Question

The derivative of digital function is defined in terms of difference. Then, which of the following defines the second order derivative ∂^2 f/∂x^2 = ___________ of a one-dimensional function f(x)?

(a) f(x+1)-f(x)

(b) f(x+1)+ f(x-1)-2f(x)

(c) All of the mentioned depending upon the time when partial derivative will be dealt along two spatial axes

(d) None of the mentioned

I had been asked this question by my school teacher while I was bunking the class.

My question comes from Sharpening Spatial Filters-2 topic in section Intensity Transformations and Spatial Filtering of Digital Image Processing

Answer 1

Right answer is (b) f(x+1)+ f(x-1)-2f(x)

Best explanation: The definition of a second order derivative of a one dimensional image f(x) is:

 (∂^2 f)/∂x^2 =f(x+1)+ f(x-1)-2f(x), where the partial derivative is used to keep notation same even for f(x, y) when partial derivative will be dealt along two spatial axes.