input image output image transform equation all pixels transform equation
TRANSCRIPT
Image Transformation
Image Transformationsimage transform:- mapping the image data from the spatial domain to the frequency (spectral) domain , were all the pixels in the input domain contribute to each value in the output domain with one – to – one correspondence between the pixels as showing in bellow figure
Input image
Output image
Transform equation
All pixels
Transform equation
• we use transform equation to make the transformation from spatial to frequency domain this equation use the inner product for getting the output image . The general form of transformation equation is A : output image matrix I : input image matrix H : transform matrix : inverse of transform matrix• The size of matrices are n*n were n is either 4 or 8 so the input image , output image and transform matrices are 4*4 block or 8*8 block
TH I HA
2. Discrete Fourier transform (DFT)
The fourier transform is the most well known and the most widely used transform .This transform allow us for the decomposition of an image into 2-D form, assuming N x N image, the equation for the 2-D discrete Fourier transform is
J is the imaginary part of the complex number equal to while
1
So we can write the discrete Fourier transform equation as
…1
Real part imaginary part…2
A complex number is a number consisting of a real part and an imaginary part
From equation )1( for DFT we can construct matrix F (u , v) as follow equation
Where u takes values 0,1,2, … N-1 along each column and v takes the same values along each raw
We need the following rules in construct the DFT matrix
Formula 1
Formula 2
Formula 3
Derive the DFT transform matrix of 4 * 4 image. The DFT of 4*4 image can be obtain with
u= 0,1,2,3 along columns , and v= 0,1,2,3 along rows
33231303
32221202
31211101
30201000
By apply formula 1 we obtain
3*34
2πj3*2
4
2πj3*1
4
2πj3*0
4
2πj
2*34
2πj2*2
4
2πj2*1
4
2πj2*0
4
2πj
1*34
2πj1*2
4
2πj1*1
4
2πj1*0
4
2πj
0*34
2πj0*2
4
2πj0*1
4
2πj0*0
4
2πj
eeee
eeee
eeee
eeee
4
1F
From apply formula 2 we obtain
2
πjjπ2
3πj
jπjπ
2
3πjjπ2
πj
eee1
e1e1
eee1
1111
2
1F
by apply formula 3 we obtain
j2
πsin j
2
πcose 2
πj
1sinπ jcosπe jπ
j2
3πsin j
2
3πcose 2
3πj
angle cos Sin
90(pi/2) 0 1
180(pi) -1 0
270(3pi/2) 0 -1
So the discrete fourier transform (DFT) matrix of 4*4 image is
j1j1
1111
j1j1
1111
4
1F
From the DFT matrix we can see the transformation matrices are symmetric this mean
TFF So the general transform equation for DFT is
F I FA
EX1: Calculate the DFT transform for the image I(4,4)
0100
0100
0100
0100
I
0100
0100
0100
0100
j1j 1
11 11
j 1j1
1 1 1 1
4
1A
j1j 1
11 11
j 1j1
1 1 1 1
F I F
j1j 1
11 11
j 1j1
1 1 1 1
4
1A
1111
1111
1111
1111 F IF F IF
0000
0000
0000
4444
A4
1
0000
0000
0000
1111
Compute the real and imaginary parts of discrete Fourier transform of the image
I
Derive the DFT transform matrix of 8 * 8 image
by apply formula 2 we obtain
by apply formula 1 we obtain
F =
F=
4πj-
e 2πj-
e 43πj-
e jπ-
e 45πj-
e 23πj-
e 47πj-
e 1
2πj-
e jπ-
e 23πj-
e 1 2πj-
e jπ-
e 23πj-
e 1
43πj-
e 23πj-
e 4πj-
e jπ-
e 47πj-
e 2πj-
e 45πj-
e 1
jπ-e
jπ-e 1
jπ-e 1
jπ-e 1
45πj-
e 2πj-
e 47πj-
ejπ-
e
23πj-
e jπ-
e 2πj-
e1
47πj-
e 23πj-
e 45πj-
e jπ-
e
1111
4πj-
e 23πj-
e
43πj-
e 1
23πj-
e jπ-
e2πj-
e1
43πj-
e 2πj-
e4πj-
e1
1111
8
1F
1
by apply formula 3 we obtain
anglecos
Sin
45 (pi/4)
90(pi/2) 0 1
135(3 pi/4)
180(pi) -1 0
225(5pi/4)
270(3pi/2)
0 -1
315(7pi/4)
21
21
21
21
21
21
21
21
2
j
2
1jsin(45)cos(45)e 4
πj
2
j
2
1jsin(135)cos(135)e 4
3πj
2
j
2
1jsin(225)cos(225)e 4
5πj
2
j
2
1jsin(315)cos(315)e 4
7πj
2
j
2
1
2
j
2
1
2
j
2
1
2
j
2
1
2
j
2
1
2
j
2
11-
2
j
2
1
2
j
2
1
1- 1 2
j
2
1j-
2
j
2
1j-
2
j
2
1
2
j
2
1
2
j
2
1j
2
j
2
11- j-
2
j
2
1j
2
j
2
1
j- - 1- - j 1
j- 1- j 1 j- 1- j 1
-j j- -1
1- 1 1- 1 1- 1
1-
j1-1
j- 1-
1111
- 1
j1
- 1
1 1 1 1
8
1F
So the discrete fourier transform (DFT) matrix of 8*8 image is
Original image DFT
Inverse discrete Fourier transform (IDFT)
The inverse of discrete Fourier transform use to get the original image from transform image by the equation
Nuvj2π
eN
1v)(u,F
So the IDFT matrix if N=4 is
j 1j-1
11 11
j-1 j1
1111
1F
4
And the general inverse transform
F AFI
j 1j-1
11 11
j-1 j1
1111
1I4
Calculate the IDFT of example 1 when N=4
0000
0000
0000
1111
j 1j-1
11 11
j-1 j1
1111
F A F
j 1j-1
11 11
j-1 j1
1111
1I4
0000
0000
0000
0400
0400
0400
0400
0400
4
1I
0100
0100
0100
0100
I