On the Link between Gaussian Homotopy Continuation and Convex Envelopes

Abstract. The continuation method is a popular heuristic in computer vision for nonconvex optimization. The idea is to start from a simpli-fied problem and gradually deform it to the actual task while tracking the solution. It was first used in computer vision under the name of graduated nonconvexity. Since then, it has been utilized explicitly or im-plicitly in various applications. In fact, state-of-the-art optical flow and shape estimation rely on a form of continuation. Despite its empirical success, there is little theoretical understanding of this method. This work provides some novel insights into this technique. Specifically, there are many ways to choose the initial problem and many ways to progres-sively deform it to the original task. However, here we show that when this process is constructed by Gaussian smoothing, it is optimal in a specific sense. In fact, we prove that Gaussian smoothing emerges from the best affine approximation to Vese’s nonlinear PDE. The latter PDE evolves any function to its convex envelope, hence providing the optimal convexification

Multi-scale active shape description in medical imaging

Shape description in medical imaging has become an increasingly important research field in recent years. Fast and high-resolution image acquisition methods like Magnetic Resonance (MR) imaging produce very detailed cross-sectional images of the human body - shape description is then a post-processing operation which abstracts quantitative descriptions of anatomically relevant object shapes. This task is usually performed by clinicians and other experts by first segmenting the shapes of interest, and then making volumetric and other quantitative measurements. High demand on expert time and inter- and intra-observer variability impose a clinical need of automating this process. Furthermore, recent studies in clinical neurology on the correspondence between disease status and degree of shape deformations necessitate the use of more sophisticated, higher-level shape description techniques. In this work a new hierarchical tool for shape description has been developed, combining two recently developed and powerful techniques in image processing: differential invariants in scale-space, and active contour models. This tool enables quantitative and qualitative shape studies at multiple levels of image detail, exploring the extra image scale degree of freedom. Using scale-space continuity, the global object shape can be detected at a coarse level of image detail, and finer shape characteristics can be found at higher levels of detail or scales. New methods for active shape evolution and focusing have been developed for the extraction of shapes at a large set of scales using an active contour model whose energy function is regularized with respect to scale and geometric differential image invariants. The resulting set of shapes is formulated as a multiscale shape stack which is analysed and described for each scale level with a large set of shape descriptors to obtain and analyse shape changes across scales. This shape stack leads naturally to several questions in regard to variable sampling and appropriate levels of detail to investigate an image. The relationship between active contour sampling precision and scale-space is addressed. After a thorough review of modem shape description, multi-scale image processing and active contour model techniques, the novel framework for multi-scale active shape description is presented and tested on synthetic images and medical images. An interesting result is the recovery of the fractal dimension of a known fractal boundary using this framework. Medical applications addressed are grey-matter deformations occurring for patients with epilepsy, spinal cord atrophy for patients with Multiple Sclerosis, and cortical impairment for neonates. Extensions to non-linear scale-spaces, comparisons to binary curve and curvature evolution schemes as well as other hierarchical shape descriptors are discussed

Dynamical models and machine learning for supervised segmentation

This thesis is concerned with the problem of how to outline regions of interest in medical images, when the boundaries are weak or ambiguous and the region shapes are irregular. The focus on machine learning and interactivity leads to a common theme of the need to balance conflicting requirements. First, any machine learning method must strike a balance between how much it can learn and how well it generalises. Second, interactive methods must balance minimal user demand with maximal user control. To address the problem of weak boundaries,methods of supervised texture classification are investigated that do not use explicit texture features. These methods enable prior knowledge about the image to benefit any segmentation framework. A chosen dynamic contour model, based on probabilistic boundary tracking, combines these image priors with efficient modes of interaction. We show the benefits of the texture classifiers over intensity and gradient-based image models, in both classification and boundary extraction. To address the problem of irregular region shape, we devise a new type of statistical shape model (SSM) that does not use explicit boundary features or assume high-level similarity between region shapes. First, the models are used for shape discrimination, to constrain any segmentation framework by way of regularisation. Second, the SSMs are used for shape generation, allowing probabilistic segmentation frameworks to draw shapes from a prior distribution. The generative models also include novel methods to constrain shape generation according to information from both the image and user interactions. The shape models are first evaluated in terms of discrimination capability, and shown to out-perform other shape descriptors. Experiments also show that the shape models can benefit a standard type of segmentation algorithm by providing shape regularisers. We finally show how to exploit the shape models in supervised segmentation frameworks, and evaluate their benefits in user trials

Quelques extensions des level sets et des graph cuts et leurs applications à la segmentation d'images et de vidéos

Image processing techniques are now widely spread out over a large quantity of domains: like medical imaging, movies post-production, games... Automatic detection and extraction of regions of interest inside an image, a volume or a video is challenging problem since it is a starting point for many applications in image processing. However many techniques were developed during the last years and the state of the art methods suffer from some drawbacks: The Level Sets method only provides a local minimum while the Graph Cuts method comes from Combinatorial Community and could take advantage of the specificity of image processing problems. In this thesis, we propose two extensions of the previously cited methods in order to soften or remove these drawbacks. We first discuss the existing methods and show how they are related to the segmentation problem through an energy formulation. Then we introduce stochastic perturbations to the Level Sets method and we build a more generic framework: the Stochastic Level Sets (SLS). Later we provide a direct application of the SLS to image segmentation that provides a better minimization of energies. Basically, it allows the contours to escape from local minimum. Then we propose a new formulation of an existing algorithm of Graph Cuts in order to introduce some interesting concept for image processing community: like initialization of the algorithm for speed improvement. We also provide a new approach for layer extraction from video sequence that retrieves both visible and hidden layers in it.Les techniques de traitement d'image sont maintenant largement répandues dans une grande quantité de domaines: comme l'imagerie médicale, la post-production de films, les jeux... La détection et l'extraction automatique de régions d'intérêt à l'intérieur d'une image, d'un volume ou d'une vidéo est réel challenge puisqu'il représente un point de départ pour un grand nombre d'applications en traitement d'image. Cependant beaucoup de techniques développées pendant ces dernières années et les méthodes de l'état de l'art souffrent de quelques inconvénients: la méthode des ensembles de niveaux fournit seulement un minimum local tandis que la méthode de coupes de graphe vient de la communauté combinatoire et pourrait tirer profit de la spécificité des problèmes de traitement d'image. Dans cette thèse, nous proposons deux prolongements des méthodes précédemment citées afin de réduire ou enlever ces inconvénients. Nous discutons d'abord les méthodes existantes et montrons comment elles sont liées au problème de segmentation via une formulation énergétique. Nous présentons ensuite des perturbations stochastiques a la méthode des ensembles de niveaux et nous établissons un cadre plus générique: les ensembles de niveaux stochastiques (SLS). Plus tard nous fournissons une application directe du SLS à la segmentation d'image et montrons qu'elle fournit une meilleure minimisation des énergies. Fondamentalement, il permet aux contours de s'échapper des minima locaux. Nous proposons ensuite une nouvelle formulation d'un algorithme existant des coupes de graphe afin d'introduire de nouveaux concepts intéressant pour la communauté de traitement d'image: comme l'initialisation de l'algorithme pour l'amélioration de vitesse. Nous fournissons également une nouvelle approche pour l'extraction de couches d'une vidéo par segmentation du mouvement et qui extrait à la fois les couches visibles et cachées présentes

Motion Tracking for Medical Applications using Hierarchical Filter Models

A medical intervention often requires relating treatment to the situation, which it was planned on. In order to circumvent undesirable effects of motion during the intervention, positional differences must be detected in real-time. To this end, in this thesis a hierarchical Particle Filter based tracking algorithm is developed in three stages. Initially, a model description of the individual nodes in the aspired hierarchical tree is presented. Using different approaches, properties of such a node are derived and approximated, leading to a parametrization scheme. Secondly, transformations and appearance of the data are described by a fixed hierarchical tree. A sparse description for typical landmarks in medical image data is presented. A static tree model with two levels is developed and investigated. Finally, the notion of 'association' between landmarks and nodes is introduced in order to allow for dynamic adaptation to the underlying structure of the data. Processes for tree maintenance using clustering and sequential reinforcement are implemented. The function of the full algorithm is demonstrated on data of abdominal breathing motion

Motor control and strategy discovery for physically simulated characters

In physics-based character animation, motions are realized through control of simulated characters along with their interactions with the virtual environment. In this thesis, we study the problem of character control on two levels: joint-level motor control which transforms control signals to joint torques, and high-level motion control which outputs joint-level control signals given the current state of the character and the environment and the task objective. We propose a Modified Articulated-Body Algorithm (MABA) which achieves stable proportional-derivative (PD) low-level motor control with superior theoretical time complexity, practical efficiency and stability than prior implementations. We further propose a high-level motion control framework based on deep reinforcement learning (DRL) which enables the discovery of appropriate motion strategies without human demonstrations to complete a task objective. To facilitate the learning of realistic human motions, we propose a Pose Variational Autoencoder (P-VAE) to constrain the DRL actions to a subspace of natural poses. Our learning framework can be further combined with a sample-efficient Bayesian Diversity Search (BDS) algorithm and novel policy seeking to discover diverse strategies for tasks with multiple modes, such as various athletic jumping tasks

Search-based system architecture development using a holistic modeling approach

This dissertation presents an innovative approach to system architecting where search algorithms are used to explore design trade space for good architecture alternatives. Such an approach is achieved by integrating certain model construction, alternative generation, simulation, and assessment processes into a coherent and automated framework. This framework is facilitated by a holistic modeling approach that combines the capabilities of Object Process Methodology (OPM), Colored Petri Net (CPN), and feature model. The resultant holistic model can not only capture the structural, behavioral, and dynamic aspects of a system, allowing simulation and strong analysis methods to be applied, it can also specify the architectural design space. Both object-oriented analysis and design (OOA/D) and domain engineering were exploited to capture design variables and their domains and define architecture generation operations. A fully realized framework (with genetic algorithms as the search algorithm) was developed. Both the proposed framework and its suggested implementation, including the proposed holistic modeling approach and architecture alternative generation operations, are generic. They are targeted at systems that can be specified using object-oriented or process-oriented paradigm. The broad applicability of the proposed approach is demonstrated on two examples. One is the configuration of reconfigurable manufacturing systems (RMSs) under multi-objective optimization and the other is the architecture design of a manned lunar landing system for the Apollo program. The test results show that the proposed approach can cover a huge number of architecture alternatives and support the assessment of several performance measures. A set of quality results was obtained after running the optimization algorithm following the proposed framework --Abstract, page iii

Decoupled Deformable Model For 2D/3D Boundary Identification

The accurate detection of static object boundaries such as contours or surfaces and dynamic tunnels of moving objects via deformable models is an ongoing research topic in computer vision. Most deformable models attempt to converge towards a desired solution by minimizing the sum of internal (prior) and external (measurement) energy terms. Such an approach is elegant, but frequently mis-converges in the presence of noise or complex boundaries and typically requires careful semi-dependent parameter tuning and initialization. Furthermore, current deformable model based approaches are computationally demanding which precludes real-time use. To address these limitations, a decoupled deformable model (DDM) is developed which optimizes the two energy terms separately. Essentially, the DDM consists of a measurement update step, employing a Hidden Markov Model (HMM) and Maximum Likelihood (ML) estimator, followed by a separate prior step, which modifies the updated deformable model based on the relative strengths of the measurement uncertainty and the non-stationary prior. The non-stationary prior is generated by using a curvature guided importance sampling method to capture high curvature regions. By separating the measurement and prior steps, the algorithm is less likely to mis-converge; furthermore, the use of a non-iterative ML estimator allows the method to converge more rapidly than energy-based iterative solvers. The full functionality of the DDM is developed in three phases. First, a DDM in 2D called the decoupled active contour (DAC) is developed to accurately identify the boundary of a 2D object in the presence of noise and background clutter. To carry out this task, the DAC employs the Viterbi algorithm as a truncated ML estimator, curvature guided importance sampling as a non-stationary prior generator, and a linear Bayesian estimator to fuse the non-stationary prior with the measurements. Experimental results clearly demonstrate that the DAC is robust to noise, can capture regions of very high curvature, and exhibits limited dependence on contour initialization or parameter settings. Compared to three other published methods and across many images, the DAC is found to be faster and to offer consistently accurate boundary identification. Second, a fast decoupled active contour (FDAC) is proposed to accelerate the convergence rate and the scalability of the DAC without sacrificing the accuracy by employing computationally efficient and scalable techniques to solve the three primary steps of DAC. The computational advantage of the FDAC is demonstrated both experimentally and analytically compared to three computationally efficient methods using illustrative examples. Finally, an extension of the FDAC from 2D to 3D called a decoupled active surface (DAS) is developed to precisely identify the surface of a volumetric 3D image and the tunnel of a moving 2D object. To achieve the objectives of the DAS, the concepts of the FDAC are extended to 3D by using a specialized 3D deformable model representation scheme and a computationally and storage efficient estimation scheme. The performance of the DAS is demonstrated using several natural and synthetic volumetric images and a sequence of moving objects