 Phase retrieval

Phase retrieval reconstructs a signal from measurements that discard its complex wave phase. A basic diffraction model measures $I(\xi )=|\hat{f}(\xi ){|}^2$, 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.^[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.

Phase retrieval is a mathematical and computational technique used to recover the phase information of a signal or wave from intensity measurements. This is particularly relevant in fields such as optics, imaging, and signal processing, where the amplitude (intensity) of a signal can often be measured directly, but the phase information is lost or difficult to obtain.