Non linear modelling of scattered multivariate data and its application to shape change

Abstract : We are given a set of points in a space of high dimension. For instance, this set may represent many visual appearances of an object, a face or a hand. We address the problem of approximating this set by a manifold in order to have a compact representation of the object appearance. When the scattering of this set is approximately an ellipsoid, then the problem has a well-known solution given by Principal Components Analysis (PCA). However, in some situations like object displacement learning or face learning this linear technique may be ill-adapted and nonlinear approximation has to be introduced. The method we propose can be seen as a Non Linear PCA (NLPCA), the main difficulty being that the data are not ordered. We propose an index which favours the choice of axes preserving the neighborhood of the nearest neighbours. These axes determine an order for visiting all the points when smoothing. Finally a new criterion, called "generalization error", is introduced to determine the smoothing rate, that is the knot number of the spline fitting. Experimental results conclude this paper: the method is tested on artificial data and on two data bases used in visual learning.
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IEEE Transactions on Pattern Analysis and Machine Intelligence, Institute of Electrical and Electronics Engineers, 1999, 21 (5), pp.422-432. 〈10.1109/34.765654〉
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Bernard Chalmond, Stephane Girard. Non linear modelling of scattered multivariate data and its application to shape change. IEEE Transactions on Pattern Analysis and Machine Intelligence, Institute of Electrical and Electronics Engineers, 1999, 21 (5), pp.422-432. 〈10.1109/34.765654〉. 〈hal-00990742〉

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