Mission

My main research area is computational image analysis with an emphasis on studying statistical models from a Bayesian perspective. My research covers object localization and recognition, data alignment, and shape representation, with a particular focus on neuroimaging. It is my long term goal to enhance patient care by creating algorithms for automatically quantifying and generalizing the information latent in images for tasks such as disease analysis and surgical planning.

Logarithm of Odds

The Logarithm of the Odds ratio (LogOdds) is frequently used in areas such as artificial neural networks, economics, and biology, as an alternative representation of probabilities. Here, we use LogOdds to place probabilistic atlases in a linear vector space. This representation has several useful properties for imaging. For example, it not only encodes the shape of multiple objects but also captures some information concerning uncertainty. Furthermore, the resulting vector space operations of addition and scalar multiplication have natural probabilistic interpretations.

In this example, a conventional likelihood model is combined with a curve length prior on boundaries. We then approximate posterior distribution on labels via the Mean Field approach. Optimizing the resulting estimator by gradient descent leads to a level set style algorithm where the level set functions are the logarithm-of-odds encoding of the posterior label probabilities in an unconstrained linear vector space.

For more information, please read K.M. Pohl El Ab. Active mean fields: Solving the mean field approximation in the level set framework , IPMI, 2007.

The movie shows the interpolation between two bell curved distributions over time. As shown in Goal, the bell curve should move from the left (time t=0) to the right (t=1). However, when computing the convex combination of the distributions defined at t=0 and t=1 the uni-modal distribution turns into a bimodal one at t=0.5 (see Prob.). We can address this issue by performing the convex combination in the LogOdds space (see LogOdds).

For more information, please read K.M. Pohl et al. Using the logarithm of odds to define a vector space on probabilistic atlases , MedIA, 2007, which was awarded the MedIA - MICCAI 06 Best Paper Prize

Prior Information in an Expectation-Maximization Framework

Many neuroscience applications require the identification of structures with weakly visible boundaries in Magnetic Resonance (MR) images. One of my interests is the development of probabilistic models for the segmentation of MR images under the assumption that the accuracy of the measurements (i.e. image data) is insufficient for the proper partition of the image data into anatomical structures. The examples below illustrate different models whose solution is determined via instances of the Expectation-Maximization (EM) algorithm.

We developed a statistical model combining the registration of an atlas with the segmentation of magnetic resonance images. Unlike other voxel-based classification methods, this framework models these problems as a single Maximum A Posteriori estimation problem, where the registration is defined by an object-specific affine mapping representation. A study empirically demonstrates the utility of simultaneously performing segmentation and registration over addressing these tasks sequentially.

For more information, please read K.M. Pohl et al. A Bayesian Model for Joint Segmentation and Registration,, NeuroImage, 2007.

The algorithm is based on a probabilistic model with a prior defined by a statistical shape atlas. The atlas is built through Principal Component Analysis (PCA) on a set of LogOdds, which captures covariant shape deformations of neighboring structures. Structure boundaries, anatomical labels, and image inhomogeneities are estimated simultaneously within an Expectation-Maximization formulation.

For more information, please read K.M. Pohl et al. Using the logarithm of odds to define a vector space on probabilistic atlases , MedIA, 2007.

The algorithm is guided by prior information represented within a tree structure. The tree mirrors the hierarchy of anatomical structures and the sub-trees correspond to limited segmentation problems. The solution to each problem is estimated via a conventional classifier. Our algorithm can be adapted to a wide range of segmentation problems by modifying the tree structure or replacing the classifier.

For more information, please read K.M. Pohl et al. A Hierarchical Algorithm for MR Brain Image Parcellation, IEEE TMI, 2007.

Applications

The images below are brief survey of the applications that my algorithms have been applied to over the years. Most of my software is publicly available and is distributed via the 3D Slicer. I would like to especially thank my collaborators for providing me with these images.