By my understanding, a transform T is homogeneous if T = 0.
Then to prove that a linear transformation is homogeneous we say that:
T[ax(n1, n2) + bx(n1, n2)] = aT[x(n1, n2)] + bT[x(n1, n2)]
What I want to know is the difference between the following two proofs and which one is "more correct" or is there perhaps a even better answer?
let x(n1, n2) = 0
then we know that,
ax(n1,n2)- ax(n1,n2) = 0, T[ax(n1, n2) + ax(n1, n2)] = aT[x(n1, n2)] + aT[x(n1, n2)] = 0
Simply showing that both sides are equal to 0 if x(n1, n2) = 0 through algebra.
On a side note, can someone also comment on why we would even care if a transformation is homogeneous vs heterogeneous? In reference to a transformation applied onto a 2D signal (such as an image transformation).
Maybe the better question to ask would be to prove if a transformation is heterogeneous, is it linear? (if so can someone help me define what it means for a transformation to be heterogeneous? Like how a homogeneous transform means T = 0.