 Open Access
 Total Downloads : 22
 Authors : Dr. Sandeep Mathur, Dr. Anjali Mathur, Nitesh Agarwal
 Paper ID : IJERTCONV4IS12009
 Volume & Issue : ETRASECT – 2016 (Volume 4 – Issue 12)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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.

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 rubbersheet 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 OneDimensional DCT:
If we have oneD 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 2D 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 2D DCT [7].
2D 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

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

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 OneDimensional DST:
For oneD sequence of signal value of length N then its equivalent DST can be expressed as

Using 8*8 IDCT or IDST blocks we create original pixel matrix to get original image.
N 1
2x 1(u 1)

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 2D FDST can be defined as
if image is 8 bit then MAX=255.

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:
Frequency Transformation 
Input Image 
Output Image 
FDCT 

IDCT 

FDST 

IDST 
Frequency Transformation 
Input Image 
Output Image 
FDCT 

IDCT 

FDST 

IDST 
& 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

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

Create pixel matrix of the image.

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

Store transformed matrix as output of geometric transformation.

Using this transformed pixel matrix get transformed image.


Implementation of Frequency Transformation Steps involved in this implementation

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

Table 1: Forward & inverse frequency transformation
MSE 
PSNR 

2D DST 
0.37 
52.47 
2D DCT 
0.29 
53.52 
Table 2: MSE & PSNR value of input image after inverse transformation
Geometric Transformation 
Input Image 
Output image 
Identity 

Scaling 
155% 

Rotation 
1800 & 450 

Translation 
tx = 50, ty = 10 

Shear(vertical) 
sV = 1 

Shear(horizontal) 
sh = 1 
Table 3: Different Geometric Transformation on an image
3. CONCLUSION
The result presented in this document shows that

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

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

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

A. M. Eskicioglu, and P. S. Fisher, Image quality measures and their performance, IEEE Trans. Commun., vol. 43, no. 12, pp. 29592965, Dec. 1995.

Andrew B. Watson, Image Compression Using Discrete Cosine Transform, NASA Ames Research Centre, 4(1),
pp. 8188, 1994.

Anjali Kapoor and Dr. Renu Dhir, Image Compression Using Fast 2D DCT Technique, International Journal on Computer Science and Engineering (IJCSE), vol. 3 pp. 24152419, 6 June 2011.

Harley R. Myler and Authur R. Weeks The Pocket Handbook of Image Processing Algorithms in C, ISBN 0136422403 Prentice Hall P T R Englewood Cliffs, New Jercy 07632

Iain E.G. Richardson H.264 and MPEG4 Video Compression: Video Coding for Nextgeneration Multimedia, ISBN 0470848375, 9780470848371, Wiley, 2003.

L.Dhang, W. Dong D.Zhang and G.Shi Two stage image denoising by principal component analysis with local pixel grouping Pattern Recognition, Vol.43, pp15311549, 2010.

N.Ahmed, T.Natatarajan, and K.R. Rao, Discrete Cosine Transform, IEEE Transactions on Computers, vol. C32,
pp. 9093, Jan. 1974.

S. Malini. & R.S. Moni. Use of Discrete Sine Transform for A Noval Image Denoising Technique. International Journal of Image Processing (IJIP), Vol, 8, Issue 4, pp. 204213, 2014.

Swati Dhamija and Priyanka Jain Comparative Analysis for Discrete Sine Transform as a suitable method for noise estimation IJCSI International Journal of Computer Science Issues, Vol. 8, Issue 5, No 3pp. 162164, September 2011.

V.P.S.Naidu, Discrete cosine Transform based Image Fusion, Defense Science Journal, Vol.60, No.1, pp.4854., Jan.2010.

Rafael C. Gonzalez & Richard E. Woods Digital Image Processing, 3rd edition, ISBN: 9789332518469, Pearson Publication, 2014.