• KSII Transactions on Internet and Information Systems (TIIS)
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• v.9 no.12
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• pp.5058-5072
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• 2015

Quaternions have been commonly employed in color image processing, but when the existing pure quaternion representation for color images is used in perceptual hashing, it would degrade the robustness performance since it is sensitive to image manipulations. To improve the robustness in color image perceptual hashing, in this paper a full quaternion representation for color images is proposed by introducing the local image luminance variances. Based on this new representation, a novel Full Quaternion Discrete Cosine Transform (FQDCT)-based hashing is proposed, in which the Quaternion Discrete Cosine Transform (QDCT) is applied to the pseudo-randomly selected regions of the novel full quaternion image to construct two feature matrices. A new hash value in binary is generated from these two matrices. Our experimental results have validated the robustness improvement brought by the proposed full quaternion representation and demonstrated that better performance can be achieved in the proposed FQDCT-based hashing than that in other notable quaternion-based hashing schemes in terms of robustness and discriminability.

• The Pure and Applied Mathematics
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• v.19 no.3
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• pp.297-304
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• 2012

There are several types of orders in a Quaternion algebra. Generally, zeta functions defined on orders of a Quaternion algebra give some informations on the ideal theory of orders. In this study, we investigate functional equalities between the zeta functions defined on orders of a Quaternion algebra.

• The Pure and Applied Mathematics
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• v.22 no.1
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• pp.65-74
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• 2015

In this paper, we provide the notions of dual quaternions and their algebraic properties based on matrices. From quaternion analysis, we give the concept of a derivative of functions and and obtain a dual quaternion Cauchy-Riemann system that are equivalent. Also, we research properties of a regular function with values in dual quaternions and relations derivative with a regular function in dual quaternions.

• Journal of applied mathematics & informatics
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• v.34 no.1_2
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• pp.95-106
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• 2016

In this paper, according to the classical LSQR algorithm forsolving least squares (LS) problem, an iterative method is proposed for finding the minimum-norm pure imaginary solution of the quaternionic least squares (QLS) problem. By means of real representation of quaternion matrix, the QLS's correspongding vector algorithm is rewrited back to the matrix-form algorthm without Kronecker product and long vectors. Finally, numerical examples are reported that show the favorable numerical properties of the method.

• The Pure and Applied Mathematics
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• v.23 no.1
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• pp.97-104
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• 2016

We give representations of differential operators and rules for addition and multiplication of dual quaternions. Also, we research the notions and properties of a regular function and a corresponding harmonic function with values in dual quaternions of Clifford analysis.

• The Pure and Applied Mathematics
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• v.14 no.2 s.36
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• pp.91-98
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• 2007

We give definitions of hyperholomorphic functions of quaternionic functions of two quaternionic variables. We investigate properties of hyperholomorphic functions on quaternion analysis, and obtain equivalence relations for domains of hyperholomorphy and hyper Stein domains in a domain of $C^2{\times}C^2$.

• The Pure and Applied Mathematics
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• v.20 no.4
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• pp.251-258
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• 2013

We define an ${\varepsilon}$-regular function in dual quaternions. From the properties of ${\varepsilon}$-regular functions, we represent the Taylor series of ${\varepsilon}$-regular functions with values in dual quaternions.

• The Pure and Applied Mathematics
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• v.22 no.1
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• pp.57-63
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• 2015

We define a hyperholomorphic function with values in split quaternions, provide split hyperholomorphic mappings on ${\Omega}{\subset}\mathbb{C}^2$ and research the properties of split hyperholomorphic functions.

• Journal of Elementary Mathematics Education in Korea
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• v.17 no.1
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• pp.129-142
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• 2013

The purpose of this study was to investigate the typical characteristics of mathematics education of Japan. In order to achieve this goal, we focused the journal 'Arithmetic Education' from 2007 to 2011. This journal has published by the Japan Society of Mathematical Education and 6 issues each year. A total of 133 articles related with mathematics education were analyzed. The typical characteristics of Japan's research for mathematics education were as follows: The number of single author of article were 98 cases (74%), and those of two co-authors were 21 cases (16%). There are some unusual research topic for mathematics education such as 'combined class'. 'cultural pluralism' and 'mathematics learning disabled children'. The articles 'statistical methods related to educational evaluation', 'statistical analysis for educational evaluation' and 'the relationship between number and quantity on the quaternion number' are very interesting results to the readers who know the basics of statistics and algebra. We may find many researcher who majored pure mathematics in the University of Educations in Korea. So we hope that they may write the paper which combine the pure mathematics and mathematics education. The education survey conducted by the policy is actually very meaningful. If the researcher can connect these surveys to the field of education, then the readers can see a nice paper in the journal of elementary mathematics education in Korea. Finally, it is very difficult to find that counterstatement paper for the results of the other's.