# MLEM for Compton Imaging Motivation: An image reconstruction technique used in [[Compton Imaging]] as image reconstruction can be considered a parameter estimation problem. ## What is MLEM in the context of Compton Imaging? See its own page [[Maximum Likelihood Expectation Maximization (MLEM)]]. MLEM is an algorithm that takes an estimated parameter and considers the [[Maximum Likelihood Estimation]] as a means of reconstructing an image. This iterative process is often preferred over [[Filtered Backprojection (FBP)]] algorithms as it is able to reduce noise without sacrificing image resolution [^1]. It is generally considered an asymptotically unbiased estimator. [^1]:https://tech.snmjournals.org/content/46/2/129 In the context of photon imaging, the gamma-ray source distribution is regarded as the parameter to be estimated: $\hat{\bf{f}} = \text{argmax}_{\bf{f} \geq 0} Pr[g|\bf{f}] $ where: - $\bf{g}$ = $\{ g_1 , g_2, ... , g_I\}$ is the measurement, and $g_i$ is the number of counts measured as event $i$ - $\bf{f}$ = $\{ f_1 , f_2, ... , f_J\}$ is the source distribution that we want to estimate. - Note that there needs to be a constraint that $\hat{\bf{f}}$ must be positive. We can also see this definition of the probability referred to as the "likelihood" of an event. In order to establish this relationship, there will be some system response, $\bf{T}$, in which an element $t_{ij}$ is the probability that a photon emitted from pixel $j$ creates a recorded event $i$. ![[Pasted image 20220715101800.png]] [^2] [^2]: https://cztlab.engin.umich.edu/wp-content/uploads/sites/187/2015/03/20060907-thesis.pdf With the appropriate statistics and using a linear model, the problem is simplified to a linear system of equations: $\bf{Tf} = \bf{g} $ And by solving this relationship, we're able to adequately reconstruct the image. The mechanics of that are notably involved, and are covered in the next section. ## See also: - [[The Need for Iterations When using MLEM for Compton Imaging for applications in Radiation Measurement]] - [[Filtered Backprojection (FBP)]] - [[Simple Backprojection (SBP)]]