In this paper we present a method for the spatial analysis of complex cellular systems based on a multiscale study of neighborhood relationships. A function to measure those relationships, M, is introduced. The refined Relative Neighborhood Graph is then presented as a method to establish vicinity relationships within layered cellular structures, and particularized to epithelial cell nuclei in the mammary gland. Finally, the method is illustrated with two examples that show interactions within one population of epithelial cells and between two different populations.

We present two new methods for automatic registration of microscope images of consecutive tissue sections. They represent two possibilities for the first step in the 3-D reconstruction of histological structures from serially sectioned tissue blocks. The goal is to accurately align the sections in order to place every relevant shape contained in each image in front of its corresponding shape in the following section before detecting the structures of interest and rendering them in 3D. This is accomplished by finding the best rigid body transformation (translation and rotation) of the image being registered by maximizing a matching function based on the image content correlation. The first method makes use of the entire image information, whereas the second one uses only the information located at specific sites, as determined by the segmentation of the most relevant tissue structures. To reduce computing time, we use a multiresolution pyramidal approach that reaches the best registration transformation in increasing resolution steps. In each step, a subsampled version of the images is used. Both methods rely on a binary image which is a thresholded version of the Sobel gradients of the image (first method) or a set of boundaries manually or automatically obtained that define important histological structures of the sections. Then distance-transform of the binary image is computed. A proximity function is then calculated between the distance image of the image being registered and that of the reference image. The transformation providing a maximum of the proximity function is then used as the starting point of the following step. This is iterated until the registration error lies below a minimum value.

In this paper we present a scheme for real time segmentation of histological structures in microscopic images of normal and neoplastic mammary gland sections. Paraffin embedded or frozen tissue blocks are sliced, and sections are stained with hematoxylin and eosin (H&E). The sections are then imaged using conventional bright field microscopy. The background of the images is corrected by arithmetic manipulation using a "phantom." Then we use the fast marching method with a speed function that depends on the brightness gradient of the image to obtain a preliminary approximation to the boundaries of the structures of interest within a region of interest (ROI) of the entire section manually selected by the user. We use the result of the fast marching method as the initial condition for the level set motion equation. We run this last method for a few steps and obtain the final result of the segmentation. These results can be connected from section to section to build a three-dimensional reconstruction of the entire tissue block that we are studying.