An Eigenfunction Approach to Conversion of the Laplace Transform of Point Masses on the Real Line to the Fourier Domain

TL;DR

The paper addresses estimating parameters of a multiexponential signal $F_0(t)=\sum_{k=1}^K A_k e^{-t\\lambda_k}$ from samples, a notoriously ill-posed inverse Laplace problem. It reframes the problem in the Fourier domain by exploiting Hermite functions, which are eigenfunctions of the Fourier transform, via a weight-adjusted transform $f(t)=F(t)e^{-t^2/2}$ and a corresponding measure $\\tilde{\\mu}$ to locate the exponents $\\lambda_k$ from $\\mathfrak{F}(\\tilde{\\mu})(\\omega)$. The authors develop a discretization and least-squares framework based on Gauss-Hermite quadrature, reproduce kernel representations, and prove error and noise bounds, applying the method to preliminary numerical experiments on biexponential signals with noise. While the approach does not eliminate ill-posedness, it provides a practical route to parameter extraction with strong performance at high SNR and offers insights for MRI applications such as myelin water fraction mapping.

Abstract

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.

Figures (3)

• Figure 1: The real part of the Fourier transform of the modified biexponential signal \ref{['eq:fundarelation']}and its approximation through expansion with Hermite polynomials \ref{['eq:leastsqexpression']}. The same signal parameters detailed in Figure \ref{['fig:modgauss']} are used. The fidelity of this approximation to the FT of the signal supports our attempt to extract a $\lambda$ value. However, the error in our approximation increases rapidly outside of a relatively small frequency window (right-hand panel).
• Figure 2: Approximation of a modified biexponential \ref{['eq:completesq']} decay curve with parameters: $A_1=0.5$, $A_2=0.5$, $T_{21}=10$ms, $T_{22}=50$ms. The error for this approximation, shown in the right panel, remains small throughout the signal duration.
• Figure 3: Illustration of $T_{22}$ estimation.The same signal parameters detailed in Figure \ref{['fig:modgauss']} are used. The vertical red line shows the theoretical value of $\lambda_1$ in terms of $x$ (see Section A) while the red circle shows the location of the point of maximum value (the estimated value of $\lambda$). For these parameters, a $T_{22}$ value of approximately 49.03 was found, close to the correct underlying value of 50.

Theorems & Definitions (19)

• Proposition 3.1
• Definition 5.1
• Example 5.1
• Example 5.2
• Example 5.3
• Remark 6.1
• Remark 6.2
• Theorem 7.1
• Theorem 7.2
• Theorem 7.3