  The local histogram transform of an image is a data cube that consists of the
histograms of the pixel values that lie within a fixed neighborhood of any
given pixel location. Such transforms are useful in image processing
applications such as classification and segmentation, especially when dealing
with textures that can be distinguished by the distributions of their pixel
intensities and colors. We, in particular, use them to identify and delineate
biological tissues found in histology images obtained via digital microscopy.
In this paper, we introduce a mathematical formalism that rigorously justifies
the use of local histograms for such purposes. We begin by discussing how local
histograms can be computed as systems of convolutions. We then introduce
probabilistic image models that can emulate textures one routinely encounters
in histology images. These models are rooted in the concept of image occlusion.
A simple model may, for example, generate textures by randomly speckling opaque
blobs of one color on top of blobs of another. Under certain conditions, we
show that, on average, the local histograms of such model-generated-textures
are convex combinations of more basic distributions. We further provide several
methods for creating models that meet these conditions; the textures generated
by some of these models resemble those found in histology images. Taken
together, these results suggest that histology textures can be analyzed by
decomposing their local histograms into more basic components. We conclude with
a proof-of-concept segmentation-and-classification algorithm based on these
ideas, supported by numerical experimentation.

Functional Analysis