Mon, 20 July, 2009, 17:15-18:15, Foyer
The Diffusion Equation, used in most Diffuse Optical Tomography reconstruction algorithms, can be analytically modeled as a linear system under the Born or Rytov approximation. The problem of imaging biological tissue is often a severely ill-posed problem due to the finite number of linearly independent measurements, noise, and often complicated structure of the heterogeneities in the tissue. Here, we present a new multistage inversion technique that improves the quality of imaging.
The algorithm consists of three steps as below:
- Reduction of the domain using signal subspace technique: Considering the linear model as a transformation between the currents on the photon density sources and the measured scattered photon densities on the detectors, it can be said that the vectors containing the measured photon densities corresponding to each source lie in the span of the Green functions vectors corresponding to the radiation from the sources induced at the heterogeneous voxels. We then find and reject the voxels whose Green functions vectors are not in the range of the transformation matrix (meaning that they definitely do not contribute to the measured photon densities).
- Traditional Inversion: Here, considering the map between the differential absorption coefficient (with reference to the homogeneous background absorption coefficient) at the voxels not rejected in step 1 and the complete set of measurements, we perform the traditional truncated SVD based pseudoinverse to obtain an estimate of the differential absorption coefficient at the voxels not rejected.
- Rejection of the voxels based on the estimate from stage 2: Since the truncated SVD based pseudoinverse provides the minimum norm solution out of the infinite physically possible solutions, it assigns a negative value at the homogeneous voxels (zero differential absorption coefficient) and a positive but lesser than actual value at the heterogeneous voxels. Based on this fact, we exclude the voxels for which the assigned value is negative and repeat the steps 2 and 3 iteratively till all the voxels used in inversion have positive retrieved values.
In essence, instead of solving the inverse problem for the entire region, we have reduced the number of unknowns in the stage 1 and then iteratively in the stage 3. This has given us two advantages: 1) reduction in the computational intensity of the problem; 2) reduction in the ill-posedness of the overall inversion problem. The concept used in stage 3 enables the imaging of complicated geometries, such as an annulus, which are conventionally difficult to image with the demonstrated precision.
Presentation slides (pdf, 359 KB)
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