= Phase retrieval
<Phase retrieval> reconstructs a signal from measurements that discard its complex <wave phase>. A basic diffraction model measures $I(\xi)=|\widehat f(\xi)|^2$, rather than the complex <Fourier transform> itself. The lost phase cannot generally be recovered simply by taking a square root of the intensity.
There are immediate ambiguities. Multiplication by a constant $e^{i\theta}$ leaves every magnitude unchanged. A <translation> multiplies the transform by a unit-modulus phase factor, and conjugate reflection $f(x)\mapsto\overline{f(-x)}$ conjugates its transform. These transformations therefore preserve $I$. They need not exhaust the ambiguities of one-dimensional data.<Fourier Phase Retrieval: Uniqueness and Algorithms by Tamir Bendory and others>[§3.1, printed p6, trivial and non-trivial ambiguities]{ref}
By the <convolution theorem>, magnitude-squared data determine the <autocorrelation of a square-integrable function> under suitable inversion hypotheses, rather than automatically determining the function itself. Known support, additional masked measurements or other structural assumptions can reduce the ambiguity; their sufficiency must be checked for the actual measurement model. <Frequency-resolved optical gating> uses a family of delayed nonlinear spectra, giving more information than a single unmodified Fourier magnitude.
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