Higher-order Projected Power Iterations for Scalable Multi-Matching

The matching of multiple objects (e.g. shapes or images) is a fundamental problem in vision and graphics. In order to robustly handle ambiguities, noise and repetitive patterns in challenging real-world settings, it is essential to take geometric consistency between points into account. Computationally, the multi-matching problem is difficult. It can be phrased as simultaneously solving multiple (NP-hard) quadratic assignment problems (QAPs) that are coupled via cycle-consistency constraints. The main limitations of existing multi-matching methods are that they either ignore geometric consistency and thus have limited robustness, or they are restricted to small-scale problems due to their (relatively) high computational cost. We address these shortcomings by introducing a Higher-order Projected Power Iteration method, which is (i) efficient and scales to tens of thousands of points, (ii) straightforward to implement, (iii) able to incorporate geometric consistency, (iv) guarantees cycle-consistent multi-matchings, and (iv) comes with theoretical convergence guarantees. Experimentally we show that our approach is superior to existing methods

Geometric deep learning

The goal of these course notes is to describe the main mathematical ideas behind geometric deep learning and to provide implementation details for several applications in shape analysis and synthesis, computer vision and computer graphics. The text in the course materials is primarily based on previously published work. With these notes we gather and provide a clear picture of the key concepts and techniques that fall under the umbrella of geometric deep learning, and illustrate the applications they enable. We also aim to provide practical implementation details for the methods presented in these works, as well as suggest further readings and extensions of these ideas

Non-rigid Point Cloud Registration with Neural Deformation Pyramid

Non-rigid point cloud registration is a key component in many computer vision and computer graphics applications. The high complexity of the unknown non-rigid motion make this task a challenging problem. In this paper, we break down this problem via hierarchical motion decomposition. Our method called Neural Deformation Pyramid (NDP) represents non-rigid motion using a pyramid architecture. Each pyramid level, denoted by a Multi-Layer Perception (MLP), takes as input a sinusoidally encoded 3D point and outputs its motion increments from the previous level. The sinusoidal function starts with a low input frequency and gradually increases when the pyramid level goes down. This allows a multi-level rigid to nonrigid motion decomposition and also speeds up the solving by 50 times compared to the existing MLP-based approach. Our method achieves advanced partialto-partial non-rigid point cloud registration results on the 4DMatch/4DLoMatch benchmark under both no-learned and supervised settings.Comment: Code: https://github.com/rabbityl/DeformationPyrami

On the Whitney distortion extension problem for Cm(Rn)C^m(\mathbb R^n) and C∞(Rn)C^{\infty}(\mathbb R^n) and its applications to interpolation and alignment of data in Rn\mathbb R^n

Let $n,m\geq 1$, $U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $\phi:U\to \mathbb R^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $\phi$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^m(\mathbb R^n)$ one-to-one and onto almost isometry $\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. (b) Let $\phi:U\to \mathbb R^n$ be $C^{\infty}$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $\phi$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^{\infty}(\mathbb R^n)$ one-to-one and onto almost isometry $\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. Our results complement those of [14,15,20] where there, $E$ is a finite set. In this case, the problem above is also a problem of interpolation and alignment of data in $\mathbb R^n$.Comment: This is part three of four papers with C. Fefferman (arXiv:1411.2451, arXiv:1411.2468, involve-v5-n2-p03-s.pdf) dealing with the problem of Whitney type extensions of $\delta>0$ distortions from certain compact sets $E\subset \Bbb R^n$ to $\varepsilon>0$ distorted diffeomorphisms on $\Bbb R^n

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

Probabilistic correspondence analysis for neuroimaging problems

Establecer correspondencias de forma significativas entre los objetivos como en los problemas de neuroimagen es crucial para mejorar los procesos de correspondencia. Por ejemplo, el problema de correspondencia consiste en encontrar relaciones significativas entre cualquier par de estructuras cerebrales como en el problema de registro estático, o analizar cambios temporales de una enfermedad neurodegenerativa dada a través del tiempo para un análisis dinámico de la forma del cerebro..

Multimodal Investigation of Brain Network Systems: From Brain Structure and Function to Connectivity and Neuromodulation

The field of cognitive neuroscience has benefited greatly from multimodal investigations of the human brain, which integrate various tools and neuroimaging data to understand brain functions and guide treatments for brain disorders. In this dissertation, we present a series of studies that illustrate the use of multimodal approaches to investigate brain structure and function, brain connectivity, and neuromodulation effects. Firstly, we propose a novel landmark-guided region-based spatial normalization technique to accurately quantify brain morphology, which can improve the sensitivity and specificity of functional imaging studies. Subsequently, we shift the investigation to the characteristics of functional brain activity due to visual stimulations. Our findings reveal that the task-evoked positive blood-oxygen-level dependent (BOLD) response is accompanied by sustained negative BOLD responses in the visual cortex. These negative BOLD responses are likely generated through subcortical neuromodulatory systems with distributed ascending projections to the cortex. To further explore the cortico-subcortical relationship, we conduct a multimodal investigation that involves simultaneous data acquisition of pupillometry, electroencephalography (EEG), and functional magnetic resonance imaging (fMRI). This investigation aims to examine the connectivity of brain circuits involved in the cognitive processes of salient stimuli. Using pupillary response as a surrogate measure of activity in the locus coeruleus-norepinephrine system, we find that the pupillary response is associated with the reorganization of functional brain networks during salience processing. In addition, we propose a cortico-subcortical integrated network reorganization model with potential implications for understanding attentional processing and network switching. Lastly, we employ a multimodal investigation that involves concurrent transcranial magnetic stimulation (TMS), EEG, and fMRI to explore network perturbations and measurements of the propagation effects. In a preliminary exploration on brain-state dependency of TMS-induced effects, we find that the propagation of left dorsolateral prefrontal cortex TMS to regions in the lateral frontoparietal network might depend on the brain-state, as indexed by the EEG prefrontal alpha phase. Overall, the studies in this dissertation contribute to the understanding of the structural and functional characteristics of brain network systems, and may inform future investigations that use multimodal methodological approaches, such as pupillometry, brain connectivity, and neuromodulation tools. The work presented in this dissertation has potential implications for the development of efficient and personalized treatments for major depressive disorder, attention deficit hyperactivity disorder, and Alzheimer's disease