Calculation Of Point Spread Function Research Articles

Experimental validation of numerical point spread function calculation including aberration estimation

Image reconstruction in fluorescence microscopy is highly sensitive to the accuracy of the impulse response, defined as the point spread function (PSF), of the optical system under which the image to reconstruct was acquired. In our previous work, we developed a MATLAB toolbox for accurately calculating realistic vector Fourier-based PSF accounting for any type of aberrations [arXiv, arXiv:2301.13515 (2023)10.48550/arXiv.2301.13515]. In this work, we present a fundamental experimental validation of these numerical methods. The simulated results are found to fit experimental data under different image acquisition conditions at an accuracy higher than 0.97 in normalized cross-correlation. These methods enable a relative contrast of up to 95%.

Point Spread Function Calculation for Unfocused Plenoptic Camera by Random Forest

A novel calculation scheme is proposed for the point spread function (PSF) of the unfocused plenoptic camera, which describes the response of image patterns to the point light source locations. It is based on the data-driven machine learning method, random forest (RF). Both synthetic and experimental images are obtained to build the datasets for training and test. By feature construction, transformed variables are selected as the inputs, the intensity values of the illuminated pixels are predicted by the RF. PSF estimation errors of the current scheme are compared with that of the widely used geometrical ray tracing method. The results demonstrate that the method by RF is obviously more accurate and robust, especially for the regions close to the focal plane, mitigating the estimation errors induced by optical aberrations.

Explicit calculation of Point Spread Function of optical system

The paper presents novel formulas for an explicit calculation of the Point Spread Function for an optical system with a circular pupil and the case of axial-point imaging, which were derived with the use of Sonin’s integral. In such a case, it is not needed to perform integration in the pupil, as it is usual with different methods. The approach is presented and verified with examples. The presented apparatus wide-spreads the theory of calculation of the Point Spread Function, and one can use it as an alternative solution of the mentioned task.

Improved preclinical hyperpolarized 129 Xe ventilation imaging with constant flip angle 3D radial golden means acquisition and keyhole reconstruction.

Hyperpolarized (HP) 129 Xe MRI is increasingly used to noninvasively probe regional lung structure and function in the preclinical setting. As in human imaging, the primary barrier to quantitative imaging with HP gases is nonequilibrium magnetization, which is depleted by T1 relaxation and radio frequency excitation. Preclinical HP gas imaging commonly involves mechanically ventilating small animals and encoding k-space over tens or hundreds of breaths, with small subsets of k-space data collected within each breath. Breath-to-breath magnetization renewal enables the use of large flip angles, but the resulting magnetization decay generates large view-to-view differences in within-breath signal intensity, leading to artifacts and degraded image quality. This deleterious signal decay has motivated the use of variable flip angle (VFA) sampling schemes, in which the flip angle is progressively increased to maintain constant view-to-view signal intensity. However, VFA imaging complicates data acquisition and provides only a global correction that fails to compensate for regional differences in signal dynamics. When constant flip angle (CFA) imaging is used alongside 3D radial golden means acquisition, the center of k-space is sampled with every excitation, thereby encoding signal dynamics alongside imaging data. Here, keyhole reconstruction is used to generate multiple images to capture in-breath HP 129 Xe signal dynamics in mice and thus provide flip angle maps to quantitatively correct images without extra data collection. These CFA images display SNR that is not significantly different from VFA images, and further, high frequency k-space scaling can be used to mitigate decay-induced image artifacts. Results are supported by point spread function calculations and simulations of radial imaging with preclinical signal dynamics. Together, these results show that CFA 3D radial golden means ventilation imaging provides comparable image quality with VFA in small animals and allows for keyhole reconstruction, which can be used to generate flip angle maps and correct images for signal depletion.

Fourier propagation tool for aberration analysis and a point spread function calculation of systems with curved focal planes

This paper describes a new Fourier propagator for computing the impulse response of an optical system with a curved focal plane array, while including terms ignored in Fresnel and Fraunhofer calculations. The propagator includes a Rayleigh-Sommerfeld diffraction formula calculation from a distant point through the optical system to its image point predicted by geometric optics on a spherical surface. The propagator then approximates the neighboring field points via the traditional binomial approximation of the Taylor series expansion around that field point. This technique results in a propagator that combines the speed of a Fourier transform operation with the accuracy of the Rayleigh-Sommerfeld diffraction formula calculation and extends Fourier optics to cases where the receiver plane is a curved surface. Bounds on the phase error introduced by the approximations are derived, which show it should be more widely applicable than traditional Fresnel propagators. Guidance on how to sample the pupil and detector planes of a simulated imaging system is provided. This report concludes by showing examples of the diffraction patterns computed by the new technique compared to those computed using the Rayleigh-Sommerfeld technique in order to demonstrate the utility of the propagator.

Evaluation of quantitative, efficient image reconstruction for VersaPET, a compact PET system.

Previously we developed a high-resolution positron emission tomography(PET) system-VersaPET-characterized by a block geometry with relatively large axial and transaxial interblock gaps and a compact geometry susceptible to parallax blurring effects. In this work, we report the qualitative and quantitative evaluation of a graphic processing unit (GPU)-accelerated maximum-likelihood by expectation-maximization (MLEM) image reconstruction framework for VersaPET which features accurate system geometry and projection space point-spread-function (PSF) modeling. We combined the ray-tracing module from software for tomographic image reconstruction (STIR),an open-source PET image reconstruction package, with VersaPET's exact block geometry for the geometric system matrix. Point-spread-function modeling of crystal penetration and scattering was achieved by a custom Monte-Carlo simulation for projection space blurring in all dimensions. We also parallelized the reconstruction in GPU taking advantage of the system's symmetry for PSF computation. To investigate the effects of PSF width, we generated and studied multiple kernels between one that reflects the true LYSO density in the MC simulation and another that reflects geometry only (no PSF). GATE simulations of hot and cold-sphere phantoms with spheres of different sizes, real microDerenzo phantom, and human blood vessel data were used to characterize the quantitative and qualitative performances of the reconstruction. Reconstruction with an accurate system geometry effectively improved image quality compared to STIR (version 3.0) which assumes an idealized system geometry. Reconstructions of GATE-simulated hot-sphere phantom data showed that all PSF kernels achieved superior performance in contrast recovery and bias reduction compared to using no PSF, but may introduce edge artifact and lumped background noise pattern depending on the width of PSF kernels. Cold-sphere phantom simulation results also indicated improvement in contrast recovery and quantification with PSF modeling (compared to no PSF) for 5 and 10mm cold spheres. Real microDerenzo phantom images with the PSF kernel that reflects the true LYSO density showed degraded resolving power of small sectors that could be resolved more clearly by underestimated PSF kernels, which is consistent with recent literature despite differences in scanner geometries and in approaches to system model estimation. The human vessel results resemble those of the hot-sphere phantom simulation with the PSF kernel that reflects the true LYSO density achieving the highest peak in the time activity curve (TAC) and similar lumped noise pattern. We fully evaluated a practical MLEM reconstruction framework that we developed for VersaPET in terms of qualitative and quantitative performance. Different PSF kernels may be adopted for improving the results of specific imaging tasks but the underlying reasons for the variation in optimal kernel for the real and simulation studies requires further study.

Space-variant point spread function measurement and interpolation at any depth based on single-pixel imaging.

Point spread function (PSF) is important for evaluating an optical system and image deblurring. In this paper, we proposed a method to measure space-variant PSF at any depth based on single-pixel imaging (SPI), and we initiated a depth-variant PSF interpolation model. In our method, we regarded space-variant PSF as light transport coefficients from object points to image pixels. By applying SPI to each image pixel to obtain these light transport coefficients at different depths, the PSF of each object point can be extracted. The depth calculation of PSF is based on multi-frequency heterodyne phase-shifting principles and perspective-n-point (PnP) algorithm. In our PSF interpolation model, we interpolated the light transport coefficients from different object points to an image pixel first. We then obtained the interpolated PSF indirectly from the interpolated coefficients. With simple experimental facilities containing a digital camera and a liquid crystal display (LCD) screen to display and capture specific patterns, which relative distance is changed, the proposed method accurately obtained the space-variant PSF at any depth. Without complicated calculation, PSF at a certain depth can be interpolated from the PSF measured data at another depth with our PSF interpolation method. Significant similarities exist between the interpolated PSF and directly measured PSF. Our work is a successful attempt in using SPI to solve traditional optical problems.

Efficient computation of 2D point-spread functions for diffractive lenses.

Diffractive lenses, such as Fresnel zone plates, photon sieves, and their modified versions, have been of significant recent interest in high-resolution imaging applications. As the advent of diffractive lens systems with different configurations expands, the fast and accurate simulation of these systems becomes crucial for both the design and image reconstruction tasks. Here we present a fast and accurate method for computing the 2D point-spread function (PSF) of an arbitrary diffractive lens. The method is based on the recently derived closed-form mathematical formula for the PSF and the transfer function of a diffractive lens. In the method, first, the samples of the transfer function are computed using the transmittance function of the diffractive lens, and then the inverse Fourier transform of this transfer function is computed to obtain the PSF. For accurate computation, the selection of the sampling parameters is handled with care, and simple selection rules are provided for this purpose. The developed method requires a single fast Fourier transform, and, therefore, has little computational complexity. Moreover, it is also applicable to any diffractive lens configuration with arbitrary-shaped structures and modulation. As a result, this fast and accurate PSF computation method enables efficient simulation, analysis, and development of diffractive lens systems under both focused and defocused settings.

A method for numerical calculation of point spread function of an irregular aperture objective lens

We have presented an efficient numerical method for calculation of point-spread function (PSF) of a high aperture objective lens based on Cartesian coordinates. We have showed that, as compared to the cylindrical coordinates, the use of the Cartesian coordinates allows to calculate the EM field distribution in focal region with complex incident beam profile, such as, annular, non-uniform or asymmetric, etc. We have also showed the benefits of using Cartesian coordinates for deriving isotropic PSF in focal region, which is very important for three-dimensional imaging and fabrication applications.

X-ray beam-shaping via deformable mirrors: surface profile and point spread function computation for Gaussian beams using physical optics.

X-ray mirrors with high focusing performances are commonly used in different sectors of science, such as X-ray astronomy, medical imaging and synchrotron/free-electron laser beamlines. While deformations of the mirror profile may cause degradation of the focus sharpness, a deliberate deformation of the mirror can be made to endow the focus with a desired size and distribution, via piezo actuators. The resulting profile can be characterized with suitable metrology tools and correlated with the expected optical quality via a wavefront propagation code or, sometimes, predicted using geometric optics. In the latter case and for the special class of profile deformations with monotonically increasing derivative, i.e. concave upwards, the point spread function (PSF) can even be predicted analytically. Moreover, under these assumptions, the relation can also be reversed: from the desired PSF the required profile deformation can be computed analytically, avoiding the use of trial-and-error search codes. However, the computation has been so far limited to geometric optics, which entailed some limitations: for example, mirror diffraction effects and the size of the coherent X-ray source were not considered. In this paper, the beam-shaping formalism in the framework of physical optics is reviewed, in the limit of small light wavelengths and in the case of Gaussian intensity wavefronts. Some examples of shaped profiles are also shown, aiming at turning a Gaussian intensity distribution into a top-hat one, and checks of the shaping performances computing the at-wavelength PSF by means of the WISE code are made.

Fast and accurate three-dimensional point spread function computation for fluorescence microscopy.

The point spread function (PSF) plays a fundamental role in fluorescence microscopy. A realistic and accurately calculated PSF model can significantly improve the performance in 3D deconvolution microscopy and also the localization accuracy in single-molecule microscopy. In this work, we propose a fast and accurate approximation of the Gibson-Lanni model, which has been shown to represent the PSF suitably under a variety of imaging conditions. We express the Kirchhoff's integral in this model as a linear combination of rescaled Bessel functions, thus providing an integral-free way for the calculation. The explicit approximation error in terms of parameters is given numerically. Experiments demonstrate that the proposed approach results in a significantly smaller computational time compared with current state-of-the-art techniques to achieve the same accuracy. This approach can also be extended to other microscopy PSF models.

Nonparaxial Fourier propagation tool for aberration analysis and point spread function calculation

This paper describes a Fourier propagator for computing the impulse response of an optical system, while including terms ignored in Fresnel and Fraunhofer calculations. The propagator includes a Rayleigh–Sommerfeld diffraction formula calculation from a distant point through the optical system to its image point predicted by geometric optics. The propagator then approximates the neighboring field points via the traditional binomial approximation of the Taylor series expansion around that field point. This technique results in a propagator that combines the speed of a Fourier transform operation with the accuracy of the Rayleigh–Sommerfeld diffraction formula calculation and extends Fourier optics to cases that are nonparaxial. The proposed propagator facilitates direct calculation of aberration coefficients, making it more versatile than the angular spectrum propagator. Bounds on the phase error introduced by the approximations are derived, which show that it should be more widely applicable than the Fresnel propagator. Guidance on how to sample the pupil and detector planes of a simulated imaging system is provided. This report concludes by showing examples of diffraction calculations for a laboratory setup and comparing them to measured diffraction patterns to demonstrate the utility of the propagator.

Simulating multiple diffraction in imaging systems using a path integration method.

We present a method for simulating multiple diffraction in imaging systems based on the Huygens-Fresnel principle. The method accounts for the effects of both aberrations and diffraction and is entirely performed using Monte Carlo ray tracing. We compare the results of this method to those of reference simulations for field propagation through optical systems and for the calculation of point spread functions. The method can accurately model a wide variety of optical systems beyond the exit pupil approximation.

Point-Spread Function Computation under the Conditions of Birefringence and Partial Coherence of Lighting using GPUs

Point-Spread Function Computation under the Conditions of Birefringence and Partial Coherence of Lighting using GPUs

Visible-wavelength two-photon excitation microscopy for fluorescent protein imaging

The simultaneous observation of multiple fluorescent proteins (FPs) by optical microscopy is revealing mechanisms by which proteins and organelles control a variety of cellular functions. Here we show the use of visible-light based two-photon excitation for simultaneously imaging multiple FPs. We demonstrated that multiple fluorescent targets can be concurrently excited by the absorption of two photons from the visible wavelength range and can be applied in multicolor fluorescence imaging. The technique also allows simultaneous single-photon excitation to offer simultaneous excitation of FPs across the entire range of visible wavelengths from a single excitation source. The calculation of point spread functions shows that the visible-wavelength two-photon excitation provides the fundamental improvement of spatial resolution compared to conventional confocal microscopy.

Investigation of the numerics of point spread function integration in single molecule localization.

The computation of point spread functions, which are typically used to model the image profile of a single molecule, represents a central task in the analysis of single molecule microscopy data. To determine how the accuracy of the computation affects how well a single molecule can be localized, we investigate how the fineness with which the point spread function is integrated over an image pixel impacts the performance of the maximum likelihood location estimator. We consider both the Airy and the two-dimensional Gaussian point spread functions. Our results show that the point spread function needs to be adequately integrated over a pixel to ensure that the estimator closely recovers the true location of the single molecule with an accuracy that is comparable to the best possible accuracy as determined using the Fisher information formalism. Importantly, if integration with an insufficiently fine step size is carried out, the resulting estimates can be significantly different from the true location, particularly when the image data is acquired at relatively low magnifications. We also present a methodology for determining an adequate step size for integrating the point spread function.

Mirrors for X-ray telescopes: Fresnel diffraction-based computation of point spread functions from metrology

The imaging sharpness of an X-ray telescope is chiefly determined by the optical quality of its focusing optics, which in turn mostly depends on the shape accuracy and the surface finishing of the grazing-incidence X-ray mirrors that compose the optical modules. To ensure the imaging performance during the mirror manufacturing, a fundamental step is predicting the mirror point spread function (PSF) from the metrology of its surface. Traditionally, the PSF computation in X-rays is assumed to be different depending on whether the surface defects are classified as figure errors or roughness. [...] The aim of this work is to overcome this limit by providing analytical formulae that are valid at any light wavelength, for computing the PSF of an X-ray mirror shell from the measured longitudinal profiles and the roughness power spectral density (PSD), without distinguishing spectral ranges with different treatments. The method we adopted is based on the Huygens-Fresnel principle for computing the diffracted intensity from measured or modeled profiles. In particular, we have simplified the computation of the surface integral to only one dimension, owing to the grazing incidence that reduces the influence of the azimuthal errors by orders of magnitude. The method can be extended to optical systems with an arbitrary number of reflections - in particular the Wolter-I, which is frequently used in X-ray astronomy - and can be used in both near- and far-field approximation. Finally, it accounts simultaneously for profile, roughness, and aperture diffraction. We describe the formalism with which one can self-consistently compute the PSF of grazing-incidence mirrors, [...] Finally, we validate this by comparing the simulated PSF of a real Wolter-I mirror shell with the measured PSF in hard X-rays.

Predicting the angular resolution of x-ray mirrors

The angular resolution of a grazing incidence focusing mirror is an important parameter in x-ray optics. It is often quantified by either the point spread function (PSF)—the distribution of the focused rays as a function of the distance from the center of the focal spot—or the half-energy width, which is double the PSF median value. The focusing of x-rays (with wavelengths 1000 shorter than visible light) is very sensitive to both a mirror’s surface profile and roughness (see Figure 1), i.e., surface relief on a variety of scales down to a few angstroms in height. PSF calculations for a focusing mirror must therefore account for both these factors. The surface roughness typically causes an x-ray scattering interference effect, which generally increases with x-ray energy. There are, however, several problems with computing the PSF by considering profile and roughness separately. It has long been known that x-ray focusing degradation caused by long-period deformations—e.g., Figure 1(a)—and surface roughness of mirrors—see Figure 1(c)—can be analyzed separately by treating the x-rays as rays or waves, respectively. When treating the x-rays as rays, geometric optics are applied. A well-established x-ray scattering (perturbation) theory1 is used when they are treated as waves, but it can be applied only to very smooth surfaces and to short-period features. However, the boundary between these two regimes2 is difficult to define and not clear-cut. Degradations in optical behavior caused by surface features in an intermediate spatial frequency range—see Figure 1(b)—cannot always be treated with either of the two methods mentioned.3 In addition, neither the geometric optics nor the scattering approach account for the effect of aperture diffraction. It can also be difficult to combine the separately derived geometry and scattering PSF values to obtain the total PSF of the mirror. We have created a new and self-consistent method that does not require boundaries between mirror geometries, profile frequencies, or surface roughnesses to be set.3 Our method is based Figure 1. Behavior of x-rays interacting with non-smooth grazing focusing mirrors. (a) Geometric optics can be applied to mirror defects with long spatial periods. (b) At mid-frequencies (i.e., centimeter to millimeter wavelength range) x-ray behavior is more uncertain. (c) Surface diffraction from micro-roughness is described by perturbation theory. (d) All three defects, as in (a), (b), and (c), are included and are difficult to treat consistently.

Evidence for anomalous optical transition radiation linear polarization effects in beam-profile monitors

Investigations of the effects of optical transition radiation (OTR) polarization components on beam profiles are presented. The transverse profiles are examined using the OTR perpendicular and parallel polarization components with respect to the dimension of interest. We observed $\ensuremath{\sim}15%$ projected profile size reductions with the perpendicularly polarized components on a $65\mathrm{\text{\ensuremath{-}}}\ensuremath{\mu}\mathrm{m}$ beam image size case at 14 MeV, a $150\mathrm{\text{\ensuremath{-}}}\ensuremath{\mu}\mathrm{m}$ beam image size at 4.5 GeV, and a $1100\mathrm{\text{\ensuremath{-}}}\ensuremath{\mu}\mathrm{m}$ beam image size at 7 GeV. These effects are all several times larger than expected (and anomalous in this sense) when compared to the standard OTR point-spread function calculations. We propose the time-averaged induced-current distribution which generates the OTR represents the actual beam size more faithfully with the perpendicular polarization component and recommend its routine use and subsequent deconvolution.

Accurate 3D Fluorophore Localization using Phase Retrieved Pupil Functions

When near the focus, the image of a point emitter has little information about the emitter's position along the optical axis. To improve the 3D localization precision, several methods such as astigmatic imaging, double helix point spread functions, and multiple focal plane strategies have been developed to encode the axial position in one or more 2D images of the same emitter. In each of these strategies, a proper model for the 3D microscope point spread function (PSF) is essential to avoid inaccuracies in localization. Generating PSF models from phase retrieved (PR) pupil functions has the advantage that the model images are noise free, can be calculated for any focal plane, and include aberrations specific to the optical system. This representation can also be extremely compact by representing the pupil function as a set of Zernike polynomials. Here we demonstrate the use of PSFs generated from PR pupil functions for 3D single molecule localization based super-resolution imaging using a dual focal plane setup. We show the phase retrieval process, as well as the details of fast PSF calculation by implementation on graphics processing unit hardware. We show a comparison between fitting with ideal, aberration free PSF models and that generated from PR pupil functions using a variety of cellular structures including the ER, microtubules and the membrane proteins STIM1 and ORAI1.