Current medical imaging modalities like Computed Tomography (CT), Magnetic Resonance Imaging (MRI) or Positron Emission Tomography (PET) has allowed minimally-invasive imaging of internal organs. Rapid advancement in these technologies have lead to an influx of data, which, along with rising clinical need, has lead towards a need for quantitative image interpretation in routine practice. Some of the applications include volumetric measurements of regions of the brain, surgery or radiotherapy planning using CT/MRI images.

This influx of data have opened up avenues for using multi-modality images for making decisions. However this is not always straightforward as the imaging parameters, scanner types, patient movement, or anatomical movement make the images miss-aligned against each other, making direct comparison between, say, as CT and MRI image tricky. This is formally known as the problem of image registration, and to this end, numerous computational methods have been proposed, which this paper surveys. 

Out of the methods proposed for both inter- and intra-patient modality registration, Mutual Information based objective maximization strategy has been extremely successful at computing the registration between 3D multi-modal medical images of various organs from the images. Mutual Information (MI) stems from the field of information theory, pioneered by Shannon, which when applied in the context of medical image registration, postulates that the MI between two images (say CT and MRI) is maximum when the images are aligned. 

The basic formulation of a MI based registration algorithm is as follows: Let $\mathcal{A}$ and $\mathcal{B}$ be two images which are geometrically related according to a transformation $T_\alpha$, such that voxels $p$ in $\mathcal{A}$ with intensity $a$ physically correspond to voxels $T_\alpha(p)$ in $\mathcal{B}$ with intensity $b$. The relationship between $p_{AB}(a,b)$ between $a$ and $b$, and hence their MI depends on $T_\alpha$. The MI criterian postulates that the MI for images that are geometrically aligned is maximum:

\begin{equation}
    \alpha^* = argmax_{\alpha} I(A,B)
\end{equation}

Where $A$ and $B$ are two discrete random variables, and $I(A,B) = \sum_{a,b}p_{AB}(a,b) log \frac{p_{AB}(a,b)}{p_A(a).p_B(b)}$. 

An optimization algorithm is utilized to find a parameter set $\alpha^*$ that maximizes the MI between $A$ and $B$. Classically, Powell's multidimensional direction set method is used to otipimize the objective function, but other methods do exist as well. 

Although the formulation of MI criterion suggests that spatial dependence of image intensities are not taken into account, is in fact essential for the criterion to be well-behaved around the registration solution. MI does not rely on pure intensity values to measure correspondence between images, but rather on their joint distribution and the relationship of occurrence. It also does not impose any modality specific constraints which makes it general enough to be applied to any problem formulation (inter- or intra modality). 

Some of the areas where MI based image registration may fail is when there is insufficient information in images due to low resolution, low number of images, images not spatially invariant, images with shading artifacts. In some of these cases, MI based criterion will have multiple local optimals.