In most imaging situations, preprocessing is a fundamental necessity. Images need to be registered to one another, segmented to find relevant structures, and then these final images suitably analyzed. Each of these steps consists of many possible algorithms but in almost all cases, the analysis proceeds linearly from one step to another, without taking into account the fact that errors are propagated from step to step. In addition, the use of automated segmentation and quantification methods in clinical practice requires their thorough mathematical analysis and statistical tuning. Automated methods can only be used in medical practice if they work robustly and reliably within a wide range of data sets arising in a particular application. Therefore, accurate modelling, statistical analysis, testing and validation of automated image analysis methods is essential.
Clinically this has particular relevance to cardiac imaging. In addition to CT-derived measures of risk in atherosclerosis we have acquired a large dataset of intra-vascular ultrasound images from the coronary arteries of patients with recent heart attack. Using finite element analysis on this dataset has yielded stress maps that can be superimposed onto the ultrasound data. However, current approaches to this problem require four hours of processing time. What is now needed is real-time co-registration of these two modalities to quickly permit the physician to determine how dangerous a plaque is at the time of image acquisition. In addition, the level of difficulty of and confidence in the registration should be incorporated into the modelling. This requires automation of multiple mathematical steps, including transformation, segmentation, mesh formation, border and overlap correction, and image shrinkage.
This project will look at joint modelling and analysis of such issues from both a computational analysis and statistical point of view. Variational techniques and metric based procedures can be used to align and analyse the data jointly [1,2], while incorporating such metrics into the subsequent statistical modelling is also possible . A dominant difficulty in joint models is the complexity of the latter, often resulting into nonlinear and non-convex variational problems whose analysis and robust computation are challenging. Also the well-posedness and the statistical validity of a joint analysis is non-trivial, requiring sophisticated techniques from applied analysis and statistics. The project could take on various connotations, from very theoretical considerations to computational algorithm development and investigation of different applications in image analysis.
In addition, all these methods, while being able to produce qualitatively good results if solved correctly, their numerical solution is challenging due to non-smooth and non-convex terms in the model. Convex relaxation is a popular approach to overcome this problem . Moreover, their performance depends on the correct choice of a set of parameters. Therefore there is a strong need for the tuning of parameters and error analysis. To do so, we will combine variational image segmentation techniques with statistical analysis and learning approaches  that can help to determine an optimal and reliable choice of parameters dependent on the image data at hand. For validating segmentation and quantification results statistical error bounds and confidence ranges will be derived.
This project could appeal to a candidate with background and interest in one or more of the following areas: inverse problems, applied / computational analysis and statistics.
 S. Ozere, C. Le Guyader, C. Gout, Joint segmentation/registration model by shape alignment via weighted total variation minimization and nonlinear elasticity, SIAM J. on Imaging Sciences 8(3), 1981-2020, 2015.
 H. Dirks, Variational Methods for Joint Motion Estimation and Image Reconstruction, PhD thesis, Institute for Computational and Applied Mathematics, University of Muenster, http://wwwmath.uni-muenster.de/num/publications/2015/Dir15/.
 S Kurtek, H Drira. A comprehensive statistical framework for elastic shape analysis of 3D faces Computers & Graphics 51, 52-59, 2015.
 X. Cai, R. Chan, and T. Zeng. A two-stage image segmentation method using a convex variant of the Mumford–Shah model and thresholding. SIAM Journal on Imaging Sciences , 6(1):368–390, 2013.
 L. Calatroni, C. Cao, J. C. De Los Reyes, C.-B. Schönlieb, and T. Valkonen, Bilevel approaches for learning of variational imaging models , to appear in Radon Book Series – Variational Methods in Image Analysis, 36 p., 2015. arXiv:1505.02120.