Graphical model based segmentation of massive numbers of irregular small objects in images, with application to axon characterization in histological sections

Segmentation and classification of images into desired components is a fundamental problem in biomedical image processing. In this work we address the particular problem of automated identification and characterization of very large numbers of small objects of interest, where the objects have similar but variable structure, are embedded in a complex cluttered background, and may have low contrast and other imaging aberrations. The motivating application is the analysis of microscopy images of stained histological sections of brain or spinal cord tissue, where quantitative measurements from closely packed axons are useful to elucidate possible physiological mechanisms underlying contrast in diffusion-weighted magnetic resonance (DW-MR) images. Our initial solution to the DW-MR analysis emplyed a pipeline of standard image processing techniques to achieve axon segmentation, applied to spinal cord sections that were first imaged with DW-MR and then sectioned, stained, and subjected to microscopic examination. A statistical analysis was carried out to relate axon number and density estimated from the stained images, to a common measure of diffusion, fractional anisotropy. However, the limitations of this initial pipeline with regard to variability in the images, inability to flexibly incorporate all relevant information, and parameter sensitivity of the pipeline components, led us to develop a model-based approach to this segmentation problem. Our framework employs a probabilistic graphical model that encodes objects based on their color and intensity, the information in their local neighborhood, and the information in their boundaries. At the pixel level, we use a class of undirected graphical models, discriminative random fields (DRFs), to represent the posterior class-conditional probability density functions. Using level set functions of these pixel model probability density maps, we construct graph nodes of homogeneous regions and their corresponding boundaries to represent each object. This allows us to combine, in a principled way, the two types of information available about objects of interest - information in local regions and information in boundaries. In addition, we introduce an overall object label for each pair of candidate region and boundary node. We then construct a graph comprised of separate DRF-based graphical models for axon regions and boundaries, and unify these two models using Bayesian Networks (BNs). In the unified graph, the goal is to find the state of all candidate region, boundary, and overall object nodes, given the region and boundary information and the relationship between region and boundary nodes, by maximizing the joint posterior probability distribution over all hidden nodes in the graph. We present results of candidate axon region and boundary labeling, using separate region and boundary models, and compare them with manually labeled regions and boundaries, respectively. We then show the results of the combined model by performing label inference, i.e. finding labels for all candidate regions, boundaries, and overall object nodes, and compare them with manually labeled data based on the overall model. Our results indicate, in particular, that when the individual models fail, the combined model performance is more robust.