In computed tomography, sparse reconstruction can be directly applied to Radon
projections from samples with piecewiseconstant brightness values, since they
have a sparse gradient. Therefore it is appropriate to minimize total variation
of the reconstructed slice which corresponds to the L1norm of the gradient.
The
constraint in this case requires that measured Radon projections agree with
reprojections of the reconstructed slice. In addition, total variation is
nondifferentiable and has to be replaced by a second, separable, augmented
Lagrangian for an auxiliary variable, wherefor variable splitting needs to be
done. The talk will also explain a singlestep minimization of the second,
separable, augmented Lagrangian using shrinkage, aka soft thresholding.
Reconstructed tomography slices obtained by filtered backprojection (FBP) will
be compared with those based on the compressed sensing concept.
In the more common case where tomography specimens are not granted to have
piecewiseconstant brightness values, the equality constraint can be relaxed by
minimizing a weighted sum of the total variation and a quadratic data fidelity
term instead. While a sparse reconstruction is not required in this case, it
will be shown that TV regularization results in removal of artifacts found in
standard reconstruction by FBP.
