Question

Problem: A 2D filter is considered separable if this filter can be written as a product of a colun vector (also called vertical filter) and a row vector (also called horizontal filter). For the two filt given below, answer the following questions.

[ -1 -2 -1] [ -1 -3 -1 ]
H₁=1/4[ -2 16 -2] , H₂=1/4[ 0 0 0]
[ -1 -2 -1] [ 1 3 1]

a) Is H₁ a separable filter? If yes, present the horizontal and vertical filters.

Answer 1

Yes, H₁ is separable with the horizontal filter (-2, 16, -2) and the vertical filter (1/4)(-1 -2 -1).

Step-by-step explanation:

A 2D filter is considered separable if it can be written as a product of a column vector (vertical filter) and a row vector (horizontal filter). To determine if H₁ is separable, we can rewrite it as a product of two vectors:

H₁ = (1/4)[-1 -2 -1][(1/4)(-2) (1/4)(16) (1/4)(-2)]

Therefore, H₁ is separable with the horizontal filter (-2, 16, -2) and the vertical filter (1/4)(-1 -2 -1).