Given two spatial masks \(S_{1}= \begin{bmatrix} 0&1&0 \\ 1&-4&0 \\ 0&1&0…
2014
Given two spatial masks
\(S_{1}= \begin{bmatrix} 0&1&0 \\ 1&-4&0 \\ 0&1&0 \end{bmatrix}\) and \(S_{2} = \begin{bmatrix} 1&1&1 \\ 1&-8&1 \\ 1&1&1 \end{bmatrix}\)
The Laplacian of an image at all points (x, y) can be implemented by convolving the image with spatial mask. Which of the following can be used as the spatial mask ?
- A.
only S1
- B.
only S2
- C.
Both S1 and S2
- D.
None of these
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Correct answer: B
Key check: a discrete Laplacian mask must be symmetric (cover all neighbors appropriately) and its elements must sum to zero.
Sum check for the first mask S1: 0 + 1 + 0 + 1 + (-4) + 0 + 0 + 1 + 0 = -1, which is not zero, so S1 is not a valid Laplacian.
Sum check for the second mask S2: eight 1's plus (-8) at the centre gives 8 + (-8) = 0, so S2 satisfies the sum-to-zero requirement.
Neighbour coverage and symmetry:
The first mask S1 omits one neighbouring term (it is asymmetric in the 3x3 window), so it does not represent the standard finite-difference Laplacian.
The second mask S2 is symmetric and covers all eight neighbours with equal weight, giving the 8-neighbour Laplacian.
Conclusion: only the second mask S2 is a valid spatial mask to implement the Laplacian.