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A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.

The clinical recognition of abnormal vascular tortuosity, or excessive bending, twisting, and winding, is important to the diagnosis of many diseases. Automated detection and quantitation of abnormal vascular tortuosity from three-dimensional (3-D) medical image data would, therefore, be of value. However, previous research has centered primarily upon two-dimensional (2-D) analysis of the special subset of vessels whose paths are normally close to straight. This report provides the first 3-D tortuosity analysis of clusters of vessels within the normally tortuous intracerebral circulation. We define three different clinical patterns of abnormal tortuosity. We extend into 3-D two tortuosity metrics previously reported as useful in analyzing 2-D images and describe a new metric that incorporates counts of minima of total curvature. We extract vessels from MRA data, map corresponding anatomical regions between sets of normal patients and patients with known pathology, and evaluate the three tortuosity metrics for ability to detect each type of abnormality within the region of interest. We conclude that the new tortuosity metric appears to be the most effective in detecting several types of abnormalities. However, one of the other metrics, based on a sum of curvature magnitudes, may be more effective in recognizing tightly coiled, "corkscrew" vessels associated with malignant tumors.

The purpose of this project was to determine whether Contrast Limited Adaptive Histogram Equalization (CLAHE)improves detection of simulated spiculations in dense mammograms. Lines simulating the appearance of spiculations, a common marker of malignancy when visualized with masses, were embedded in dense mammograms digitized at 50 micron pixels, 12 bits deep. Film images with no CLAHE applied were compared to film images with nine different combinations of clip levels and region sizes applied. A simulated spiculation was embedded in a background of dense breast tissue, with the orientation of the spiculation varied. The key variables involved in each trial included the orientation of the spiculation, contrast level of the spiculation and the CLAHEsettings applied to the image. Combining the 10 CLAHE conditions, 4 contrast levels and 4 orientations gave 160 combinations. The trials were constructed by pairing 160 combinations of key variables with 40 backgrounds. Twenty student observers were asked to detect the orientation of the spiculation in the image. There was a statistically significant improvement in detection performance for spiculations with CLAHE over unenhanced images when the region size was set at 32 with a clip level of 2, and when the region size was set at 32 with a clip level of 4. The selected CLAHEsettings should be tested in the clinic with digital mammograms to determine whether detection of spiculations associated with masses detected at mammography can be improved. Copyright © 1998 by W.S. Saunders Company KEY WORDS: mammography, image processing, contrast limited adaptive histogram equalization, observer studies, breast cancer, spiculations."PPROXIMATELY 10% to 15% of palpable ft malignancies are not visible mammographically.1 It is highly likely that many nonpalpable cancers are also not visible with current technology. Digital mammography might allow for greater contrast and improved detection of small and early tumors over standard film screen technology, especially if image processing is used to improve image contrast.2-5We have previously published two articles reporting laboratory results that show improved performance by students in finding simulated masses and simulated clustered calcifications embedded in dense mammographic background when Intensity Win-

Abstract. Representation of object shape by medial structures has been an important aspect of image analysis. Methods for describing objects in a binary image by medial axes are well understood. Many attempts have been made to construct similar medial structures for objects in gray scale images. In particular, researchers have studied images by analyzing the graphs of the intensity data and identifying ridge and valley structures on those surfaces. In this paper we review many of the definitions for ridges. Computational vision models require that medial structures should remain invariant under certain transformations of the spatial locations and intensities. For each ridge definition we point out which invariances the definition satisfies. We also give extensions of the concepts so that we can locate d-dimensional ridge structures within n-dimensional images. A comparison of the ridge structures produced by the different definitions is given both by mathematical examples and by an application to a 2-dimensional MR image of a head. Keywords. 1 The Need for Ridges in Image AnalysisMethods for representing shapes of objects in gray-scale images have typically fallen into two categories: edge-based or region-based. Edgebased algorithms are developed under the assumption that large gradients of image intensity indicate the presence of an edge. The property of edgeness at a pixel is determined by measuring the dissimilarity between the pixel intensity and its neighbors' intensities, for example, by us-*Research supported by National Science Foundation Grant DMS-9003037.~Research supported by NIH grant # P01 CA 47982.ing the magnitude of the gradient of intensity. These algorithms additionally must handle edge orientation, edge strength, and edge connectivity. The method of edge detection essentially consists of following r/dges of edgeness. Figure 1 illustrates this for a simple object.Many edge-based methods are deficient since the presence of noise can make it difficult to detect an edge and determine its orientation.Moreover, the characterization of the global structure and shape of an object by its boundary depends greatly on the correctness of the edge connectivity scheme.Region-based algorithms are developed under

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A model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives. Metrics are described that compute an object representation's prior probability of local geometry by reflecting variabilities in the net's node and link parameter values and that compute a likelihood function measuring the degree of match of an image to that object representation. A paradigm for image analysis of deforming such a model to optimize a posterior probability is described, and this paradigm is shown to be usable as a uniform approach for object definition, object-based registration between images of the same or different imaging modalities, and measurement of shape variation of an abnormal anatomical object compared with a normal. Examples of applications of these methods in radiotherapy, surgery, and psychiatry are given.

Adaptive histogram equalization (AHE) and intensity windowing have been compared using psychophysical observer studies. Experienced radiologists were shown clinical CT (computerized tomographic) images of the chest. Into some of the images, appropriate artificial lesions were introduced; the physicians were then shown the images processed with both AHE and intensity windowing. They were asked to assess the probability that a given image contained the artificial lesion, and their accuracy was measured. The results of these experiments show that for this particular diagnostic task, there was no significant difference in the ability of the two methods to depict luminance contrast; thus, further evaluation of AHE using controlled clinical trials is indicated.