1. Applications of Frequency Domain Processing
Convolution in the frequency domain
useful when the image is larger than 1024x1024 and the template size is greater than 16x16
Template and image must be the same size
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Image Processing

2. Use FHT or FFT instead of DHT or DFT
Number of points should be kept small
The transform is periodic
zeros must be padded to the image and the template
minimum image size must be (N+n-1) x (M+m-1)
Convolution in frequency domain is “real convolution”
Normal convolution
Real convolution
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Image Processing

3. Convolution using the Fourier transform
Technique 1: Convolution using the Fourier transform
USE: To perform a convolution
OPERATION:
zero-padding both the image (MxN) and the template (m x n) to the size (N+n-1) x (M+m-1)
Applying FFT to the modified image and template
Multiplying element by element of the transformed image against the transformed template
Multiplication is done as follows:
F(image) F(template) F(result)
(r1,i1) (r2, i2) (r1r2 - i1i2, r1i2+r2i1)
i.e. 4 real multiplications and 2 additions
Performing Inverse Fourier transform
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Image Processing

4. Hartley convolution Technique 2: Hartley convolution
USE: To perform a convolution
OPERATION:
zero-padding both the image (MxN) and the template (m x n) to the size (N+n-1) x (M+m-1)
image template
Applying Hartley transform to the modified image and template
image template
Multiplying them by evaluating:
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Image Processing

5. Hartley convolution: Cont’d
Giving:
Performing Inverse Hartley transform, gives:
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Image Processing

6. Deconvolution Convolution R = I * T Deconvolution I = R *-1 T
Deconvolution of R by T = convolution of R by some ‘inverse’ of the template T (T’)
Consider periodic convolution as a matrix operation. For example
is equivalent to
A B C
AB = C
ABB-1 = CB-1
A = CB-1
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Image Processing