Directly Invertible Nonlinear Divisive Normalization Pyramid for Image Representation Roberto Valerio 1 , Eero P. Simoncelli 2 , and Rafael Navarro 1 1 Instituto de Óptica “Daza de Valdés” - CSIC, Madrid, Spain, 28006 {r.valerio, r.navarro}@io.cfmac.csic.es 2 Howard Hughes Medical Institute, Center for Neural Science, and Courant Institute for Mathematical Sciences - New York University, New York, USA, NY 10003 eero.simoncelli@nyu.edu Abstract. We present a multiscale nonlinear image representation that permits an efficient coding of natural images. The input image is first decomposed into a set of subbands at multiple scales and orientations using near-orthogonal symmetric quadrature mirror filters. This is followed by a nonlinear “divisive normalization” stage, in which each linear coefficient is divided by a value computed from a small set of neighboring coefficients in space, orientation and scale. This neighborhood is chosen to allow this nonlinear operation to be effi- ciently inverted. The parameters of the normalization operation are optimized in order to maximize the independence of the normalized responses for natural images. We demonstrate the near-independence of these nonlinear responses, and suggest a number of applications for which this representation should be well suited. 1 Introduction The choice of an appropriate image representation is often driven by the goal of re- moving statistical redundancy in the input signal. The general problem is extremely difficult, and thus one typically must restrict it by constraining the form of the de- composition and/or by simplifying the description of the input statistics. A classical solution is to consider only linear decompositions, and the second-order (i.e. covari- ance) properties of the input signal. This technique, known as Principal Components Analysis (PCA), has several drawbacks. First, the solution is not unique if one does not impose additional constraints. Moreover, although PCA can be used to recover a set of statistically independent axes for representing Gaussian data, the technique of- ten fails when the data are non-Gaussian (as is the case of natural images [1]). More recently, a number of authors have shown that one may use higher-order statistics to uniquely constrain the choice of linear decomposition. These procedures are com- monly known as Independent Components Analysis (ICA). The resulting basis func- tions of such decompositions are similar to cortical receptive fields [2, 3] and the as- sociated coefficients are generally more independent than principal components. Nevertheless, linear decompositions cannot completely eliminate higher-order sta- tistical dependencies [e.g. 4, 5], basically due to the fact that natural images are not Presented in: Proc. 8th Annual Workshop on Very Low Bitrate Video Coding, Visual Content Processing and Representation, Eds. N Garcia, JM Martinez and L Salgado. Madrid Spain, 18-19 Sept 2003. DOI 10.1007/b13938 Published as: Lecture Notes in Computer Science, vol 2849, pp 331-340, Springer, 2003.