Phase Retrieval and Hilbert Integral Equations – Beyond MinimumPhase
Abstract
The Fourier transform (spectrum) of a signal is a complex function and is characterized by the magnitude and phase spectra. Phase retrieval is the reconstruction of the phase spectrum from the measurements of the magnitude spectrum. Such problems are encountered in imaging modalities such as Xray crystallography, frequencydomain optical coherence tomography (FDOCT), quantitative phase microscopy, digital holography, etc., where only the magnitudes of the wavefront are detected by the sensors. The phase retrieval problem is illposed in general, since an in nite number of signals can have the same magnitude spectrum. Typical phase retrieval techniques rely on certain prior knowledge about the signal, such as its support or sparsity, to reconstruct the signal. A classical result in phase retrieval is that minimumphase signals have logmagnitude and phase spectra that satisfy the Hilbert integral equations, thus facilitating exact phase retrieval.
In this thesis, we demonstrate that there exist larger classes of signals beyond minimumphase signals, for which exact phase retrieval is possible. We generalize Hilbert integral equations to 2D, and also introduce a variant that we call the composite Hilbert transform in the context of 2D periodic signals.
Our first extension pertains to a particular type of parametric modelling of 2D signals. While 1D minimumphase signals have a parametric representation, in terms of poles and zeros, there exists no such 2D counterpart. We introduce a new class of parametric 2D signals that possess the exact phase retrieval property, that is, their magnitude spectrum completely characterizes the signal. Starting from the magnitude spectrum, a sequence of nonlinear operations lead us to a sumofexponentials signal, from which the parameters are computed employing concepts from highresolution spectral estimation such as the annihilating filter and algebraically coupled matrixpencil methods. We demonstrate that, for this new class of signals, our method outperforms existing techniques even in the presence of noise.
Our second extension is to continuousdomain signals that lie in a principal shiftinvariant space spanned by a known basis. Such signals are characterized by the basis combining coefficients. These signals need not be minimumphase, but certain conditions on the coefficients lead to exact phase retrieval of the continuousdomain signal. In particular, we introduce the concept of causal, delta dominant (CDD) sequences, and show that such signals are characterized by their magnitude spectra. This condition pertains to the time/spatialdomain description of the signal, in contrast to the minimumphase condition, which is described in the spectral domain. We show that there exist CDD sequences that are not minimumphase, and vice versa. However, finitelength CDD sequences are always minimumphase. Our method reconstructs the signal from the magnitude spectrum up to machine precision. We thus have a class of continuousdomain signals that are neither causal nor minimum phase, and yet allow for exact phase retrieval. The shiftinvariant structure is applicable to modelling signals encountered in imaging modalities such as FDOCT.
We next present an application of 2D phase retrieval to continuousdomain CDD signals in the context of quantiative phase microscopy. We develop sufficient conditions on the interfering reference wave for exact phase retrieval from magnitude measurements. In particular, we show that when the reference wave is a plane wave with magnitude greater that the intensity of the object wave, and when the carrier frequency is larger than the bandwidth of the object wave, we can reconstruct the object wave exactly. We demonstrate highresolution reconstruction of our method on USAF target images.
Our final and perhaps the most unifying contribution is in developing Hilbert integral equations for 2D firstquadrant signals and in introducing the notion of generalized minimumphase signals for both 1D and 2D signals. For 2D continuousdomain, firstquadrant signals, we establish partial Hilbert transform relations between the real and imaginary parts of the spectrum. In the context of 2D discretedomain signals, we show that the partial Hilbert transform does not suffice and introduce the notion of composite Hilbert transform and establish the integral equations. We then introduce four classes of signals (combinations of 1D/2D and continuous/discretedomain) that we call generalized minimumphase signals, which satisfy corresponding Hilbert integral equations between logmagnitude and phase spectra, hence facilitating exact phase retrieval. This class of generalized minimumphase signals subsumes the well known class of minimumphase signals. We further show that, akin to minimumphase signals, these signals also have stable inverses, which are also generalized minimumphase signals.
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