Abstract
Recently, there has been a great deal of interest in modeling the non-Gaussian structures of natural images. However, despite the many advances in the direction of sparse coding and multi-resolution analysis, the full probability distribution of pixel values in a neighborhood has not yet been described. In this study, we explore the space of data points representing the values of 3 × 3 high-contrast patches from optical and 3D range images. We find that the distribution of data is extremely “sparse” with the majority of the data points concentrated in clusters and non-linear low-dimensional manifolds. Furthermore, a detailed study of probability densities allows us to systematically distinguish between images of different modalities (optical versus range), which otherwise display similar marginal distributions. Our work indicates the importance of studying the full probability distribution of natural images, not just marginals, and the need to understand the intrinsic dimensionality and nature of the data. We believe that object-like structures in the world and the sensor properties of the probing device generate observations that are concentrated along predictable shapes in state space. Our study of natural image statistics accounts for local geometries (such as edges) in natural scenes, but does not impose such strong assumptions on the data as independent components or sparse coding by linear change of bases.
Highlights
A number of recent attempts have been made to describe the non-Gaussian statistics of natural images (Field, 1987; Ruderman and Bialek, 1994; Olshausen and Field, 1996; Huang and Mumford, 1999; Simoncelli, 1999b; Grenander and Srivastava, 2001)
The research in natural image statistics can roughly be divided into two related directions
There are studies of image statistics which try to find an “optimal” set of linear projections or basis functions in the state space defined by the image data (8 × 8 patches, for example, define a distribution in R64)
Summary
A number of recent attempts have been made to describe the non-Gaussian statistics of natural images (Field, 1987; Ruderman and Bialek, 1994; Olshausen and Field, 1996; Huang and Mumford, 1999; Simoncelli, 1999b; Grenander and Srivastava, 2001). There are studies of image statistics which try to find an “optimal” set of linear projections or basis functions in the state space defined by the image data (8 × 8 patches, for example, define a distribution in R64). Our analysis is divided into three parts: In Section 4 we study the distribution of our data with respect to a Voronoi tessellation of the space of data points This first part, is a model-free first exploration of the state space of contrast-normalized patches.
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