Generative modeling for computer vision has shown immense progress in the last few years, revolutionizing the way we perceive, understand, and manipulate visual data. This rapidly evolving field has witnessed advancements in image generation, 3D animation, and video prediction that unlock diverse applications across multiple fields including entertainment, design, healthcare, and education. As the demand for sophisticated computer vision systems continues to grow, this dissertation attempts to drive innovation in the field by exploring novel formulations of conditional generative models, and innovative applications in images, 3D animations, and video. Our research focuses on architectures that offer reversible transformations of noise and visual data, and the application of encoder-decoder architectures for generative tasks and 3D content manipulation. In all instances, we incorporate conditional information to enhance the synthesis of visual data, improving the efficiency of the generation process as well as the generated content. Prior successful generative techniques which are reversible between noise and data include normalizing flows and denoising diffusion models. The continuous variant of normalizing flows is powered by Neural Ordinary Differential Equations (Neural ODEs), and have shown some success in modeling the real image distribution. However, they often involve huge number of parameters, and high training time. Denoising diffusion models have recently gained huge popularity for their generalization capabilities especially in text-to-image applications. In this dissertation, we introduce the use of Neural ODEs to model video dynamics using an encoder-decoder architecture, demonstrating their ability to predict future video frames despite being trained solely to reconstruct current frames. In our next contribution, we propose a conditional variant of continuous normalizing flows that enables higher-resolution image generation based on lower-resolution input. This allows us to achieve comparable image quality to regular normalizing flows, while significantly reducing the number of parameters and training time. Our next contribution focuses on a flexible encoder-decoder architecture for accurate estimation and editing of full 3D human pose. We present a comprehensive pipeline that takes human images as input, automatically aligns a user-specified 3D human/non-human character with the pose of the human, and facilitates pose editing based on partial input information. We then proceed to use denoising diffusion models for image and video generation. Regular diffusion models involve the use of a Gaussian process to add noise to clean images. In our next contribution, we derive the relevant mathematical details for denoising diffusion models that use non-isotropic Gaussian processes, present non-isotropic noise, and show that the quality of generated images is comparable with the original formulation. In our final contribution, devise a novel framework building on denoising diffusion models that is capable of solving all three video tasks of prediction, generation, and interpolation. We perform ablation studies using this framework, and show state-of-the-art results on multiple datasets. Our contributions are published articles at peer-reviewed venues. Overall, our research aims to make a meaningful contribution to the pursuit of more efficient and flexible generative models, with the potential to shape the future of computer vision. / La modélisation générative pour la vision par ordinateur a connu d’immenses progrès ces dernières années, révolutionnant notre façon de percevoir, comprendre et manipuler les données visuelles. Ce domaine en constante évolution a connu des avancées dans la génération d’images, l’animation 3D et la prédiction vidéo, débloquant ainsi diverses applications dans plusieurs domaines tels que le divertissement, le design, la santé et l’éducation. Alors que la demande de systèmes de vision par ordinateur sophistiqués ne cesse de croître, cette thèse s’efforce de stimuler l’innovation dans le domaine en explorant de nouvelles formulations de modèles génératifs conditionnels et des applications innovantes dans les images, les animations 3D et la vidéo. Notre recherche se concentre sur des architectures offrant des transformations réversibles du bruit et des données visuelles, ainsi que sur l’application d’architectures encodeur-décodeur pour les tâches génératives et la manipulation de contenu 3D. Dans tous les cas, nous incorporons des informations conditionnelles pour améliorer la synthèse des données visuelles, améliorant ainsi l’efficacité du processus de génération ainsi que le contenu généré. Les techniques génératives antérieures qui sont réversibles entre le bruit et les données et qui ont connu un certain succès comprennent les flux de normalisation et les modèles de diffusion de débruitage. La variante continue des flux de normalisation est alimentée par les équations différentielles ordinaires neuronales (Neural ODEs) et a montré une certaine réussite dans la modélisation de la distribution d’images réelles. Cependant, elles impliquent souvent un grand nombre de paramètres et un temps d’entraînement élevé. Les modèles de diffusion de débruitage ont récemment gagné énormément en popularité en raison de leurs capacités de généralisation, notamment dans les applications de texte vers image. Dans cette thèse, nous introduisons l’utilisation des Neural ODEs pour modéliser la dynamique vidéo à l’aide d’une architecture encodeur-décodeur, démontrant leur capacité à prédire les images vidéo futures malgré le fait d’être entraînées uniquement à reconstruire les images actuelles. Dans notre prochaine contribution, nous proposons une variante conditionnelle des flux de normalisation continus qui permet une génération d’images à résolution supérieure à partir d’une entrée à résolution inférieure. Cela nous permet d’obtenir une qualité d’image comparable à celle des flux de normalisation réguliers, tout en réduisant considérablement le nombre de paramètres et le temps d’entraînement. Notre prochaine contribution se concentre sur une architecture encodeur-décodeur flexible pour l’estimation et l’édition précises de la pose humaine en 3D. Nous présentons un pipeline complet qui prend des images de personnes en entrée, aligne automatiquement un personnage 3D humain/non humain spécifié par l’utilisateur sur la pose de la personne, et facilite l’édition de la pose en fonction d’informations partielles. Nous utilisons ensuite des modèles de diffusion de débruitage pour la génération d’images et de vidéos. Les modèles de diffusion réguliers impliquent l’utilisation d’un processus gaussien pour ajouter du bruit aux images propres. Dans notre prochaine contribution, nous dérivons les détails mathématiques pertinents pour les modèles de diffusion de débruitage qui utilisent des processus gaussiens non isotropes, présentons du bruit non isotrope, et montrons que la qualité des images générées est comparable à la formulation d’origine. Dans notre dernière contribution, nous concevons un nouveau cadre basé sur les modèles de diffusion de débruitage, capable de résoudre les trois tâches vidéo de prédiction, de génération et d’interpolation. Nous réalisons des études d’ablation en utilisant ce cadre et montrons des résultats de pointe sur plusieurs ensembles de données. Nos contributions sont des articles publiés dans des revues à comité de lecture. Dans l’ensemble, notre recherche vise à apporter une contribution significative à la poursuite de modèles génératifs plus efficaces et flexibles, avec le potentiel de façonner l’avenir de la vision par ordinateur.

4D Imaging Radar is increasingly becoming a critical component in various industries due to beamforming technology and hardware advancements. However, it does not replace visual data in the form of 2D images captured by an RGB camera. Instead, 4D radar point clouds are a complementary data source that captures spatial information and velocity in a Doppler dimension that cannot be easily captured by a camera's view alone. Some discriminative features of the scene captured by the two sensors are hypothesized to have a shared representation. Therefore, a more interpretable visualization of the radar output can be obtained by learning a mapping from the empirical distribution of the radar to the distribution of images captured by the camera. To this end, the application of deep generative models to generate images conditioned on 4D radar data is explored. Two approaches that have become state-of-the-art in recent years are tested, generative adversarial networks and diffusion models. They are compared qualitatively through visual inspection and by two quantitative metrics: mean squared error and object detection count. It is found that it is easier to control the generative adversarial network's generative process through conditioning than in a diffusion process. In contrast, the diffusion model produces samples of higher quality and is more stable to train. Furthermore, their combination results in a hybrid sampling method, achieving the best results while simultaneously speeding up the diffusion process. / 4D bildradar får en alltmer betydande roll i olika industrier tack vare utveckling inom strålformningsteknik och hårdvara. Det ersätter dock inte visuell data i form av 2D-bilder som fångats av en RGB-kamera. Istället utgör 4D radar-punktmoln en kompletterande datakälla som representerar spatial information och hastighet i form av en Doppler-dimension. Det antas att vissa beskrivande egenskaper i den observerade miljön har en abstrakt representation som de två sensorerna delar. Därmed kan radar-datan visualiseras mer intuitivt genom att lära en transformation från fördelningen över radar-datan till fördelningen över bilderna. I detta syfte utforskas tillämpningen av djupa generativa modeller för bilder som är betingade av 4D radar-data. Två metoder som har blivit state-of-the-art de senaste åren testas: generativa antagonistiska nätverk och diffusionsmodeller. De jämförs kvalitativt genom visuell inspektion och med kvantitativa metriker: medelkvadratfelet och antalet korrekt detekterade objekt i den genererade bilden. Det konstateras att det är lättare att styra den generativa processen i generativa antagonistiska nätverk genom betingning än i en diffusionsprocess. Å andra sidan är diffusionsmodellen stabil att träna och producerar generellt bilder av högre kvalité. De bästa resultaten erhålls genom en hybrid: båda metoderna kombineras för att dra nytta av deras respektive styrkor. de identifierade begränsningarna i de enskilda modellerna och kurera datan för att jämföra hur dessa modeller skalar med större datamängder och mer variation.

<p dir="ltr">In Chapter 1, we introduce geometric motions from the general perspective of gradient flows. Here we develop the basic framework in which to pose the two main results of this thesis.</p><p dir="ltr">In Chapter 2, we examine the pinch-off phenomenon for a tubular surface moving by surface diffusion. We prove the existence of a one parameter family of pinching profiles obeying a long wavelength approximation of the dynamics.</p><p dir="ltr">In Chapter 3, we study a diffusion-based numerical scheme for curve shortening flow. We prove that the scheme is one time-step consistent.</p>

While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.

While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.