Leo Grady

Random Walks for Image Segmentation

A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with user-defined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a high-quality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs

A Seeded Image Segmentation Framework Unifying Graph Cuts And Random Walker Which Yields A New Algorithm

In this work, we present a common framework for seeded image segmentation algorithms that yields two of the leading methods as special cases - The graph cuts and the random walker algorithms. The formulation of this common framework naturally suggests a new, third, algorithm that we develop here. Specifically, the former algorithms may be shown to minimize a certain energy with respect to either an l1 or an l2 norm. Here, we explore the segmentation algorithm defined by an linfin norm, provide a method for the optimization and show that the resulting algorithm produces an accurate segmentation that demonstrates greater stability with respect to the number of seeds employed than either the graph cuts or random walker methods.

A multilevel banded graph cuts method for fast image segmentation

In the short time since publication of Boykov and Jollys seminal paper [2001], graph cuts have become well established as a leading method in 2D and 3D semi-automated image segmentation. Although this approach is computationally feasible for many tasks, the memory overhead and supralinear time complexity of leading algorithms results in an excessive computational burden for high-resolution data. In this paper, we introduce a multilevel banded heuristic for computation of graph cuts that is motivated by the well-known narrow band algorithm in level set computation. We perform a number of numerical experiments to show that this heuristic drastically reduces both the running time and the memory consumption of graph cuts while producing nearly the same segmentation result as the conventional graph cuts. Additionally, we are able to characterize the type of segmentation target for which our multilevel banded heuristic yields different results from the conventional graph cuts. The proposed method has been applied to both 2D and 3D images with promising results.

Isoperimetric graph partitioning for image segmentation

Spectral graph partitioning provides a powerful approach to image segmentation. We introduce an alternate idea that finds partitions with a small isoperimetric constant, requiring solution to a linear system rather than an eigenvector problem. This approach produces the high quality segmentations of spectral methods, but with improved speed and stability.

Power Watershed: A Unifying Graph-Based Optimization Framework

In this work, we extend a common framework for graph-based image segmentation that includes the graph cuts, random walker, and shortest path optimization algorithms. Viewing an image as a weighted graph, these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes. Introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watershed in this same framework. We then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm, which we term the power watershed. In particular, when q=2, the power watershed leads to a multilabel, scale and contrast invariant, unique global optimum obtained in practice in quasi-linear time. Placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation.

Multilabel random walker image segmentation using prior models

The recently introduced random walker segmentation algorithm by Grady and Funka-Lea (2004) has been shown to have desirable theoretical properties and to perform well on a wide variety of images in practice. However, this algorithm requires user-specified labels and produces a segmentation where each segment is connected to a labeled pixel. We show that incorporation of a nonparametric probability density model allows for an extended random walkers algorithm that can locate disconnected objects and does not require user-specified labels. Finally, we show that this formulation leads to a deep connection with the popular graph cuts method by Boykov et al. (2001) and Wu and Leahy (1993).

Power watersheds: A new image segmentation framework extending graph cuts, random walker and optimal spanning forest

In this work, we extend a common framework for seeded image segmentation that includes the graph cuts, random walker, and shortest path optimization algorithms. Viewing an image as a weighted graph, these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes. Introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watersheds in this same framework. We then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm, which we term power watersheds. Placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watersheds to optimize more general models of use in application beyond image segmentation.

Random walks for interactive organ segmentation in two and three dimensions: implementation and validation

A new approach to interactive segmentation based on random walks was recently introduced that shows promise for allowing physicians more flexibility to segment arbitrary objects in an image. This report has two goals: To introduce a novel computational method for applying the random walker algorithm in 2D/3D using the Graphics Processing Unit (GPU) and to provide quantitative validation studies of this algorithm relative to different targets, imaging modalities and interaction strategies.

The Piecewise Smooth Mumford–Shah Functional on an Arbitrary Graph

The Mumford-Shah functional has had a major impact on a variety of image analysis problems, including image segmentation and filtering, and, despite being introduced over two decades ago, it is still in widespread use. Present day optimization of the Mumford-Shah functional is predominated by active contour methods. Until recently, these formulations necessitated optimization of the contour by evolving via gradient descent, which is known for its overdependence on initialization and the tendency to produce undesirable local minima. In order to reduce these problems, we reformulate the corresponding Mumford-Shah functional on an arbitrary graph and apply the techniques of combinatorial optimization to produce a fast, low-energy solution. In contrast to traditional optimization methods, use of these combinatorial techniques necessitates consideration of the reconstructed image outside of its usual boundary, requiring additionally the inclusion of regularization for generating these values. The energy of the solution provided by this graph formulation is compared with the energy of the solution computed via traditional gradient descent-based narrow-band level set methods. This comparison demonstrates that our graph formulation and optimization produces lower energy solutions than the traditional gradient descent based contour evolution methods in significantly less time. Finally, we demonstrate the usefulness of the graph formulation to apply the Mumford-Shah functional to new applications such as point clustering and filtering of nonuniformly sampled images.

Random walks based multi-image segmentation: Quasiconvexity results and GPU-based solutions

We recast the Cosegmentation problem using Random Walker (RW) segmentation as the core segmentation algorithm, rather than the traditional MRF approach adopted in the literature so far. Our formulation is similar to previous approaches in the sense that it also permits Cosegmentation constraints (which impose consistency between the extracted objects from ≥ 2 images) using a nonparametric model. However, several previous nonparametric cosegmentation methods have the serious limitation that they require adding one auxiliary node (or variable) for every pair of pixels that are similar (which effectively limits such methods to describing only those objects that have high entropy appearance models). In contrast, our proposed model completely eliminates this restrictive dependence - the resulting improvements are quite significant. Our model further allows an optimization scheme exploiting quasiconvexity for model-based segmentation with no dependence on the scale of the segmented foreground. Finally, we show that the optimization can be expressed in terms of linear algebra operations on sparse matrices which are easily mapped to GPU architecture. We provide a highly specialized CUDA library for Cosegmentation exploiting this special structure, and report experimental results showing these advantages.