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.

The spectral centroid of a signal is the curve whose value at any given time is the centroid of the corresponding constant-time cross section of the signal's spectrogram. A spectral centroid provides a noise-robust estimate of how the dominant frequency of a signal changes over time. As such, spectral centroids are an increasingly popular tool in several signal processing applications, such as speech processing. We provide a new, fast and accurate algorithm for the real-time computation of the spectral centroid of a discretetime signal. In particular, by exploiting discrete Fourier transforms, we show how one can compute the spectral centroid of a signal without ever needing to explicitly compute the signal's spectrogram. We then apply spectral centroids to an emerging biometrics problem: to determine a person's heart and breath rates by measuring the Doppler shifts their body movements induce in a continuous wave radar signal. We apply our algorithm to real-world radar data, obtaining heart-and breath-rate estimates that compare well against ground truth.

We introduce a rigorous mathematical theory for the analysis of local histograms, and consider the appropriateness of their use in the automated classification of textures commonly encountered in images of H&E stained tissues. We first discuss some of the many image features that pathologists indicate they use when classifying tissues, focusing on simple, locally-defined features that essentially involve pixel counting: 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 then introduce a probabilistic, occlusion-based model for textures that exhibit these features, in particular demonstrating how certain tissue-similar textures can be built up from simpler ones. After considering the basic notions and properties of local histogram transforms, we then formally demonstrate that such transforms are natural tools for analyzing the textures produced by our model. In particular, 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.