For the following 8-bit grayscale image matrix: ... Calculate the output produced if the 3 x 3 Gaussian kernel is applied at coordinate (2, 3). What is the maximum possible output... For the following 8-bit grayscale image matrix: ... Calculate the output produced if the 3 x 3 Gaussian kernel is applied at coordinate (2, 3). What is the maximum possible output value at a given position of the Gaussian filter kernel? For the next grayscale matrix, calculate the output using a sharpen kernel at coordinate (2, 3).
Understand the Problem
The questions are asking to calculate specific outputs using Gaussian and sharpening kernels on given grayscale image matrices. The first and third questions require computing the output from applying a Gaussian kernel at a specified coordinate, while the second question asks for the maximum possible output value from a given Gaussian kernel. Key concepts include image processing techniques commonly used in convolution operations with kernels.
Answer
The output is $114$.
Answer for screen readers
The output is $114$.
Steps to Solve
- Extract the Relevant Matrix Elements To calculate the output for the Gaussian kernel at coordinate (2, 3), extract the 3x3 pixel window from the image matrix. The window is centered at (2, 3):
$$ \begin{bmatrix} 82 & 57 & 87 \ 48 & 159 & 225 \ 130 & 73 & 232 \end{bmatrix} $$
- Define the Gaussian Kernel The 3x3 Gaussian kernel given is:
$$ k = \begin{bmatrix} \frac{1}{16} & \frac{1}{8} & \frac{1}{16} \ \frac{1}{8} & \frac{1}{4} & \frac{1}{8} \ \frac{1}{16} & \frac{1}{8} & \frac{1}{16} \end{bmatrix} $$
- Compute the Convolution Calculate the convolution by performing element-wise multiplication between the pixel values and the kernel, then sum the results:
$$ \text{Output} = (82 \times \frac{1}{16}) + (57 \times \frac{1}{8}) + (87 \times \frac{1}{16}) + (48 \times \frac{1}{8}) + (159 \times \frac{1}{4}) + (225 \times \frac{1}{8}) + (130 \times \frac{1}{16}) + (73 \times \frac{1}{8}) + (232 \times \frac{1}{16}) $$
- Perform the Calculations Compute the result of the summation:
- $82 \times \frac{1}{16} = 5.125$
- $57 \times \frac{1}{8} = 7.125$
- $87 \times \frac{1}{16} = 5.4375$
- $48 \times \frac{1}{8} = 6$
- $159 \times \frac{1}{4} = 39.75$
- $225 \times \frac{1}{8} = 28.125$
- $130 \times \frac{1}{16} = 8.125$
- $73 \times \frac{1}{8} = 9.125$
- $232 \times \frac{1}{16} = 14.5$
Adding these together yields:
$$ 5.125 + 7.125 + 5.4375 + 6 + 39.75 + 28.125 + 8.125 + 9.125 + 14.5 = 113.6875 $$
- Round the Result Finally, round the output to the nearest integer, which gives:
$$ \text{Output} = 114 $$
The output is $114$.
More Information
The computed value of $114$ represents the intensity at the specified coordinate after applying the Gaussian smoothing process on the original grayscale image matrix.
Tips
- Ignoring the correct window: Make sure to correctly extract the 3x3 window centered at the specified coordinates.
- Mistaking the kernel values: Always double-check kernel values to avoid incorrect multiplication.
- Incorrect rounding: Remember to round correctly to the nearest integer as specified.
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