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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 0,1 in the given table?
What is the value of F 2,2 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 largest frequency in the given table?
What is the power corresponding to the frequency 3.6?
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?
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?
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?
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?
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?
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?
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.