1. Image Enhancement (Frequency Domain)

2. Frequency-Domain Filtering
Compute the Fourier Transform of the image
Multiply the result by filter transfer function
Take the inverse transform
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EE Image Analysis I

3. Frequency-Domain Filtering
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EE Image Analysis I

4. Frequency-Domain Filtering
Ideal Lowpass Filters
Non-separable
>> [f1,f2] = freqspace(256,'meshgrid');
>> H = zeros(256,256); d = sqrt(f1.^2 + f2.^2) < 0.5;
>> H(d) = 1;
>> figure; imshow(H);
Separable
>> [f1,f2] = freqspace(256,'meshgrid');
>> H = zeros(256,256); d = abs(f1)<0.5 & abs(f2)<0.5;
>> H(d) = 1;
>> figure; imshow(H);
Bahadir K. Gunturk
EE Image Analysis I

5. Frequency-Domain Filtering
Butterworth Lowpass Filter
As order increases the frequency response approaches ideal LPF
Bahadir K. Gunturk
EE Image Analysis I

6. Frequency-Domain Filtering
Butterworth Lowpass Filter
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EE Image Analysis I

7. Frequency-Domain Filtering
Gaussian Lowpass Filter
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EE Image Analysis I

8. Frequency-Domain Filtering
Ideal LPF
Butterworth LPF
Gaussian LPF
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EE Image Analysis I

9. Example
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EE Image Analysis I

10. Highpass Filters
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EE Image Analysis I

11. Example
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EE Image Analysis I

12. Homomorphic Filtering
Consider the illumination and reflectance components of an image
Illumination
Reflectance
Take the ln of the image
In the frequency domain
Bahadir K. Gunturk
EE Image Analysis I

13. Homomorphic Filtering
The illumination component of an image shows slow spatial variations.
The reflectance component varies abruptly.
Therefore, we can treat these components somewhat separately in the frequency domain. 1
With this filter, low-frequency components are attenuated, high-frequency components are emphasized.
Bahadir K. Gunturk
EE Image Analysis I

14. Homomorphic Filtering
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EE Image Analysis I

15. Summary Digital Image Fundamentals: Pixel, resolution, bit depth, ...
Linear Systems: Shift invariance, causality, convolution, impulse response, ...
Fourier Transform: 2D Fourier Transform of continuous and discrete signals, 2D FT properties (linearity, shifting, modulation, convolution, multiplication, energy conservation, etc.), Dirac delta function, Kronecker delta function, …
2D Sampling: Aliasing, anti-aliasing filter, downsampling, interpolation, …
Discrete Fourier Transform: Periodicity, other properties, frequency-domain filtering, …
Discrete Cosine Transform: Properties (real basis functions, good energy compaction), relationship with DFT, matrix representation of DCT, …
Image Enhancement: Image enhancement by point processing (intensity transformation, histogram equalization, histogram specification, etc.), Image enhancement by spatial-domain filtering (lowpass filtering, highpass filtering, median filtering, high-boost filtering, gradient and laplacian operators, etc.), Image enhancement by frequency-domain filtering (lowpass/highpass filters, homomorphic filtering, etc.)
Bahadir K. Gunturk
EE Image Analysis I