FiW
The function FiW computes the defect FFqmF = FF(qROI) - FmeasDelta and the adjoint of the Frechet derivative applied to the defect in the context of discrete wavelet transform.
Contents
Syntax
[FFqmF, DH] = FiW(qROI,FmeasDelta,seti,iW,iWStar)
Description
[FFqmF, DH] = FiW(qROI,FmeasDelta,seti,iW,iWStar) computes the defect FFqmF = FF(qROI) - FmeasDelta and the adjoint of the Fréchet derivative applied to the defect. In this routine the latter is in dependence of the transformed contrast qROI by discrete wavelet transform; afterwards it is transformed back. Therefore it is the analog of mimo (in the case of 'adjOfDer'). The details are in "More About".
Input Arguments
- qROI : Contrast
- FmeasDelta : Data with noise
- seti : Structure array (mutable data structure of IPscatt)
- iW : Inverse wavelet transform (currently unused in this routine)
- iWStar : Adjoint of the inverse wavelet transform
Output Arguments
- FFqmF : Defect, i.e. the difference between the predicted data
, that bases on the reconstructed contrast
, and the given data FmeasDelta.
- DH : Adjoint of the Fréchet derivative applied to the defect in the context of discrete wavelet transform, i.e. defect H(Wq) := | F(iW( Wq )) - y |_HS, see below for details.
More About
The function FiW is used in the routine minShrink.
The input arguments iW and iWStar may be weighted in the routine setWavelet.
FiW as analog of mimo
As already mentioned the function FiW is the analog of mimo (in the case of 'adjOfDer'), i.e. computes the defect and the adjoint of the Fréchet derivative applied to the defect. In contrast to mimo the defect is in dependence of the transformed contrast qROI by discrete wavelet transform and transformed back after the computation of the adjoint.
To make this point clear we compare both formulations in the following.
As notation we use:
- y := FmeasDelta
- q := qROI
- F := FF (forward operator
)
- HS : Hilbert-Schmidt norm
- W : discrete wavelet transform
- iW : inverse wavelet transform
In the case of mimo (in the case 'adjOfDer') the defect is:
G(q) := | F( q ) - y |_HS
In the case of FiW the defect is in the context of discrete wavelet transform, i.e.
H(Wq) := | F(iW( Wq )) - y |_HS
The adjoint of the derivative is
iWStar[DF(iWq(Wq))]^*[ F(iW(Wq)) - y ]
respectively
iWStar[DF(q)]^*[ F(q) - y ] = iWStar(mimo(q)(2)) = DH
Of course, the defect in dependence of Wq is dWq = F(iW( Wq )) - y = F(q) - y.
Note that mimo(q)(2) describes the secound output argument of the routine mimo.
See Also
Code
function [FFqmF, DH] = FiW(qROI,FmeasDelta,seti,iW,iWStar) % % Update information: % % FFWqy = FF(q)-F (because input q and not Wq followed by q = iW(Wq) any more). % That is why |iW| is needless. % if nargout == 2 [FFqmF, DG] = mimo(seti, qROI, FmeasDelta,'adjOfDer'); DH = iWStar(DG); else FFqMeas = mimo(seti, qROI, 'simo'); FFqmF = FFqMeas - FmeasDelta; end end