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 ℓ 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.

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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). Like the analytic signal, 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 Preprint submitted to Multidimensional Systems and Signal Processing arXiv:1607.00097v2 [math.CA] 16 Dec 2016 competing methods on real-world images consistently show the superiority of the proposed methods.1. We give the solution of the problem: if the higher dimensional signal is not intrinsically 1D signal, we derive the relationship between the local phase-vector and and local attenuation.2. We proposed the local attenuation (LA) method for edge detection operator. We establish the theoretical and experiment results on the newly methods.3. We proposed the modified differential phase congruency (MDPC) method for the edge detection operator. We establish the theoretical and experiment results on the newly methods.4. We show that in higher dimensional space, the instantaneous frequency

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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 timelimiting data.

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.

One of the fundamental problems in communications is finding the energy distribution of signals in time and frequency domains. It should therefore be of great interest to find the quaternionic signal whose time‐frequency energy distribution is most concentrated in a given time‐frequency domain. The present paper finds a new kind of quaternionic signals whose energy concentration is maximal in both time and frequency under the quaternionic Fourier transform. The new signals are a generalization of the classical prolate spheroidal wave functions to a quaternionic space, which are called the quaternionic prolate spheroidal wave functions. The purpose of this paper is to present the definition and fundamental properties of the quaternionic prolate spheroidal wave functions and to show that they can reach the extreme case within the energy concentration problem both from the theoretical and experimental description. The superiority of the proposed results can be widely applied to the application of 4D valued problems. In particular, these functions are shown as an effective method for bandlimited quaternionic signals relying on the extrapolation problem. Copyright © 2017 John Wiley & Sons, Ltd.

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