A Review on Affine Transformation

ISSN (Online) 2393-8021 IARJSET ISSN (Print) 2394-1588

International Advanced Research Journal in Science, Engineering and Technology ISO 3297:2007 Certified Vol. 3, Issue 8, August 2016

S. C. Shrivastava Department of Applied Mathematics, Rungta College of Engineering and Technology, Bhilai (C.G.), India

Abstract: The growth of multimedia applications, Fractal image compression techniques is very important for the efficient transmission and storage of digital data. The purpose of this paper, we described fractal image compression using affine transformation correspond to translations, scaling, rotations and reflections in metric space.

Keywords: Fractal image compression, affine transformation, image coding and decoding.

1. INTRODUCTION

In geometry, an affine transformation in latin “affinis” Where the parameter a11, a12, a21, a22form the linear part means “related” is a transformation introduced by Euler in which determines the rotation, skew and scaling and the 1748 in his book “Introduction in analysis in infinitorum parameters 푡푥 , 푡푦 are the translation distances in x and y then 1827 August Mobius wrote on affine geometry. directions, respectively[1], [2]. In geometry, an affine transformation is a function that maps an object from an affine space to another and which x′ x a11 a12 tx x preserve structures. Indeed, an affine transformation y′ = T y = a21 a22 ty y preserves lines or distance ratios but change the 1 1 0 0 1 1 ′ orientation, size or position of the object. The set of affine 푥 = 푥 푎11 + 푦 푎12 + 푡푥 ′ transformation is composed of various operations. 푦 = 푥 푎21 + 푦 푎22 + 푡푦 Translations which modify object position in the image. Homothetic transformations composed of the contraction The general affine transformation can be defined with only and dilatation of an object, both scaling operations. The six parameter: transvection (shear mapping) which shifts every point of an object. Rotation which to allows to rotate an object θ ∶ the rotation angle. according to its axis[3]. Affine transformation improved of tx : the x component of the translation vector. compression ratio of fractal image compression. The ty ∶ the y component of the translation vector. improvement of compression rates is done by applying the Sx ∶ the x component of the scaling vector. lossy compression techniques on the parameters of the S ∶ the y component of the scaling vector. affine transformation of the fractal compressed image. y Sh ∶ the x component of the shearing vector. x Sh ∶ the y component of the shearing vector[3]. 2. AFFINE TRANSFORMATION y

The use of homogeneous coordinates is the central point of In the other words, The Fractal is made up of the union of affine transformation which allow us to use the several copies of itself, where each copy is transformed by mathematical properties of matrices to perform a function Ti, such a function is 2D affine transformation, transformations. So to transform an image, we use a so the IFS is defined by a finite number of affine transformation which characterized by Translation, matrix 푇 ∈ M3(R) providing the changes to apply scaling, shearing and rotation.

a11 a12 tx 2.1 Translation T = a21 a22 ty A translation is a function that moves every point with p p 1 x y constant distance in a specified direction by the vector 푇 = [푇 , 푇 ] which provide the orientation and the distance The vector [T , T ] represent the translation vector 푥 푦 x y according to axis x and y respectively[3]. according the canonical vectors. the vector [Px , Py ] represents the projection vector on the basis. The square The mathematical transformation is the following[4], [6]: matrix composed by the aij elements is the affine transformation matrix [3], [5]. An affine transformation 푥′ 1 0 푡푥 푥 2 2 ′ T ∶ R → R is a transformation of the form T ∶ Ax + B 푇 = 푦′ = 0 1 푡푦 푦 defined by 1 0 0 1 1 x a11 a12 x tx ′ T y = a21 a22 y + ty 푥 = 푥 + 푡푥 ′ 1 0 0 1 1 푦 = 푦 + 푡푦

International Advanced Research Journal in Science, Engineering and Technology ISO 3297:2007 Certified Vol. 3, Issue 8, August 2016

2.2 Transvection 2.4 Scaling A transvection is a function that shifts evry point with Scaling is a linear transformation that enlarge or shrinks constant distance in a basis direction x or y[3]. objects by a scale factor that is the same in the The mathematical vertical and horizontal transvection direction[3]. respectively is the following[4], [6]: The mathematical transformation is the following[4], [6]:

푥′ 1 푠푣 0 푥 푥′ 푠푥 0 0 푥 ′ ′ 푇 = 푦′ = 0 1 0 푦 푇 = 푦′ = 0 푠푦 0 푦 1 0 0 1 1 1 0 0 1 1 ′ ′ 푥 = 푥 + 푦 푠푣 푥 = 푥 푠푥 ′ ′ 푦 = 푦 푦 = 푦 푠푦

푥′ 1 0 0 푥 ′ 푇 = 푦′ = 푠ℎ 1 0 푦 1 0 0 1 1 푥′ = 푥 ′ 푦 = 푥 푠ℎ + 푦

REFERENCES

[1] Barnsely M. F., “ Fractal Everywhere”, Acadmic Press, San Diego, 1993. [2] Barnsely M. F. and Hurd L.P., “ Fractal Image Compression”, AK Peters Ltd, Wellesely, Massachuretts, 1992. 2.3 Rotation [3] Coste, “Affine Transformation, Landmarks registration”, Nonlinear warping, October, 2012. Rotationis a circular trnsformation around a point or an [4] David F. Rogers and J. Alan Admas, “Mathematical elements for axis. A rotation around an axis defined by the normal computer graphics”, 2nd Ed, NewYork, 1990. vector to the image plane located at the center of the [5] Yan Bin Jia, “Transformaions in Homogeneous coordinates”, Notes image[3]. 2014. The mathematical rotation about clockwise and counter [6] University of Texas at Austin, coputer graphics, 2010. clockwise is the following[4], [6]:

푥′ 푐표푠휃 푠𝑖푛휃 0 푥 푇′ = 푦′ = −푠𝑖푛휃 푐표푠휃 1 푦 1 0 0 1 1 푥′ = 푥 푐표푠휃 + 푦 푠𝑖푛휃 푦′ = −푥푠𝑖푛휃 + 푦 푐표푠휃

푥′ 푐표푠휃 −푠𝑖푛휃 0 푥 푇′ = 푦′ = 푠𝑖푛휃 푐표푠휃 0 푦 1 0 0 1 1 푥′ = 푥 푐표푠휃 – 푦 푠𝑖푛휃 푦′ = 푥 푠𝑖푛휃 + 푦 푐표푠휃