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Questions and Answers
What is the value of F 0,1 in the given table?
What is the value of F 2,2 in the given table?
What is the largest frequency in the given table?
What is the power corresponding to the frequency 3.6?
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What percentage of the remaining power is calculated after removing the frequency 4.2?
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What is the cut-off frequency chosen in the solution?
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What is the value of F 1,2 in the given table?
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What is the power calculated by removing the frequency 4.2 and adding the frequencies 3.6?
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What is the value of F 3,2 in the given table?
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What is the percentage of the remaining power calculated after removing the frequency 4.2 and adding the frequencies 3.6?
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Study Notes
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
, whereD
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
, whereD
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
, whereu
andv
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, ...
- Calculate the frequency matrix using the equation:
- Solution (2):
- Calculate the frequency matrix using the equation:
f = u^2 + v^2
, whereu
andv
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.
- Calculate the frequency matrix using the equation:
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Description
This quiz covers filtering in the frequency domain, including the convolution theorem, types of filtering such as ideal and Butterworth filtering, and their applications in image processing.