Digital Image Processing 12282021 1 Digital Image Processing

Digital Image Processing 12/28/2021 1

Digital Image Processing Fourier Transforms: Implementation and basic filtering 12/28/2021 2

Masking, Correlation and Convolution Masking Cross Correlation: Convolution: 12/28/2021 3

Masking, Correlation and Convolution § Masking § Correlation 12/28/2021 4

Masking, Convolution and Correlation • Convolution 12/28/2021 5

Properties of Fourier Transform 11. Convolution Theorem 12. Cross correlation theorem Autocorrelation theorem The power spectrum of an image is the Fourier Transform of the spatial autocorrelation of that image. 12/28/2021 6

Properties of Fourier Transform 12. Computing the inverse transform using forward transform Considering 1 -D DFT Taking complex conjugate on both sides of inverse DFT equation For real functions as in images therefore the inverse transform can be obtained by again doing the forward transform of conjugate of Fourier Transform. For 2 -D images the inverse transform is found by this technique as 12/28/2021 7

Filtering using Fourier transforms 12/28/2021 8

Filtering using Fourier transforms 12/28/2021 9

Filtering using Fourier transforms 12/28/2021 10

Gaussian low pass and high pass filters 12/28/2021 11

Example of Gaussian LPF and HPF Original image LPF applied DFT of the image HPF applied 12/28/2021 12

Example of modified HPF 12/28/2021 13

Need of padding due to symmetrical properties of DFT 12/28/2021 14

Need of padding due to symmetrical properties of DFT To overcome this problem due periodicity of DFT Extended/padded functions are used, given by (in 1 -D) where A and B are the total number of samples for f(x) and h(x), respectively. 12/28/2021 15

Need of padding due to symmetrical properties of DFT Extended/padded function 12/28/2021 16

Need of padding due to symmetrical properties of DFT For 2 -D images f(x, y) and h(x, y) with sizes Ax. B and Cx. D, the extended/padded function is given by Here P and Q are given by 12/28/2021 17

Need of padding due to symmetrical properties of DFT 12/28/2021 18

Padding during filtering in Frequency domain 12/28/2021 19