Image Processing: Filtering in Frequency Domain

Study Notes

Filtering in the frequency domain is possible due to the convolution theorem, which states that a spatial convolution can be performed by element-wise multiplication of the Fourier transform by a suitable filter matrix.

Ideal filtering involves using filters that allow specific frequencies and set other frequencies to zero.
Types of ideal filters:
- Ideal low pass filter: removes noise from the image by filtering out high-frequency components.
- Ideal high pass filter: highlights fine details in the image by filtering out low-frequency components.

Expressed by the equation: f(x) = 1 if x ≤ D, 0 if x > D, where D is the cutoff frequency.
Only passes frequencies below the cutoff frequency D and replaces all other frequencies with zero.
Can be applied by transforming the image from the spatial domain to the frequency domain and then selecting the appropriate cutoff frequency D.
Example: A circle with a diameter of 15 can be used as the required matrix for ideal low pass filtering.

Expressed by the equation: f(x) = 1 if x ≥ D, 0 if x < D, where D is the cutoff frequency.
Only passes frequencies above the cutoff frequency D and replaces all other frequencies with zero.
Can be applied by transforming the image from the spatial domain to the frequency domain and then selecting the appropriate cutoff frequency D.
Example: A circle with a diameter of 5 can be used as the required matrix for ideal high pass filtering.

Solution (1):
- Calculate the frequency matrix using the equation: f = u^2 + v^2, where u and v are the indices of the elements in the matrix.
- Apply the ideal low pass filter with a cutoff frequency of 1.99.
- Resultant matrix: F(0,0) = 1, F(0,1) = 0, F(0,2) = 0, ...
Solution (2):
- Calculate the frequency matrix using the equation: f = u^2 + v^2, where u and v are the indices of the elements in the matrix.
- Choose the cutoff frequency to be 4.2, which corresponds to a power of 18.
- The remaining power is 96% after removing the frequency 4.2.