Fast Phase Retrieval Research Articles

Fast phase retrieval in slightly off-axis digital holography

In this study, three efficient algorithms are proposed for fast phase retrieval in slightly off-axis digital holography using spectrum cropping, spatial multiplexing, and complex encoding. In the first algorithm, the real spectral order of the subtracted hologram is filtered and cropped, and the number of pixels is decreased in the subsequent retrieval operations. In the second algorithm, two sequential subtracted holograms are digitally phase shifted and spatial multiplexed into one synthetic hologram, and thus only one inverse Fourier transformation is then required. In the third algorithm, two sequential subtracted holograms are encoded separately into the real part and the imaginary part of a complex hologram. Two cross-correlations can be used to reconstruct the phase, thereby improving the utilization of the spectrum. The three new algorithms speed up our previously proposed retrieval method with the assistance of specimen-free holograms. Our experiments demonstrated the validity and improved time requirements of the proposed methods.

Simple, flexible, and accurate phase retrieval method for generalized phase-shifting interferometry.

This paper presents a non-iterative phase retrieval method from randomly phase-shifted fringe images. By combining the hyperaccurate least squares ellipse fitting method with the subspace method (usually called the principal component analysis), a fast and accurate phase retrieval algorithm is realized. The proposed method is simple, flexible, and accurate. It can be easily coded without iteration, initial guess, or tuning parameter. Its flexibility comes from the fact that totally random phase-shifting steps and any number of fringe images greater than two are acceptable without any specific treatment. Finally, it is accurate because the hyperaccurate least squares method and the modified subspace method enable phase retrieval with a small error as shown by the simulations. A MATLAB code, which is used in the experimental section, is provided within the paper to demonstrate its simplicity and easiness.

Non-Interferometric Tomography of Phase Objects Using Spatial Light Modulators

Quantitative 3D phase retrieval techniques are based on either interferometric techniques such as holography or noninterferometric intensity-based techniques such as the transport of intensity equation (TIE). Interferometric techniques are vibration-sensitive and often use a reference beam requiring complicated optical alignment. In this work we develop a simple, fast, and noninterferometric tomographic 3D phase retrieval technique based on the TIE which does not suffer from such drawbacks. The optical setup is a modified 4f TIE system which uses an SLM to replace the slow translation of the CCD required to record several diffraction patterns in a traditional TIE system. This novel TIE setup is suitable for dynamical events such as imaging biological processes. A rotating mechanical stage is constructed to obtain tomographic phase images of the object. The tomographic reconstruction algorithm is based on the Fourier slice theorem (backprojection algorithm) which applies to objects with a small refractive index span. Simulation and experimental results are shown as part of this work. A graphical user interface is developed to perform the TIE tomographic reconstruction algorithm and to synchronize the captured intensities by the CCD, the phase patterns displayed on the SLM, and the Arduino controlled rotating stage assembly.

Fast Phase Retrieval from Local Correlation Measurements

We develop a fast phase retrieval method which can utilize a large class of local phaseless correlation-based measurements in order to recover a given signal ${\bf x} \in \mathbb{C}^d$ (up to an unknown global phase) in near-linear $\mathcal{O} \left( d \log^4 d \right)$-time. Accompanying theoretical analysis proves that the proposed algorithm is guaranteed to deterministically recover all signals ${\bf x}$ satisfying a natural flatness (i.e., non-sparsity) condition for a particular choice of deterministic correlation-based measurements. A randomized version of these same measurements is then shown to provide nonuniform probabilistic recovery guarantees for arbitrary signals ${\bf x} \in \mathbb{C}^d$. Numerical experiments demonstrate the method's speed, accuracy, and robustness in practice -- all code is made publicly available. Finally, we conclude by developing an extension of the proposed method to the sparse phase retrieval problem; specifically, we demonstrate a sublinear-time compressive phase retrieval algorithm which is guaranteed to recover a given $s$-sparse vector ${\bf x} \in \mathbb{C}^d$ with high probability in just $\mathcal{O}(s \log^5 s \cdot \log d)$-time using only $\mathcal{O}(s \log^4 s \cdot \log d)$ magnitude measurements. In doing so we demonstrate the existence of compressive phase retrieval algorithms with near-optimal linear-in-sparsity runtime complexities.

Improved quantitative phase imaging in lensless microscopy by single-shot multi-wavelength illumination using a fast convergence algorithm.

We report on a novel algorithm for high-resolution quantitative phase imaging in a new concept of lensless holographic microscope based on single-shot multi-wavelength illumination. This new microscope layout, reported by Noom et al. along the past year and named by us as MISHELF (initials incoming from Multi-Illumination Single-Holographic-Exposure Lensless Fresnel) microscopy, rises from the simultaneous illumination and recording of multiple diffraction patterns in the Fresnel domain. In combination with a novel and fast iterative phase retrieval algorithm, MISHELF microscopy is capable of high-resolution (micron range) phase-retrieved (twin image elimination) biological imaging of dynamic events. In this contribution, MISHELF microscopy is demonstrated through qualitative concept description, algorithm implementation, and experimental validation using both a synthetic object (resolution test target) and a biological sample (swine sperm sample) for the case of three (RGB) illumination wavelengths. The proposed method becomes in an alternative instrument improving the capabilities of existing lensless microscopes.

Parallel two-step spatial carrier phase-shifting interferometric phase microscopy with fast phase retrieval

A method of parallel two-step spatial carrier phase-shifting two-window interferometer with fast phase retrieval is applied for microscopic imaging. The method prior captures interferograms offline without any sample only once. As a consequence, the interferograms for a phase sample can be acquired by camera in a single shot, and the phase distribution of the sample can be then reconstructed without phase unwrapping process if the phase variance is less than 2π. This method can eliminate the aberration and noise of the system and phase tilt at the same time, and thus improve the phase retrieval speed. Experiments are demonstrated to prove the validation of the presented method.

Fast phase retrieval method based on twice derivatives in phase-shifting interferometry with a blind phase shift

A derivative method for phase retrieval in two-step phase-shifting interferometry (PSI) with a blind phase shift is proposed in this Letter. By numerically calculating the first-order and second-order derivatives of two PSI images, one can directly determine the phase shift and then obtain the quantitative phase image. We illustrate the method with both a theoretical analysis and some simulations of an average-sized red blood cell in different interferometry recording modes. From the results, the validity, accuracy, as well as efficiency of this method are verified. Moreover, this method can be applied to any quantitative phase imaging, including on-axis, off-axis, and slight off-axis interferometry.

Linear phase retrieval for real-time adaptive optics

We developed a fast phase retrieval algorithm that is suitable for real-time applications such as adaptive optics. The phase retrieval model is developed by linearising the pupil function in the approximation of small aberrations and is valid for low-NA focused field. The linear model in conjunction with a particular choice for the position of the single out-of-focus measurement plane and an efficient control algorithm, significantly reduces the computation time for phase retrieval. The experimental results demonstrate the validity of the described approach for fast correction of aberrations.

An ameliorated fast phase retrieval iterative algorithm based on the angular spectrum theory

For designing the light beam shaper in the optical transmitting antenna in inter-satellite optical communication system, the essential problem is, according to the input optical field and ideal output light field, to determine the shaper of phase distribution, in which its core is in phase recovery. Based on the traditional Gerchberg-Saxton (G-S) iterative algorithm, using the angular spectrum propagation theory, a kind of amplitude gradient addition iterative algorithm is proposed, and the detailed algorithm process and analysis are given. Compared with G-S algorithm, the new one using iterative process, constructs the light field amplitude feedback loop, and searches out the optimal iteration path using the gradient; their joint action accelerates the convergence process. Numerical simulation shows that the iterative error-drop speed caused by unit iterations in the new algorithm is 1.7 times that of the G-S algorithm, its convergence rate is obviously superior to the G-S algorithm; for different random initial phase, the new algorithm can effectively carry out iteration, show the advantages of the strong adaptability and the good convergent consistency. Amplitude gradient addition iterative algorithm gives a new effective way of recovering the complex optical field phase, and provides technical support for designing all kinds of diffraction optical element.

Novel Fourier-domain constraint for fast phase retrieval in coherent diffraction imaging

Coherent diffraction imaging (CDI) for visualizing objects at atomic resolution has been realized as a promising tool for imaging single molecules. Drawbacks of CDI are associated with the difficulty of the numerical phase retrieval from experimental diffraction patterns; a fact which stimulated search for better numerical methods and alternative experimental techniques. Common phase retrieval methods are based on iterative procedures which propagate the complex-valued wave between object and detector plane. Constraints in both, the object and the detector plane are applied. While the constraint in the detector plane employed in most phase retrieval methods requires the amplitude of the complex wave to be equal to the squared root of the measured intensity, we propose a novel Fourier-domain constraint, based on an analogy to holography. Our method allows achieving a low-resolution reconstruction already in the first step followed by a high-resolution reconstruction after further steps. In comparison to conventional schemes this Fourier-domain constraint results in a fast and reliable convergence of the iterative reconstruction process.

Quantitative comparison of direct phase retrieval algorithms in in‐line phase tomography

A well-known problem in x-ray microcomputed tomography is low sensitivity. Phase contrast imaging offers an increase of sensitivity of up to a factor of 10(3) in the hard x-ray region, which makes it possible to image soft tissue and small density variations. If a sufficiently coherent x-ray beam, such as that obtained from a third generation synchrotron, is used, phase contrast can be obtained by simply moving the detector downstream of the imaged object. This setup is known as in-line or propagation based phase contrast imaging. A quantitative relationship exists between the phase shift induced by the object and the recorded intensity and inversion of this relationship is called phase retrieval. Since the phase shift is proportional to projections through the three-dimensional refractive index distribution in the object, once the phase is retrieved, the refractive index can be reconstructed by using the phase as input to a tomographic reconstruction algorithm. A comparison between four phase retrieval algorithms is presented. The algorithms are based on the transport of intensity equation (TIE), transport of intensity equation for weak absorption, the contrast transfer function (CTF), and a mixed approach between the CTF and TIE, respectively. The compared methods all rely on linearization of the relationship between phase shift and recorded intensity to yield fast phase retrieval algorithms. The phase retrieval algorithms are compared using both simulated and experimental data, acquired at the European Synchrotron Radiation Facility third generation synchrotron light source. The algorithms are evaluated in terms of two different reconstruction error metrics. While being slightly less computationally effective, the mixed approach shows the best performance in terms of the chosen criteria.

In-line holography and coherent diffractive imaging with x-ray waveguides

A Fresnel coherent diffraction imaging experiment with hard x rays is here presented, using two planar crossed waveguides as optical elements, leading to a virtual pointlike source. The coherent wave field obtained with this setup is used to illuminate a micrometric single object having the shape of a butterfly. A digital two-dimensional in-line holographic reconstruction of the unknown object at low resolution (200 nm) has been obtained directly via fast Fourier transform (FFT) of the raw data. The object and its twin image are well separated because suitable geometrical conditions are satisfied. A good estimate of the incident wave field phase has been extracted directly from the FFT of the raw data. A partial object reconstruction with 50 nm spatial resolution was achieved by fast iterative phase retrieval, the major limitation for a full reconstruction being the nonideal structure of the guided beam. The method offers a route for fast and reliable phase retrieval in x-ray coherent diffraction.

Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials

A new technique is proposed for the recovery of optical phase from intensity information. The method is based on the decomposition of the transport-of-intensity equation into a series of Zernike polynomials. An explicit matrix formula is derived, expressing the Zernike coefficients of the phase as functions of the Zernike coefficients of the wave-front curvature inside the aperture and the Fourier coefficients of the wave-front boundary slopes. Analytical expressions are given, as well as a numerical example of the corresponding phase retrieval matrix. This work lays the basis for an effective algorithm for fast and accurate phase retrieval.