Section:
New Results
Sparse 3D reconstruction in fluorescence imaging
Participants :
Emmanuel Soubies, Laure Blanc-Féraud, Sébastien Schaub, Gilles Aubert.
Sparse reconstruction
Super-resolution microscopy techniques allow to overstep the
diffraction limit of conventional optics. Theses techniques are very
promising since they give access to the visualisation of finer
structures which is of fundamental importance in biology. In this work
we deal with Multiple-Angle Total Internal Reflection Microscopy
(MA-TIRFM) which allows reconstructing 3D sub-cellular structures of a
single layer of behind the glass coverslip with a high
axial resolution. The 3D volume reconstruction from a set of 2D
measurements is an ill-posed inverse problem and requires some
regularization. Our aim in this work is to propose a new
reconstruction method for sparse structures that is robust to Poisson
noise and background fluorescence. The sparse property of the solution
can be seen as a regularizer using the -norm. Let us denote
the unknown fluorophore density, then
the problem states as
where is defined from the likelihood function of the
observation given , is a weight parameter and
denotes the -norm (which counts the number of
nonzero components of ). In order to solve this
combinatorial problem, we propose a new algorithm based on a smoothed
-norm allowing minimizing the non-convex
energy (1 ). Following [20] , the
idea is to approach the -norm by a suitable continuous
function depending on a positive parameter and tending to the
-norm when the parameter tends to zero. Then the algorithm
solves a sequence of functionals which starts with a convex one (on a
large convex set) and introduce progressively the non-convexity of the
-norm (Graduated Non Convexity
approach). Figure 1 shows the accuracy of the method
on a simulated membrane.
Figure
1. From left to right: Simulated membrane, Microscope acquisition (numerical simulations) with two different incident angles. The two images on the right represent position errors (nm) in the axial direction of the reconstructed membrane obtained with different algorithms: Richardson-Lucy algorithm without regularization (left) and our algorithm with (right).
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Axial profile calibration
In order to turn on real sample reconstructions we need to perform a calibration of the TIRF microscope. Its principle is based on an evanescent wave with an exponential theoretical decay. However this decay is generally not a pure exponential in practice and we need to have a good knowledge about it. Then based on a phantom specimen of known geometry (bead) we are working on a method to estimate experimentally/numerically this decay profile and calibrate all parameters of the system.