Variational analysis applied to the problem of optical phase retrieval

Luke, D. Russell and Burke, James V. (2003) Variational analysis applied to the problem of optical phase retrieval. [Preprint]

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    We apply nonsmooth analysis to a well known optical inverse problem, phase retrieval. The phase retrieval problem arises in many different modalities of electromagnetic imaging and has been studied in the optics literature for over forty years. The state of the art for this problem in two dimensions involves iterated projections for solving a nonconvex feasibility problem. Despite widespread use of these algorithms, current mathematical theory cannot explain their success. At the heart of projection algorithms is a nonconvex, nonsmooth optimization problem. We obtain some insight into these algorithms by applying techniques from nonsmooth analysis. In particular, we show that the weak closure of the set of directions toward the projection generate the subdifferential of the corresponding squared set distance function. Following a pattern of proof described in F.H. Clarke, Yu.S. Ledyaev, R.J. Stern, and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer (1998), this result is generalized to provide conditions under which the subdifferential of an integral function equals the integral of the subdifferential.

    Item Type: Preprint
    Additional Information: pubdom FALSE
    Uncontrolled Keywords: phase retrieval, least squares, nonsmooth analysis, variational analysis
    Subjects: 78-xx Optics, electromagnetic theory > 78Axx General
    49-xx Calculus of variations and optimal control; optimization > 49Jxx Existence theories
    93-xx Systems theory; control > 93Exx Stochastic systems and control
    Faculty: UNSPECIFIED
    Depositing User: Users 1 not found.
    Date Deposited: 20 Nov 2003
    Last Modified: 21 Apr 2010 11:13

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