Question

While implementing the digital Laplacian filter mask, which of the following is/are the possible value(s) of the center pixel?

• A. -4
• B. -8
• C. -2
• D. -6

Hint:

A Laplacian mask's coefficients must sum to zero; work out the 4-neighbour and 8-neighbour cases.

Answer 1

The Correct Option is A, B

Step 1: Write the general discrete Laplacian and where the center coefficient comes from.
The Laplacian of \(f(x,y)\) is \(\nabla^2 f=\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}\). Using standard second-difference approximations along each axis and adding them gives \[ \nabla^2 f \approx \left[f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)\right]-4f(x,y) \] so the center coefficient in this standard 4-neighbour (cross-shaped) mask is \(-4\), with the four axis-aligned neighbours each having coefficient \(+1\) and all corners \(0\). This matches option (A).
Step 2: Extend to the 8-neighbour Laplacian mask.
An extended mask also includes the diagonal second-derivative terms, giving all 8 surrounding pixels coefficient \(+1\). Since any Laplacian-type mask must have all its coefficients sum to zero (a constant-intensity region must give zero response), the center must balance \(1\times8=8\), so the center coefficient is \(-8\), matching option (B).
Step 3: Check why (C) -2 and (D) -6 are not standard values.
With unit neighbour weights at 4 or 8 positions, the only center values consistent with a zero mask-sum are \(-4\) or \(-8\); \(-2\) and \(-6\) do not correspond to any standard Laplacian mask construction with unit neighbour weights.
Step 4: Conclusion.
The two possible center-pixel values for the standard digital Laplacian masks are \(-4\) (4-neighbour mask) and \(-8\) (8-neighbour mask).\[ \boxed{-4 \text{ and } -8} \]