Many tasks in computer vision involve assigning a label (such as disparity) to every pixel. A common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist, e.g., at object boundaries. These tasks are naturally stated in terms of energy minimization. The authors consider a wide class of energies with various smoothness constraints. Global minimization of these energy functions is NP-hard even in the simplest discontinuity-preserving case. Therefore, our focus is on efficient approximation algorithms. We present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves, namely expansion moves and swap moves. These moves can simultaneously change the labels of arbitrarily large sets of pixels. In contrast, many standard algorithms (including simulated annealing) use small moves where only one pixel changes its label at a time. Our expansion algorithm finds a labeling within a known factor of the global minimum, while our swap algorithm handles more general energy functions. Both of these algorithms allow important cases of discontinuity preserving energies. We experimentally demonstrate the effectiveness of our approach for image restoration, stereo and motion. On real data with ground truth, we achieve 98 percent accuracy.

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Fast approximate energy minimization via graph cuts

We present a novel approach to measuring similarity between shapes and exploit it for object recognition. In our framework, the measurement of similarity is preceded by: (1) solving for correspondences between points on the two shapes; (2) using the correspondences to estimate an aligning transform. In order to solve the correspondence problem, we attach a descriptor, the shape context, to each point. The shape context at a reference point captures the distribution of the remaining points relative to it, thus offering a globally discriminative characterization. Corresponding points on two similar shapes will have similar shape contexts, enabling us to solve for correspondences as an optimal assignment problem. Given the point correspondences, we estimate the transformation that best aligns the two shapes; regularized thin-plate splines provide a flexible class of transformation maps for this purpose. The dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points, together with a term measuring the magnitude of the aligning transform. We treat recognition in a nearest-neighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image. Results are presented for silhouettes, trademarks, handwritten digits, and the COIL data set.

Shape matching and object recognition using shape contexts

In this paper we address the problem of minimizing a large class of energy functions that occur in early vision. The major restriction is that the energy function's smoothness term must only involve pairs of pixels. We propose two algorithms that use graph cuts to compute a local minimum even when very large moves are allowed. The first move we consider is an /spl alpha/-/spl beta/-swap: for a pair of labels /spl alpha/,/spl beta/, this move exchanges the labels between an arbitrary set of pixels labeled a and another arbitrary set labeled /spl beta/. Our first algorithm generates a labeling such that there is no swap move that decreases the energy. The second move we consider is an /spl alpha/-expansion: for a label a, this move assigns an arbitrary set of pixels the label /spl alpha/. Our second algorithm, which requires the smoothness term to be a metric, generates a labeling such that there is no expansion move that decreases the energy. Moreover, this solution is within a known factor of the global minimum. We experimentally demonstrate the effectiveness of our approach on image restoration, stereo and motion.

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In this paper we present a computationally efficient framework for part-based modeling and recognition of objects. Our work is motivated by the pictorial structure models introduced by Fischler and Elschlager. The basic idea is to represent an object by a collection of parts arranged in a deformable configuration. The appearance of each part is modeled separately, and the deformable configuration is represented by spring-like connections between pairs of parts. These models allow for qualitative descriptions of visual appearance, and are suitable for generic recognition problems. We address the problem of using pictorial structure models to find instances of an object in an image as well as the problem of learning an object model from training examples, presenting efficient algorithms in both cases. We demonstrate the techniques by learning models that represent faces and human bodies and using the resulting models to locate the corresponding objects in novel images.

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Pictorial Structures for Object Recognition

There is a growing need to be able to accurately and efficiently search visual data sets, and in particular, 3D shape data sets. This paper proposes a novel technique, called Topology Matching, in which similarity between polyhedral models is quickly, accurately, and automatically calculated by comparing Multiresolutional Reeb Graphs (MRGs). The MRG thus operates well as a search key for 3D shape data sets. In particular, the MRG represents the skeletal and topological structure of a 3D shape at various levels of resolution. The MRG is constructed using a continuous function on the 3D shape, which may preferably be a function of geodesic distance because this function is invariant to translation and rotation and is also robust against changes in connectivities caused by a mesh simplification or subdivision. The similarity calculation between 3D shapes is processed using a coarse-to-fine strategy while preserving the consistency of the graph structures, which results in establishing a correspondence between the parts of objects. The similarity calculation is fast and efficient because it is not necessary to determine the particular pose of a 3D shape, such as a rotation, in advance. Topology Matching is particularly useful for interactively searching for a 3D object because the results of the search fit human intuition well.

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Topology matching for fully automatic similarity estimation of 3D shapes

Curve matching is one instance of the fundamental correspondence problem. Our flexible algorithm is designed to match curves under substantial deformations and arbitrary large scaling and rigid transformations. A syntactic representation is constructed for both curves and an edit transformation which maps one curve to the other is found using dynamic programming. We present extensive experiments where we apply the algorithm to silhouette matching. In these experiments, we examine partial occlusion, viewpoint variation, articulation, and class matching (where silhouettes of similar objects are matched). Based on the qualitative syntactic matching, we define a dissimilarity measure and we compute it for every pair of images in a database of 121 images. We use this experiment to objectively evaluate our algorithm. First, we compare our results to those reported by others. Second, we use the dissimilarity values in order to organize the image database into shape categories. The veridical hierarchical organization stands as evidence to the quality of our matching and similarity estimation.

Flexible syntactic matching of curves and its application to automatic hierarchical classification of silhouettes

We present a stochastic clustering algorithm which uses pairwise similarity of elements and show how it can be used to address various problems in computer vision, including the low-level image segmentation, mid-level perceptual grouping, and high-level image database organization. The clustering problem is viewed as a graph partitioning problem, where nodes represent data elements and the weights of the edges represent pairwise similarities. We generate samples of cuts in this graph, by using Karger's contraction algorithm (1996), and compute an "average" cut which provides the basis for our solution to the clustering problem. The stochastic nature of our method makes it robust against noise, including accidental edges and small spurious clusters. The complexity of our algorithm is very low: O(|E| log/sup 2/ N) for N objects, |E| similarity relations, and a fixed accuracy level. In addition, and without additional computational cost, our algorithm provides a hierarchy of nested partitions. We demonstrate the superiority of our method for image segmentation on a few synthetic and real images, both B&W and color. Our other examples include the concatenation of edges in a cluttered scene (perceptual grouping) and the organization of an image database for the purpose of multiview 3D object recognition.

Self-organization in vision: stochastic clustering for image segmentation, perceptual grouping, and image database organization

We present a stochastic clustering algorithm which uses pairwise similarity of elements, based on a new graph theoretical algorithm for the sampling of cuts in graphs. The stochastic nature of our method makes it robust against noise, including accidental edges and small spurious clusters. We demonstrate the robustness and superiority of our method for image segmentation on a few synthetic examples where other recently proposed methods (such as normalized-cut) fail. In addition, the complexity of our method is lower. We describe experiments with real images showing good segmentation results.

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Stochastic image segmentation by typical cuts

We present a novel pairwise clustering method Given a proximity matrix of pairwise relations (ie pairwise similarity or dissimilarity estimates) between data points, our algorithm extracts the two most prominent clusters in the data set The algorithm, which is completely nonparametric, iteratively employs a two-step transformation on the proximity matrix The first step of the transformation represents each point by its relation to all other data points, and the second step re-estimates the pairwise distances using a statistically motivated proximity measure on these representations Using this transformation, the algorithm iteratively partitions the data points, until it finally converges to two clusters Although the algorithm is simple and intuitive, it generates a complex dynamics of the proximity matrices Based on this bipartition procedure we devise a hierarchical clustering algorithm, which employs the basic bipartition algorithm in a straightforward divisive manner The hierarchical clustering algorithm copes with the model validation problem using a general cross-validation approach, which may be combined with various hierarchical clustering methods

We further present an experimental study of this algorithm We examine some of the algorithm's properties and performance on some synthetic and ‘standard’ data sets The experiments demonstrate the robustness of the algorithm and indicate that it generates a good clustering partition even when the data is noisy or corrupted

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A New Nonparametric Pairwise Clustering Algorithm Based on Iterative Estimation of Distance Profiles

One of the key problems in appearance-based vision is understanding how to use a set of labeled images to classify new images. Classification systems that can model human performance, or that use robust image matching methods, often make use of similarity judgments that are non-metric; but when the triangle inequality is not obeyed, most existing pattern recognition techniques are not applicable. We note that exemplar-based (or nearest-neighbor) methods can be applied naturally when using a wide class of non-metric similarity functions. The key issue, however, is to find methods for choosing good representatives of a class that accurately characterize it. We show that existing condensing techniques for finding class representatives are ill-suited to deal with non-metric dataspaces. We then focus on developing techniques for solving this problem, emphasizing two points: First, we show that the distance between two images is not a good measure of how well one image can represent another in non-metric spaces. Instead, we use the vector correlation between the distances from each image to other previously seen images. Second, we show that in non-metric spaces, boundary points are less significant for capturing the structure of a class than they are in Euclidean spaces. We suggest that atypical points may be more important in describing classes. We demonstrate the importance of these ideas to learning that generalizes

Yoram Gdalyahu

Papers

Flexible syntactic matching of curves and its application to automatic hierarchical classification of silhouettes

Self-organization in vision: stochastic clustering for image segmentation, perceptual grouping, and image database organization

Stochastic image segmentation by typical cuts

A New Nonparametric Pairwise Clustering Algorithm Based on Iterative Estimation of Distance Profiles

Class Representation and Image Retrieval with Non-Metric Distances