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Applications of Frequency Domain Processing

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- 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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- 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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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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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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Giving:

- Performing Inverse Hartley transform, gives:

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- Convolution R = I * T
- Deconvolution I = R *-1T
- 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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