An alternating optimization approach for phase retrieval

We develop for the first time a mathematical framework in which the class of projection algorithms can be applied to high numerical aperture (NA) phase retrieval. Within this framework, we first analyze the basic steps of solving the high-NA phase retrieval problem by projection algorithms and establish the closed forms of all the relevant projection operators. We then study the geometry of the high-NA phase retrieval problem and the obtained results are subsequently used to establish convergence criteria of projection algorithms in the presence of noise. Making use of the vectorial point-spread-function (PSF) is, on the one hand, the key difference between this paper and the literature of phase retrieval mathematics which deals with the scalar PSF. The results of this paper, on the other hand, can be viewed as extensions of those concerning projection methods for low-NA phase retrieval. Importantly, the improved performance of projection methods over the other classes of phase retrieval algorithms in the low-NA setting now also becomes applicable to the high-NA case. This is demonstrated by the accompanying numerical results which show that available solution approaches for high-NA phase retrieval are outperformed by projection methods.

Phase-contrast x-ray computed tomography (PCCT) is currently investigated as an interesting extension of conventional CT, providing high soft-tissue contrast even if examining weakly absorbing specimen. Until now, the potential for dose reduction was thought to be limited compared to attenuation CT, since meaningful phase retrieval fails for scans with very low photon counts when using the conventional phase retrieval method via phase stepping. In this work, we examine the statistical behaviour of the reverse projection method, an alternative phase retrieval approach and compare the results to the conventional phase retrieval technique. We investigate the noise levels in the projections as well as the image quality and quantitative accuracy of the reconstructed tomographic volumes. The results of our study show that this method performs better in a low-dose scenario than the conventional phase retrieval approach, resulting in lower noise levels, enhanced image quality and more accurate quantitative values. Overall, we demonstrate that the lower statistical limit of the phase stepping procedure as proposed by recent literature does not apply to this alternative phase retrieval technique. However, further development is necessary to overcome experimental challenges posed by this method which would enable mainstream or even clinical application of PCCT.

In this paper, we present a method to characterize not only shape but also depth of defects in line and space mask patterns. Features in a mask are too fine for conventional imaging system to resolve them and coherent imaging system providing only the pattern diffracted by the mask are used. Then, phase retrieval methods may be applied, but the accuracy it too low to determine the exact shape of the defect. Deterministic methods have been proposed to characterize accurately the defect, but it requires a reference pattern. We propose to use successively phase retrieval algorithm to retrieve the general shape of the mask and then deterministic approach to characterize precisely the defects detected.

Efficient phase retrieval based on dark fringe extraction and phase pattern construction with a good anti-noise capability

The classical problem of phase retrieval has found a wide array of applications in optics, imaging and signal processing. In this paper, we consider the phase retrieval problem in a one-bit setting, where the signals are sampled using one-bit analog-to-digital converters (ADCs). A significant advantage of deploying one-bit ADCs in signal processing systems is their superior sampling rates as compared to their high-resolution counterparts. This leads to an enormous amount of one-bit samples gathered at the output of the ADC in a short period of time. We demonstrate that this advantage pays extraordinary dividends when it comes to convex phase retrieval formulations, namely that the often encountered matrix semi-definiteness constraints as well as rank constraints (that are computationally prohibitive to enforce), become redundant for phase retrieval in the face of a growing sample size. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.

The problem of phase retrieval underlies various imaging methods from astronomy to nanoscale imaging. Traditional phase retrieval methods are iterative and are therefore computationally expensive. Deep learning (DL) models have been developed to either provide learned priors or completely replace phase retrieval. However, such models require vast amounts of labeled data, which can only be obtained through simulation or performing computationally prohibitive phase retrieval on experimental datasets. Using 3D X-ray Bragg coherent diffraction imaging (BCDI) as a representative technique, we demonstrate AutoPhaseNN, a DL-based approach which learns to solve the phase problem without labeled data. By incorporating the imaging physics into the DL model during training, AutoPhaseNN learns to invert 3D BCDI data in a single shot without ever being shown real space images. Once trained, AutoPhaseNN can be effectively used in the 3D BCDI data inversion about 100× faster than iterative phase retrieval methods while providing comparable image quality.

There are several important remote sensing applications where the development of Ground Penetrating Synthetic Aperture Radar (GPENSAR) is the logical approach, e.g., searching for buried military facilities, minefield mapping, survey of underground pipelines. Penetration of sufficient soil depth for useful results require a SAR to operate at VHF/UHF frequencies, e.g., 200 - 300 MHz. At these frequencies a satellite SAR will encounter substantial distortion in the double passage of the SAR signal through the ionosphere. One of the ionospheric distortions is equivalent the phase aberrations caused in imaging through the turbulent atmosphere, and the problem of phase retrieval for the GPENSAR becomes a necessity. For GPENSR there are imaging concepts that exploit dual polarization radiation of the SAR pulse. The phase retrieval problem then becomes one of compensation for the phase aberrations induced in each of the polarization components returned to the satellite receiver. We discuss the use of the two polarizations to cancel the ionospheric phase aberrations. Unfortunately, the resulting signal has only relative phase of the two polarizations. We discuss an algorithm for the retrieval of the absolute phase. The algorithm is based on an optimization approach. Although phase retrieval by optimization is difficult because of local minima, the retrieval of absolute phase in the dual polarization case is substantially less difficult, because the two polarizations constrain the solution sufficiently to eliminate many local minima.

Advances in radiotherapy irradiation techniques have led to very complex treatments requiring for a more stringent control. The dosimetric properties of electronic portal imaging devices (EPID) encouraged their use for treatment verification. Two main approaches have been proposed: the forward approach, where measured portal dose images are compared to predicted dose images and the backward approach, where EPID images are used to estimate the dose delivered to the patient. Both approaches need EPID images to be converted into a fluence distribution by deconvolution. However, deconvolution is an ill-posed problem which is very sensitive to small variations on input data. This study presents the application of a deconvolution method based on l1-norm minimization; this is a method known for being very stable while working with noisy data. The algorithm was first evaluated on synthetic images with different noise levels, the results were satisfactory. Deconvolution algorithm was then applied to experimental portal images; the required EPID response kernel and energy fluence images were computed by Monte-Carlo calculation, accelerator treatment head and EPID models had already been commissioned in a previous work. The obtained fluence images were in good agreement with simulated fluence images. This deconvolution algorithm may be generalized to an inverse problem with a general operator, where image formation is not longer modeled by a convolution but by a linear operation that might be seen as a position-dependent convolution. Moreover, this procedure would be detector independent and could be used for any detector type provided its response function is known.

In this paper the analysis and comparison of diffraction spectra of amplitude-phase type fractal screens are presented. Amplitude-phase type fractal screens based on well-known fractals, possess exact or statistical self-similarity, but have managed amplitude transmittance and phase shift. Modeling results show that diffraction spectrum of amplitude-phase type fractal screens possess prevailing power of high frequencies in comparison with spectra of fractal structures with binary transmittance and phase shift. Averaging scattering indicatrices for fractal screens with different parameters are presented. The problem of phase retrieval arises in various fields of science and engineering, including electron microscopy, biology, crystallography, astronomy, and optical imaging. At presence there is a great number of phase retrieval methods without reference wave based on iterative algorithms and additional intensity distribution registration. In this paper we propose the new phase retrieval method with using amplitude-phase type fractal screens.

Phase retrieval is a promising method for optical system and surface metrology that makes use of intensity measurements of diffraction patterns. An iterative algorithm is used to solve the inverse problem to find the phase of the field producing the measured intensity distributions. For practical reasons, such as the reduction of coherent artifacts or to improve the signal-to-noise ratio of the measured data, it is often desirable to measure intensity distributions using broadband illumination. It is possible to perform phase retrieval with broadband data by incorporating a broadband model of the system into the phase retrieval algorithm. To do this, the system is modeled at several discrete wavelengths and the results from each are summed incoherently to produce a broadband result. This significantly increases the computational load. We show here that when aberrations are small, accurate estimates of the OPD distribution, on the level of λ/1000 RMS error, can be achieved using data with bandwidth up to about 10% as the input to a phase retrieval algorithm that assumes monochromatic data.

This paper considers two-dimensional phase retrieval using a window function. First, we address the uniqueness and reconstruction of a two-dimensional signal from the Fourier intensities of the three signals: the original signal, the signal windowed by a window w(m,n) and the signal windowed by its complementary window w/sub c/(m,n)=1-w(m,n). Then we consider the phase retrieval without a complementary window. We develop conditions under which a signal can be uniquely specified from the Fourier intensities of the original signal and the windowed signal by w(m,n). We also present a reconstruction algorithm.

Compressive phase retrieval refers to the problem of recovering a structured n-dimensional complex-valued vector from its phase-less under-determined linear measurements. The non-linearity of the measurement process makes designing theoretically-analyzable efficient phase retrieval algorithms challenging. As a result, to a great extent, existing recovery algorithms only take advantage of simple structures such as sparsity and its convex generalizations. The goal of this article is to move beyond simple models through employing compression codes. Such codes are typically developed to take advantage of complex signal models to represent the signals as efficiently as possible. In this work, it is shown how an existing compression code can be treated as a black box and integrated into an efficient solution for phase retrieval. First, COmpressive PhasE Retrieval (COPER) optimization, a computationally-intensive compression-based phase retrieval method, is proposed. COPER provides a theoretical framework for studying compression-based phase retrieval. The number of measurements required by COPER is connected to κ, the α-dimension (closely related to the ratedistortion dimension) of a given family of compression codes. To finds the solution of COPER, an efficient iterative algorithm called gradient descent for COPER (GD-COPER) is proposed. It is proven that under some mild conditions on the initialization and the compression code, if the number of measurements is larger than Cκ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> n, where C is a constant, GD-COPER obtains an accurate estimate of the input vector in polynomial time. In the simulation results, JPEG2000 is integrated in GD-COPER to confirm the state-of-the-art performance of the resulting algorithm on real-world images.

The article presents a novel technological procedure for speech signal processing based on the empirical mode decomposition, being an adaptive time-frequency analysis method. The proposed procedure is based on the uniform splitting of the original speech signal into fragments, the decomposition of fragments into empirical modes, and the formation of new mode speech signals. The goal of the technological procedure elaboration is to expand the space for informatively significant amplitude, time, frequency, and energy characteristics of the original speech signal. A brief description of various types of empirical mode decomposition has been presented, and their advantages and disadvantages have been revealed. The functionality of the proposed technological procedure has been detailed, and the research outcomes have been reported. An analysis of the research results has evidenced that the minimum time for the formation of a set of modal speech signals is afforded when analyzing 300–1000 ms fragments; the minimum error in the formation of a set of mode speech signals is obtained when the fragments are decomposed into 8–10 empirical modes, and the difference between the original and reconstructed signals being less than 0.001 V (0.1 %). It has been concluded that the proposed technological procedure actually provides an expansion of the space for informatively significant amplitude, time, frequency, and energy characteristics due to the formation of a set of new mode speech signals. Thus, it can be efficiently used in the formation of an optimal set of speech parameters relevant to naturally expressed human emotions.

An automated method for modifying a speech signal in a telephone network by applying a gain factor which is a function of the level of background noise at a given destination, and transmitting the modified speech signal to the destination. The gain applied may be a function of both the background noise level and the original speech signal. Either a linear or a non-linear (e.g., compressed) amplification of the original speech signal may be performed, where a compressed amplification results in the higher level portions of the speech signal being amplified by a smaller gain factor than lower level portions. The speech signal may be separated into a plurality of subbands, each resultant subband signal being individually modified in accordance with the present invention. In this case, each subband speech signal is amplified by a gain factor based on a corresponding subband noise signal, generated by separating the background noise signal into a corresponding plurality of subbands. The individual modified subband signals may then be combined to form the resultant modified speech signal.

Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input-output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.