A Theoretically Guaranteed Quaternion Weighted Schatten p-norm Minimization Method for Color Image Restoration

Inspired by the fact that the matrix formulated by nonlocal similar patches in a natural image is of low rank, the rank approximation issue have been extensively investigated over the past decades, among which weighted nuclear norm minimization (WNNM) and weighted Schatten $p$-norm minimization (WSNM) are two prevailing methods have shown great superiority in various image restoration (IR) problems. Due to the physical characteristic of color images, color image restoration (CIR) is often a much more difficult task than its grayscale image counterpart. However, when applied to CIR, the traditional WNNM/WSNM method only processes three color channels individually and fails to consider their cross-channel correlations. Very recently, a quaternion-based WNNM approach (QWNNM) has been developed to mitigate this issue, which is capable of representing the color image as a whole in the quaternion domain and preserving the inherent correlation among the three color channels. Despite its empirical success, unfortunately, the convergence behavior of QWNNM has not been strictly studied yet. In this paper, on the one side, we extend the WSNM into quaternion domain and correspondingly propose a novel quaternion-based WSNM model (QWSNM) for tackling the CIR problems. Extensive experiments on two representative CIR tasks, including color image denoising and deblurring, demonstrate that the proposed QWSNM method performs favorably against many state-of-the-art alternatives, in both quantitative and qualitative evaluations. On the other side, more importantly, we preliminarily provide a theoretical convergence analysis, that is, by modifying the quaternion alternating direction method of multipliers (QADMM) through a simple continuation strategy, we theoretically prove that both the solution sequences generated by the QWNNM and QWSNM have fixed-point convergence guarantees.Comment: 46 pages, 10 figures; references adde

Robust online subspace learning

In this thesis, I aim to advance the theories of online non-linear subspace learning through the development of strategies which are both efficient and robust. The use of subspace learning methods is very popular in computer vision and they have been employed to numerous tasks. With the increasing need for real-time applications, the formulation of online (i.e. incremental and real-time) learning methods is a vibrant research field and has received much attention from the research community. A major advantage of incremental systems is that they update the hypothesis during execution, thus allowing for the incorporation of the real data seen in the testing phase. Tracking acts as an attractive and popular evaluation tool for incremental systems, and thus, the connection between online learning and adaptive tracking is seen commonly in the literature. The proposed system in this thesis facilitates learning from noisy input data, e.g. caused by occlusions, casted shadows and pose variations, that are challenging problems in general tracking frameworks. First, a fast and robust alternative to standard L2-norm principal component analysis (PCA) is introduced, which I coin Euler PCA (e-PCA). The formulation of e-PCA is based on robust, non-linear kernel PCA (KPCA) with a cosine-based kernel function that is expressed via an explicit feature space. When applied to tracking, face reconstruction and background modeling, promising results are achieved. In the second part, the problem of matching vectors of 3D rotations is explicitly targeted. A novel distance which is robust for 3D rotations is introduced, and formulated as a kernel function. The kernel leads to a new representation of 3D rotations, the full-angle quaternion (FAQ) representation. Finally, I propose 3D object recognition from point clouds, and object tracking with color values using FAQs. A domain-specific kernel function designed for visual data is then presented. KPCA with Krein space kernels is introduced, as this kernel is indefinite, and an exact incremental learning framework for the new kernel is developed. In a tracker framework, the presented online learning outperforms the competitors in nine popular and challenging video sequences. In the final part, the generalized eigenvalue problem is studied. Specifically, incremental slow feature analysis (SFA) with indefinite kernels is proposed, and applied to temporal video segmentation and tracking with change detection. As online SFA allows for drift detection, further improvements are achieved in the evaluation of the tracking task.Open Acces

Latent variable regression and applications to planetary seismic instrumentation

The work presented in this thesis is framed by the concept of latent variables, a modern data analytics approach. A latent variable represents an extracted component from a dataset which is not directly measured. The concept is first applied to combat the problem of ill-posed regression through the promising method of partial least squares (PLS). In this context the latent variables within a data matrix are extracted through an iterative algorithm based on cross-covariance as an optimisation criterion. This work first extends the PLS algorithm, using adaptive and recursive techniques, for online, non-stationary data applications. The standard PLS algorithm is further generalised for complex-, quaternion- and tensor-valued data. In doing so it is shown that the multidimensional algebras facilitate physically meaningful representations, demonstrated through smart-grid frequency estimation and image-classification tasks. The second part of the thesis uses this knowledge to inform a performance analysis of the MEMS microseismometer implemented for the InSight mission to Mars. This is given in terms of the sensor's intrinsic self-noise, the estimation of which is achieved from experimental data with a colocated instrument. The standard coherence and proposed delta noise estimators are analysed with respect to practical issues. The implementation of algorithms for the alignment, calibration and post-processing of the data then enabled a definitive self-noise estimate, validated from data acquired in ultra-quiet, deep-space environment. A method for the decorrelation of the microseismometer's output from its thermal response is proposed. To do so a novel sensor fusion approach based on the Kalman filter is developed for a full-band transfer-function correction, in contrast to the traditional ill-posed frequency division method. This algorithm was applied to experimental data which determined the thermal model coefficients while validating the sensor's performance at tidal frequencies 1E-5Hz and in extreme environments at -65C. This thesis, therefore, provides a definitive view of the latent variables perspective. This is achieved through the general algorithms developed for regression with multidimensional data and the bespoke application to seismic instrumentation.Open Acces

Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Nearest neighbor search for cyro-electron microscopy images

Cryo-electron microscopy (EM) technology enables researchers to determine the structure of a molecule at a much higher resolution than ever before. The single particle reconstruction (SPR) is one of the most widespread techniques used to reconstruct the 3D model of a molecule from its large set of cryo-EM projection images with unknown viewing angles. Classifying those projection images with similar views is regarded as one of the significant steps in the SPR problem. The goal in our research thesis is to find an effective way to identify and classify cryo-EM images solely based on their viewing angles. In the first part of the thesis, we try four different k-nearest neighbors search (KNNS) algorithms on cryo-EM projection images classifying them based on the viewing directions. Brute force search (BFS) could be very time consuming when the dataset is large. Therefore, strategies like locality sensitive hashing (LSH) and some neural network methods are taken into consideration. The LSH method maps similar items into the same cell in a hash table, which effectively shortens the query time of searching for k-nearest neighbors, while the neural network methods are able to capture more expressive features, and then the classification is performed according to these learned features. The four KNNS algorithms include hyperplane LSH, cross-polytope LSH, unsupervised stochastic generative hashing (SGH) as well as unsupervised deep hashing (DH). Their performances are analyzed by the accuracy for finding 100 near neighbors for each query data, construction time and query time. However, all four algorithms fail to accurately classify projection images under similar views because of the orientation difference. Thus, further preprocessing techniques are required. One of the techniques we take is augmenting the image dataset by generating more planar rotation copies for each projection image in a way that allows more image candidates under the same view observable. It shows that with image augmentation, the accuracies of all four methods increase. The LSH methods provide rather good performances both in computation time and the accuracy of searching for 100 nearest neighbors if the dataset is clean. However, for the noisy dataset, the traditional PCA is not working, and other preprocessing techniques should be applied to avoid color noise caused by the rotational image copies. In the second part of the thesis, we attempt to devise a neural network model for learning rotation invariant features of the cryo-EM images as well as reconstructing them from the invariant features. After obtaining the desired rotation-invariant features, we can use the LSH method to do the classification since the trained features are distinguished solely depending on the viewing direction. In addition, this model possibly could be used in recovering the 3D molecule structure directly from 2D projection images, which will be explained in details in Section 3.3.7. However, since the image is often of large size, this 3D model reconstruction could be a highly complex computation problem. Currently, we begin with building up a simple version of this model and test with 1D signals. It shows that this model works well for small-length signals. However, when the signal length gets larger, it becomes harder to optimize the model and the result is less satisfying. Therefore, in the future we may need to adjust the architecture of the neural network model and tune the hyper-parameters to make the model adapt to a more general case