9 research outputs found
Mathematical Modeling for 2D Light-Sheet Fluorescence Microscopy image reconstruction
We study an inverse problem for Light Sheet Fluorescence Microscopy (LSFM),
where the density of fluorescent molecules needs to be reconstructed. Our first
step is to present a mathematical model to describe the measurements obtained
by an optic camera during an LSFM experiment. Two meaningful stages are
considered: excitation and fluorescence. We propose a paraxial model to
describe the excitation process which is directly related with the Fermi
pencil-beam equation. For the fluorescence stage, we use the transport equation
to describe the transport of photons towards the detection camera. For the
mathematical inverse problem that we obtain after the modeling, we present a
uniqueness result, recasting the problem as the recovery of the initial
condition for the heat equation in from
measurements in a space-time curve. Additionally, we present numerical
experiments to recover the density of the fluorescent molecules by discretizing
the proposed model and facing this problem as the solution of a large and
sparse linear system. Some iterative and regularized methods are used to
achieve this objective. The results show that solving the inverse problem
achieves better reconstructions than the direct acquisition method that is
currently used
Lipschitz stability for Backward Heat Equation with application to Fluorescence Microscopy
In this work, we study a Lipschitz stability result in the reconstruction of
a compactly supported initial temperature for the heat equation in
, from measurements along a positive time interval and over an
open set containing its support. We take advantage of the explicit dependency
of solutions to the heat equation with respect to the initial condition. By
means of Carleman estimates we obtain an analogous result for the case when the
observation is made along an exterior region , such that
the unobserved part is bounded. In the latter
setting, the method of Carleman estimates gives a general conditional
logarithmic stability result when initial temperatures belong to a certain
admissible set, and without the assumption of compactness of support.
Furthermore, we apply these results to deduce a similar result for the heat
equation in for measurements available on a curve contained in
, from where a stability estimate for an inverse
problem arising in 2D Fluorescence Microscopy is deduced as well. In order to
further understand this Lipschitz stability, in particular, the magnitude of
its stability constant with respect to the noise level of the measurements, a
numerical reconstruction is presented based on the construction of a linear
system for the inverse problem in Fluorescence Microscopy. We investigate the
stability constant with the condition number of the corresponding matrix
PET Reconstruction with non-Negativity Constraint in Projection Space: Optimization Through Hypo-Convergence
International audienceStandard positron emission tomography (PET) reconstruction techniques are based on maximum-likelihood (ML) optimization methods, such as the maximum-likelihood expectation-maximization (MLEM) algorithm and its variations. Most of these methodologies rely on a positivity constraint on the activity distribution image. Although this constraint is meaningful from a physical point of view, it can be a source of bias for low-count/high-background PET, which can compromise accurate quantification. Existing methods that allow for negative values in the estimated image usually utilize a modified log-likelihood, and therefore break the data statistics. In this work we propose to incorporate the positivity constraint on the projections only, by approximating the (penalized) log-likelihood function by an adequate sequence of objective functions that are easily maximized without constraint. This sequence is constructed such that there is hypo-convergence (a type of convergence that allows the convergence of the maximizers under some conditions) to the original log-likelihood, hence allowing us to achieve maximization with positivity constraint on the projections using simple settings. A complete proof of convergence under weak assumptions is given. We provide results of experiments on simulated data where we compare our methodology with the alternative direction method of multipliers (ADMM) method, showing that our algorithm converges to a maximizer which stays in the desired feasibility set, with faster convergence than ADMM. We also show that this approach reduces the bias, as compared with MLEM images, in necrotic tumors-which are characterized by cold regions surrounded by hot structures-while reconstructing similar activity values in hot regions