Distance Metric Between 3D Models and 3D Images for Recognition and Classification

Similarity measurements between 3D objects and 2D images are useful for the tasks of object recognition and classification. We distinguish between two types of similarity metrics: metrics computed in image-space (image metrics) and metrics computed in transformation-space (transformation metrics). Existing methods typically use image and the nearest view of the object. Example for such a measure is the Euclidean distance between feature points in the image and corresponding points in the nearest view. (Computing this measure is equivalent to solving the exterior orientation calibration problem.) In this paper we introduce a different type of metrics: transformation metrics. These metrics penalize for the deformatoins applied to the object to produce the observed image. We present a transformation metric that optimally penalizes for "affine deformations" under weak-perspective. A closed-form solution, together with the nearest view according to this metric, are derived. The metric is shown to be equivalent to the Euclidean image metric, in the sense that they bound each other from both above and below. For Euclidean image metric we offier a sub-optimal closed-form solution and an iterative scheme to compute the exact solution.

Trajectory Mapping ("TM''): A New Non-Metric Scaling Technique

Trajectory Mapping "TM'' is a new scaling technique designed to recover the parameterizations, axes, and paths used to traverse a feature space. Unlike Multidimensional Scaling (MDS), there is no assumption that the space is homogenous or metric. Although some metric ordering information is obtained with TM, the main output is the feature parameterizations that partition the given domain of object samples into different categories. Following an introductory example, the technique is further illustrated using first a set of colors and then a collection of textures taken from Brodatz (1966).

Direct Computation of 3D Shape Invariants and the Focus of Expansion

Structure from motion often refers to the computation of 3D structure from a matched sequence of images. However, a depth map of a surface is difficult to compute and may not be a good representation for storage and recognition. Given matched images, I will first show that the sign of the normal curvature in a given direction at a given point in the image can be computed from a simple difference of slopes of line-segments in one image. Using this result, local surface patches can be classified as convex, concave, parabolic (cylindrical), hyperbolic (saddle point) or planar. At the same time the translational component of the optical flow is obtained, from which the focus of expansion can be computed.

The Alignment of Objects With Smooth Surfaces: Error Analysis of the Curvature Method

The recognition of objects with smooth bounding surfaces from their contour images is considerably more complicated than that of objects with sharp edges, since in the former case the set of object points that generates the silhouette contours changes from one view to another. The "curvature method", developed by Basri and Ullman [1988], provides a method to approximate the appearance of such objects from different viewpoints. In this paper we analyze the curvature method. We apply the method to ellipsoidal objects and compute analytically the error obtained for different rotations of the objects. The error depends on the exact shape of the ellipsoid (namely, the relative lengths of its axes), and it increases a sthe ellipsoid becomes "deep" (elongated in the Z-direction). We show that the errors are usually small, and that, in general, a small number of models is required to predict the appearance of an ellipsoid from all possible views. Finally, we show experimentally that the curvature method applies as well to objects with hyperbolic surface patches.

Convergence Results for the EM Approach to Mixtures of Experts Architectures

The Expectation-Maximization (EM) algorithm is an iterative approach to maximum likelihood parameter estimation. Jordan and Jacobs (1993) recently proposed an EM algorithm for the mixture of experts architecture of Jacobs, Jordan, Nowlan and Hinton (1991) and the hierarchical mixture of experts architecture of Jordan and Jacobs (1992). They showed empirically that the EM algorithm for these architectures yields significantly faster convergence than gradient ascent. In the current paper we provide a theoretical analysis of this algorithm. We show that the algorithm can be regarded as a variable metric algorithm with its searching direction having a positive projection on the gradient of the log likelihood. We also analyze the convergence of the algorithm and provide an explicit expression for the convergence rate. In addition, we describe an acceleration technique that yields a significant speedup in simulation experiments.

(Semi-)Predictive Discretization During Model Selection

In this paper, we present an approach to discretizing multivariate continuous data while learning the structure of a graphical model. We derive the joint scoring function from the principle of predictive accuracy, which inherently ensures the optimal trade-off between goodness of fit and model complexity (including the number of discretization levels). Using the so-called finest grid implied by the data, our scoring function depends only on the number of data points in the various discretization levels. Not only can it be computed efficiently, but it is also independent of the metric used in the continuous space. Our experiments with gene expression data show that discretization plays a crucial role regarding the resulting network structure.

A Constant-Factor Approximation Algorithm for Embedding Unweighted Graphs into Trees

We present a constant-factor approximation algorithm for computing an embedding of the shortest path metric of an unweighted graph into a tree, that minimizes the multiplicative distortion.

Perceptually-based Comparison of Image Similarity Metrics

The image comparison operation ??sessing how well one image matches another ??rms a critical component of many image analysis systems and models of human visual processing. Two norms used commonly for this purpose are L1 and L2, which are specific instances of the Minkowski metric. However, there is often not a principled reason for selecting one norm over the other. One way to address this problem is by examining whether one metric better captures the perceptual notion of image similarity than the other. With this goal, we examined perceptual preferences for images retrieved on the basis of the L1 versus the L2 norm. These images were either small fragments without recognizable content, or larger patterns with recognizable content created via vector quantization. In both conditions the subjects showed a consistent preference for images matched using the L1 metric. These results suggest that, in the domain of natural images of the kind we have used, the L1 metric may better capture human notions of image similarity.

Sparse Correlation Kernel Analysis and Reconstruction

This paper presents a new paradigm for signal reconstruction and superresolution, Correlation Kernel Analysis (CKA), that is based on the selection of a sparse set of bases from a large dictionary of class- specific basis functions. The basis functions that we use are the correlation functions of the class of signals we are analyzing. To choose the appropriate features from this large dictionary, we use Support Vector Machine (SVM) regression and compare this to traditional Principal Component Analysis (PCA) for the tasks of signal reconstruction, superresolution, and compression. The testbed we use in this paper is a set of images of pedestrians. This paper also presents results of experiments in which we use a dictionary of multiscale basis functions and then use Basis Pursuit De-Noising to obtain a sparse, multiscale approximation of a signal. The results are analyzed and we conclude that 1) when used with a sparse representation technique, the correlation function is an effective kernel for image reconstruction and superresolution, 2) for image compression, PCA and SVM have different tradeoffs, depending on the particular metric that is used to evaluate the results, 3) in sparse representation techniques, L_1 is not a good proxy for the true measure of sparsity, L_0, and 4) the L_epsilon norm may be a better error metric for image reconstruction and compression than the L_2 norm, though the exact psychophysical metric should take into account high order structure in images.

Investigating shape representation in area V4 with HMAX: Orientation and Grating selectivities

The question of how shape is represented is of central interest to understanding visual processing in cortex. While tuning properties of the cells in early part of the ventral visual stream, thought to be responsible for object recognition in the primate, are comparatively well understood, several different theories have been proposed regarding tuning in higher visual areas, such as V4. We used the model of object recognition in cortex presented by Riesenhuber and Poggio (1999), where more complex shape tuning in higher layers is the result of combining afferent inputs tuned to simpler features, and compared the tuning properties of model units in intermediate layers to those of V4 neurons from the literature. In particular, we investigated the issue of shape representation in visual area V1 and V4 using oriented bars and various types of gratings (polar, hyperbolic, and Cartesian), as used in several physiology experiments. Our computational model was able to reproduce several physiological findings, such as the broadening distribution of the orientation bandwidths and the emergence of a bias toward non-Cartesian stimuli. Interestingly, the simulation results suggest that some V4 neurons receive input from afferents with spatially separated receptive fields, leading to experimentally testable predictions. However, the simulations also show that the stimulus set of Cartesian and non-Cartesian gratings is not sufficiently complex to probe shape tuning in higher areas, necessitating the use of more complex stimulus sets.