# Timetable (INVW03)

## Analytic and Geometric Methods in Medical Imaging

Monday 22nd August 2011 to Friday 26th August 2011

 09:00 to 09:45 Registration 09:45 to 10:30 T Chan ([HKUST])Computational Conformal / Quasi-conformal Geometry and Its Applications Conformal (C) / Quasi-conformal (QC) geometry has a long history in pure mathematics, and is an active field in both modern geometry and modern physics. Recently, with the rapid development of 3D digital scanning technology, the demand for effective geometric processing and shape analysis is ever increasing. Computational conformal / quasi-conformal geometry plays an important role for these purposes. Applications can be found in different areas such as medical imaging, computer visions and computer graphics. In this talk, I will first give an overview of how conformal geometry can be applied in medical imaging and computer graphics. Examples include brain registration and texture mapping, where the mappings are constructed to be as conformal as possible to reduce geometric distortions. In reality, most registrations and surface mappings involve non-conformal distortions, which require more general theories to study. A direct generalization of conformal mapping is quasiconformal mapping, where the mapping is allowed to have bounded conformality distortions. In the second part of my talk, theories of quasicoformal geometry and its applications will be presented. In particular, I will talk about how QC can be used for registration of biological surfaces, shape analysis, medical morphometry and the inpainting of surface diffeomorphism. INI 1 10:30 to 11:00 Coffee / Tea 11:00 to 11:45 K Astala (University of Helsinki)Calderon's inverse problem in 2D Electrical Impedance Tomography Calderon's problem asks if and how one can determine the conductivity structure of a material from boundary current-voltage measurements. In two dimensions the problem admits a complete solution. This includes the uniqueness proof for (very) rough coefficients, developing new reconstruction algorithms and their computer implementation.In this talk we give an overview of the recent progress on the EIT problem in two dimensions. The talk is based on joint works with M. Lassas, L. Päivärinta, S. Siltanen, J. Müller and A. Perämäki. INI 1 11:45 to 12:30 G Uhlmann ([Washington & UC, Irvine])Travel Time Tomography and Tensor Tomography We will give a survey on some recent results on travel tomography which consists in determining the index of refraction of a medium by measuring the travel times of sound waves going through the medium. We will also consider the related problem of tensor tomography which consists in determining a function, a vector field or tensors of higher rank from their integrals along geodesics. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 14:45 R Lai (University of Southern California)Laplace-Beltrami Eigen-Geometry and Applications to 3D Medical Imaging Rapid development of 3D data acquisition technologies stimulates researches on 3D surface analysis. Intrinsic descriptors of 3D surfaces are crucial to either process or analyze surfaces. In this talk, I will present our recent work on 3D surfaces analysis by using Laplace-Beltrami (LB) eigen-system. The intrinsically defined LB operator provides us a powerful tool to study surface geometry through its LB eigen-system. By combining with other variational PDEs on surfaces, I will show our results on skeleton construction, feature extraction, pattern identification and surface mapping in 3D brain imaging by using LB eigen-geometry. The nature of LB eigen-system guarantee that our methods are robust to surfaces rotation and translation variations. INI 1 14:45 to 15:30 N Paragios (École Centrale de Paris)Graphical Models and Discrete Optimization in Biomedical Imaging: Theory and Applications Image-based bio-markers have become powerful diagnostic tools due to the rapid and amazing development of medical hardware. In such a context, efficient processing and understanding of the corresponding images has gained significant attention over the past decade. The task to be addressed is extremely challenging due to: (i) curse of non-linearity (images and desired bio-markers exhibit a non-linear relationship), (ii) curse of dimensionality (number of degrees of freedom versus their inference), (iii) curse of non-convexity (designed objective functions present numerous local minima) and (iv) curse of modularity (variability of organs, imaging modalities). In this talk, we will provide some preliminary answers to the aforementioned challenges by exploiting through graphical models and discrete optimization algorithms. Furthermore, concrete examples will be presented towards addressing fundamental problems in biomedical perception like knowledge-based segmentation and deformable image fusion. INI 1 15:30 to 16:00 Coffee / Tea 16:00 to 16:30 A Ciomaga (École Normale Supérieure)Image Visualization and Restoration by Curvature Motions The role of curvatures in visual perception goes back to '54 and is due to Attneave. It can be argued on neurological grounds that human brain could not possible use all the information provided by states of simulation. But information that stimulates the retina, is located at regions where color changes abruptly (contours), and furthermore at angles and peaks of curvature. Yet, a direct computation of curvatures on a raw image is impossible. We show in this presentation how curvatures can be accurately estimated, at subpixel resolution, by a direct computation on level lines after their independent smoothing. This view towards shape analysis requires a representation of an image in terms of its level lines. At the same time, it involves short time smoothing (in occurrence Curve Shortening or Af?ne Shortening) simultaneously for level lines and images. In this setting, we found an explicit connection between the geometric approach for Curve / Af?ne Shortening and the viscosity approach for the Mean / Af?ne Curvature Motion, based on a complete image processing pipeline, that we term Level Lines (Af?ne) Shortening, shortly LL(A)S. We show that LL(A)S provides an accurate visualization tool of image curvatures, that we call an Image Curvature Microscope. As an application we give some illustrative examples of image visualization and restoration: noise, JPEG artifacts, and aliasing will be shown to be nicely smoothed out by the subpixel curvature motion. INI 1 16:30 to 17:00 E Konukoglu (Microsoft Research)Efficient Probabilistic Model Personalization Integrating Uncertainty on Data and Parameters: Application to Eikonal-Diffusion Models in Cardiac Electrophysiolo Biophysical models are increasingly used for medical applications at the organ scale. However, model predictions are rarely associated with a confidence measure although there are important sources of uncertainty in computational physiology methods. For instance, the sparsity and noise of the clinical data used to adjust the model parameters (personalization), and the difficulty in modeling accurately soft tissue physiology. The recent theoretical progresses in stochastic models make their use computationally tractable, but there is still a challenge in estimating patient-specific parameters with such models. In this talk I will describe an efficient Bayesian inference method for model personalization (parameter estimation) using polynomial chaos and compressed sensing. I will demonstrate the method in the context of cardiac electrophysiology and show how this can help in quantifying the impact of the data characteristics and uncertainty on the personalization (and thus prediction) results. Described method can be beneficial for the clinical use of personalized models as it explicitly takes into account the uncertainties on the data and the model parameters while still enabling simulations that can be used to optimize treatment. Such uncertainty handling can be pivotal for the proper use of modeling as a clinical tool, because there is a crucial requirement to know the confidence one can have in personalized models. INI 1 17:00 to 17:30 P Athavale (University of Toronto)Variational Methods in images processing, integro-differential equations and applications to histology and MR imaging Multiscale analysis can give useful insight into various natural and manmade phenomena. In this talk, we will discuss some new techniques of multiscale analysis in the context of digital images. We will discuss multiscale image processing using variational and partial dierential equations. We will describe novel integro-dierential equations based on the Rudin-Osher- Fatemi decomposition and its variants. In the second part of the talk, we will discuss the problem of tracing blood vessel boundaries in placental histology images using a combination of global/local registration and Chan-Vese segmentation. INI 1 17:30 to 18:30 Welcome Drinks Reception 18:45 to 19:30 Dinner at Wolfson Court
 09:00 to 09:45 L Cohen (Université Paris-Dauphine)Geodesic methods for Biomedical Image Segmentation Tubular and tree structures appear very commonly in biomedical images like vessels, microtubules or neuron cells. Minimal paths have been used for long as an interactive tool to segment these structures as cost minimizing curves. The user usually provides start and end points on the image and gets the minimal path as output. These minimal paths correspond to minimal geodesics according to some adapted metric. They are a way to find a (set of) curve(s) globally minimizing the geodesic active contours energy. Finding a geodesic distance can be solved by the Eikonal equation using the fast and efficient Fast Marching method. In the past years we have introduced different extensions of these minimal paths that improve either the interactive aspects or the results. For example, the metric can take into account both scale and orientation of the path. This leads to solving an anisotropic minimal path in a 2D or 3D+radius space. On a different level, the user interaction can be minimized by adding iteratively what we called the keypoints, for example to obtain a closed curve from a single initial point. The result is then a set of minimal paths between pairs of keypoints. This can also be applied to branching structures in both 2D and 3D images. We also proposed different criteria to obtain automatically a set of end points of a tree structure by giving only one starting point. More recently, we introduced a new general idea that we called Geodesic Voting or Geodesic Density. The approach consists in computing geodesics between a given source point and a set of points scattered in the image. The geodesic density is defined at each pixel of the image as the number of geodesics that pass over this pixel. The target structure corresponds to image points with a high geodesic density. We will illustrate different possible applications of this approach. The work we will present involved as well F. Benmansour, Y. Rouchdy and J. Mille at CEREMADE. INI 1 09:45 to 10:30 M Lassas (University of Helsinki)X-ray Tomography and Discretization of Inverse Problems In this talk we consider the question how inverse problems posed for continuous objects, for instance for continuous functions, can be discretized. This means the approximation of the problem by infinite dimensional inverse problems. We will consider linear inverse problems of the form $m=Af+\epsilon$. Here, the function $m$ is the measurement, $A$ is a ill-conditioned linear operator, $u$ is an unknown function, and $\epsilon$ is random noise. The inverse problem means determination of $u$ when $m$ is given. In particular, we consider the X-ray tomography with sparse or limited angle measurements where $A$ corresponds to integrals of the attenuation function $u(x)$ over lines in a family $\Gamma$. The traditional solutions for the problem include the generalized Tikhonov regularization and the estimation of $u$ using Bayesian methods. To solve the problem in practice $u$ and $m$ are discretized, that is, approximated by vectors in an infinite dimensional vector space. We show positive results when this approximation can successfully be done and consider examples of problems that can appear. As an example, we consider the total variation (TV) and Besov norm penalty regularization, the Bayesian analysis based on total variation prior and Besov priors. INI 1 10:30 to 11:00 Coffee / Tea 11:00 to 11:45 A Trouve (École Normale Supérieure)Shape Analysis of Population of Manifolds in Computational Anatomy The accelerated development of imaging techniques in biomedical engineering is challenging mathematicians and computer scientists to develop appropriate methods for the representation and the statistical analysis of various geometrically structured data like submanifolds. We will first explain how the concepts of homogeneous spaces and riemannian manifolds embedded in the large deformation diffeomorphic metric mapping setting (LDDMM) and the introduction of mathematical currents by Glaunes and Vaillant in this setting have been a powerful and effective framework to support local statistical analysis in more and more complex shape spaces. We will then discuss a new extension when the submanifolds are the supports of informative fields that need to be also analyzed in a common geometrical-functional representation (joint work with Nicolas Charon). INI 1 11:45 to 12:30 D Hawkes (University College London)Challenges of combining image derived information across modalities, over scale, over time and across populations INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 14:30 A Moradifam (University of Toronto)Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions We consider the problem of recovering an isotropic conductivity outside some perfectly conducting or insulating inclusions from the interior measurement of the magnitude of one current density field $|J|$. We prove that the conductivity outside the inclusions, and the shape and position of the perfectly conducting and insulating inclusions are uniquely determined (except in an exceptional case) by the magnitude of the current generated by imposing a given boundary voltage. We have found an extension of the notion of admissibility to the case of possible presence of perfectly conducting and insulating inclusions. This makes it possible to extend the results on uniqueness of the minimizers of the least gradient problem $F(u)=\int_{Omega}a | \nabla u|$ with $u|_{\partial \Omega}=f$ to cases where $u$ has flat regions (is constant on open sets). This is a joint work with Adrian Nachman and Alexandru Tamasam. INI 1 14:30 to 15:00 N Hoell (University of Toronto)The attenuated X-ray transform on curves We discuss inversion formulae for the attenuated X-ray transform on curves in the two-dimensional unit disc. This tomographic problem has applications in the medical imaging modality SPECT, and has more recently arisen in the problem of determining the internal permittivity and permeability parameters from a conductive body based on external measurements. INI 1 15:00 to 15:30 K Chen (University of Liverpool)A new multi-modality model for effective intensity standardization and image registration Image registration and segmentation tasks lie in the heart of Medical Imaging. In registration, our concern is to align two or more images using deformable transforms that have desirable regularities. In a multimodal image registration scenario, where two given images have similar features, but non-comparable intensity variations, the sum of squared differences is not suitable to measure image similarities. In this talk, we first propose a new variational model based on combining intensity and geometric transformations, as an alternative to using mutual information and an improvement to the work by Modersitzki and Wirtz (2006, LNCS, vol.4057), and then develop a fast multigrid algorithm for solving the underlying system of fourth order and nonlinear partial differential equations. We can demonstrate the effective smoothing property of the adopted primal-dual smoother by a local Fourier analysis. An earlier use of mean curvature to regulairse image denosing models was in T F Chan and W Zhu (2008) and the previous work of developing a multigrid algorithm for the Chan-Zhu model was by Brito-Chen (2010). Numerical tests will be presented to show both the improvements achieved in image registration quality as well as multigrid efficiency. Joint work with Dr Noppadol Chumchob. INI 1 15:30 to 16:00 Coffee / Tea 18:45 to 19:30 Dinner at Wolfson Court