In this paper, we present a wavelet-based approach to solve the reconstruction problem encountered in
diffuse optical tomography (DOT). The DOT reconstruction problem using model based iterative image
reconstruction (MoBIIR) procedure involves repeated implementation of the three steps: (i) solution to the
diffusion equation (DE) to generate the simulated data from the current optical properties, (ii) estimation of
the Jacobian for the current optical property values and (iii) inversion of the perturbation equation leading
to the update vectors for the optical properties. Consequently, there are three approaches to a wavelet
based solution to the DOT problem: (i) waveletization of the perturbation equation, (ii) application of
wavelets for computation of the Jacobian for use in the perturbation equation and (iii) the solution to the
DE in the wavelet domain. While the first of these approaches has been addressed earlier, the other two
have not been attempted to the best of our knowledge. In this work, we have attempted the second
approach, i.e., waveletization of the perturbation equation for each measurement, which requires computing
the Jacobian in the wavelet domain. Our results show that this method outperforms the earlier method.
With each measurement appropriately represented in wavelet domain, the localization and de-noising
property of wavelets are exploited. Our simulation results show that the mean-square-error at convergence
is not affected by the increase in noise in data (up to 4% additive Gaussian noise). In addition, the usual V-cycle
strategy of wavelets is attempted.
In this paper, we present a wavelet-based approach to solve the non-linear perturbation equation encountered in optical tomography. A particularly suitable data gathering geometry is used to gather a data set consisting of differential changes in intensity owing to the presence of the inhomogeneous regions. With this scheme, the unknown image, the data, as well as the weight matrix are all represented by wavelet expansions, thus yielding the representation of the original non-linear perturbation equation in the wavelet
domain. The advantage in use of the non-linear perturbation equation is that there is no need to recompute the derivatives during the entire reconstruction process. Once the derivatives are computed, they are transformed into the wavelet domain. The purpose of going to the wavelet domain, is that, it has an inherent localization and de-noising property. The use of approximation coefficients, without the detail coefficients, is ideally suited for diffuse optical tomographic reconstructions, as the diffusion equation removes most of the high frequency information and the reconstruction appears low-pass filtered. We demonstrate through numerical simulations, that through solving merely the approximation coefficients one can reconstruct an image which has the same information content as the reconstruction from a nonwaveletized procedure. In addition we demonstrate a better noise tolerance and much reduced computation
time for reconstructions from this approach.
Reconstructions in optical tomography involve obtaining the images of absorption and reduced scattering coefficients. The
integrated intensity data has greater sensitivity to absorption coefficient variations than scattering coefficient. However, the
sensitivity of intensity data to scattering coefficient is not zero. We considered an object with two inhomogeneities (one in
absorption and the other in scattering coefficient). The standard iterative reconstruction techniques produced results, which
were plagued by cross talk, i.e., the absorption coefficient reconstruction has a false positive corresponding to the location
of scattering inhomogeneity, and vice-versa. We present a method to remove cross talk in the reconstruction, by generating
a weight matrix and weighting the update vector during the iteration. The weight matrix is created by the following method:
we first perform a simple backprojection of the difference between the experimental and corresponding homogeneous
intensity data. The built up image has greater weightage towards absorption inhomogeneity than the scattering
inhomogeneity and its appropriate inverse is weighted towards the scattering inhomogeneity. These two weight matrices are
used as multiplication factors in the update vectors, normalized backprojected image of difference intensity for absorption
inhomogeneity and the inverse of the above for the scattering inhomogeneity, during the image reconstruction procedure.
We demonstrate through numerical simulations, that cross-talk is fully eliminated through this modified reconstruction
procedure.
Optical tomography (OT) recovers the cross-sectional distribution of optical parameters inside a highly scattering medium from information contained in measurements that are performed on the boundaries of the medium. The image reconstruction problem in OT can be considered as a large-scale optimization problem, in which an appropriately defined objective functional needs to be minimized. Most of earlier work is based on a forward model based iterative image reconstruction (MOBIIR) method. In this method, a Taylor series expansion of the forward propagation operator around the initial estimate, assumed to be close to the actual solution, is terminated at the first order term. The linearized perturbation equation is solved iteratively, re-estimating the first order term (or Jacobian) in each iteration, until a solution is reached. In this work we consider a nonlinear reconstruction problem, which has the second order term (Hessian) in addition to the first order. We show that in OT the Hessian is diagonally dominant and in this work an approximation involving the diagonal terms alone is used to formulate the nonlinear perturbation equation. This is solved using conjugate gradient search (CGS) without re-estimating either the Jacobian or the Hessian, resulting in reconstructions better than the original MOBIIR reconstruction. The computation time in this case is reduced by a factor of three.
We describe here a method for tomographic reconstruction of the optical properties of a diffuse scattering object. The data are the integrated intensity (for absorption coefficient μa) and the mean arrival time (for reduced scattering coefficient μs), obtained from the temporal point spread function (tpsf) measurements. In both the cases we used a nonlinear optimization method to minimize the mean-squared error between the experimental data and the computed data. For computing the forward data, a diffusion equation is used as the model for light propagation. The main contribution of this work is the generation of the approximate location of the inhomogeneity in the object and its approximate value through a simple backpropagation of the data, which are input as a priori known starting information to the iterative reconstruction algorithm. Two advantages follow: (i) with background optical properties assumed to be known, the number of unknowns reduces to those inside the known inhomogeneity, which reduces the dimension of the inversion problem; (ii) initial estimate of the μa or μs values helps to get a quicker convergence to the actual solution. It is shown that without the a priori information from the backprojection algorithm, the inversion either took a much longer time or failed to converge.
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