A Review of basic Mathematical Transformation used in Image Processing

A Review of basic Mathematical Transformation used in Image Processing

Dr. Sandeep Mathur1

Department of Mathematics

Jodhpur Institute of Engineering & Technology Jodhpur, India

Dr. Anjali Mathur2

Nitesh Agarwal3

Department of Mathematics

Jodhpur Institute of Engineering & Technology Jodhpur, India

Department of Computer Science Jodhpur Institute of Engineering & Technology

Jodhpur, India

Abstract Mathematics is a very important tool to solve many engineering problems. Almost in every field of engineering mathematics play an important role. Digital Image Processing also such a field that use mathematics as an important tool such as array operation to store image in digital format, to perform geometrical changes in image geometrical transformations are used, in lossy image compression process frequency transformation play an important role. This paper deals with study of basic mathematical transformation (Geometric

Input Image

Input Image

a b c

m n

p q r

Input matrix

Transformed Image

Transformed Image

Transformation Method

Transformation Method

a b c

m n

p q r

Transformed matrix

Transformation, Frequency Transformation) used in image processing.

Keywords: DCT & DST, DST, Transformation.

1. INTRODUCTION

   An image basically a two dimensional figure & can be represented by a 2D pixel matrix in other words an image mathematically can be defined as a two dimensional function f(x, y) where x, y are spatial coordinate & f define the color value of image at x, y in the form of pixel [11].

   Fig 1: Mathematical Transformation Process in Image Processing

   1.1 Geometric Transformation

   Geometrical transformation deals with geometrical changes of coordinate values of an input image. It deals with modify spatial relationship between pixels in an image. The geometrical transformation is known as rubber-sheet transformation because all the geometrical transformation can be seen by printing an image on sheet of rubber and then stretching the sheet according to predefined set of rules. In digital image processing geometric transformation consists of

   f(x, y) =

   f(0,0) f(0,1)

   f(1,0) f(1,1)

   f(0, N 1)

   f(1, N 1)

   two basic operations (1) spatial transformation of coordinate

   (2) intensity interpolation that assign intensity values to the spatially transformed pixels. The Transformed coordinate can

   f(M 1,0) f(M 1,1)

   f(M 1, N 1)

   be expressed as

   (x, y) = T{(x, y)} (1)

   Mathematical transformation cannot be performed directly on an image to perform transformation on image, image is stored in the form of a 2D matrix. In image processing system image acquisition is done by some sensing device & then its each coordinate & amplitude value is digitalize using some mathematical sampling & quantization techniques, digitalization of coordinate is known as sampling & digitalization of amplitude is known as quantization. Both these digitalize value stored in the form of 2D matrix on which mathematical transformations are performed. This paper deals with two basic mathematical transformation like geometrical transformation & frequency transformation. The process of basic mathematical transformation is shown in fig. 1

   Where (x, y) are pixel coordinate in the original image and (x, y) are the pixel coordinate in transformed image. The affine transform is one of the most commonly used spatial coordinate transformations which is define as

   t11 t12 0

   [x y 1] = [x y 1]T = [x y 1] t21 t22 0 . . (2)

   t31 t32 1

   This transformations can scale, rotate, translate or sheer an

   input matrix & can be defined as-

   Identity x = u

   y = w (3)

   Scaling x = cxv

   y = cyw (4)

   Rotation x = v cos 0 w sin 0

   y = v sin 0 + w cos 0 (5)

   Translation x = v + tx

   image of 24 bit we can use N=24 but using block size N=24 time complexity may increase hence we operate DCT & DST on individual color component for a color image. Color image consist of 8 bit red + 8 bit green + 8 bit blue hence we apply DCT & DST on each color component (Red, Green, Blue) using block size N=8.

   1.2.1 One-Dimensional DCT:

   If we have one-D sequence of signal value of length N then its equivalent DCT can be expressed as

   y = w + ty (6)

   N 1

   2x 1u

   Shear(vertical) x = v + s w

   Cu u f xcos

   …9

   V

   y = w

   Shear(horizontal) x = v

   (7)

   x0 2N

   for u = 0,1,2,,N 1.

   y = shv + w (8)

   & inverse transformation is defined as

   In the form of affine matrix these transformation can be written as [11]

   N 1

   N 1

   f x ucu

   u 0

   cos 2x 1u …10

   2N

   Identity		0	0
   	0	1	0
   	0	0	1
   Scaling	cx	0	0
   	0	cy	0
   	0	0	1

   Identity		0	0
   	0	1	0
   	0	0	1
   Scaling	cx	0	0
   	0	cy	0
   	0	0	1

   Transformation Name Affine Matrix T

   Where

   f x is signal value at point x & u is transform

   1

   coefficient for value u.

   (u)

   1 for N

   2

   u 0

   …11

   for u 0

   Rotation

   cos 0 sin 0 0

   N

   Translation

   sin 0 cos 0 0 0 0 1

   1 0 0

   0 1 0

   tx ty 1

   1.2.1.1 Two Dimensional DCT

   An image is 2-D pixel matrix where each position (i,j) represents a color value for that particular point or position. Hence to transform an image into its equivalent DCT matrix we use 2-D DCT [7].

   2-D FDCT can be defined as

   Shear(vertical) 1 0 0

   N1 N1

   2x1u

   2y1v

   sv 1 0

   Cu,vu(v) fx, ycos

   cos

   …12

   Shear(horizontal)

   0 0 1

   1 s 0

   x0 y0

   for u, v = 0,1,2,,N 1.

   2N

   2N

   0 1 0

   0 0 1

   & inverse transformation is defined as (IDCT)

   N1 N1

   N1 N1

   f x, y u(v)cu,vcos 2x1u os 2y1v ..13

   Fig 2: Basic Affine Transformation on pixel matrix

   u0 v0

   2N

   c

   .

   2N

   1.2 Frequency Transformation:

   Frequency Transformation deals with transform the spatial domain of image into its equivalent frequency domain using some sine & cosine functions. Present paper deals with two

   Where Cu, v represents frequency value for u, v &

   f x, y represents pixel color value at position ( x, y ).

   basic transformations DCT & DST. Both transformation convert a signal into its equivalent frequency omain & can work with single & multiple variable. DCT & DST convert an

   (u)

   1 for u 0

   N

   2

   …14

   image into its equivalent frequency domain by partitioning image pixel matrix into blocks of size N*N, N depends upon the type of image. For example if we used a black & white image of 8 bit then all shading of black & white color can be expressed into 8 bit hence we use N=8, similarly for color

   for u 0

   N

   (v)

   1 for N

   v 0

   …15

2. Apply FDCT (Forward Discrete Cosine Transform) or FDST on each 8*8 block of pixel matrix to get equivalent 8*8 DCT or DST blocks respectively.

   2

   N

   for v 0

3. To get Original image we apply IDCT (Inverse Discrete Cosine Transform) or IDST on each 8*8 block DCT or DST respectively & get its equivalent 8*8 IDCT or IDST block respectively.

   1.2.2 One-Dimensional DST:

   For one-D sequence of signal value of length N then its equivalent DST can be expressed as

4. Using 8*8 IDCT or IDST blocks we create original pixel matrix to get original image.

   N 1

   2x 1(u 1)

5. Now we Find MSE (Mean Squared Error) & PSNR

   su u f xsin

   …16

   (Peak Signal To Noise Ratio) to determine quality of

   x0 2N

   for u = 0,1,2,,N 1.

   image obtain by IDST. MSE & PSNR calculated by following formulas

   1 H 1 W 1 2

   & inverse transformation is defined as

   N 1

   N 1

   f x u susin 2x 1(u 1)

   …17

   MSE

   H * W

   x 0

   y 0

   [o ( x, y ) m ( x, y )]

   20

   u 0

   2N

   PSNR=20*log10 (MAX) – 10*log10 (MSE) (21)

   Where H=Height of Image, W= Width of Image,

   Where

   f x is signal value at point x & u is transform

   variable MAX shows max value of a pixel for example

   coefficient for value u & define as same as one dimensional DCT.

   1.1 Two Dimensional DST 2-D FDST can be defined as

   if image is 8 bit then MAX=255.

6. Quality of image obtain by IDCT or IDST is depend on MSE & PSNR value. If as the MSE value increases PSNR value decreases then we get a bad quality of image by IDST or IDCT & if as the MSE value

decreases PSNR value increases we get a batter quality

N1 N1

2×1(u1)

2y1(v1)

image hence a best suitable transformation like DCT,

su,vu(v)fx,ysin

2N sin

2N …18

DST, DFT is taken on the basis of this MSE & PSNR

x0 y0

for u, v = 0,1,2,,N 1.

value.

2.3 Outputs:

& inverse transformation is defined as (IDST)

N1 N1

2×1(u1)

2y1(v1)

fx,yu(v)su,vsin

sin

…19

u0 v0

2N

2N

Where

su, v represents frequency value for u, v , f x, y

represents pixel color value at position ( x, y ) & defines as same as Two Dimensional DCT [8].

2. MAIN RESULTS & OUTPUTS

(u) is

1. Implementation of Geometric Transformation on an Image Steps involved in this implementation

   1. Create pixel matrix of the image.

   2. Apply different geometric transformation on each pixel of image as per requirement.

   3. Store transformed matrix as output of geometric transformation.

   4. Using this transformed pixel matrix get transformed image.

2. Implementation of Frequency Transformation Steps involved in this implementation

   1. Create pixel matrix of the image & divided it into blocks of size 8*8

Table 1: Forward & inverse frequency transformation

Table 2: MSE & PSNR value of input image after inverse transformation

Table 3: Different Geometric Transformation on an image

3. CONCLUSION

The result presented in this document shows that

1. The results shows that both DCT & DST transformation add some error to input image.

2. DCT is more efficient then DST to transform an image into frequency domain because it add less error then DST in input image.

3. In the rotation transformation output pixel value goes out of bound from the range of image area.

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