Abstract: Medical imaging feeds research in many computer vision and image processing fields: image filtering, segmentation, shape recovery, registration, retrieval and pattern matching. Because of their low contrast changes and large variety of artifacts and noise, medical imaging processing techniques relying on an analysis of the geometry of image level sets rather than on intensity values result in more robust treatment. From the starting point of treatment of intravascular images, this PhD thesis ad- dresses the design of differential image operators based on geometric principles for a robust shape modelling and restoration. Among all fields applying shape recovery, we approach filtering and segmentation of image objects. For a successful use in real images, the segmentation process should go through three stages: noise removing, shape modelling and shape recovery. This PhD addresses all three topics, but for the sake of algorithms as automated as possible, techniques for image processing will be designed to satisfy three main principles: a) convergence of the iterative schemes to non-trivial states avoiding image degeneration to a constant image and representing smooth models of the originals; b) smooth asymptotic behav- ior ensuring stabilization of the iterative process; c) fixed parameter values ensuring equal (domain free) performance of the algorithms whatever initial images/shapes. Our geometric approach to the generic equations that model the different processes approached enables defining techniques satisfying all the former requirements. First, we introduce a new curvature-based geometric flow for image filtering achieving a good compromise between noise removing and resemblance to original images. Sec- ond, we describe a new family of diffusion operators that restrict their scope to image level curves and serve to restore smooth closed models from unconnected sets of points. Finally, we design a regularization of snake (distance) maps that ensures its smooth convergence towards any closed shape. Experiments show that performance of the techniques proposed overpasses that of state-of-the-art algorithms.

Abstract: Early diagnosis and accurate treatment of Left Ventricle (LV) dysfunction significantly increases the patient survival. Impairment of LV contractility due to cardiovascular diseases is reflected in its motion patterns. Recent advances in medical imaging, such as Magnetic Resonance (MR), have encouraged research on 3D simulation and modelling of the LV dynamics. Most of the existing 3D models consider just the gross anatomy of the LV and restore a truncated ellipse which deforms along the cardiac cycle. The contraction mechanics of any muscle strongly depends on the spatial orientation of its muscular fibers since the motion that the muscle undergoes mainly takes place along the fibers. It follows that such simplified models do not allow evaluation of the heart electro-mechanical function and coupling, which has recently risen as the key point for understanding the LV functionality . In order to thoroughly understand the LV mechanics it is necessary to consider the complete anatomy of the LV given by the orientation of the myocardial fibres in 3D space as described by Torrent Guasp. We propose developing a 3D patient-sensitive model of the LV integrating, for the first time, the ven- tricular band anatomy (fibers orientation), the LV gross anatomy and its functionality. Such model will represent the LV function as a natural consequence of its own ventricular band anatomy. This might be decisive in restoring a proper LV contraction in patients undergoing pace marker treatment. The LV function is defined as soon as the propagation of the contractile electromechanical pulse has been modelled. In our experiments we have used the wave equation for the propagation of the electric pulse. The electromechanical wave moves on the myocardial surface and should have a conductivity tensor oriented along the muscular fibers. Thus, whatever mathematical model for electric pulse propa- gation [4] we consider, the complete anatomy of the LV should be extracted. The LV gross anatomy is obtained by processing multi slice MR images recorded for each patient. Information about the myocardial fibers distribution can only be extracted by Diffusion Tensor Imag- ing (DTI), which can not provide in vivo information for each patient. As a first approach, we have computed an average model of fibers from several DTI studies of canine hearts. This rough anatomy is the input for our electro-mechanical propagation model simulating LV dynamics. The average fiber orientation is updated until the simulated LV motion agrees with the experimental evidence provided by the LV motion observed in tagged MR (TMR) sequences. Experimental LV motion is recovered by applying image processing, differential geometry and interpolation techniques to 2D TMR slices [5]. The pipeline in figure 1 outlines the interaction between simulations and experimental data leading to our patient-tailored model.

Abstract: In this paper we introduce a novel application of the diffusion tensor for anisotropic image processing. The Anisotropic Contour Completion (ACC) we suggest consists in extending the characteristic function of the open curve by means of a degenerated diffusion tensor that prevents any diffusion in the normal direction. We show that ACC is equivalent to a dilation with a continuous elliptic structural element that takes into account the local orientation of the contours to be closed. Experiments on contours extracted from real images show that ACC produces shapes able to adapt to any curve in an active contour framework. 1.