D. Keysers, J. Dahmen, and H. Ney, "A Probabilistic View on Tangent Distance," Proc. 22. DAGM Symp. Mustererkennung, pp. 107-114, Sept. 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... algorithm [7] While this approach was initially aimed at solving classification tasks with a nearest neighbor paradigm, some work has already been done in developing it into a probabilistic interpretation for mixtures with a few gaussians, as well as for full fledged kernel density estimation [8, 9]. The main difference between our approach and the above is that the Manifold Parzen estimator does not require prior knowledge, as it infers the local directions directly from the data, although it should be easy to also incorporate prior knowledge if available. We should also mention ....

D. Keysers, J. Dahmen, and H. Ney. A probabilistic view on tangent distance. In 22nd Symposium of the German Association for Pattern Recognition, Kiel, Germany, 2000.

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D. Keysers, J. Dahmen, and H. Ney, "A Probabilistic View on Tangent Distance," Proc. 22. DAGM Symp. Mustererkennung, pp. 107-114, Sept. 2000.

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D. Keysers, J. Dahmen, H. Ney, "A Probabilistic View on Tangent Distance", Proceedings of the 22. Symposium of the German Association for Pattern Recognition (DAGM), Kiel, Germany, September 2000, this volume.

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D. Keysers, J. Dahmen, and H. Ney. A Probabilistic View on Tangent Distance. In 22. DAGM Symposium Mustererkennung 2000, Springer, Kiel, Germany, pages 107--114, September 2000.

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D. Keysers, J. Dahmen, and H. Ney, "A Probabilistic View on Tangent Distance," Proc. 22. DAGM Symp. Mustererkennung, pp. 107-114, Sept. 2000.

....in a Probabilistic Framework To embed the TD into a statistical framework we will focus on the one sided TD, assuming that the references are subject to variations. A more detailed presentation including the remaining cases of variation of the observations and the two sided TD can be found in [8]. We restrict our considerations here to the case where the observations x are normally distributed with expectation and covariance matrix . The extension to Gaussian mixtures or kernel densities is straightforward using maximum approximation or the EM algorithm. In order to simplify the ....

.... 1 (11) Note that the exponent in Eq. 10) leads to the conventional Mahalanobis distance for 0 and to TD for 1. Thus, the incorporation of tangent vectors adds a corrective term to the Mahalanobis distance that only a ects the covariance matrix which can be interpreted as structuring [8]. For the limiting case = I , a similar result was derived in [6] The probabilistic interpretation of TD can also be used for a more reliable estimation of the parameters of the distribution [2, 8] Note furthermore that det( 0 ) 1 2 ) L det( 5, pp. 38 . which is independent of ....

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D. Keysers, J. Dahmen, and H. Ney. A Probabilistic View on Tangent Distance. In 22. DAGM Symposium Mustererkennung 2000, Springer, Kiel, Germany, pages 107-114, September 2000.

....the parameters (with = 2 I in the experiments) Note that T = 1 N Z p( N X n=1 x n; d = 18) Thus, the empirical sample mean does not change in the presence of tangent vectors. More information on the probabilistic interpretation of tangent distance can be found in [4, 16]. 4.3. The image distortion model Computation of tangent distance as given in Eq. 13) still requires the calculation of the (squared) Euclidean distance between the optimally transformed image x and the reference image . Although small global transformations have been compensated for by the ....

D. Keysers, J. Dahmen, and H. Ney, \A Probabilistic view on tangent distance ", in Proc. 22nd Symposium German Association for Pattern Recognition, Kiel, Germany, 2000, pp. 107-114.

....2.1. A probabilistic framework for tangent distance For the purpose of embedding TD into a statistical framework we will focus on the consideration of one sided TD, assuming that only the references are subject to variations. A detailed overview including the two sided TD can be found in [6]. For the moment we assume that the tangent vectors l are known. The observations x shall be normal distributed with expectation and covariance matrix . In order to simplify the notation, class indices are omitted. Using the first order approximation of the manifold M for a mean vector ....

....of the pdf in Eq. 8) can be assumed as stochastically independent since they form a basis of the tangent subspace. Hence, it is always possible to decorrelate the tangent vectors using e.g. a singular value decomposition. The evaluation of the integral in Eq. 9) leads to the following expression [6]: p(x j ; 2 L X l=1 l T l 1 2 exp 1 2 h (x ) T 1 L X l=1 [ T l 1 ] T [ T l 1 ] 1= 2 T l 1 l (x ) i (11) Note that the exponent in Eq. 11) leads to conventional Mahalanobis distance for 0 and TD ....

D. Keysers, J. Dahmen, and H. Ney, "A probabilistic view on tangent distance," in 22. DAGM Symposium Mustererkennung 2000, (Kiel, Germany), pp. 107--114, Springer, Sept. 2000.

....corpus is chosen to illustrate the e ect) Similarly, we can de ne a double sided TD approximating both manifolds. TD can lead to transformation tolerance in classi cation and can furthermore be easily incorporated into statistical classi ers as it has a well founded probabilistic interpretation [8]. By adding to the training set a number of transformed instances of the original data (virtual data) one can achieve better performance of the classi er without actually requiring more training data. For the RBC task we used rotations by multiples of =2 and ipping. For the OCR data, rotation ....

D. Keysers, J. Dahmen, and H. Ney. A Probabilistic View on Tangent Distance. In 22. DAGM Symposium Mustererkennung 2000, Springer, Kiel, Germany, pages 107-114, September 2000.

.... Sigma cannot be used explicitly (as it does not exist for 1) yet calculating single sided tangent distance is equivalent to using Sigma . As the required calculations to prove this statement are rather lengthy, they are omitted here. A detailed discussion of this topic can be found in [8]. 5 Results We started our experiments by applying the kernel density based classifier to the USPS task. As the solution of the bilinear equation system (5) is very time consuming, the USPS images were scaled down to a size of 8 Theta 8 pixels. Experiments were done using the following ....

D. Keysers, J. Dahmen, H. Ney, "A Probabilistic View on Tangent Distance", Proceedings of the 22. Symposium of the German Association for Pattern Recognition (DAGM), Kiel, Germany, September 2000, this volume.

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