This paper describes an advancement in color image processing, using geometric algebra. This is achieved using a compact representation of vectors within $n$ dimensional space. Geometric Algebra (GA) is a preferred framework for signal representation and image representation. In this context the R, G, B color channels are not defined separately but as a single entity. As GA provides a rich set of operations, the signal and image processing operations becomes straightforward and the algorithms intuitive. From the experiments described in this paper, it is also possible to conclude that the convolution operation with the rotor masks within GA belong to a class of linear vector filters and can be applied to image or speech signals. The usefulness of the introduced approach has been demonstrated by analyzing and implementing two different types of edge detection schemes

Al-Hashimi, Bashir

Mishra, Biswajit

Wilson, Peter

English

Southampton (e-Prints Soton)

ADVANCEMENT IN COLOR IMAGE PROCESSING USING GEOMETRICALGEBRABiswajit Mishra, Peter Wilson and Bashir M. Al-HashimiElectronic Systems and Devices, University Of SouthamptonHighﬁeld, SO17 1BJ, Southampton, United Kingdomphone: + (44) 238059 6665, fax: + (44) 238059 2901, email: {bm2,prw,bmah}@ecs.soton.ac.ukweb: www.ecs.soton.ac.ukABSTRACTDue to the compact representation of vectors within n dimen-sional space Geometric Algebra (GA) is a preferred frame-work for signal representation and image representation. Inthis context the R, G, B color channels are not deﬁned sep-arately but as a single entity. As GA provides a rich set ofoperations, the signal and image processing operations be-comes straightforward and the algorithms intuitive. Fromthe experiments it is also concluded that the convolution op-eration with the rotor masks within GA belong to a class oflinearvectorﬁltersandcanbeappliedtoimageorspeechsig-nals. The usefulness of the introduced approach was demon-strated by analyzing and implementing two different types ofedge detection schemes.1. INTRODUCTIONOver the past few years, there has been a signiﬁcant progressin color image processing [1] [2] [3]. Color signal represen-tations also have undergone a sea change from being mere al-gebraic equations and matrix representation to more sophisti-cated vector valued representation. This is because the colorimage consists of three color planes in RGB and is inher-ently vector in nature. Therefore for color images the vec-tor valued image processing techniques are more suitable.Though these techniques are crucial and appropriate, are notvery popular because of the complexity and lack of a uniﬁedframework. They rely on scalar image processing functionsbased on monochrome techniques.The RGB color vector has a magnitude and direction andhence can be represented as an entity having a luminanceand chrominance component. However this is not a directconversion and involves a transformation of one vector space(RGB) to another vector space (Hue-Saturation-Intensity orHSI) and the transformation is nonlinear in nature. Most ofthe conversions within the color space are lossy conversionsdue to this nonlinearity, so the algorithms are not portableand use a lot of vector additions and complex vector multi-plications that also pose fundamental limitation.In both gray scale and color images edge detection isone of the most basic operation in image processing. Theedge in gray scale images is deﬁned in an achromatic way asthe discontinuity in the brightness function. The color edgeis an extension to this i.e. to ﬁnd the discontinuities alongthe adjacent regions of a color image in the 3D color spacedependent on some function of metric distance of the colorspace, some function of merging of the derived edges andsome function of imposing uniformity constraints to utilizeall three independent edges concurrently. However, the fun-damentalproblemassociatedwithcoloredgedetectionisdueto the uncorrelated monochrome techniques applied to thethree correlatedcolor channels. For example tosmooth alonga particular color component within the image, component-wise ﬁltering will give an incorrect result. The reason is dueto the nonlinear ﬁltering of the smoothing operation. Withcomponent-wise ﬁltering the median pixel resulting fromeach component-wise operation may not yield the same me-dian pixel on each color plane (R, G or B) and hence theresults may contain noise and/or incorrect pixel value. Butwith median ﬁlter [4], the operation picks the correct pixelbecause it chooses the correct median value of the pixel us-ing the distance information as it treats the color as a 3Dvector. There also exist alternative methods where vectormethods are used for non linear ﬁltering [5] operations whichgive better results and is preferred to component wise ﬁlter-ing. In this paper, a new method using Geometric Algebrais discussed which deﬁnes color as a single vector entity forcolor image processing. Based on it a linear ﬁlter is proposedwhich is applied to edge detection algorithms.This paper is organized as follows: Section 2 discussessome previous work on vector ﬁltering methods. GeometricAlgebra (GA) is introduced in Section 3. In Section 4, theconcepts of rotations and rotors in GA are discussed. In Sec-tion 5 the color is deﬁned as a single entity as a vector con-sisting of the individual elements of the multivector deﬁnedwithin GA. The physical signiﬁcance of the transformationof the color space is derived and discussed in Section 6. Twodifferent edge detection techniques using GA are discussedin Section 7. Finally in Section 8, the results are summarized.2. COLOR IMAGE PROCESSING AND VECTORFILTERINGThe standard technique to identify the critical features of theimage in image processing applications is to use the rota-tional and curvature properties of the vector ﬁelds [6]. There-fore, the convolution and correlation techniques are quitecommon to image processing algorithms on scalar ﬁelds.These techniques have been extended to vector ﬁelds for vi-sualization, signalanalysisandapplicationssuchasdetectionof patterns within DNA images [2]. Independently in [7], theauthor developed a hypercomplex fourier transform of a vec-tor ﬁeld for image processing. This was extended in [8] [9]for image processing by GA methods. In [10] the authorshave developed the GA or Clifford convolution fourier trans-form method for pattern matching on vector ﬁelds whichare used for visualization of 3D vector ﬁeld ﬂows. It hasbeen shown that the conventional feature detection fouriermethods based on scalar techniques are simple extensions offourier transforms based on GA. It has also been proven inthese works that the GA convolution techniques are superiorbecause of the uniﬁed notation for convolution of scalar, vec-tor and multivector ﬁelds.3. GEOMETRIC ALGEBRA IN 3DThis section describes the basics of GA in 3D. Expressionswithin GA embed and extend existing theories and meth-ods to express geometric relations without the need for spe-cial case considerations in higher dimensions with simple yetpowerful techniques [11] [12] [13].Let the Euclidean vector space E3 be deﬁned by the or-thonormal basis vectors e1,e2 and e3. In GA this 3D space inE3 can be decomposed into a linear space spanned by thefollowing elements 1,e1,e2,e3,e1 ∧ e2,e2 ∧ e3,e3 ∧ e1,e1 ∧e2 ∧ e3 in G3. The individual element of this linear sub-space are called the blades and the elements of the algebraare multivectors. The blade signiﬁes the subspace i.e scalar,vector or bivector. Hence the linear subspace consists ofa scalar (1) a 0 grade blade element, vectors (e1,e2,e3) 1grade blade element, bivector (e1 ∧e2, e2 ∧e3, e3 ∧e1) a 2grade blade element (the directed area segment formed fromsweeping e1 along e2 for e1 ∧e2 which has orientation andis a 2-dimensional subspace.) and e1 ∧e2 ∧e3 a trivector orthe pseudoscalar. The collection of the subspaces deﬁne thecomplete space in G3. The multiplication is associative, bi-linear and is deﬁned by the following rules:1ei = 1ei; eiei = 1; eiej = −ejeiIn GA it is possible to add different grade vectors to forma multivector. This reveals a great power within it by ex-pressing the different grade vectors in a single product. Forexample, the multivectors in G2 in E2 would be the linearcombination of the blades and will contain a scalar, vectorand bivector parts:a0 |{z}scalar+a1e1+a2e2 | {z }vector+a3e1∧e2 | {z }bivectorIn GA the multiplication of any two vectors (or multivec-tor) a and b yields the geometric product given by: ab =a·b+a∧b. which consists of the inner product(or dot prod-uct) and the outer product. It gives the information about themagnitude and orientation of the vector. For example, if aand b are collinear, then a∧b = 0 and the geometric productwhich now has only the dot product, gives the magnitude ofthe vectors. If a and b are perpendicular, then a·b = 0, thegeometric product now consisting of only the outer producta∧b gives the orientation of the bivector. If the vectors a andb are neither of the two extremes then the geometric productwill give information something in between. In essence thisis the most important element of this algebra and all the othermeaningful operations are derived from this geometric prod-uct.4. ROTATIONS IN THE GEOMETRIC ALGEBRA3-D SPACEThe rotations in GA are represented by a pair of reﬂections.For any vector a reﬂected in a plane perpendicular to an unitvector n is −nan and followed by another reﬂection in an-other plane perpendicular to m results in another vector givenby −m(−nan)m=(mn)a(nm)=Ra ˜ R [14]. This product R isa multivector and is called a rotational element or Rotor andsatisﬁes R ˜ R =1, where ˜ R is the conjugate of R. The equationRa ˜ R works for any dimension, any grade and any objects.Thereforetakingonlythescalarandbivector parts, agen-eral rotation in 3-D can be written as shown in eqn (1):R = exp(nθ2) = cosθ2+nsinθ2(1)where θ represents a rotation about an axis parallel to unitvectornandtherotationaxisnisgivenby: n1e2e3+n2e3e1+n3e1e2 is spanned by the bivector basis.5. COLOR AS A VECTORIt is fair to assume that the human eye doesn’t process differ-ent colors in RGBimages (ﬁg1) separately but processes likea continuous vector valued approach in the 3-D Euclideanspace. In this regard the bivector representation of color vec-tors in GA ﬁts neatly for the 3-D Euclidean space. Then thecolor information of (r,g,b)T vector of the color image cm,ncan be written as:cm,n = rm,ne2e3+gm,ne3e1+bm,ne1e2. (2)where rm,n, gm,n and bm,n are the RGB vectors of the imagecm,n. Alternately, the following analysis and derivations inthe next few sections holds good for the color vector cm,n =rm,ne1+gm,ne2+bm,ne3 deﬁnition.￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿Figure 1: R-G-B plane6. TRANSFORMATION AND DIFFERENCESUBSPACEThough most of the algorithms extend the conventional grayscale algorithms to the color images, very few have treatedcolor as a vector or a single entity. In this paper the coloris deﬁned as a single entity (eqn 2) and treat the image asa superset of this entity. Therefore the color edge detectionand image processing algorithms with this type of approachbecomes straightforward and intuitive.Let µ be the rotation axis spanned by the bivector basise1e2+e2e3+e3e1 in the color cube (ﬁg 2) given by:µ = e1e2+e2e3+e3e1For a unit transformation on the color vector the normalizedcolor representation is chosen. This ensures that the orienta-tioninformationiskeptwhilethedistanceinformationisnor-malized. For a unit transformation on the normalized color,the rotation vector is given by:R = cosθ +1√3µsinθ = cosθ +e12+e23+e31 √3sinθ˜ R = cosθ −1√3µsinθ = cosθ −e12+e23+e31 √3sinθThese rotations given by R and ˜ R rotates any vector by anangle θ in 3D about an axis parallel to the rotation axis.Hence the unit transformation or the convolution operationon a color elementC is given by:RC ˜ R = (cθ +1√3µsθ)(C)(cθ −1√3µsθ)= c2θ(C)| {z }I−cθsθ√3(C)µ| {z }II+cθsθ√3µ(C)| {z }III+s2θ√3µ(C)µ| {z }IVwhere C = re23 +ge31 +be12 and cθ, sθ is cosθ and sinθrespectively.The part of II in the above equation simpliﬁes to−r−g−b−(b−g)e23−(r−b)e31−(g−r)e12and similarly the part of III equates to−r−g−b+(b−g)e23+(r−b)e31+(g−r)e12Finally, the part IV equates tore23+ge31+be12−2(re31+re12+ge23+ge12+be23+be31)Putting it altogether, eqn 3 reduces to as follows:cos2θ(re23+ge31+be12)| {z }A+23sin2θµ(r+g+b)| {z }B+1√3sin2θ[(b−g)e23+(r−b)e31+(g−r)e12]| {z }CThe ﬁrst element ‘A’ of the above equation is the RGB spacecomponent, the second element ‘B’ is the intensity compo-nent and ‘C’ element is the color difference or the projectionof the tristimuli in the Maxwell triangle. This is also knownas the chromaticity (due to hue and saturation) of the vector.Interestingly from the above equations if we rotate thevector by an angle θ = π4, the above equation reduces toonly two components as shown in eqn 3 and eqn 4, wherethe space component ‘A’ is cancelled. This transformationis similar to a RGB space being transformed to an HSI likespace and the remaining two components describes the lumi-nance and chrominance of the image. For θ = π4RC ˜ R =13µ(r+g+b)| {z }luminance+1√3[(b−g)e23 +(r−b)e31 +(g−r)e12]| {z }chrominance(3)and similarly the opposite rotation is given by:˜ RCR =13µ(r+g+b)| {z }luminance−1√3[(b−g)e23 +(r−b)e31 +(g−r)e12]| {z }chrominance(4)Ifthecolorvectorishomogeneousthenadditionoftwotrans-forms equates to:RC ˜ R+ ˜ RCR =23µ(r+g+b) (5)which yields the equation for the intensity of the image.Subtracting the two transforms gives:RC ˜ R− ˜ RCR =2√3[(b−g)e23+(r−b)e31+(g−r)e12] (6)which is the difference of the color vector in the image or thechange in chromaticity or shift in hue.From the above it can be concluded that if the color vec-tors are different in both the unit transformations then theydon’t cancel out and that would result in a hue or chromaticshift. On the other hand if the color is same then only theintensity component will be left out. This phenomenon gen-erates edges where sharp changes of color occurs.7. EXPERIMENTAL RESULTS7.1 Color Difference Edge Detection by Rotor Convolu-tionThe edge detection process involves convolving masksmL(x,y) (for left) and mR(x,y) (for right) of the size X ×Ywith the image c(m,n) of dimension (m×n). In a rotor basedapproach the convolution involves the geometric product ofthe vector with the rotor as shown in the convolution eqn 7:ˆ c(m,n) =X−1∑x=0Y−1∑y=0mL[x,y]c[m−x,n−y]mR[x,y] (7)The horizontal left and right masks for the rotor convolutionare deﬁned in eqn 8. The vertical masks are obtained byinterchanging the rows and columns.mL =" R R R0 0 0˜ R ˜ R ˜ R#, mR =" ˜ R ˜ R ˜ R0 0 0R R R#(8)And the rotors are given by:R = senπ/4 = s[cos(π/4)+nsin(π/4)] (9)where n is the unit vector and is given by: n=(e2e3+e3e1+e1e2)/√3 and s = 1/√6 is the scale factor.As shown in the ﬁg 2 the color vector is split into twocomponents, (ck is the component parallel to the gray axisand the perpendicular component is c⊥). When the masksare applied on the color vector only the perpendicular com-ponent c⊥ is affected but the parallel component ck is un-changed [14]. After the convolution the perpendicular com-ponentisrotatedbyanamountspeciﬁedbytherotorRandbyan angle 2θ. As R is a rotor in a right hand screw sense, therotor Ra˜ R would rotate the color vector by the same amountas would the rotor ˜ RaR by the same amount but opposite indirection. Hence, if the color vectors are homogeneous thenafter the masks are applied, both the components would can-cel out (as derived in eqn 5) and the pixel point would fallsomewhere on the gray axis or simply perceived as the in-tensity of the image. However if the color components are

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Advancement in Color Image Processing using Geometric Algebra

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Advancement in Color Image Processing using Geometric Algebra

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