Abstract
In this chapter we apply non-commutative Fourier analysis in order to build numerically efficient algorithms for heat diffusion on groups and its application to image reconstruction. The first section of the chapter is devoted to recalling definition and some basic properties of hypoelliptic diffusions on Lie groups. These constructions are then extended to the semi-discrete semi-direct products that are the focus of the monograph. In particular, we present an efficient approach for the heat diffusion on lifted functions and on almost-periodic functions. These results are then particularized to the case of SE(2, N), and an associated image reconstruction algorithm is presented.
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Notes
- 1.
The Mathieu equation is
$$\begin{aligned} \partial _x^2 f(x) + (a - 2q\cos (2x))f(x) = 0, \qquad a, q\in {\mathbb R}. \end{aligned}$$(6.12)For fixed \(q\in {\mathbb R}\), there exist two ordered discrete sets of characteristic values, \(\{a_n\}_{n\in {\mathbb N}}\) and \(\{b_n\}_{n\in {\mathbb N}}\), such that the Mathieu equation with \(a=a_n\) (resp. \(a=b_n\)) admits a unique even (resp. odd) \(2\pi \)-periodic solution with \(L^2\) norm equal to 1, the Mathieu cosine \(\mathrm{ce}_n(x, q)\) (resp. the Mathieu sine \(\mathrm{se}_n(x, q)\).
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Prandi, D., Gauthier, JP. (2018). Image Reconstruction. In: A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-78482-3_6
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DOI: https://doi.org/10.1007/978-3-319-78482-3_6
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