Jumping Manifolds: Geometry Aware Dense Non-Rigid Structure from Motion

Given dense image feature correspondences of a non-rigidly moving object across multiple frames, this paper proposes an algorithm to estimate its 3D shape for each frame. To solve this problem accurately, the recent state-of-the-art algorithm reduces this task to set of local linear subspace reconstruction and clustering problem using Grassmann manifold representation \cite{kumar2018scalable}. Unfortunately, their method missed on some of the critical issues associated with the modeling of surface deformations, for e.g., the dependence of a local surface deformation on its neighbors. Furthermore, their representation to group high dimensional data points inevitably introduce the drawbacks of categorizing samples on the high-dimensional Grassmann manifold \cite{huang2015projection, harandi2014manifold}. Hence, to deal with such limitations with \cite{kumar2018scalable}, we propose an algorithm that jointly exploits the benefit of high-dimensional Grassmann manifold to perform reconstruction, and its equivalent lower-dimensional representation to infer suitable clusters. To accomplish this, we project each Grassmannians onto a lower-dimensional Grassmann manifold which preserves and respects the deformation of the structure w.r.t its neighbors. These Grassmann points in the lower-dimension then act as a representative for the selection of high-dimensional Grassmann samples to perform each local reconstruction. In practice, our algorithm provides a geometrically efficient way to solve dense NRSfM by switching between manifolds based on its benefit and usage. Experimental results show that the proposed algorithm is very effective in handling noise with reconstruction accuracy as good as or better than the competing methods.Comment: New version with corrected typo. 10 Pages, 7 Figures, 1 Table. Accepted for publication in IEEE Conference on Computer Vision and Pattern Recognition (CVPR) 2019. Acknowledgement added. Supplementary material is available at https://suryanshkumar.github.io

A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold

Although Deep Learning (DL) has achieved success in complex Artificial Intelligence (AI) tasks, it suffers from various notorious problems (e.g., feature redundancy, and vanishing or exploding gradients), since updating parameters in Euclidean space cannot fully exploit the geometric structure of the solution space. As a promising alternative solution, Riemannian-based DL uses geometric optimization to update parameters on Riemannian manifolds and can leverage the underlying geometric information. Accordingly, this article presents a comprehensive survey of applying geometric optimization in DL. At first, this article introduces the basic procedure of the geometric optimization, including various geometric optimizers and some concepts of Riemannian manifold. Subsequently, this article investigates the application of geometric optimization in different DL networks in various AI tasks, e.g., convolution neural network, recurrent neural network, transfer learning, and optimal transport. Additionally, typical public toolboxes that implement optimization on manifold are also discussed. Finally, this article makes a performance comparison between different deep geometric optimization methods under image recognition scenarios.Comment: 41 page

Study of compression techniques for partial differential equation solvers

Partial Differential Equations (PDEs) are widely applied in many branches of science, and solving them efficiently, from a computational point of view, is one of the cornerstones of modern computational science. The finite element (FE) method is a popular numerical technique for calculating approximate solutions to PDEs. A not necessarily complex finite element analysis containing substructures can easily gen-erate enormous quantities of elements that hinder and slow down simulations. Therefore, compression methods are required to decrease the amount of computational effort while retaining the significant dynamics of the problem. In this study, it was decided to apply a purely algebraic approach. Various methods will be included and discussed, ranging from research-level techniques to other apparently unrelated fields like image compression, via the discrete Fourier transform (DFT) and the Wavelet transform or the Singular Value Decomposition (SVD)

Sparsity Preserving Discriminant Projections with Applications to Face Recognition

Dimensionality reduction is extremely important for understanding the intrinsic structure hidden in high-dimensional data. In recent years, sparse representation models have been widely used in dimensionality reduction. In this paper, a novel supervised learning method, called Sparsity Preserving Discriminant Projections (SPDP), is proposed. SPDP, which attempts to preserve the sparse representation structure of the data and maximize the between-class separability simultaneously, can be regarded as a combiner of manifold learning and sparse representation. Specifically, SPDP first creates a concatenated dictionary by classwise PCA decompositions and learns the sparse representation structure of each sample under the constructed dictionary using the least square method. Secondly, a local between-class separability function is defined to characterize the scatter of the samples in the different submanifolds. Then, SPDP integrates the learned sparse representation information with the local between-class relationship to construct a discriminant function. Finally, the proposed method is transformed into a generalized eigenvalue problem. Extensive experimental results on several popular face databases demonstrate the feasibility and effectiveness of the proposed approach

Unifying the Visible and Passive Infrared Bands: Homogeneous and Heterogeneous Multi-Spectral Face Recognition

Face biometrics leverages tools and technology in order to automate the identification of individuals. In most cases, biometric face recognition (FR) can be used for forensic purposes, but there remains the issue related to the integration of technology into the legal system of the court. The biggest challenge with the acceptance of the face as a modality used in court is the reliability of such systems under varying pose, illumination and expression, which has been an active and widely explored area of research over the last few decades (e.g. same-spectrum or homogeneous matching). The heterogeneous FR problem, which deals with matching face images from different sensors, should be examined for the benefit of military and law enforcement applications as well. In this work we are concerned primarily with visible band images (380-750 nm) and the infrared (IR) spectrum, which has become an area of growing interest.;For homogeneous FR systems, we formulate and develop an efficient, semi-automated, direct matching-based FR framework, that is designed to operate efficiently when face data is captured using either visible or passive IR sensors. Thus, it can be applied in both daytime and nighttime environments. First, input face images are geometrically normalized using our pre-processing pipeline prior to feature-extraction. Then, face-based features including wrinkles, veins, as well as edges of facial characteristics, are detected and extracted for each operational band (visible, MWIR, and LWIR). Finally, global and local face-based matching is applied, before fusion is performed at the score level. Although this proposed matcher performs well when same-spectrum FR is performed, regardless of spectrum, a challenge exists when cross-spectral FR matching is performed. The second framework is for the heterogeneous FR problem, and deals with the issue of bridging the gap across the visible and passive infrared (MWIR and LWIR) spectrums. Specifically, we investigate the benefits and limitations of using synthesized visible face images from thermal and vice versa, in cross-spectral face recognition systems when utilizing canonical correlation analysis (CCA) and locally linear embedding (LLE), a manifold learning technique for dimensionality reduction. Finally, by conducting an extensive experimental study we establish that the combination of the proposed synthesis and demographic filtering scheme increases system performance in terms of rank-1 identification rate

Principal Component Analysis

This book is aimed at raising awareness of researchers, scientists and engineers on the benefits of Principal Component Analysis (PCA) in data analysis. In this book, the reader will find the applications of PCA in fields such as image processing, biometric, face recognition and speech processing. It also includes the core concepts and the state-of-the-art methods in data analysis and feature extraction

Robust Face Recognition based on Color and Depth Information

One of the most important advantages of automatic human face recognition is its nonintrusiveness property. Face images can sometime be acquired without user's knowledge or explicit cooperation. However, face images acquired in an uncontrolled environment can appear with varying imaging conditions. Traditionally, researchers focus on tackling this problem using 2D gray-scale images due to the wide availability of 2D cameras and the low processing and storage cost of gray-scale data. Nevertheless, face recognition can not be performed reliably with 2D gray-scale data due to insu_cient information and its high sensitivity to pose, expression and illumination variations. Recent rapid development in hardware makes acquisition and processing of color and 3D data feasible. This thesis aims to improve face recognition accuracy and robustness using color and 3D information.In terms of color information usage, this thesis proposes several improvements over existing approaches. Firstly, the Block-wise Discriminant Color Space is proposed, which learns the discriminative color space based on local patches of a human face image instead of the holistic image, as human faces display different colors in different parts. Secondly, observing that most of the existing color spaces consist of at most three color components, while complementary information can be found in multiple color components across multiple color spaces and therefore the Multiple Color Fusion model is proposed to search and utilize multiple color components effectively. Lastly, two robust color face recognition algorithms are proposed. The Color Sparse Coding method can deal with face images with noise and occlusion. The Multi-linear Color Tensor Discriminant method harnesses multi-linear technique to handle non-linear data. Experiments show that all the proposed methods outperform their existing competitors.In terms of 3D information utilization, this thesis investigates the feasibility of face recognition using Kinect. Unlike traditional 3D scanners which are too slow in speed and too expensive in cost for broad face recognition applications, Kinect trades data quality for high speed and low cost. An algorithm is proposed to show that Kinect data can be used for face recognition despite its noisy nature. In order to fully utilize Kinect data, a more sophisticated RGB-D face recognition algorithm is developed which harnesses theColor Sparse Coding framework and 3D information to perform accurate face recognition robustly even under simultaneous varying conditions of poses, illuminations, expressionsand disguises