## Section: Research Program

### Discrete Biomedical Image Perception

A wide variety of tasks in medical image analysis can be formulated as discrete labeling problems. In very simple terms, a discrete optimization problem can be stated as follows: we are given a discrete set of variables $\mathcal{V}$, all of which are vertices in a graph $\mathcal{G}$. The edges of this graph (denoted by $\mathcal{E}$) encode the variables' relationships. We are also given as input a discrete set of labels $\mathcal{L}$. We must then assign one label from $\mathcal{L}$ to each variable in $\mathcal{V}$. However, each time we choose to assign a label, say, ${x}_{{p}_{1}}$ to a variable ${p}_{1}$, we are forced to pay a price according to the so-called *singleton* potential function ${g}_{p}\left({x}_{p}\right)$, while each time we choose to assign a pair of labels, say, ${x}_{{p}_{1}}$ and ${x}_{{p}_{2}}$ to two interrelated variables ${p}_{1}$ and ${p}_{2}$ (two nodes that are connected by an edge in the graph $\mathcal{G}$), we are also forced to pay another price, which is now determined by the so called *pairwise* potential function ${f}_{{p}_{1}{p}_{2}}({x}_{{p}_{1}},{x}_{{p}_{2}})$. Both the singleton and pairwise potential functions are problem specific and are thus assumed to be provided as input.

Our goal is then to choose a labeling which will allow us to pay the smallest total price. In other words, based on what we have mentioned above, we want to choose a labeling that minimizes the sum of all the MRF potentials, or equivalently the MRF energy. This amounts to solving the following optimization problem:

$arg\underset{\left\{{x}_{p}\right\}}{min}\mathcal{P}(g,f)=\sum _{p\in \mathcal{V}}{g}_{p}\left({x}_{p}\right)+\sum _{({p}_{1},{p}_{2})\in \mathcal{E}}{f}_{{p}_{1}{p}_{2}}({x}_{{p}_{1}},{x}_{{p}_{2}}).$ | (8) |

The use of such a model can describe a number of challenging problems in medical image analysis. However these simplistic models can only account for simple interactions between variables, a rather constrained scenario for high-level medical imaging perception tasks. One can augment the expression power of this model through higher order interactions between variables, or a number of cliques $\{{C}_{i},i\in [1,n]=\left\{\{{p}_{{i}^{1}},\cdots ,{p}_{{i}^{|{C}_{i}|}}\}\right\}$ of order $|{C}_{i}|$ that will augment the definition of $\mathcal{V}$ and will introduce hyper-vertices:

$arg\underset{\left\{{x}_{p}\right\}}{min}\mathcal{P}(g,f)=\sum _{p\in \mathcal{V}}{g}_{p}\left({x}_{p}\right)+\sum _{({p}_{1},{p}_{2})\in \mathcal{E}}{f}_{{p}_{1}{p}_{2}}({x}_{{p}_{1}},{x}_{{p}_{2}})+\sum _{{C}_{i}\in \mathcal{E}}{f}_{{p}_{1}\cdots {p}_{n}}({x}_{{p}_{{i}^{1}}},\cdots ,{p}_{{x}_{{i}^{|{C}_{i}|}}}).$ | (9) |

where ${f}_{{p}_{1}\cdots {p}_{n}}$ is the price to pay for associating the labels $({x}_{{p}_{{i}^{1}}},\cdots ,{p}_{{x}_{{i}^{|{C}_{i}|}}})$ to the nodes $({p}_{1}\cdots {p}_{{i}^{|{C}_{i}|}})$. Parameter inference, addressed by minimizing the problem above, is the most critical aspect in computational medicine and efficient optimization algorithms are to be evaluated both in terms of computational complexity as well as of inference performance. State of the art methods include deterministic and non-deterministic annealing, genetic algorithms, max-flow/min-cut techniques and relaxation. These methods offer certain strengths while exhibiting certain limitations, mostly related to the amount of interactions which can be tolerated among neighborhood nodes. In the area of medical imaging where domain knowledge is quite strong, one would expect that such interactions should be enforced at the largest scale possible.