. In a previous paper, the author proposes to see the deformations of a common pattern as the action of an infinite dimensional group. We show in this paper that this approach can be applied numerically for pattern matching in image analysis of digital images. Using Lie group ideas, we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations. Then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub-optimal pattern matching. Contents 1. Introduction 1 2. Algorithmic side: gradient descent on AB 5 3. Numerical scheme 7 4. Numerical results 8 5. Conclusion 15 References 15 1. Introduction In [6, 5], we proposed an infinite dimensional group approach for physics based models in pattern recognition. Let us recall the outlines of this approach in the particular case of image analysis we are interested in. Consider that the set of gray The author would like to thank professor Robert...

. Non rigid deformations of patterns can be interpreted as the action of an infinite dimensional group A(n) on a given set P of patterns. Following Lie group ideas, a small deformation can be well described by an element y of the tangent space at identity T e A(n). Given a metric n on T e A(n), which brings the cost of a small deformation, we show that we can define on A(n) a left invariant distance d A(n) which gives the distance between two arbitrary large deformations. This allows to reformulate in a unified framework many pattern recognition tasks. Finally, we propose a sub-optimal algorithm to solve three important classes of pattern recognition problems through a gradient algorithm on A(n) whose convergence is rigorously established. Contents 1. Introduction 2 2. The abstract construction of A(n) 7 2.1. Control lemmas on A 2.2. Definition of A(n) 3. A weak differentiable structure on A(n) 16 3.1. Tangent spaces 3.2. Exponential 3.3. Differentiable applications 3.4. Important...