By Sami Romdhani, Volker Blanz, Thomas Vetter (auth.), Anders Heyden, Gunnar Sparr, Mads Nielsen, Peter Johansen (eds.)
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Extra info for Computer Vision — ECCV 2002: 7th European Conference on Computer Vision Copenhagen, Denmark, May 28–31, 2002 Proceedings, Part IV
2) 22 A. Barla, F. Odone, and A. Verri As in the case of SVMs for binary classiﬁcation, the objective function is quadratic, the Hessian positive semi-deﬁnite, the inequality constraints are box constraints. The two main diﬀerences are the form of the linear term in the objective function and the equality constraint (the Lagrange Multipliers sum to 1 instead than 0). Like in the case of SVMs, the training points for which αi > 0 are the support vectors for this learning problem. The sphere center x0 is found as a weighted sum of the examples as x0 = αi xi while the radius R can be determined from the Kuhn-Tucker condition associated to any training point xi for which 0 < αi < C as R2 = (xi − x0 )2 .
Hence, δβ is obtained by solving a simple system of linear equations. Similarly, α, pi and pr are updated separately while maintaining the other parameters constant. This process is repeated iteratively until global convergence is obtained. The recovery of each of the parameters is detailed in the next two sections. It should be noted that each parameter update is biased because the other parameters are kept constant. 3, is used. Updating the shape parameters α from an image difference δt is difficult due to the combined influence of: (i) the shape error (ii) the normals which depends non-linearly on the shape and (iii) the texture error (see Equation 9 and 11).
Then, we determine the conditions under which this measure deﬁnes a legitimate kernel function. 1 23 Hausdorﬀ Distances Given two ﬁnite point sets A and B (both subsets of IRN ), the directed Hausdorﬀ distance h, can be written as h(A, B) = max min ||a − b||. a∈A b∈B Clearly the directed Hausdorﬀ distance is not symmetric and thus not a “true” distance, but it is very useful to measure the degree of mismatch of one set with respect to another. To obtain a distance in the mathematical sense, symmetry can be restored by taking the maximum between h(A, B) and h(B, A).
Computer Vision — ECCV 2002: 7th European Conference on Computer Vision Copenhagen, Denmark, May 28–31, 2002 Proceedings, Part IV by Sami Romdhani, Volker Blanz, Thomas Vetter (auth.), Anders Heyden, Gunnar Sparr, Mads Nielsen, Peter Johansen (eds.)