Stephen M. Pizer

Principal geodesic analysis for the study of nonlinear statistics of shape

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.

Segmentation, registration, and measurement of shape variation via image object shape

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 representations prior probability of local geometry by reflecting variabilities in the nets 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 posteriori 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 anatomical object. Examples of applications of these methods in radiotherapy, surgery, and psychiatry are given.

Ridges for Image Analysis

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 located-dimensional ridge structures withinn-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.

Deformable M-Reps for 3D Medical Image Segmentation

M-reps (formerly called DSLs) are a multiscale medial means for modeling and rendering 3D solid geometry. They are particularly well suited to model anatomic objects and in particular to capture prior geometric information effectively in deformable models segmentation approaches. The representation is based on figural models, which define objects at coarse scale by a hierarchy of figures—each figure generally a slab representing a solid region and its boundary simultaneously. This paper focuses on the use of single figure models to segment objects of relatively simple structure.A single figure is a sheet of medial atoms, which is interpolated from the model formed by a net, i.e., a mesh or chain, of medial atoms (hence the name m-reps), each atom modeling a solid region via not only a position and a width but also a local figural frame giving figural directions and an object angle between opposing, corresponding positions on the boundary implied by the m-rep. The special capability of an m-rep is to provide spatial and orientational correspondence between an object in two different states of deformation. This ability is central to effective measurement of both geometric typicality and geometry to image match, the two terms of the objective function optimized in segmentation by deformable models. The other ability of m-reps central to effective segmentation is their ability to support segmentation at multiple levels of scale, with successively finer precision. Objects modeled by single figures are segmented first by a similarity transform augmented by object elongation, then by adjustment of each medial atom, and finally by displacing a dense sampling of the m-rep implied boundary. While these models and approaches also exist in 2D, we focus on 3D objects.The segmentation of the kidney from CT and the hippocampus from MRI serve as the major examples in this paper. The accuracy of segmentation as compared to manual, slice-by-slice segmentation is reported.

A multiresolution hierarchical approach to image segmentation based on intensity extrema

A computer algorithm which segments gray-scale images into regions of interest (objects) has been developed. These regions can provide the basis for scene analysis (including shape-parameter calculation) or surface-based, shaded-graphics display. The algorithm creates a tree structure for image description by defining a linking relationship between pixels in successively blurred versions of the initial image. The image is described in terms of nested light and dark regions. This algorithm, successfully implemented in one, two, and three dimensions, can theoretically work with any number of dimensions. The interactive postprocessing developed technique selects regions from the descriptive tree for display in several ways: pointing to a branch of the image description tree, specifying by sliders the range of scale and/or intensity of all regions which should be displayed, and pointing (on the original image) to any pixel in the desired region. The algorithm has been applied to approximately 15 computer tomography (CT) images of the abdomen. >

An evaluation of the effectiveness of adaptive histogram equalization for contrast enhancement

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.

Contrast Limited Adaptive Histogram Equalization image processing to improve the detection of simulated spiculations in dense mammograms

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 CLAHE settings 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 CLAHE settings should be tested in the clinic with digital mammograms to determine whether detection of spiculations associated with masses detected at mammography can be improved.

Multiscale Medial Loci and Their Properties

Blums medial axes have great strengths, in principle, in intuitively describing object shape in terms of a quasi-hierarchy of figures. But it is well known that, derived from a boundary, they are damagingly sensitive to detail in that boundary. The development of notions of spatial scale has led to some definitions of multiscale medial axes different from the Blum medial axis that considerably overcame the weakness. Three major multiscale medial axes have been proposed: iteratively pruned trees of Voronoi edges (Ogniewicz, 1993; Székely, 1996; Näf, 1996), shock loci of reaction-diffusion equations (Kimia et al., 1995; Siddiqi and Kimia, 1996), and height ridges of medialness (cores) (Fritsch et al., 1994; Morse et al., 1993; Pizer et al., 1998). These are different from the Blum medial axis, and each has different mathematical properties of generic branching and ending properties, singular transitions, and geometry of implied boundary, and they have different strengths and weaknesses for computing object descriptions from images or from object boundaries. These mathematical properties and computational abilities are laid out and compared and contrasted in this paper.

Object representation by cores: Identifying and representing primitive spatial regions

We propose a model of the spatial visual processes underlying the identification and representation of the shape of primitive spatial regions. We propose that a regions boundaries are sensed at multiple scales by boundariness detectors that give graded responses, that stimulated boundariness detectors of similar scale, sigma, connect to one another across a distance that is proportional to their scale, and that they connect via cores, where a core encodes the middles and widths of the region and hence is a trace in (chi, gamma, sigma), i.e. 3-D scale space.

Multiscale deformable model segmentation and statistical shape analysis using medial descriptions

This paper presents a multiscale framework based on a medial representation for the segmentation and shape characterization of anatomical objects in medical imagery. The segmentation procedure is based on a Bayesian deformable templates methodology in which the prior information about the geometry and shape of anatomical objects is incorporated via the construction of exemplary templates. The anatomical variability is accommodated in the Bayesian framework by defining probabilistic transformations on these templates. The transformations, thus, defined are parameterized directly in terms of natural shape operations, such as growth and bending, and their locations. A preliminary validation study of the segmentation procedure is presented. We also present a novel statistical shape analysis approach based on the medial descriptions that examines shape via separate intuitive categories, such as global variability at the coarse scale and localized variability at the fine scale. We show that the method can be used to statistically describe shape variability in intuitive terms such as growing and bending.