Calculate the output produced if it is subject to the unsharp mask (USM) process. Use a 3 x 3 mean filter kernel, defined below, for the blurring step. Show the output for the cent... Calculate the output produced if it is subject to the unsharp mask (USM) process. Use a 3 x 3 mean filter kernel, defined below, for the blurring step. Show the output for the center position of this 3 x 3 image matrix. Round any answer to be a natural number and round .5 upwards. Do not rescale or limit your answer to be in the range [0..255].
Understand the Problem
The question is asking us to perform a calculation using the unsharp mask (USM) process on an 8-bit grayscale image matrix. We need to apply a 3 x 3 mean filter kernel to the provided matrix, specifically focusing on the center position of the 3 x 3 matrix. The output needs to be rounded and presented according to specific guidelines without rescaling.
Answer
The output value for the center position is 0.
Answer for screen readers
The output value for the center position is 0.
Steps to Solve
- Define the image and kernel We have the following 8-bit grayscale image matrix:
$$ L = \begin{bmatrix} 189 & 13 & 63 \ 127 & 23 & 11 \ 141 & 55 & 24 \end{bmatrix} $$
The kernel for the mean filter is:
$$ k = \begin{bmatrix} \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \ \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \ \frac{1}{9} & \frac{1}{9} & \frac{1}{9} \end{bmatrix} $$
- Calculate the convolution of the image with the kernel To find the output value for the center position (23 in this case), we perform the convolution using the kernel.
The computation is as follows:
$$ L * k = \frac{1}{9} \left(189 + 13 + 63 + 127 + 23 + 11 + 141 + 55 + 24\right) $$
- Simplify the summation Calculating the sum of the elements:
$$ 189 + 13 + 63 + 127 + 23 + 11 + 141 + 55 + 24 = 612 $$
Now apply the kernel factor:
$$ L * k = \frac{1}{9} \cdot 612 = 68 $$
- Apply the USM formula Using the USM formula:
$$ usm(L) = L + (L - L * k) $$
Substituting (L = 23) (the center value) and (L * k = 68):
$$ usm(23) = 23 + (23 - 68) $$
- Compute the final value Now we compute:
$$ usm(23) = 23 + (23 - 68) = 23 - 45 = -22 $$
Since we can’t have negative pixel values, we will take the minimum of 0.
- Final rounding Round and convert to an integer, since pixel values must be whole numbers:
The output value for the center position will be 0.
The output value for the center position is 0.
More Information
This method effectively uses an unsharp mask technique to enhance image details by sharpening. The convolution with a mean kernel blurs the image slightly and helps identify areas that can be sharpened.
Tips
- Not applying boundary conditions correctly, leading to incorrect pixel values.
- Forgetting to round appropriately when required.
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