F. de la Torre, M.J. Black, Robust principal component analysis for computer vision, in: Internat. Conf. on Computer Vision, 2001, pp. 362--369.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....weighting map will be generated. Since we treat the pixels with zero weights as missing data, we will introduce how to retrain the model using missing data modeling techniques in the next section. 3. MISSING DATA MODELING Many researchers have proposed different methods for missing data modeling [11] [12] As motivated by the research in error conceahnent for video sequences [13] we propose the following algorithm to train a statistical model for data with missing pixels. First, given the unwrapped image S ,S 2,A S , and their corresponding weighting map W,W2,A W, we compute the mean by ....

De la Torre, F. and Black, M. J., Robust principal component analysis for computer vision, Int. Conf. on Computer Vision, Vancouver, 2001.

....accordingly. Finally, the algorithm for weighted learning is embedded in this incremental framework, resulting in a weighted incremental method. A number of methods for estimation of principal axes in the presence of data with varying reliability and missing data have already been proposed [4, 16, 3, 12], however, all of them operate in a batch mode. Several algorithms for incremental learning have been proposed as well [5, 6, 2] but they are not suitable for weighted learning. One exception is the method for incremental learning with temporal weighting proposed by Liu and Chen [9] However, ....

....However, the improvements of the results in the scenarios where the selective treatment of individual images and pixels is required, justifies this extra computational power. In addition, the algorithm could be sped up by using alternative techniques for weighted least squares minimization [3] or by performing SVD on a weighted covariance matrix [13] Furthermore, the algorithm could also be completely parallelized, since each pixel image is processed independently. Weighted PCA is suitable to use in situations when an additional knowledge about significance of individual images as ....

F. De la Torre and M. Black. Robust principal component analysis for computer vision. In ICCV'01, pages I: 362--369, 2001.

....Layer maps by increasing the window radius of mean shift algorithm. In this paper, we used SVD to compute the subspace. Given Gaussian noise, SVD achieves global optimality in the sense of least square error. If the data contain outliers, robust algorithm can be used for deriving the subspace [22]. Acknowledgements Thanks go to Harry Shum, Simon Baker, Martial Hebert, and Alan Lipton for helpful discussions and comments on the paper. We would also like to thank the anonymous reviewers for their feedback. This work was supported in part by DiamondBack Vision, Inc. ....

F. Torre and M. J. Black. Robust principal component analysis for computer vision. In ICCV2001.

.... approaches have also focused on this technique to overcome frequent computer vision problems such as the recognition of objects taken under a wide range of conditions (several viewpoints and illumination conditions) 12] or dealing with partial occlusions by using robust estimation techniques [1, 5]. However, PCA based techniques suffer from several difficulties. Mainly, an image projection to a PCA based space depends on the precise position of relevant objects, on the intensity and shape of background zones, and on intensity and color of illumination. Since PCA treats its inputs globally, ....

F. de la Torre and M. Black. Robust principal component analysis for computer vision. In Proc. of ICCV'2001.

....and recognition of classes. Nevertheless, PCA is often used directly for pattern and object recognition tasks. In the computer vision community for example it has been used for recognition of faces [7] and 3D objects [6] or dealing with partial occlusions by using robust estimation techniques [1, 2]. Recently, Lee and Seung [3] proposed a new technique, called Non negative Matrix Factorization (NMF) to obtain # This work was supported by IST project IST 1999 20188CORKINSPECT, sponsored by the European Comission and by Comissionat per a Universitats i Recerca de la Generalitat de Catalunya ....

F. de la Torre and M. Black. Robust principal component analysis for computer vision. In Proc. of ICCV'2001.

....cluster a data set under an affine invariant distance function including priors on the affine transformation. Invariant approaches to unsupervised clustering have taken a number of routes. In a vector space, the techniques which have been used for robustification of principal components analysis [5] and to include some transformation invariance [9] could be applied to clustering, but these solutions are expensive to compute, and many interesting computer vision problems do not have data which may be linearly combined. In a metric space, attention must concentrate on the distance function in ....

F. De la Torre and M. J. Black. Robust principal component analysis for computer vision. In Proc. International Conference on Computer Vision, 2001.

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F. de la Torre, M.J. Black, Robust principal component analysis for computer vision, in: Internat. Conf. on Computer Vision, 2001, pp. 362--369.

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F. de la Torre, M.J. Black, Robust principal component analysis for computer vision, in: Internat. Conf. on Computer Vision, 2001, pp. 362--369.

....community) They did not address the general problem of robustly learning the principal components in the first place. Here we address the more general problem which involves learning both the basis vectors and linear coefficients robustly. Preliminary results of this work have been presented in [18]. 2.1 Energy Functions and PCA PCA is a statistical technique that is useful for dimensionality reduction. Let # : n ## : be a matrix d#n # , where each column i is a data sample (or image) n is the number of training images, and d is the number of pixels in each ....

F. de la Torre and M. J. Black. Robust principal component analysis for computer vision. In International Conference on Computer Vision, pages 362--369, 2001.

....maximum variation of the data D. Although a closed form solution for computing the principal components (B) can be achieved by computing the k largest eigenvectors of the covariance matrix DD [18] here it is useful to exploit work that formulates PCA as the minimization of an energy function [14,18]. Related formulations have been studied in various communities Bold capital letters denote a matrix D, bold lower case letters a column vector d. d j represents the j th column of the matrix D and d is a column vector representing the j th row of the matrix D. d ij denotes the scalar in row ....

..... 2 denotes the L2 norm of the vector d, that is d d. W denotes the weighted L2 norm of the vector d, that is d Wd, D F is the Frobenius norm of a matrix, tr(D D) tr(DD ) D1 D2 denotes the Hadamard (point wise) product between two matrices of equal dimension. see [14]) machine learning, statistics, neural networks and computer vision. In spirit, all these approaches essentially minimize the following energy function (although with different noise models, deterministic or Bayesian frameworks, or different metrics) E pca (B, C) D BC F = Bc ....

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F. de la Torre and M. J. Black. Robust principal component analysis for computer vision. In International Conference on Computer Vision, pages 362--369, 2001.

....the method as discussed in the conclusions. Finally, by making the coupling coecients explicit, we can extend the energy minimization approach to account for temporal dependencies in dynamic data sets. The approach complements recent work on robust PCA and robust Singular Value Decomposition (SVD) [7]. We illustrate the method by learning a coupled model of the faces in Fig. 1. This model can be used to animate one face using an image sequence of the other. Also, we illustrate the results by learning a coupled, dynamic model of two people swing dancing. Then given a new sequence of one of the ....

....of maximum variation of the data D. Although a closed form solution for computing the principal components (B) can be achieved by computing the k largest eigenvectors of the covariance matrix DD [8] here it is useful to exploit work that formulates PCA as the minimization of an energy function [7, 8, 9, 11, 21, 25, 27]. Related formulations have been studied in various communities: machine learning [21, 25] statistics [9, 11] neural networks [8] and computer vision [7, 27] In spirit, all these approaches essentially minimize the following energy function (although with di erent noise models, deterministic or ....

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F. De la Torre and M. Black. Robust principal component analysis for computer vision. ICCV, pp. 362-369, 2001.

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F. De la Torre and M. Black. Robust principal component analysis for computer vision. In ICCV '01, volume I, pages 362--369, 2001.

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F. de la Torre and M.J. Black. Robust principal component analysis for computer vision. In ICCV, pages I: 362--369, Vancouver, Canada, July 2001.

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F. De la Torre and M. J. Black. Robust principal component analysis for computer vision. In Proc. Eighth International Conference on Computer Vision, Vancouver, pages 362--369, 2001.

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F. De la Torre and M. Black. Robust principal component analysis for computer vision. In Proc. ICCV, pages 362--369, 2001.

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F. de la Torre and M.J. Black. Robust principal component analysis for computer vision. Proc. ICCV, pp. 362369, 2001.

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F. De la Torre and M. J. Black. Robust principal component analysis for computer vision. In CVPR, pages 362--369, 2001.

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F. de la Torre and M.J. Black. Robust principal component analysis for computer vision. Proc. ICCV, pp. 362369, 2001.

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F. Torre and M. J. Black. Robust principal component analysis for computer vision. In Intl. Conf. on Computer Vision (ICCV), pages 362--369, 2001.

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F. De la Torre and M. Black. Robust principal component analysis for computer vision. In Proc. ICCV, pages 362--369, 2001.

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F. De la Torre and M. Black. Robust principal component analysis for computer vision. In ICCV '01, volume I, pages 362--369, 2001.

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F.Torre and M. J. Black. Robust principal component analysis for computer vision. In 8th International Conference on Computer Vision, volume I, pages 362--349, Vancouver, Canada, July 2001.

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F. De la Torre and M. Black, "Robust principal component analysis for computer vision," in IEEE International Conference on Computer Vision, Vancouver, Canada, 2001, vol. 1, pp. 362--369.

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