Applications of Frequency Domain Processing

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 240-373 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 240-373 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 240-373 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: 240-373 Image Processing

5. Hartley convolution: Cont’d Giving: • Performing Inverse Hartley transform, gives: 240-373 Image Processing

6. 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