Approximation by Group Invariant Spaces

BARBIERI, DAVIDE; CABRELLI, CARLOS A.; HERNÁNDEZ, EUGENIO; MOLTER, URSULA M.

Bordeaux

Congreso; SamPTA 2019 Sampling Theory and Applications; 2019

SampTA

In this talk we will look at approximation properties of spaces invariant under the action of a crystal group. We show how to characterize these spaces by a property of the range function. Using this fact and the results for shift invariant spaces, we show how to solve the following problem:Let $\mathcal F := \{f_1, \dots, f_m\}$ (the data) be given vectors of a Hilbert space $\mathcal H$. Which is the crystal invariant subspace $S \subset \mathcal H$ of $k$ generators that minimizes the error to the data, in the sense that $$ \sum_{i=1}^m \|f_i - P_{S}(f_i)\|^2$$is minimal, where $P_{S}$ is the orthogonal projection onto $S$.This provides a rotational invariant model for images.