Question

A Laplacian for an image f(x, y) is defined as:   is given by ________

(a) [f(x + 1, y) + f(x – 1, y) – 2f(x, y)] and [f(x, y + 1) + f(x, y – 1) – 2f(x, y)] respectively

(b) [f(x + 1, y + 1) + f(x, y – 1) – 2f(x, y)] and [f(x , y + 1) + f(x – 1, y) – 2f(x, y)] respectively

(c) [f(x, y + 1) + f(x, y – 1) – 2f(x, y)] and [f(x + 1, y) + f(x – 1, y) – 2f(x, y)] respectively

(d) None of the mentioned

I had been asked this question in homework.

I'd like to ask this question from Use of Second Order Derivative for Enhancement in section Image Enhancement of Digital Image Processing

Answer 1

The correct answer is (a) [f(x + 1, y) + f(x – 1, y) – 2f(x, y)] and [f(x, y + 1) + f(x, y – 1) – 2f(x, y)] respectively

The explanation is: For a Laplacian given by:∇^2 f=

Applying second order derivative in x direction (∂^2 f)/∂x^2  = [f(x + 1, y) + f(x – 1, y) – 2f(x, y)], and

Applying second order derivative in y direction (∂^2 f)/∂y^2 = [f(x, y + 1) + f(x, y – 1) – 2f(x, y)].