Model reduction using wavelet multiresolution technique applied to fluorescence diffuse optical tomography.

Fluorescence diffuse optical tomography is a powerful tool for the investigation of molecular events in studies for new therapeutic developments. Here, the stress is put on the mathematical problem of tomography, which can be formulated in terms of an estimation of physical parameters appearing as a set of partial differential equations and solved by the finite element method. This method is well known to be time consuming, and our principal objective is to reduce the model in order to speed up computation. A method based on a wavelet multiresolution technique is presented in detail. A validation study was conducted on synthetic data and experiments.

Reconstruction method and optimal design of an interferometric spectrometer.

In this paper, we present a method to estimate the power spectral distribution of a source from input data acquired by an interferometric-based spectrometer. Our spectrometer shows distortions in the fringe pattern and a lack of data, making it impossible to apply the Fourier transform approach, which is the gold standard as a spectral recovery method for interferometric spectrometers. We combined linear inverse problem solving and iterative methods instead, considering that each detector of the spectrometer has a specific and known spectral response. Iterative methods are used to overcome problems caused by lack of input data. We show that a good spectral estimation of relatively simple spectra having a resolution of 400 points is typically achieved using fewer than 10 detectors with such a method. Since the quality of spectral restitution with such an approach relies both on signal processing and an optimal selection of the detector's spectral responses, the paper also shows that some sets of spectral responses selected for the detection and consequently a spectral repartition of the detectors are more successful than others in the spectral recovery process. We chose the condition number of the inversion matrix as an optimization criterion and evaluated how this criterion can be used within this framework. We found that maximizing it achieves better spectral restitution, within a range where noise remains low.

Application of a wavelet-Galerkin method to the forward problem resolution in fluorescence diffuse optical tomography.

Fluorescence diffuse optical tomography is a powerful tool for the investigation of molecular events in studies for new therapeutic developments. Here, the emphasis is put on the mathematical problem of tomography, which can be formulated in terms of an estimation of physical parameters appearing as a set of Partial Differential Equations (PDEs). The standard polynomial Finite Element Method (FEM) is a method of choice to solve the diffusion equation because it has no restriction in terms of neither the geometry nor the homogeneity of the system, but it is time consuming. In order to speed up computation time, this paper proposes an alternative numerical model, describing the diffusion operator in orthonormal basis of compactly supported wavelets. The discretization of the PDEs yields to matrices which are easily computed from derivative wavelet product integrals. Due to the shape of the wavelet basis, the studied domain is included in a regular fictitious domain. A validation study and a comparison with the standard FEM are conducted on synthetic data.