Bayesian model of computational anatomy

Computational anatomy is a discipline within medical imaging focusing on the study of anatomical shape and form at the visible or gross anatomical scale of morphology. The field is broadly defined and includes foundations in anatomy, applied mathematics and pure mathematics, including medical imaging, neuroscience, physics, probability, and statistics. It focuses on the anatomical structures being imaged, rather than the medical imaging devices.

Main article: The Bayesian model of computational anatomy

Source-channel model showing the source of images the deformable template $I\doteq \phi \cdot I_{\mathrm {temp} }\in {\mathcal {I}}$ and channel output associated with MRI sensor $I^{D}\in {\mathcal {I}}^{\mathcal {D}}$

The central statistical model of Computational Anatomy in the context of medical imaging has been the source-channel model of Shannon theory; the source is the deformable template of images $I\in {\mathcal {I}}$, the channel outputs are the imaging sensors with observables $I^{D}\in {\mathcal {I}}^{\mathcal {D}}$ (see Figure). The importance of the source-channel model is that the variation in the anatomical configuration are modelled separated from the sensor variations of the Medical imagery. The Bayes theory dictates that the model is characterized by the prior on the source, $\pi _{\mathcal {I}}(\cdot )$ on $I\in {\mathcal {I}}$, and the conditional density on the observable $p(\cdot |I)\ {\text{on}}\ I^{D}\in {\mathcal {I}}^{\mathcal {D}}$ conditioned on $I\in {\mathcal {I}}$.

For image action $I(g)\doteq g\cdot I_{\mathrm {temp} },g\in {\mathcal {G}}$, then the prior on the group $\pi _{\mathcal {G}}(\cdot )$ induces the prior on images $\pi _{\mathcal {I}}(\cdot )$, written as densities the log-posterior takes the form

$\log p(I(g)\mid I^{D})\simeq \log p(I^{D}\mid I(g))+\log \pi _{\mathcal {G}}(g)\ .$

MAP estimation in the multiple-atlas orbit model

Maximum a posteriori estimation (MAP) estimation is central to modern statistical theory. Parameters of interest $\theta \in \Theta$ take many forms including (i) disease type such as neurodegenerative or neurodevelopmental diseases, (ii) structure type such as cortical or subcorical structures in problems associated to segmentation of images, and (iii) template reconstruction from populations. Given the observed image $I^{D}$, MAP estimation maximizes the posterior:

${\hat {\theta }}\doteq \arg \max _{\theta \in \Theta }\log p(\theta \mid I^{D}).$

This requires computation of the conditional probabilities $p(\theta \mid I^{D})={\frac {p(I^{D},\theta )}{p(I^{D})}}$. The multiple atlas orbit model randomizes over the denumerable set of atlases $\{I_{a},a\in {\mathcal {A}}\}$. The model on images in the orbit take the form of a multi-modal mixture distribution

$p(I^{D},\theta )=\textstyle \sum _{a\in {\mathcal {A}}}p(I^{D},\theta \mid I_{a})\pi _{\mathcal {A}}(a)\ .$

The conditional Gaussian model has been examined heavily for inexact matching in dense images and for alndmark matching.

• Dense Image Matching: Model $I^{D}(x),x\in X$ as a conditionally Gaussian random field conditioned, mean field, $\phi _{1}\cdot I\doteq I(\phi _{1}^{-1}),\phi _{1}\in Diff_{V}$. For uniform variance the endpoint error terms plays the role of the log-conditional (only a function of the mean field) giving the endpoint term:

$-\log p(I^{D}\mid I(g))\simeq E(\phi _{1})\doteq {\frac {1}{2\sigma ^{2}}}\|I^{D}-I\circ \phi _{1}^{-1}\|^{2}.$

Conditional-Gaussian

• Landmark Matching: Model $Y=\{y_{1},y_{2},\dots \}$ as conditionally Gaussian with mean field $\phi _{1}(x_{i}),i=1,2,\dots ,\phi _{1}\in Diff_{V}$, constant noise variance independent of landmarks. The log-conditional (only a function of the mean field) can be viewed as the endpoint term:

$-\log p(I^{D}\mid I(g))\simeq E(\phi _{1})\doteq {\frac {1}{2\sigma ^{2}}}\sum _{i}\|y_{i}-\phi _{1}(x_{i})\|^{2}.$

MAP segmentation based on multiple atlases

The random orbit model for multiple atlases models the orbit of shapes as the union over multiple anatomical orbits generated from the group action of diffeomorphisms, ${\mathcal {I}}=\textstyle \bigcup _{a\in {\mathcal {A}}}\displaystyle \operatorname {Diff} _{V}\cdot I_{a}$, with each atlas having a template and predefined segmentation field $(I_{a},W_{a}),a=a_{1},a_{2},\ldots$. incorporating the parcellation into anatomical structures of the coordinate of the MRI.. The pairs are indexed over the voxel lattice $I_{a}(x_{i}),W_{a}(x_{i}),x_{i}\in X\subset {\mathbb {R} }^{3}$ with an MRI image and a dense labelling of every voxel coordinate.The anatomical labelling of parcellated structures are manual delineations by neuroanatomists.

The Bayes segmentation problem is given measurement $I^{D}$ with mean field and parcellation $(I,W)$, the anatomical labelling $\theta \doteq W$. mustg be estimated for the measured MRI image. The mean-field of the observable $I^{D}$ image is modelled as a random deformation from one of the templates $I\doteq \varphi \cdot I_{a}$, which is also randomly selected, $A=a$,. The optimal diffeomorphism $\varphi \in {\mathcal {G}}$ is hidden and acts on the background space of coordinates of the randomly selected template image $I_{a}$. Given a single atlas $a$, the likelihood model for inference is determined by the joint probability $p(I^{D},W\mid A=a)$; with multiple atlases, the fusion of the likelihood functions yields the multi-modal mixture model with the prior averaging over models.

The MAP estimator of segmentation $W_{a}$ is the maximizer $\max _{W}\log p(W\mid I^{D})$ given $I^{D}$, which involves the mixture over all atlases.

${\hat {W}}\doteq \arg \textstyle \max _{W}\displaystyle \log p(I^{D},W){\text{ with }}p(I^{D},W)=\textstyle \sum _{a\in {\mathcal {A}}}\displaystyle p(I^{D},W\mid A=a)\pi _{A}(a).$

The quantity $p(I^{D},W)$ is computed via a fusion of likelihoods from multiple deformable atlases, with $\pi _{A}(a)$ being the prior probability that the observed image evolves from the specific template image $I_{a}$.

The MAP segmentation can be iteratively solved via the expectation-maximization(EM) algorithm

$W^{\text{new}}\doteq \arg \max _{W}\int \log p(W,{I^{D}},A,\varphi )\,dp(A,\varphi |W^{\text{old}},I^{D}).$