2014 Fifth International Conference on Intelligent Systems, Modelling and Simulation Color Image Representation Using Multivector Somasis Roy Anirban Mitra Sanjit Kumar Setua Dept. of Computer Science Dept. of Computer Science Dept. of Computer Science University of Calcutta Academy of Technology, WBUT University of Calcutta Abstract – This paper discusses about Clifford Algebra and its scalar that’s why a RGB color image will use three times significance in a color image representation. A color image as much memory as a gray-scale image of the same pixel following RGB color model is described by a multivector in dimensions. The main objective is representation of an 3D space. The color blades in the multivector define the image as independent of coordinate system. colors. Different shades of color can be obtained by applying There are several mathematical tools independent of rotor operator on those blades. Clifford color space is introduced to define each color against a pixel of an image in coordinate system to represent an image. By the form of a color blade. A multivector function is derived from traditional approach of linear algebra, the concept of this space for explaining the color image. In simpler way, it is homogeneous coordinate is successful for developing proposed that a color image is stored in the Clifford color computer graphics packages [2]. The dissimilarity space. This paper is also useful in the reduction of between matrix computation and geometric computation time for color image processing as every Grade – interpretation leads a problem in this approach. As k vectors are colors. mentioned earlier the safest and efficient way to get rid – of coordinate system is the adoption of vector. A color Keywords Clifford algebra, Wedge, Multivector, Color blade, image is defined as the summation of vectors against Grade, Color vector, Clifford color space each pixel exists in that image [3]. It is observed that I. INTRODUCTION higher grade of vector is also responsible to define a color of a pixel in an image. In that case, Quaternion An image is defined as the finite set of spatial algebra [4] and Clifford algebra shows their power as a coordinates with attributes. It is described by the geometric mathematical tool in the development of computer shapes and their relations using the coordinates. Each graphics. Both of these algebra is different from coordinate is represented by a two- traditional linear algebra and both carefully handle the dimensional function at , where the function returns subspaces of any dimensions. A subspace may act as a a value known as the intensity of the coordinate. Each point (scalar), line (vector), plane (bivector), volume coordinate holds data of an image in various forms such as (trivector) etc. It is developed in different vector space binary, gray-scale or color form. A digital image which is for representing a certain geometric object. Clifford described by color is said to be a color image. It holds data algebra provides a strong framework to describe a color in color form, and contains the highest level of information in RGB model by using subspaces [5]. compare to other forms [1]. A color model or a color William K. Clifford (1845-1879) introduced Clifford representation system is needed to analyze the data present Algebra, also called Geometric Algebra where geometric in color image. product is the combination of the dot product and the There are several existing models widely used in outer product. Clifford’s motivation was to combine Computer Graphics like RGB model, HSI model, CMY Grassman’s “associativity” and Hamilton’s model etc. Every distinct color model has the distinguished “anticommutativity” features into a single matter. nature to define a pixel of an image. The most convenient Clifford also united the several advantages of quaternion color model used in digital image processing is the RGB algebra by introducing the subspaces. Clifford Algebra is color model. It is the additive color model whereas CMY used as a powerful mathematical tool in areas as (Cyan Magenta Yellow) is the subtractive color model. computer vision, computer graphics, robotics, etc. As it HSI (Hue-Saturation-Intensity) model provides best color is a coordinate-free tool it is easier to model geometric description for human interpretation. In RGB model each objects and their several transformations. pixel is represented by three values, the amount of Red, There are many color spaces by which an image is Green and Blue. It is spanned over three dimensional represented but RGB color space is considered here. Euclidean space and every coordinate of this model defines Geometrically, this RGB space is represented by a cube a color. A color in this color model is expressed as a where each of the axis representing unique color vector instead of coordinate system. In case of a gray-scale components [4]. According to pure quaternion every image a pixel holds a scalar (gray) value derived from the pixel in a color image contains three components RGB vector. Mathematically, a pixel of an image holds therefore the function is defined as: data in form of a vector or scalar to make coordinate free. A vector holds higher amount of actual data compare to 2166-0662/14 $31.00 © 2014 IEEE 357 DOI 10.1109/ISMS.2014.66 where , and are the imaginary part of the quaternion is empowered by two powerful operations like reflection [6]. The imaginary parts are actually mathematical and rotation. A rotor is responsible for representing a color at a particular pixel. Rotor edge detection based on representation of bivector. In Clifford algebra for - dimension Euclidean space number of subspace are RGB color space replaces traditional edge detection formed. The function for the color of a pixel techniques effectively [9]. An edge detection algorithm was introduced that accepts color value triples as vectors in three [10]. Clifford algebra framework is used because it dimensional Euclidean space where , and are extends the traditional convolution and Fourier transform the unit vectors. As the function shows each of the pixel is to vector fields. represented by a vector, therefore to represent an image number of vectors are needed, where and are III. CLIFFORD ALGEBRA the dimensions of the image. The wedge operator of A. Definition Clifford Algebra has a significant role in handling n A vector space is is considered for dimension . A subspaces. Each of these subspaces is defined as a wedge set of orthonormal basis vectors of n is { }. product or outer product. According to Clifford algebra every color is a subspace in RGB vector space. The A new element can be obtained from the geometric motivation of this paper is to define each of the color in product of the basis elements. Assuming any two basis RGB space by appropriate subspaces and representation of vectors and , the geometric product is formed color image in form of a multivector, combination of and it is anticommutative. Mathematically, it is subspaces. formulated as, = = - = - , This paper gives an idea to represent an image in form Squaring on the basis vectors results +1, -1 or 0. It of a multivector by using Clifford algebra. The rest of the suggests that there are nonnegative integers and paper is organized as follows. Section II discusses about such that = + + and the Clifford Algebra applications as the related works. Clifford Algebra is introduced in section III. In section IV, the geometrical transformations like reflection and rotation = is discussed. Section V explores about image representation by homogenous vector. In section VI we discuss about the proposal, how to represent a color image These operations give proof of the associativity of by using multivector. Finally section VII contains linear algebra with identity to define Clifford Algebra conclusions and future scope from this work. ( ) of dimension = + + generated by the vector space n. II. RELATED WORK The elements belong to the Clifford Algebra or Clifford Algebra is applied in Computer Science to Geometric Algebra is called multivectors. The space show new horizons in research and applications. It is a which contains all the dimensional multivectors is coordinate free algebra providing efficient framework to called the dimensional Clifford Algebra ( ). A those applications. Image processing is one of the applied multivector is generated as an entity containing several areas of Clifford Algebra. Color image representation and grades of elements of the basis set of . Such as, edge detection are the most inevitable part of color image processing. Several measures and approaches were + (1) proposed for the improvement of both the cases. This algebra authorizes number of operators to represent a color is expressed as the ג image effectively. Generally, image representation using where the multivectors RGB color space increases the dimensionality, non summation of as 0-vector (scalar), as 1- linearity, redundancy and reduces the efficiency of color vector (vector), as 2-vector (bivector), as 3- image processing [1]. Clifford algebra resolves these vector (trivector) and so on up to n-vector constraints while considering RGB color space by (pseudoscalar). introducing vector and multivector. It has invoked the vector concept to describe a color [7]. The colors in three B. Two dimensional Approach dimensional RGB color space are manipulated and the Vectors are the elements of the dimensional vector color components are the vector parts in the multivector space . In Clifford Algebra, the higher dimensional [3]. It solves real life problems such as removal of facial oriented subspaces are the basic elements of computation make up disturbances where a color is a multivector [8]. [12]. is represented as an extension of the Clifford Algebra is used to define different color spaces and act as a natural unified language to solve multicolor dimensional Euclidean vector space . For = 2, the image processing and pattern recognition problems [5].