1. Digital Image Processing
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2. Digital Image Processing
Fourier Transforms:
Implementation and basic filtering
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3. Masking, Correlation and Convolution
Cross Correlation:
Convolution:
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4. Masking, Correlation and Convolution
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5. Masking, Convolution and Correlation
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6. 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.
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7. 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
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8. Filtering using Fourier transforms
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9. Filtering using Fourier transforms
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10. Filtering using Fourier transforms
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11. Gaussian low pass and high pass filters
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12. Example of Gaussian LPF and HPF
Original image
LPF applied
DFT of the image
HPF applied
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13. Example of modified HPF
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14. Need of padding due to symmetrical properties of DFT
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15. 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.
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16. Need of padding due to symmetrical properties of DFT
Extended/padded function
Extended/padded function
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17. Need of padding due to symmetrical properties of DFT
For 2-D images f(x,y) and h(x,y) with sizes AxB and CxD, the extended/padded function is given by
Here P and Q are given by
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18. Need of padding due to symmetrical properties of DFT
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19. Padding during filtering in Frequency domain
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