James P. Williams

Analysis of the Pulmonary Vascular Tree Using Differential Geometry Based Vector Fields

We extract topological and local structural information from X-ray-computed tomography (CT) volume images of the vascular tree of the lung. We then produce an abstract model of the topology and geometry of the tree which is suitable for generating physical models. Physiologists model the vascular tree as a connected network of tubes and, to date, the only accurate data available to produce such models has come from dissection and measurement. Modeling the dynamic behavior of the tree in a living organism has been, until now, impossible. Building models from CT scans will increase the volume and quality of information available for both the study of physiology and diagnosis. We present an efficient, accurate analysis technique for volume images of tube networks. Using local operations that link techniques based on morphology to differential geometry, we transform the volume into a vector field which resembles idealized, axis-parallel fluid flow through the tube network. This field provides information to condense the salient features of the image into an augmented Euclidean minimum spanning tree (EMST). This augmented EMST proves to be a convenient and logical abstract representation of the vascular tree.

Photometric surface analysis in a tri-luminal environment

Methods for the analysis of images of the same scene taken under three different lighting conditions are illustrated. A technique that separates the effects of geometry and surface coloration/texture in this tri-luminal environment is developed and experimental results are shown. Exploiting this technique to isolate geometric information, two methods which extract differential geometric properties of surfaces (the sign of Gaussian curvature and its magnitude to within a multiplicative factor) directly from tri-luminal photometric data are derived and demonstrated.

Rational discrete generalized cylinders and their application to shape recovery in medical images

Generalized cylinders (GCs) are a popular representational tool in computer vision. In medical imaging, the curved axis GC is particularly applicable to a number of elongated physical structures such as vasculature, bone and bronchi. In many of these instances, it is necessary to recover curved-axis GCs with arbitrary cross-sections. It is also vital that these structures, once recovered, can be analyzed and visualized with off-the-shelf algorithms and software packages. Such tools are usually designed to operate on the domain of polynomial or rational surfaces. Unfortunately most extant, suitably versatile GC representations do not admit rational parameterizations. We develop an entirely rational B-spline representation for generalized cylinders with curved axes and arbitrary cross-section functions. We demonstrate how our representation can be used as a deformable model by extracting a rational GC from pre-segmented spinal data using a discrete dynamic surface fit.

A rational model of the surface swept by a curve

This paper shows how to construct a rational Bezier model of a swept surface that interpolates N frames (i.e., N position/orientation pairs) of a fixed rational space curve c(s) and maintains the shape of the curve at all intermediate points of the sweep. Thus, the surface models an exact sweep of the curve, consistent with the given data. The primary novelty of the method is that this exact modeling of the sweep is achieved without sacrificing a rational representation for the surface. Through a simple extension, we also allow the sweeping curve to change its size through the sweep. The position, orientation, and size of the sweeping curve can change with arbitrary continuity (we use C2 continuity in this paper). Our interpolation between frames has the classical properties of Bezier interpolation, such as the convex hull property and linear precision.

A curvature based descriptor invariant to pose and albedo derived from photometric data

Gaussian curvature is an invariant local descriptor of smooth surfaces. We present an object signature which is a condensed representation of the distribution of Gaussian curvature information at visible object points. An invariant related to Gaussian curvature at a point is derived from the covariance matrix of the photometric values in a neighborhood about that point. In addition, we introduce an albedo-normalization method that is capable of cancelling albedo on Lambertian surfaces. We use three illumination conditions, two of which are unknown. The three-tuple of intensity values at a point is related via a one-to-one mapping to the surface normal at that point. The determinant of the covariance matrix of the local three-tuples is invariant to albedo, rotation and translation. The collection of determinants over mutually illuminated object points is combined into a signature distribution which is albedo, rotation, translation, and scale invariant. An object recognition methodology using these signatures is proposed.

Curvature Based Signatures for Object Description and Recognition

An invariant related to Gaussian curvature at an object point is developed based upon the covariance matrix of photometric values related to surface normals within a local neighborhood about the point. We employ three illumination conditions, two of which are completely unknown. We never need to explicitly know the surface normal at a point. The determinant of the covariance matrix of these three-tuples in the local neighborhood of an object point is shown to be invariant with respect to rotation and translation. A way of combining these determinants to form a signature distribution is formulated that is rotation, translation, and, scale invariant. This signature is shown to be invariant over large ranges of poses of the same objects, while being significantly different between distinctly shaped objects. A new object recognition methodology is proposed by compiling signatures for only a few poses of a given object.

Form from function: a vector field based approach to the analysis of CT images of the vascular tree

We extract topological and local structural information from x-ray computed tomography (CT) volume images of the vascular tree and present it in a form that facilitates creation of physiological models. Physiologists model the vascular tree as a connected network of tubes and, to date, the only accurate data available to produce such models has come from dissection and measurement. Modeling the dynamic behavior of the tree in a living organism has been, until now, impossible. Building models from CT scans will increase the volume and quality of informution available for both the study of physiology and diagnosis. We present an efJicient, accurate analysis technique for volume images of tube networks. Using local operations, the volume image is transformed into a vectorfield which resembles idealized fluid flow through the tube network. This field provides information to condense the salient features of the image into an augmented Euclidean minimum spanning tree (EMST). This augmented EMSTproves to be an information-rich and logical abstract representation of the vascular tree.

Curvature-based signatures for object description and recognition

An invariant related to Gaussian curvature at an object point is developed based upon the covariance matrix of photometric values within a local neighborhood about the point. We employ three illumination conditions, two of which are completely unknown. We never need to explicitly know the surface normal at a point. The determinant of the covariance matrix of the intensity three-tuples in the local neighborhood of an object point is shown to be invariant with respect to rotation and translation. A way of combing these determinant to form a signature distribution is formulated that is rotation, translation, and scale invariant. This signature is shown to be invariant over large ranges of poses of the same objects, while being significantly different between distinctly shaped objects. A new object recognition methodology is proposed by compiling signatures for only a few viewpoints of a given object.