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Description
The expectation maximization (EM) algorithm, widely used in medical image tomographic reconstruction, iteratively estimates the values of a discrete distribution of emission sources $\rho_{t=1,2,..,T}$ from a set of detector measurements $A_{k=1,2,..,K}$. The two steps of an iteration are:
$A_k^{(n)} = \sum_{t} P_{k \leftarrow t} \cdot \rho_t^{(n)}$ (E) and $\rho_t^{(n+1)} = \rho_t^{(n)} \sum_k \left[ \frac{A_k}{A_k^{(n)}} P_{k \leftarrow t} \right]$ (M)
where superscript indicates estimated values at step n, and $P_{k \leftarrow t}$ is the probability that an emission from source t is detected by detector k.
Using analytical tools, it has been shown that the algorithm progresses to a fixed point, but further description of the ideal algorithm behavior is incomplete.
We propose to view measurements $A_k$ as defining hyperplanes in the finite-dimensional linear space defined by the emission source values ($\rho_{t=1,2,..,T}$) (equation E). In imaging the system is overdetermined (more measurements than emission sources), thus, when the attached linear system is consistent, these hyperplanes have a single common point (the solution).
Because all values appearing in equations (E) and (M) are positive, we argue adjustments to $\rho_t$ in step (M) induced by measurements k with $\frac{A_k}{A_k^{(n)}} > 1$ cannot cancel adjustments induced by measurements k with $\frac{A_k}{A_k^{(n)}}<1$. We show the two convex cones defined by the direction vectors to the $\frac{A_k}{A_k^{(n)}} > 1$ hyperplanes and the direction vectors to the $\frac{A_k}{A_k^{(n)}}<1$ hyperplanes, respectively, with the vertex in the solution point, have only that point as intersection. Thus, the corresponding adjustments cannot cancel each other and, for a consistent overdetermined problem, the only fixed point for the EM algorithm is the solution.
The linear geometry view of the EM algorithm promises a path for better understanding of this important algorithm.