Toward a computational theory of shape: An overview

Fully automated liver segmentation from SPIR image series

Accurate liver segmentation is an important component of surgery planning for liver transplantation, which enables patients with liver disease a chance to survive. Spectral pre-saturation inversion recovery (SPIR) image sequences are useful for liver vessel segmentation because vascular structures in the liver are clearly visible in these sequences. Although level-set based segmentation techniques are frequently used in liver segmentation due to their flexibility to adapt to different problems by incorporating prior knowledge, the need to initialize the contours on each slice is a common drawback of such techniques. In this paper, we present a fully automated variational level set approach for liver segmentation from SPIR image sequences. Our approach is designed to be efficient while achieving high accuracy. The efficiency is achieved by (1) automatically defining an initial contour for each slice, and (2) automatically computing weight values of each term in the applied energy functional at each iteration during evolution. Automated detection and exclusion of spurious structures (e.g. cysts and other bright white regions on the skin) in the pre-processing stage increases the accuracy and robustness. We also present a novel approach to reduce computational cost by employing binary regularization of level set function. A signed pressure force function controls the evolution of the active contour. The method was applied to ten data sets. In each image, the performance of the algorithm was measured using the receiver operating characteristics method in terms of accuracy, sensitivity and specificity. The accuracy of the proposed method was 96%. Quantitative analyses of results indicate that the proposed method can accurately, efficiently and consistently segment liver images.

TurboPixels: fast superpixels using geometric flows

We describe a geometric-flow-based algorithm for computing a dense oversegmentation of an image, often referred to as superpixels. It produces segments that, on one hand, respect local image boundaries, while, on the other hand, limiting undersegmentation through a compactness constraint. It is very fast, with complexity that is approximately linear in image size, and can be applied to megapixel sized images with high superpixel densities in a matter of minutes. We show qualitative demonstrations of high-quality results on several complex images. The Berkeley database is used to quantitatively compare its performance to a number of oversegmentation algorithms, showing that it yields less undersegmentation than algorithms that lack a compactness constraint while offering a significant speedup over N-cuts, which does enforce compactness.

A regularized curvature flow designed for a selective shape restoration

Among all filtering techniques, those based exclusively on image level sets (geometric flows) have proven to be the less sensitive to the nature of noise and the most contrast preserving. A common feature to existent curvature flows is that they penalize high curvature, regardless of the curve regularity. This constitutes a major drawback since curvature extreme values are standard descriptors of the contour geometry. We argue that an operator designed with shape recovery purposes should include a term penalizing irregularity in the curvature rather than its magnitude. To this purpose, we present a novel geometric flow that includes a function that measures the degree of local irregularity present in the curve. A main advantage is that it achieves non-trivial steady states representing a smooth model of level curves in a noisy image. Performance of our approach is compared to classical filtering techniques in terms of quality in the restored image/shape and asymptotic behavior. We empirically prove that our approach is the technique that achieves the best compromise between image quality and evolution stabilization.

On the psychophysics of the shape triangle

We earlier introduced an approach to categorical shape description based on the singularities (shocks) of curve evolution equations. We now consider the simplest compositions of shocks, and show that they lead to three classes of parametrically ordered shape sequences, organized along the sides of a shape triangle. By conducting several psychophysical experiments we demonstrate that shock-based descriptions are predictive of performance in shape perception. Most significantly, the experiments reveal a fundamental difference between perceptual effects dominated by when shocks form with respect to one another, versus those dominated by where they form. The shock-based theory provides a foundation for unifying tasks as diverse as shape bisection, recognition, and categorization.

Geometric active deformable models in shape modeling

This paper analyzes the problem of shape modeling using the principle of active geometric deformable models. While the basic modeling technique already exists in the literature, we highlight many of its drawbacks and discuss their source and steps to overcome them. We propose a new stopping criterion to address the stopping problem. We also propose to apply a level set algorithm to implement the active geometric deformable models, thereby handling topology changes automatically. To alleviate the numerical problems associated with the implementation of the level set algorithm, we propose a new adaptive multigrid narrow band algorithm. All the proposed new changes have been illustrated with experiments with synthetic images and medical images.

Area and length minimizing flows for shape segmentation

A number of active contour models have been proposed that unify the curve evolution framework with classical energy minimization techniques for segmentation, such as snakes. The essential idea is to evolve a curve (in two dimensions) or a surface (in three dimensions) under constraints from image forces so that it clings to features of interest in an intensity image. The evolution equation has been derived from first principles as the gradient flow that minimizes a modified length functional, tailored to features such as edges. However, because the flow may be slow to converge in practice, a constant (hyperbolic) term is added to keep the curve/surface moving in the desired direction. We derive a modification of this term based on the gradient flow derived from a weighted area functional, with image dependent weighting factor. When combined with the earlier modified length gradient flow, we obtain a partial differential equation (PDE) that offers a number of advantages, as illustrated by several examples of shape segmentation on medical images. In many cases the weighted area flow may be used on its own, with significant computational savings.

Salience of visual parts

Many objects have component parts, and these parts often differ in their visual salience. In this paper we present a theory of part salience. The theory builds on the minima rule for defining part boundaries. According to this rule, human vision defines part boundaries at negative minima of curvature on silhouettes, and along negative minima of the principal curvatures on surfaces. We propose that the salience of a part depends on (at least) three factors: its size relative to the whole object, the degree to which it protrudes, and the strength of its boundaries. We present evidence that these factors influence visual processes which determine the choice of figure and ground. We give quantitative definitions for the factors, visual demonstrations of their effects, and results of psychophysical experiments.

A geometric snake model for segmentation of medical imagery

In this note, we employ the new geometric active contour models formulated in [25] and [26] for edge detection and segmentation of magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound medical imagery. Our method is based on defining feature-based metrics on a given image which in turn leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus, the snake is attracted very quickly and efficiently to the desired feature.

Optical flow: a curve evolution approach

A novel approach for the computation of optical flow based on an L (1) type minimization is presented. It is shown that the approach has inherent advantages since it does not smooth the flow-velocity across the edges and hence preserves edge information. A numerical approach based on computation of evolving curves is proposed for computing the optical flow field. Computations are carried out on a number of real image sequences in order to illustrate the theory as well as the numerical approach.

M-lattice: from morphogenesis to image processing

The paper is based on reaction-diffusion, a nonlinear mechanism first proposed by Turing in 1952 to account for morphogenesis, the formation of shape and pattern in nature. One of the key limitations of reaction-diffusion systems is that they are generally unbounded, making them awkward for digital image processing. In this paper we introduce the "M-lattice", a system that preserves the pattern-formation properties of reaction-diffusion and is bounded. On the theoretical front, we establish how the M-lattice is closely related to the analog Hopfield network and the cellular neural network, but has more flexibility in how its variables interact. Like many "neurally inspired" systems, the bounded M-lattice also enables computer or analog VLSI implementations to simulate a variety of partial and ordinary differential equations. On the practical front, we demonstrate two novel applications of reaction-diffusion formulated as the new M-lattice. These are adaptive filtering, applied to the restoration and enhancement of fingerprint images, and nonlinear programming, applied to image halftoning in both "faithful" and "special effects" styles.