Destination-Constrained Linear Dynamical System Modeling in Set-Valued Frameworks

Directional motion towards a specified destination is a common occurrence in physical processes and human societal activities. Utilizing this prior information can significantly improve the control and predictive performance of system models. This paper primarily focuses on reconstructing linear dynamic system models based on destination constraints in the set-valued framework. We treat destination constraints as inherent information in the state evolution process and employ convex optimization techniques to construct a coherent and robust state model. This refined model effectively captures the impact of destination constraints on the state evolution at each time step. Furthermore, we design an optimal weight matrix for the reconstructed model to ensure smoother and more natural trajectories of state evolution. We also analyze the theoretical guarantee of optimality for this weight matrix and the properties of the reconstructed model. Finally, simulation experiments verify that the reconstructed model has significant advantages over the unconstrained and unoptimized weighted models and constrains the evolution of state trajectories with different starting and ending points.

Color Image Denoising Using The Green Channel Prior

Noise removal in the standard RGB (sRGB) space remains a challenging task, in that the noise statistics of real-world images can be different in R, G and B channels. In fact, the green channel usually has twice the sampling rate in raw data and a higher signal-to-noise ratio than red/blue ones. However, the green channel prior (GCP) is often understated or ignored in color image denoising since many existing approaches mainly focus on modeling the relationship among image patches. In this paper, we propose a simple and effective one step GCP-based image denoising (GCP-ID) method, which aims to exploit the GCP for denoising in the sRGB space by integrating it into the classic nonlocal transform domain denoising framework. Briefly, we first take advantage of the green channel to guide the search of similar patches, which improves the patch search quality and encourages sparsity in the transform domain. Then we reformulate RGB patches into RGGB arrays to explicitly characterize the density of green samples. The block circulant representation is utilized to capture the cross-channel correlation and the channel redundancy. Experiments on both synthetic and real-world datasets demonstrate the competitive performance of the proposed GCP-ID method for the color image and video denoising tasks. The code is available at github.com/ZhaomingKong/GCP-ID.

A Comparison of Image Denoising Methods

The advancement of imaging devices and countless images generated everyday pose an increasingly high demand on image denoising, which still remains a challenging task in terms of both effectiveness and efficiency. To improve denoising quality, numerous denoising techniques and approaches have been proposed in the past decades, including different transforms, regularization terms, algebraic representations and especially advanced deep neural network (DNN) architectures. Despite their sophistication, many methods may fail to achieve desirable results for simultaneous noise removal and fine detail preservation. In this paper, to investigate the applicability of existing denoising techniques, we compare a variety of denoising methods on both synthetic and real-world datasets for different applications. We also introduce a new dataset for benchmarking, and the evaluations are performed from four different perspectives including quantitative metrics, visual effects, human ratings and computational cost. Our experiments demonstrate: (i) the effectiveness and efficiency of representative traditional denoisers for various denoising tasks, (ii) a simple matrix-based algorithm may be able to produce similar results compared with its tensor counterparts, and (iii) the notable achievements of DNN models, which exhibit impressive generalization ability and show state-of-the-art performance on various datasets. In spite of the progress in recent years, we discuss shortcomings and possible extensions of existing techniques. Datasets, code and results are made publicly available and will be continuously updated at https://github.com/ZhaomingKong/Denoising-Comparison.

A Comprehensive Comparison of Multi-Dimensional Image Denoising Methods

Filtering multi-dimensional images such as color images, color videos, multispectral images and magnetic resonance images is challenging in terms of both effectiveness and efficiency. Leveraging the nonlocal self-similarity (NLSS) characteristic of images and sparse representation in the transform domain, the block-matching and 3D filtering (BM3D) based methods show powerful denoising performance. Recently, numerous new approaches with different regularization terms, transforms and advanced deep neural network (DNN) architectures are proposed to improve denoising quality. In this paper, we extensively compare over 60 methods on both synthetic and real-world datasets. We also introduce a new color image and video dataset for benchmarking, and our evaluations are performed from four different perspectives including quantitative metrics, visual effects, human ratings and computational cost. Comprehensive experiments demonstrate: (i) the effectiveness and efficiency of the BM3D family for various denoising tasks, (ii) a simple matrix-based algorithm could produce similar results compared with its tensor counterparts, and (iii) several DNN models trained with synthetic Gaussian noise show state-of-the-art performance on real-world color image and video datasets. Despite the progress in recent years, we discuss shortcomings and possible extensions of existing techniques. Datasets and codes for evaluation are made publicly available at https://github.com/ZhaomingKong/Denoising-Comparison.

Randomized Online CP Decomposition

CANDECOMP/PARAFAC (CP) decomposition has been widely used to deal with multi-way data. For real-time or large-scale tensors, based on the ideas of randomized-sampling CP decomposition algorithm and online CP decomposition algorithm, a novel CP decomposition algorithm called randomized online CP decomposition (ROCP) is proposed in this paper. The proposed algorithm can avoid forming full Khatri-Rao product, which leads to boost the speed largely and reduce memory usage. The experimental results on synthetic data and real-world data show the ROCP algorithm is able to cope with CP decomposition for large-scale tensors with arbitrary number of dimensions. In addition, ROCP can reduce the computing time and memory usage dramatically, especially for large-scale tensors.

Sparse Density Estimation with Measurement Errors

This paper aims to build an estimate of an unknown density of the data with measurement error as a linear combination of functions of a dictionary. Inspired by penalization approach, we propose the weighted Elastic-net penalized minimal $L_2$-distance method for sparse coefficients estimation, where the weights adaptively coming from sharp concentration inequalities. The optimal weighted tuning parameters are obtained by the first-order conditions holding with high-probability. Under local coherence or minimal eigenvalue assumptions, non-asymptotical oracle inequalities are derived. These theoretical results are transposed to obtain the support recovery with high-probability. Then, the issue of calibrating these procedures is studied by some numerical experiments for discrete and continuous distributions, it shows the significant improvement obtained by our procedure when compared with other conventional approaches. Finally, the application is performed for a meteorology data set. It shows that our method has potency and superiority of detecting the shape of multi-mode density compared with other conventional approaches.

Color Image and Multispectral Image Denoising Using Block Diagonal Representation

Filtering images of more than one channel is challenging in terms of both efficiency and effectiveness. By grouping similar patches to utilize the self-similarity and sparse linear approximation of natural images, recent nonlocal and transform-domain methods have been widely used in color and multispectral image (MSI) denoising. Many related methods focus on the modeling of group level correlation to enhance sparsity, which often resorts to a recursive strategy with a large number of similar patches. The importance of the patch level representation is understated. In this paper, we mainly investigate the influence and potential of representation at patch level by considering a general formulation with block diagonal matrix. We further show that by training a proper global patch basis, along with a local principal component analysis transform in the grouping dimension, a simple transform-threshold-inverse method could produce very competitive results. Fast implementation is also developed to reduce computational complexity. Extensive experiments on both simulated and real datasets demonstrate its robustness, effectiveness and efficiency.

A Brief Review of Real-World Color Image Denoising

Filtering real-world color images is challenging due to the complexity of noise that can not be formulated as a certain distribution. However, the rapid development of camera lens pos- es greater demands on image denoising in terms of both efficiency and effectiveness. Currently, the most widely accepted framework employs the combination of transform domain techniques and nonlocal similarity characteristics of natural images. Based on this framework, many competitive methods model the correlation of R, G, B channels with pre-defined or adaptively learned transforms. In this chapter, a brief review of related methods and publicly available datasets is presented, moreover, a new dataset that includes more natural outdoor scenes is introduced. Extensive experiments are performed and discussion on visual effect enhancement is included.

DuSK: A Dual Structure-preserving Kernel for Supervised Tensor Learning with Applications to Neuroimages

With advances in data collection technologies, tensor data is assuming increasing prominence in many applications and the problem of supervised tensor learning has emerged as a topic of critical significance in the data mining and machine learning community. Conventional methods for supervised tensor learning mainly focus on learning kernels by flattening the tensor into vectors or matrices, however structural information within the tensors will be lost. In this paper, we introduce a new scheme to design structure-preserving kernels for supervised tensor learning. Specifically, we demonstrate how to leverage the naturally available structure within the tensorial representation to encode prior knowledge in the kernel. We proposed a tensor kernel that can preserve tensor structures based upon dual-tensorial mapping. The dual-tensorial mapping function can map each tensor instance in the input space to another tensor in the feature space while preserving the tensorial structure. Theoretically, our approach is an extension of the conventional kernels in the vector space to tensor space. We applied our novel kernel in conjunction with SVM to real-world tensor classification problems including brain fMRI classification for three different diseases (i.e., Alzheimer's disease, ADHD and brain damage by HIV). Extensive empirical studies demonstrate that our proposed approach can effectively boost tensor classification performances, particularly with small sample sizes.