identical statistics through second (and even third) order that were visually distinct 10, 37]. Since then, two important developments have enabled a new generation of more powerful statistical texture models. The rst is the theory of Markov random elds, in which the full