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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.04348 |
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Table of Contents:
- Motivated by applications in magnetic resonance relaxometry, we consider the following problem: Given samples of a function $t\mapsto \sum_{k=1}^K A_k\exp(-tλ_k)$, where $K\ge 2$ is an integer, $A_k\in\mathbb{R}$, $λ_k>0$ for $k=1,\cdots, K$, determine $K$, $A_k$'s and $λ_k$'s. Unlike the case in which the $λ_k$'s are purely imaginary, this problem is notoriously ill-posed. Our goal is to show that this problem can be transformed into an equivalent one in which the $λ_k$'s are replaced by $iλ_k$. We show that this may be accomplished by approximation in terms of Hermite functions, and using the fact that these functions are eigenfunctions of the Fourier transform. We present a preliminary numerical exploration of parameter extraction from this formalism, including the effect of noise. We do not claim to have eliminated the inherent ill-posedness of the original problem, as reflected in the numerical results.