Color-Dependent Diffusion Equations Based on Quaternion Algebra

Chinese Journal of Electronics

Vol.21,No.2,Apr.2012

Color-Dependent Diffusion Equations Based on

Quaternion Algebra∗

LI Yafeng1,2,FENG Xiangchu1,WANG Weiwei1

(1.Department of Applied Mathematics,Xidian University,Xi’an710071,China)

(2.Department of Computer Science,Baoji University of Arts and Science,Baoji721016,China)

Abstract—Based on quaternion algebra,this pa-per proposes two new smoothing models called color-dependent diffusion equations.One model is designed to only smooth the components in the direction of a chosen color.The other model is designed to only smooth the com-ponents perpendicular to the direction of a chosen color.

To improve the color selectivity,a weight factor W is proposed.For example,if wefilter the parallel compo-nents of the image,greater weight is given to pixels with the color close to the chosen color.

Finally,a color-dependent diffusion scheme is pre-sented,which can detect a chosen color edge or low-passfil-ter a certain color efficiently.Experimental results demon-strate the effectiveness of this scheme applied to natural color images.

Key words—Color imagefiltering,Diffusion equation, Color-dependent,Denoising,Quaternion.

I.Introduction

Color image processing has been researched for several decades.For color image,most of the existing techniques do not consider the interactions between color channels.Tradi-tional way of smoothing color image is to process each color channel with gray imagefilter techniques.No matter what image’s color is,each color component will befiltered in the same degree.So,traditional methods are not color-dependent.

Quaternion representation of color images has been pro-posed and studied in Refs.[1,2].A quaternion has one real part and three imaginary parts and can be written as q=a+bi+cj+dk[3].The RGB color triple is represented as a pure imaginary quaternion R i+G j+B k.In this way,the color image can be processed as a whole,rather than as three sep-arate channels.Sangwine and Ell[4−9]have successfully used quaternions for color image smoothing,edge detection,corre-lation,hypercomplex Fourier transform,and compression.In Refs.[9,10],based on vector decomposition,Sangwine,etc. proposed a linear color-dependent imagefiltering.Theyfirst choose a Color of interest(COI),and resolve all the pixels into parallel/perpendicular directions by using standard vec-tor resolution.Then they apply a scalarfilter to only one of the separated image components and leave the another one untouched.Finally,they add the two images together to ob-tain thefiltered result,see Fig.1.Their aim is to apply low-passfiltering to a particular color in the image.They showed that the implementation of parallel/perpendicular separation could be made algebraically.Thefilter is implemented in the Fourier domain.To improve the color selectivity,they pro-posed a weight factor.More details were presented in Refs.[9, 10].Two remarks are worth to note.Firstly,thefiltering is calculated in the Fourier domain.Secondly,weight factor only considers the lack of directional resolution,not the magnitude of COI.

Partial differential equation(PDE)based methods have been used successfully for many image processing tasks.How-ever,most PDE models are not color-dependent[11−13].In this paper,inspired by the complex diffusion[14],we will propose two color-dependent diffusion equations.Besides the advan-tages of the PDE method,there are two more merits than Refs.[9,10].Firstly,the color-dependent diffusion schemes can be directly implemented in the spatial domain rather than in the Fourier domain.This makes thefilter process simpler. Secondly,the weight factor W is determined not only by the di-rection of COI but also by the magnitude of COI.The schemes are successful tofilter pixels which are close to the COI,while the rest pixels of the image are remained.Experimental results demonstrate the effectiveness of our

methods.

Fig.1.Color selectivefiltering

scheme used in Ref.[9]

This paper is organized

as follows.Color-dependent

diffusion equations are

presented in Section II.The

proposed weight is discussed

in Section III.Based on

color-dependent diffusion

equation and weight factor, a color-dependent diffusion scheme is presented in Section IV. Section V concludes this paper.For readers requiring some

∗Manuscript Received Apr.2010;Accepted Nov.2011.This work is supported by the National Natural Science Foundation of China(No.61105011,No.60872138,No.60773209),the Specialized Research Fund for the Doctoral Program of Higher Education (No.20100203120010),the Key Project of Chinese Ministry of618Education(No.211155),the Key Project of Baoji University of Arts and Science(No.ZK10171).