Blind deconvolution estimation by multi-exponential models and alternated least squares approximations: Free-form and sparse approach

The deconvolution process is a key step for quantitative evaluation of fluorescence lifetime imaging microscopy (FLIM) samples. By this process, the fluorescence impulse responses (FluoIRs) of the sample are decoupled from the instrument response (InstR). In blind deconvolution estimation (BDE), the FluoIRs and InstR are jointly extracted from a dataset with minimal a priori information. In this work, two BDE algorithms are introduced based on linear combinations of multi-exponential functions to model each FluoIR in the sample. For both schemes, the InstR is assumed with a free-form and a sparse structure. The local perspective of the BDE methodology assumes that the characteristic parameters of the exponential functions (time constants and scaling coefficients) are estimated based on a single spatial point of the dataset. On the other hand, the same exponential functions are used in the whole dataset in the global perspective, and just the scaling coefficients are updated for each spatial point. A least squares formulation is considered for both BDE algorithms. To overcome the nonlinear interaction in the decision variables, an alternating least squares (ALS) methodology iteratively solves both estimation problems based on non-negative and constrained optimizations. The validation stage considered first synthetic datasets at different noise types and levels, and a comparison with the standard deconvolution techniques with a multi-exponential model for FLIM measurements, as well as, with two BDE methodologies in the state of the art: Laguerre basis, and exponentials library. For the experimental evaluation, fluorescent dyes and oral tissue samples were considered. Our results show that local and global perspectives are consistent with the standard deconvolution techniques, and they reached the fastest convergence responses among the BDE algorithms with the best compromise in FluoIRs and InstR estimation errors.

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Additional Editor Comments:
I agree with the reviewer's comments. Please make sure to address them in the revised version.

RESPONSE
We sincerely thank the Editor for the evaluation of our manuscript. In the following, we have included a precise response to each reviewer comment.

Reviewer #1: Comments to the Author
This work proposed a new method for solving blind deconvolution estimation (BDE) problem in fluorescence imaging microscopy, based on multi-exponential models and alternating least squares. Blind deconvolution is a well studied problem recently, and the idea of alternating minimization idea is not quite new. The reviewer does not find much technical novelty, but an application to a new problem. The work is well-organized overall, while the presentation can be improved. Below are some comments: 1. The abbreviations for fluorescence impulse response (FIR) and instrument response (IR) might be a bit confusing for readers with signal processing background. In signal processing, FIR and IIR are typically for finite impulse response and infinite impulse response, respectively. The authors should consider how to differ from the traditional abbreviations, otherwise readers might got confused with the meaning of FIR and IR at the first glance.

RESPONSE
We acknowledge this important remark by the reviewer, so we have updated the abbreviations for fluorescence impulse response and instrument response to FluoIR and InstR, respectively to avoid a possible confusion to the reader. For consistency, the abbreviation of fluorescence decay was also updated to FluoD in the whole manuscript (see page 20). In particular, the reviewer finds the proposed method is quite close to those of [1,2], though [1,2] considered a different application in Calcium imaging. In [1,2], the authors assumed some sparsity structures on FIR, while the proposed method does not. Some more comments and detailed comparison discussions are needed. In particular, the method might be improved by imposing sparsity structure on FIRs similarly, or other structures.

RESPONSE
We thank deeply the reviewer for bringing to our attention these three relevant works [1,2,3] related to blind deconvolution estimation (BDE). We reviewed carefully the three previous contributions and we detected key differences with our BDE formulation. Contrary to [1,2], in our approach, the excitation signal does not have a spike train shape, and our observation model relies on a convolution with a multi-exponential kernel, compared to the autoregressive structure in the references. Furthermore, the main goal in our work is to reconstruct the fluorescence impulse response in each pixel of the FLIM measurement, nor to identify the spike train of the excitation signal, as in [1,2]. In addition, our formulation considers a free-form for the excitation signal, which is common to all pixels in the FLIM image, but with some sparsity condition as well. Hence, for FLIM datasets, the excitation signal is one narrow pulse without repetitions. Meanwhile, the fluorescence impulse responses have a smooth monotonic exponential decay over the whole measurement interval, so we cannot assume for them a sparsity condition, as suggested by the reviewer. With respect to [3], our formulation does not assume a short kernel and a sparse activation map. In addition, our multi-exponential kernel is not restricted to a sphere constraint, as in [3]. Also, the extension in [3] to convolutional dictionary learning considers multiple unknown kernels/motifs, i.e., multiple input signals, which is not consistent with the studied BDE formulation. Nonetheless, references [1,2,3] ([33], [34] and [35] in our manuscript) are described in the introduction and the main differences to our formulation are also discussed there (see page 2).
3. Since the problem is highly nonconvex, can the authors provide some convergence guarantees of the proposed optimization method? Some numerical and even theoretical justification is needed. Additionally, for convolution, the implementation could be faster via FFTs.

RESPONSE
We acknowledge the detailed and precise evaluation of our work by the reviewer. Since in our formulation for FLIM datasets, we assume a narrow pulse without repetitions as excitation, and each measured fluorescence decay will exhibit a sharp increase to its peak value, followed by a monotonic decrease. Moreover, all the measurements are scaled to sum-to-one, and also the instrument response is limited to sum-to-one. As a result, the scaled shift symmetry described in [3] will not hold in our BDE formulation. In addition, at each stage of the alternated least squares (ALS) scheme in the local and global approaches, a quadratic approximation problem is solved by either a non-linear least squares or linear least squares. So, at each stage, the estimation error is reduced or at least maintained. Consequently, convergence is guaranteed in the iterative scheme, but only to a local minimum. For this reason, the initialization based on processing the FLIM dataset is a crucial step in our formulation to obtain meaningful results. We include this discussion in third section, while describing the alternated least squares structure (see page 7).
In this new version, we have added a numerical convergence evaluation of the BDE schemes (BDELME and BDEGME) with synthetic FLIM datasets for different noise levels (SNR,PSNR) ∈ {(10,40), (15,45), (20,50), (25,55)} dB, and orders in the multi-exponential model of the synthetic impulse response (2 nd , 3 rd and 4 th ). The stopping threshold is set to ! = 1 × 10 "! . The resulting normalized metric Υ # in the alternated least squares iterations is described below in Fig. 1, where it is observed that in either BDE scheme, the convergence is always monotonic for any noise combination and model order, and just its final steady-state value depends on the noise level. From Fig. 1, we conclude that the local approach (BDELME) converges slower than the global scheme (BDEGME), which could be intuitively expected, since BDELME has roughly twice the free parameters to optimize. For BDELME, the most drastic reductions are achieved in the first five iterations, and for BDEGME, in the first two. In this way, our previous convergence analysis is validated numerically. This previous analysis and discussion is included in page 12. Fig. 1 Normalized metric (Υ ! ) vs iteration (t) in the alternated least squares scheme for BDELME and BDEGME by considering synthetic FLIM datasets at different noise levels (SNR,PSNR) ∈ {(10,40), (15,45), (20,50), (25,55)} dB, and impulse response orders (2 nd , 3 rd and 4 th ). Now, with respect to the suggestion of implementing the convolution via FFTs, we have chosen to use a time-domain perspective to allow a direct optimization by nonlinear and linear least squares of the parameters in the impulse and instrument responses, nor for the computation of the estimated fluorescence decays. Also, our formulation considers a linear convolution for the observation model. Nonetheless, we thank the reviewer for this interesting suggestion. [1-3], or other generic methods are needed.