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 31/07/62 240-373 Image Processing

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 31/07/62 240-373 Image Processing

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 31/07/62 240-373 Image Processing

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: 31/07/62 240-373 Image Processing

Hartley convolution: Cont’d Giving: Performing Inverse Hartley transform, gives: 31/07/62 240-373 Image Processing

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 31/07/62 240-373 Image Processing