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0. Laguerre Deconvolution Background
In the context of time-domain fluorescence techniques, the measured fluorescence response from biological tissue to a light excitation pulse stimulus is a convolution of the fluorescence impulse response function (fIRF) representing the intrinsic tissue fluorescence with the instrument impulse response function (iIRF) representing the distorted light pulse due to the instrument electronics and other delay components.
The intrinsic fIRF can be expanded onto an ordered basis set of orthonormal functions, namely Laguerre basis functions (LBFs), as illustrated below (The first 8 Laguerre basis functions are plotted). A set of finite-dimension Laguerre basis functions are completely specified by two basis parameters: α and dimension L (i.e., the number of LBFs within the ordered basis set such that the highest order of LBFs is L − 1). The l th order Laguerre functions with independent variable k is defined as:
for l = 0,...,L-1 and 0< α <1 is the discrete-time Laguerre parameter that determines the rate of exponential asymptotic decline of these functions.
In general, a high-dimensional Laguerre function set (i.e., a large number of Laguerre basis functions) is preferred for the identification of a system with a complex fluorescence signal where a broad range of decays is present; the higher the number of LBFs, the larger the degrees of freedom (DOF) available to capture the characteristics of a system response with complex decay dynamics. It was also observed that by incorporating a large number of LBFs, the least-squares deconvolution (LSD) becomes less sensitive to the choice of the scale parameters. These facts suggest that it is possible to use one Laguerre basis set with pre-specified α and L values for deconvolving fluorescence signals with a broad range of decay dynamics when many LBFs are used.
As stated above, the value of α determines the rate of exponential asymptotic decline of the LBFs. The smaller the α value, the faster the LBFs decays, which is illustrated below. As a result, for a given data truncation length, the choice of the value of α will determine the range of fluorescence lifetimes the specific set of LBFs can accurately fit. For how to choose the best α value, please refer to the section ['choosing Laguerre parameters'] (https://github.com/marculab/FLImBrushDataProcessingTool/wiki/2.-Choosing-Laguerre-parameters)
The practical use of Laguerre expansion of fIRF requires the selection of the Laguerre basis set (α and L). However, the optimal choice of the basis parameters is not trivial, since the choices of scale and dimension parameters are coupled. In principle, if the fIRF is expanded on an infinite number of LBFs (L -> ∞) over infinite time intervals, the choice of scale parameter is arbitrary (0 < α < 1). In practice, only a finite number of expansion terms can be used, and the measured time series are always truncated for numerical analysis. Thus, the appropriate selection of the Laguerre basis set is critical for the practical use of the Laguerre expansion of fIRFs.