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 nonnegativity constraint in astronomy are considered, with emphasis on the latter. The symmetric doublepass method provides autocorrelation of the retinal psf and, thus, the modulation transfer function. With the transfer function 17, one can now formulate a new. Abstract phase transfer catalysis ptc has been widely used for the synthesis of organic compounds for more than three decades. The transfer function is called a highpass function because its gain approaches zero at low frequencies. The defocused weakobject transfer function of a partially coherent brightfield microscope is calculated. 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. Transfer functions and bode plots transfer functions for sinusoidal time variations, the input voltage to a.
On the algorithmic side, as we detail further below, ideas of semide nite relaxation and sparsity have played a key role in modern phase retrieval. Murnane jila, university of colorado boulder, colorado 803090440, usa. It is applicable to many optical modalities, with phase retrieval techniques having been used in visible light, xray, electron, and neutron optics, and can be achieved using a variety of techniques 58. The retrieval of phase from intensity is one of the most important and most challenging problems in classical optics. Gureyev a a csiro manufacturing and materials technology, private bag 33, clayton south, vic. Several phase retrieval algorithms have been developed for twodimensional imaging in astronomy. Linear phase retrieval algorithms have significant performance. Partially coherent quantitative phase retrieval for euv.
Several regularization methods have been used in the literature to obtain stable solutions for the phase retrieval or the phase contrast tomography problems. Since only intensities of diffracted waves can be measured, the main mathematical challenge consists in a phaseretrieval problem in the nearfield regime. Local minima analysis of phase diverse phase retrieval. Regularization methods for phase retrieval and phase. Prtf stands for phase retrieval transfer function algorithm. By exploiting the total variation tv regularization scheme and the contrast transfer function ctf, a phase map can be retrieved from singledistance coherent diffraction images via the sparsity of the investigated object. Phase retrieval from the magnitude of the fourier transforms. Quantitative comparison of direct phase retrieval algorithms. Compressive phase retrieval via generalized approximate.
However, the ctftv phase retrieval algorithm often struggles in the presence of strong noise, since it is based on the traditional compressive sensing optimization. The purpose of cruise control is to keep the velocity of a car constant. Recovering phase information with deterministic approaches as the transport of intensity equation tie has recently emerged as an alternative tool to the interferometric techniques because it is experimentally easy to implement and provides fast and accurate results. Yet, the functional framework and the convergence properties of the methods have not been studied in detail. We consider the problem of quantitative phase retrieval from images obtained using a coherent shiftinvariant linear imaging system whose associated transfer function i.
Use of this transfer function for phase restoration quantitative phase retrieval from images of weak mixed phase amplitude objects is discussed. Phase contrast, phase retrieval and aberration balancing in shiftinvariant linear imaging systems david m. A generalization for optimized phase retrieval algorithms daniel e. Us8003947b1 system and method for magnitude and phase. Sheppard, defocused transfer function for a partially coherent microscope and application to phase retrieval, journal of the optical society of america a, vol. The bode magnitude and phase plots are shown in fig. In cdi, the intensity of the diffraction pattern scattered from a target is measured. Osa robust contrasttransferfunction phase retrieval. On the relation between the wave aberration function and the. Osa defocused transfer function for a partially coherent. This means that raar searches for the best approximate solution to the phase problem when an exact solution does not. Phase contrast, phase retrieval and aberration balancing.
The utility of phase retrieval stems from the fact that it allows the shape of transparent objects, biological cells in particular, to be quantified in two and three spatial dimensions using visible light 1, 2. Phase retrieval and support estimation in xray diffraction. The transfer function of a time delay is thus gs e. The asymmetric doublepass method provides lowfrequency partial phase information and the partial phase retrieval algorithm is applied to obtain the complete phase information, and to estimate the psf. Request pdf phase retrieval using coherent imaging systems with linear transfer functions we consider the problem of quantitative phase retrieval from images obtained using a coherent shift. Since only intensities of diffracted waves can be measured, the main mathematical challenge consists in a phase retrieval problem in the nearfield regime. Reconstruction of the pointspread function of the human. Phase retrieval with tunable phase transfer function based. Cloetens, jp guigay transfer function approach focus variation d1 d2 dn series of images recorded at different distances each distance is most sensitive to a specific range of spatial frequencies australian school. Phase retrieval via matrix completion siam journal on. In contrast, however, it can be shown that raar has. The phase retrieval for phase imaging in the 4foptical setup with the random phase modulation in the fourier plane is studied in katkovnik, et al. Phase retrieval with application to optical imaging electrical. Use of this transfer function for phase restoration quantitative phase retrieval from images of weak mixed phaseamplitude objects is discussed.
Phase retrieval of widefield microscopy point spread. How is phase retrieval transfer function algorithm abbreviated. It is shown that the phase transfer function contains information on the oddorder aberrations, which can be retrieved from coefficients to the taylor expansion of the derived quadrature relation. Pdf phase retrieval methods applied to coherent imaging. Phase retrieval with application to optical imaging arxiv. Coherence 2005, jrf, 6052 outline phase retrieval applications phase retrieval algorithms iterative transform algorithm hybrid inputoutput algorithm gradient search algorithms support constraint importance reconstructing object support from autocorrelation support tightening the support constraint several examples. Phase retrieval with tunable phase transfer function based on. Cn from the magnitudes y z of the linear often fourierbased transformation z ax0.
May 20, 20 this work reports on the retrieval of the pupil function and coherent transfer function of a coherent reflection type confocal microscope from simulated measurements of the intensity point spread function. Phase retrieval is a longstanding problem in imaging when only the intensity of the wavefield can be recorded. Phase retrieval using coherent imaging systems with linear transfer functions d. The phase retrieval from multiple experiments is for mulated as nding a complexvalued vector x 2 c n from a set of realvalued observations y s 2 r m. Phase retrieval by coherent modulation imaging nature. Us8003947b1 us12712,736 us71273610a us8003947b1 us 8003947 b1 us8003947 b1 us 8003947b1 us 71273610 a us71273610 a us 71273610a us 8003947 b1 us8003947 b1 us 8003947b1 authority. Eldarz babak hassibi ydepartment of electrical engineering, caltech zdepartment of electrical engineering, technion, israel institute of technology 1 introduction in many physical measurement systems, one can only measure the power spectral density, i. Phase retrieval methods applied to coherent imaging. The consideration is carried out in its general formulation. The problem of phase retrieval, namely the recovery of a function given the magnitude of its fourier transform arises in various. The basic principle of phasetransfer catalysis, some mechanistic aspects and important applications senthamizh selvi, r1, nanthini, r2 and sukanyaa, g3 abstract phasetransfer catalysis ptc has been widely used for the synthesis of organic compounds for more than three decades. Exploring phase retrieval in optical settings, speci. Tomographic reconstruction without phase retrieval yields the threedimensional distribution of the laplacian of the decrement of. Stability estimates for linearized nearfield phase retrieval.
Phase retrieval with application to optical imaging. Phaseretrieval transfer functions prtf computed for different ipr algorithms. However, in 1 the phase of the complexvalued object x 2cnis also unknown and actually reconstruction. Liu, a comparison of iterative algorithms and a mixed approach for inline. Phase retrieval instrument with our uperresolution microscopys prism satisfies these needs as a versatile multiplane platform with high image acquisition rates, integrating 3d fluorescence imaging anda novel whitelight phase tomography. This vectorization is conventional in this kind of the problems. The basic principle of phasetransfer catalysis, some. The resolution is estimated from the phaseretrieval transferfunction prtf2 2021 which gives a full period resolution of 125 nm fig. Inspite of the huge amount of work it has attracted, phase retrieval has been a long standing open problem. We present a method for phase retrieval in propagationbased xray imaging, based on the contrast transfer and transport of intensity equation approaches. A comparison of iterative algorithms and a mixed approach. The algorithm is iteratively extended to also apply to nonweak objects.
The steady state gain of a system is simply the ratio of the output and the input in steady state. However, the ctftv phase retrieval algorithm often struggles in the presence of strong noise, since it is based on the traditional compressive sensing optimization problem. This work reports on the retrieval of the pupil function and coherent transfer function of a coherent reflection type confocal microscope from simulated measurements of the intensity point spread function. The phase of the diffraction pattern is then obtained using phase retrieval algorithms and an image of the target is constructed. Quasianalytical quadrature formulas are derived that link the optical transfer function and the phase transfer function with the aberration terms. Phase retrieval of coherent diffractive images with global. In fact, many disciplines involve wave or particle scattered from an object, phaseretrieval approach will be. We show that the contrast transfer model does not coincide with the transport of intensity in the limit of small propagation distances, and we derive a new model that alleviates this problem. We demonstrate the algorithm on an euv multilayer defect imaged throughfocus on sharp with both a standard zone plate and a phase contrast zone plate. The driver drives the car at the desired speed, the cruise control system. Summary this work reports on the retrieval of the pupil function and.
Phase transfer catalysis ptc uses catalytic amounts of phase transfer agents which facilitate interphase transfer of species, making reactions between reagents in two immiscible phases possible. Note that the slope of the asymptotic magnitude plot rotates by. In the specific case of phase retrieval, higher spatial frequencies within the spatial bandwidth as determined by the numerical aperture na have uniform transmission modulus but are more severely scrambled by the chirped oscillations of the transfer function 3. Conventionally, the term phase retrieval is addressed to reconstruction of the missing phases of the vectors a sx. The asymmetric doublepass method provides lowfrequency partialphase information and the partial phaseretrieval algorithm is applied to obtain the completephase information, and to estimate the psf. The retrieval of the phase of electromagnetic fields is one of the most important and most challenging problems in classical optics. Convex phase retrieval without lifting via phasemax pmlr. Phase contrast tomography is a high sensitivity medical imaging technique. Optical transfer function phase of a complex value. Nov 18, 2016 phase retrieval is a longstanding problem in imaging when only the intensity of the wavefield can be recorded. Pdf defocused transfer function for a partially coherent. We study convex relaxation algorithms for phase retrieval on imaging problems.
Combined multiplane phase retrieval and superresolution. A general formulation for phase retrieval was first presented by liu and wu 14, 15, and was then rederived and experimentally proved in other groups publications 16, 3. Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region. Resolution metrics that compare the computed estimate directly with the recorded data, such as the phase retrieval transfer function, are not. Quasianalytical quadrature formulas are derived that link. Osa robust contrasttransferfunction phase retrieval via.
Phase retrieval using coherent imaging systems with linear. A comparison of iterative algorithms and a mixed approach for. Recent approaches to phase retrieval can be broadly classi ed into two categories. Kozacki phase retrieval with tunable phase transfer function based on the transport of intensity equation. We treat an often used linearized version of this problem known as the contract transfer function model. Assuming that the the input and the output of the system. Propagation based phase retrieval of simulated intensity. In the xray band, quantitative phase imaging is also useful.
Reconstruction of the pointspread function of the human eye. Defocused transfer function for a partially coherent microscope and application to phase retrieval article pdf available in journal of the optical society of america a 215. We show that exploiting structural assumptions on the signal and the observations, such as sparsity, smoothness or positivity. Propagation based phase retrieval reconstructs the exit phase of a specimen from outoffocus intensity images. When the object is known and need not be estimated, for example a known point source target such as an isolated star, pd reduces to phase diverse phase retrieval pdpr. Aug 15, 2009 a general formulation for phase retrieval was first presented by liu and wu 14, 15, and was then rederived and experimentally proved in other groups publications 16, 3. Local minima analysis of phase diverse phase retrieval using. Steady state gain the transfer function has many useful physical interpretations. Introduction reconstructing the phase of a complex. Index termsphase retrieval, transportofintensity equation,sparse reconstruction, total variation regularisation. Phasecontrast xray imaging in two and three dimensions. Early work in this area focused on using holography to capture the phase information along with magnitude measurements 12. We show that the contrast transfer model does not coincide with the transport of intensity in.
Coherent diffraction imaging is a lensless technique that uses iterative algorithms. An overview of recent developments kishore jaganathan yyonina c. Unfortunately, it is not straightforward to determine the complexvalued pupil function from the measured intensity psf, because of the lack of phase information. For weak defocus, this can be expressed analytically. Threedimensional reconstruction of the giant mimivirus particle with. A ptychographic phase retrieval process is used to recover the complex object, the unknown speckle pattern, and the coherent transfer function at the same time. Using the argument that the autocorrelation function of any sort of image is twice the size of the image itself in each dimension, bates stated that the phase information could be re. Phase retrieval algorithms are a key component in enabling cdi imaging. Phase contrast, phase retrieval and aberration balancing in. Prtf is defined as phase retrieval transfer function algorithm somewhat frequently. Two phase retrieval algorithms are presented in this vein, which incorporate the multiple pupil dependence of image formation in confocal. It was shown that this formulation covers other approaches based on the transport of intensity equation tie and the contrast transfer function ctf.
A generalization for optimized phase retrieval algorithms. Cloetens, jp guigay transfer function approach focus variation d1 d2 dn series of images recorded at different distances each distance is most sensitive to a specific range of spatial frequencies. Phase retrieval is a key component of coherent diffraction imaging cdi. On the relation between the wave aberration function and. Examples of optical setups for lensless a and 4flens b scenarios for phase retrieval. The structure of all molecules in a human body is essential to understanding biological processes, diseases and developing suitable medicine. Osa mixed transfer function and transport of intensity. The high spatial coherence of xfel data allowed us to apply a contrasttransfer function ctfbased phaseretrieval method utilizing the alternating direction method of multipliers admm 18. We present a phase recovery algorithm based on the weak object transfer function. How to find phase angle from transfer function electrical. Pdf mixed transfer function and transport of intensity. Siam journal on applied mathematics siam society for.
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