Abstract—We propose an eigenvector-based heteroscedastic linear dimension reduction (LDR) technique for multiclass data. The technique is based on a heteroscedastic two-class technique which utilizes the so-called Chernoff criterion, and successfully extends the well-known linear discriminant analysis (LDA). The latter, which is based on the Fisher criterion, is incapable of dealing with heteroscedastic data in a proper way. For the two-class case, the between-class scatter is generalized so to capture differences in (co)variances. It is shown that the classical notion of between-class scatter can be associated with Euclidean distances between class means. From this viewpoint, the between-class scatter is generalized by employing the Chernoff distance measure, leading to our proposed heteroscedastic measure. Finally, using the results from the two-class case, a multiclass extension of the Chernoff criterion is proposed. This criterion combines separation information present in the class mean as well as the class covariance matrices. Extensive experiments and a comparison with similar dimension reduction techniques are presented. Index Terms—Linear dimension reduction, linear discriminant analysis, Fisher criterion, Chernoff distance, Chernoff criterion. 1

This paper investigates the use of range images of faces for recognizing people. 3D scans of faces lead to range images that are linearly projected to low-dimensional subspaces for use in a classifier, say a nearest neighbor classifier or a support vector machine, to label people. Learning of subspaces is performed using an optimal component analysis, i.e. a stochastic optimization algorithm (on a Grassmann manifold) to find a subspace that maximizes classifier performance on the training image set. Results are presented for face recognition using FSU face database, and are compared with standard component analysis such as PCA and ICA. This provides an efficient tool for analyzing certain aspects of facial shapes while avoiding a difficult task of geometric surface modeling.

Linear representations and linear dimension reduction techniques are very common in signal and image processing. Many such applications reduce to solving problems of stochastic optimizations or statistical inferences on the set of all subspaces, i.e. a Grassmann manifold. Central to solving them is the computation of an "exponential" map (for constructing geodesics) and its inverse on a Grassmannian. Here we suggest efficient techniques for these two steps and illustrate two applications: (i) For image-based object recognition, we define and seek an optimal linear representation using a Metropolis-Hastings type, stochastic search algorithm on a Grassmann manifold. (ii) For statistical inferences, we illustrate computation of sample statistics, such as mean and variances, on a Grassmann manifold.

Abstract — Low dimensional representations of images impose equivalence relations in the image space; the induced equivalence class of an image is named as its intrinsic generalization. The intrinsic generalization of a representation provides a novel way to measure its generalization and leads to more fundamental insights than the commonly used recognition performance, which is heavily influenced by the choice of training and test data. We demonstrate the limitations of linear subspace representations by sampling their intrinsic generalization, and propose a nonlinear representation that overcomes these limitations. The proposed representation projects images nonlinearly into the marginal densities of their filter responses, followed by linear projections of the marginals. We use experiments on large datasets to show that the representations that have better intrinsic generalization also lead to better recognition performance. I.

Abstract—This paper proposes, analyzes, and illustrates several best basis search algorithms for dictionaries consisting of lapped orthogonal bases. It improves upon the best local cosine basis selection based on a dyadic tree [10], [11] by considering larger dictionaries of bases. It is shown that this can result in sparser representations and approximate shift invariance. An algorithm that is strictly shift invariant is also provided. The experiments in this paper suggest that the new dictionaries can be advantageous for time-frequency analysis, compression, and noise removal. Accelerated versions of the basic algorithm are provided that explore various tradeoffs between computational efficiency and adaptability. It is shown that the proposed algorithms are in fact applicable to any finite dictionary comprised of lapped orthogonal bases. One such novel dictionary is proposed that constructs the best local cosine representation in the frequency domain, and it is shown that the new dictionary is better suited for representing certain types of signals. Index Terms—Best basis, lapped transforms, time-frequency analysis. I.

This thesis presents new techniques which enable the automatic recognition of everyday objects like chairs and ladders in images of highly cluttered scenes. Given an image, we extract information about the shape and texture properties present in small patches of the image and use that information to identify parts of the objects we are interested in. We then assemble those parts into overall hypotheses about what objects are present in the image, and where they are. Solving this problem in a general setting is one of the central problems in computer vision, as doing so would have an immediate impact on a far-reaching set of applications in medicine, surveillance, manufacturing, robotics, and other areas.

3D face information, it is not trivial to process the large amount of facial surface data. For example, it is hard to keep the correspondences among different subjects because their models have different vertices, and a post-processing procedure needs to follow because the range

Linear techniques are widely used to reduce the dimension of image representa-tion spaces in applications such as image indexing and object recognition. Optimal Component Analysis (OCA) is a method that addresses the problem of learning an optimal linear representation for a particular classification task. The problem is formulated in the framework of optimization on a Grassmann manifold and treated with stochastic gradient methods intrinsic to the manifold. OCA has been suc-cessfully applied to image classification problems arising in a variety of contexts. However, as the search space is typically very high dimensional, OCA optimization often requires a large number of iterations, each involving extensive computations that make the algorithm somewhat costly to implement. In this paper, we propose a two-stage method, which we refer to as two-stage OCA, that improves the search efficiency by orders of magnitude without compromising the quality of the esti-mation. In fact, extensive experiments using face and object classification datasets indicate that the proposed method often leads to more accurate classification than the original OCA since it is not as prone to over-fitting. Two-stage OCA also leads to Preprint submitted to Elsevier 30 May 2007 substantial improvement in classification performance as compared to other linear dimension reduction methods.

Abstract. We present a novel method for dimensionality reduction and recognition based on Linear Discriminant Analysis (LDA), which specifically deals with the Small Sample Size (SSS) problem in Computer Vision applications. Unlike the traditional methods, which impose specific assumptions to address the SSS problem, our approach introduces a variant of bootstrap bumping technique, which is a general framework in statistics for model search and inference. An intermediate linear representation is first hypothesized from each bootstrap sample. Then LDA is performed in the reduced subspace. Lastly, the final model is selected among all hypotheses for the best classification. Experiments on synthetic and real datasets demonstrate the advantages of our Bootstrap Bumping LDA (BB-LDA) approach over the traditional LDA based methods. 1

Object detection is a part of our everyday lives, however, automatic object detection by computer is still an open question. In 30 years of research in computer vision, little progress has been made. This report is a survey on the most recent techniques in object detection research. First, we introduce the definition, challenges, applications and general components of the object detection system. This is followed by a review of various appearance based approaches and feature based approaches. Appearance based approaches are classified based on different classifiers into linear representation, distribution-based, support vector machines, sparse Winnow network. Meanwhile different feature based approaches are distinguished from each other by what features are being used- texture, shape, context and multiple features. Then a framework of an object detection system is