Non-stationary subdivision scheme with a shape parameter and application to medical imaging

Résumé

In this thesis, we study and construct non-stationary subdivision schemes (uniform or non- uniform), using a combination of trigonometric and hyperbolic functions with tension parameters. These new subdivision schemes have the ability to generate more flexible curves and surfaces. In the first step, we recall the various mathematical techniques required to understand the non-stationary subdivision schemes studied in this thesis. Following that, we propose two new uni-variate subdivision schemes using a mixture of trigonometric and hyperbolic functions. In addition, we examine the convergence of these two schemes, as well as their regularity, in both theoretical and practical context. The second step aims to extend the previous chapter schemes to the surface case. Specifically, we suggest subdivision rules for meshes with arbitrary topology. We then establish the convergence and regularity of these schemes based on analytical and algebraic tools. Then, we propose several algorithms to numerically reconstruct surfaces from medical images using the proposed rules. In the third step, we are interested in the construction of two new approaches for reverse subdivision. The first approach is based on direct computation and the second one consist in solving an optimization problem. We also present numerical tests that show the efficiency of the proposed schemes.

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