An operator H is linear if two properties hold(f1 and f2 are some functions, a is a constant):
Additivity(things sum):
-H(f1 + f2) = H(f1) + H(f2)(looks like distributive law)
Multiplicative scaling(Homogeneity of degree 1)
(constant scales):
-H(a*f1) = a*H(f1)
Because it is sums ans multiplies, the “filtering” operation we were doing are linear:
An impulse function…
it’s just a value of 1 at a single location
An impulse response
-if I have unknown system and I put in an impulse, the response is called the impulse response
-so if the black box is linear you can describe H by h(x)
Assuming center coordinate is “reference point”.
kernel is size MxM, and our image was NxN.
filter whole image is M*M*N*N
Correlation vs Convolution(flip in both dimensions, bottom to top, right to left)
Shift invariant:
linear & shift invariant
Associative
(f*g)*h = f*(g*h)
Identity:
unit impulse e = […,0,0,1,0,0,…] f*e = f
Differentiation: α/αx(f*g) = αf/αx*g
