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Help with Freeman-Durden decomposition theory
Submitted by bdeschamps on Mon, 10/19/2009 - 06:41
I am trying to understand Freeman-Durden decomposition as outlined in the paper:
Freeman, A. (1998). "A Three-Component Scattering Model for Polarimetric SAR Data", IEEE Transactions on Geoscience and Remote Sensing, 36(3).
My question is: once you have the three equations in three unknowns (equation 11 in the paper) and you set either alpha = -1 or beta = 1 depending on the sign of ShhS*vv, what are the equations for fd, fs, a and B, and how are they derived?
Any help or suggestions would be fantastic.
Benjamin
#1 freeman-durden equations
For alpha=-1, we have 3 complex equations:
|hh|^2 = fd + |b|^2 * fs
|vv|^2 = fs + fd
hh*conj(vv) = b*fs - fd
fs, fd are real. beta (b) is complex.
splitting beta into br and bi (real and imaginary), 4 equations and 4 unknown: (real)
|hh|^2 = fd + |b|^2 * fs
|vv|^2 = fs + fd
Re(hh*conj(vv)) = br*fs - fd
Im(hh*conj(vv)) = bi*fs
Setting it up this way:
x = Re(hh*conj(vv))
y = Im(hh*conj(vv))
h = |hh|^2
v = |vv|^2
Analytic solution of the 4 equations & 4 unknowns:
fd = (-x^2 - y^2 + hv) / D1
br = (hx + v^2 + hv + 3xv + 2x^2) / D2 - 1
bi = D1 * y / D2
fs = D2 / D1
where
D1 = h + v + 2x
D2 = v^2 + 2xv + x^2 + y^2
I don't have my actual solution method any more, but it wasn't too bad once
it was set up properly.
For beta=1 everything is similar:
|hh|^2 = fs + |a|^2 * fd
|vv|^2 = fs + fd
hh*conj(vv) = a*fd + fs
fs, fd are real. alpha (a) is complex.
splitting alpha into ar and ai (real and imaginary), 4 equations and 4 unknown: (real)
|hh|^2 = fs + |a|^2 * fd
|vv|^2 = fs + fd
Re(hh*conj(vv)) = ar*fd + fs
Im(hh*conj(vv)) = ai*fd
x = Re(hh*conj(vv))
y = Im(hh*conj(vv))
h = |hh|^2
v = |vv|^2
Solution:
fs = (-x^2 - y^2 + hv) / D1
ar = (hx - v^2 - hv + 3xv - 2x^2) / D2 + 1
ai = D1 * y / D2
fd = D2 / D1
where
D1 = h + v - 2x
D2 = v^2 - 2xv + x^2 + y^2
I hope this helps! I can try to dig up the solution method if you would like.