/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

dix A. Even if you don’t choose to memorize them this system aids in reference and retreival of important formulas. The book was compiled from notes developed during eight years of teaching a graduate course on the subject and was used as a text. Thus it has been student tested. Appendix F contains a number of problems, grouped to be used on a chapter by chapter basis The problems are designed to illustrate practical applications of the text material. The parts of this book of most interest and value to the EMC engineer will be the chapters on Thermal Noise, Antennas, Propagation and Transmission Lines, and  Reflection and Refraction. This is not  to downpade the chapters on Statistics and Its Applications, Signal Processing and Detection, and Some System Characteristics which also contain much potentially useful materials. Additional plus values for the book include a list of 40 references, a table of symbols used throughout the book, and a subject index. Some readers may find the condensed type and close line spacing hard to read. It was apparently set up by typewriter using an elite type face with single line spacing. When reduced down to a 6 by 9 5 inch size page it is too crowded for easy reading. In spite of this shortcoming your reviewer recommends this book as a worthwhile reference in this field of interest.

http://ieeexplore.ieee.org/iel5/5/31269/01455368.pdf

Signal analysis

In this paper, we investigate the global stability of quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of quaternion multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: $ij=-ji=k,~jk=-kj=i$ , $ki=-ik=j$ , $i^{2}=j^{2}=k^{2}=ijk=-1$ . With the Lyapunov function method, some criteria are, respectively, presented to ensure the global $\mu $ -stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of quaternion multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where quaternion self-conjugate matrices and quaternion positive definite matrices are used. Some new sufficient conditions in the form of quaternion-valued LMI are, respectively, established for the global $\mu $ -stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.

Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches

Low-rank matrix approximation (LRMA)-based methods have made a great success for grayscale image processing. When handling color images, LRMA either restores each color channel independently using the monochromatic model or processes the concatenation of three color channels using the concatenation model. However, these two schemes may not make full use of the high correlation among RGB channels. To address this issue, we propose a novel low-rank quaternion approximation (LRQA) model. It contains two major components: first, instead of modeling a color image pixel as a scalar in conventional sparse representation and LRMA-based methods, the color image is encoded as a pure quaternion matrix, such that the cross-channel correlation of color channels can be well exploited; second, LRQA imposes the low-rank constraint on the constructed quaternion matrix. To better estimate the singular values of the underlying low-rank quaternion matrix from its noisy observation, a general model for LRQA is proposed based on several nonconvex functions. Extensive evaluations for color image denoising and inpainting tasks verify that LRQA achieves better performance over several state-of-the-art sparse representation and LRMA-based methods in terms of both quantitative metrics and visual quality.

Low-Rank Quaternion Approximation for Color Image Processing

An edge detection is important for its reliability and security which delivers a better understanding of object recognition in the applications of computer vision, such as pedestrian detection, face detection, and video surveillance. This paper introduced two fundamental limitations encountered in edge detection: edge connectivity and edge thickness, those have been used by various developments in the state-of-the-art. An optimal selection of the threshold for effectual edge detection has constantly been a key challenge in computer vision. Therefore, a robust edge detection algorithm using multiple threshold approaches (B-Edge) is proposed to cover both the limitations. The majorly used canny edge operator focuses on two thresholds selections and still witnesses a few gaps for optimal results. To handle the loopholes of the canny edge operator, our method selects the simulated triple thresholds that target to the prime issues of the edge detection: image contrast, effective edge pixels selection, errors handling, and similarity to the ground truth. The qualitative and quantitative experimental evaluations demonstrate that our edge detection method outperforms competing algorithms for mentioned issues. The proposed approach endeavors an improvement for both grayscale and colored images.

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An Efficient Edge Detection Approach to Provide Better Edge Connectivity for Image Analysis

Collaborative representation-based classification (CRC) and sparse RC (SRC) have recently achieved great success in face recognition (FR). Previous CRC and SRC are originally designed in the real setting for grayscale image-based FR. They separately represent the color channels of a query color image and ignore the structural correlation information among the color channels. To remedy this limitation, in this paper, we propose two novel RC methods for color FR, namely, quaternion CRC (QCRC) and quaternion SRC (QSRC) using quaternion $\ell _{1}$ minimization. By modeling each color image as a quaternionic signal, they naturally preserve the color structures of both query and gallery color images while uniformly coding the query channel images in a holistic manner. Despite the empirical success of CRC and SRC on FR, a few theoretical results are developed to guarantee their effectiveness. Another purpose of this paper is to establish the theoretical guarantee for QCRC and QSRC under mild conditions. Comparisons with competing methods on benchmark real-world databases consistently show the superiority of the proposed methods for both color FR and reconstruction.

Quaternion Collaborative and Sparse Representation With Application to Color Face Recognition

The hypercomplex 2D analytic signal has been proposed by several authors with applications in color image processing. The analytic signal enables to extract local features from images. It has the fundamental property of splitting the identity, meaning that it separates qualitative and quantitative information of an image in form of the local phase and the local amplitude. The extension of analytic signal of linear canonical transform domain from 1D to 2D, covering also intrinsic 2D structures, has been proposed. We use this improved concept on envelope detector. The quaternion Fourier transform plays a vital role in the representation of multidimensional signals. The quaternion linear canonical transform (QLCT) is a well-known generalization of the quaternion Fourier transform. Some valuable properties of the two-sided QLCT are studied. Different approaches to the 2D quaternion Hilbert transforms are proposed that allow the calculation of the associated analytic signals, which can suppress the negative frequency components in the QLCT domains. As an application, examples of envelope detection demonstrate the effectiveness of our approach.

Envelope detection using generalized analytic signal in 2D QLCT domains

Linear regression has shown an effective tool for face recognition in recent years. Most existing linear regression based methods are devised for grayscale image based face recognition and fail to exploit the color information of color face images. To extend linear regression for color images, we propose a novel color face recognition method by formulating the color face recognition problem as a quaternion linear regression model. The proposed quaternion linear regression classification (QLRC) algorithm models each color facial image as a quaternion signal and codes multiple channels of each query color image in a holistic manner. Thus, the correlation among distinct channels of each color image is well preserved and leveraged by QLRC to further improve the recognition performance. To further improve QLRC, we propose a quaternion collaborative representation optimized classifier (QCROC) which integrates QLRC and quaternion collaborative representation based classifier into a unified framework. The experiments on benchmark datasets demonstrate the efficacy of the proposed approaches for color face recognition.

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From Grayscale to Color: Quaternion Linear Regression for Color Face Recognition

Monogenic signal is regarded as a generalization of analytic signal from one dimensional to higher dimensional space, which has been received considerable attention in the literature. It is defined by an original signal with its isotropic Hilbert transform (the combination of Riesz transform). Similar to analytic signal, the monogenic signal can be written in the polar form. Then it provides the signal features representation, such as the local attenuation and the local phase vector. The aim of the paper is twofold: first, to analyze the relationship between the local phase vector and the local attenuation in the higher dimensional spaces. Secondly, a study on image edge detection using modified differential phase congruency is presented. Comparisons with competing methods on real-world images consistently show the superiority of the proposed methods.

Edge detection methods based on modified differential phase congruency of monogenic signal

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time-limiting data. Copyright © 2016 John Wiley & Sons, Ltd.

Cuiming Zou

Papers

Quaternion Collaborative and Sparse Representation With Application to Color Face Recognition

Envelope detection using generalized analytic signal in 2D QLCT domains

From Grayscale to Color: Quaternion Linear Regression for Color Face Recognition

Edge detection methods based on modified differential phase congruency of monogenic signal

Uncertainty principle for measurable sets and signal recovery in quaternion domains