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 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 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 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 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