Motion phenomena and correction in SPECT
One of the big problems during acquisiton of MPI SPECT is the motion phenomena, which can result in acquisition readback, misdiagnosis or the evaluation of invasive procedures on patients.
Motion effect on frames
Effect of motion on the projection frames
Motion effect on the reconstructed volume
After reconstructing \(u\) with the adjugate forward (backward) operator \(F^{*}\), one arrives at a volume with incorrect cardiac activities and shape variations.
# Running reconstruction and all kinds of magicAlgorithms to detect and correct motion
1. Optical flow
Classical approach This methodology is based on the assumption of brightness constancy , which means that the intensity of a point between consecutive frames remains the same. This can be formalized as follows \[I(x, y, t) - I(x + u, y + v, t + 1) = 0, \] where \(w = (u, v)\) is the displacement vector between the frames. This leads to an underdetermined system of equations, needing further constraints to overcome the aperture problem.
Regularized models There can be various techniques to regularize the aperture problem, where most naturally the smoothness constraint can be posed, where one can optimize the following functional \[E = \iint_{\Omega} \Psi(I(x + u, y + v, t + 1) - I(x, y, t)) + \alpha \Psi(|\nabla u|) + \alpha \Psi(|\nabla v|) dx dy,\] where \(\Omega\) is the extent of the entire image \(I(x, y, t)\), \(\nabla\) is the nabla (gradient) operator, \(\alpha\) is a constant and \(\Psi(.)\) is a loss function. One approach to solve this optimization problem to use the calculus of variations to compute the first variation and as a result get a necessary condition for the extremum. The BCC can be approximated by the Taylor series expansion, more specifically \[\frac{\delta I}{\delta x}u + \frac{\delta I}{\delta y}v + \frac{\delta I}{\delta t} = 0\], wherewith the optimization problem can be rewritten as \[E = \iint_{\Omega} \Psi(I_{x}u + I_{y}v + I_{t}) + \alpha \Psi(|\nabla u|) + \alpha \Psi(|\nabla v|) dx dy.\]
Parametric models Instead of posing regularity constraints point-by-point on the pixels of the projection frames, one can group pixels into regions and estimate the motion of the regions themselves. This way one assumes that the motion can be estimated by a set of parameters, therefore the model looks as follows \[\hat{\alpha}= \underset{\alpha}{\operatorname{argmin}} \sum_{(x,y)\in R} g(x, y) \rho(x, y, I_{1}, I_{2}, u_{\alpha}, v_{\alpha}),\] where \(\alpha\) is the set of parameters determining the motion of the region \(R()\), data cost term is \(\rho()\), \(g()\) is the weighting funtion that determines the influence of pixel \((x, y)\) on the total cost, where \(I_{1}\) and \(I_{2}\) are the consecutive frames.