| We introduce a rigorous mathematical theory for the analysis of local histograms, and study how they interact with textures that can be modeled as occlusions of simpler components. We first show how local histograms can be computed as a system of convolutions and discuss some basic local histogram properties. We then introduce a probabilistic, occlusion-based model for textures and formally demonstrate that local histogram transforms are natural tools for analyzing the textures produced by our model. Next, we characterize all nonlinear transforms which satisfy the three key properties of local histograms and consider the appropriateness of local histogram features in the automated classification of textures commonly encountered in histological images. When classifying tissues, pathologists indicate they focus on simple, locally-defined features that essentially involve pixel counting, such as the number of cells in a region of given size, the size of the nuclei within these cells, and the distribution of color within both. We discuss how local histogram transforms can be used to produce numerical features that, when fed into mainstream classification schemes, mimic the baser aspects of a pathologist's thought process. |
