(i) Show that if A, S and E are, respectively, isotropic smoothing (averaging), sharpening (high-boost) and edge-detection (high-pass) filters described in the spatial domain, and I is the identity operator, the following are possible relationships:

(a) S = aI + (1 - a)A,

(b) E = b(I - S).

Discuss the range of values that a and b can take, and any further modification of (b) that may be needed in a practical implementation.
(ii) The kernel of a 3 x 3 Laplace filter is shown below:

0        1        0
1       -4        1
0        1        0

Explain why this is an edge-detection filter, and derive two sharpening and two smoothing filters from it using the equations in (i). Explain briefly the differences between the two filters in each case.