Image Processing: Filtering in Frequency Domain

Filtering in the Frequency Domain

• 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

• 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.

Ideal Low Pass Filter

• 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.

Ideal High Pass Filter

• 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.

Example Solutions

• 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.