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# Applications of Frequency Domain Processing PowerPoint PPT Presentation

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. Use FHT or FFT instead of DHT or DFT Number of points should be kept small

Applications of Frequency Domain Processing

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### 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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• 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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### 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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### 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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### Hartley convolution: Cont’d

Giving:

• Performing Inverse Hartley transform, gives:

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### Deconvolution

• 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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