In SDP based phase retrieval we have intensity measurement (abs(FFT2(x))^2) of the formA(xo) = |ak,x|^2 =b^2, phase retrieval is then find x that obeys A(xo)=bThe quadratic measurements can be lifted and interpreted as linear about rank 1 matrix X =xx*|ak,x|^2 = b^2 = Tr(AX), with X =xx* and we want to recover the phase (x - signal to recover).In the standard approach named as phase lift we imagine lift of dimension X =xx* and we look to minimize rank of Tr(X), subject to A(X)=b, X>=0And we need roughly 4n measurement for n length x signal if I understand it correctly.(X dimmension is nxn)This is fine for 1D problems. When we have 2D pahse retrieval going to 4D in X doesnt look good approach.What is the trick for phase recover in the case of 2D phase retrieval problems? Just few reference links.I would like to ask how to construct phase retrieval (with matrix completion) of test picture 2D, let say phantom (phantom('Modified Shepp-Logan',50) in matlab) with semidefinite programing like phaselift.To retrieve a phase from diffraction pattern where only the square of amplitudes are recorded one needs to rely on phase retrieval with projection algorithms like Saxton-Gerchberg or solve the problem casting it into matrix completion problem (where the SDP comes to minimize rank of higher dimension matrix, which is considered NP hard). http://web.stanford.edu/~mahdisol/PhaseRetrieval_CDP.pdf Phase lift methodExamples in 1D online.http://web.stanford.edu/~mahdisol/PhaseRetrieval_CDP.htmlvia semidefinite method like phaselifthttp://web.stanford.edu/~mahdisol/PRcode.html codeor http://arxiv.org/pdf/1111.6323v3.pdfhttp://users.isy.liu.se/en/rt/ohlsson/code.html codeThank you.
Phase retrieval via SDP (semidefinite program) of 2D test image (Matrix completion)
matlab;sparse;convex optimization;inverse problem;semidefinite programming