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

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Applications of frequency domain processing l.jpg
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

240-373 Image Processing


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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

240-373 Image Processing


Convolution using the fourier transform l.jpg
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

240-373 Image Processing


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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:

240-373 Image Processing


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Hartley convolution: Cont’d

Giving:

  • Performing Inverse Hartley transform, gives:

240-373 Image Processing


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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

240-373 Image Processing


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