3D Point Spread Function Research Articles

Rotating point spread function via pupil-phase engineering

A simple approach based on the use of a properly designed pupil-phase profile can yield a 3D point-spread function (PSF) that rotates with changing defocus, while keeping its transverse shape approximately invariant over ±3-4 waves of defocus. Unlike Gauss-Laguerre mode-based approaches, it generalizes readily for encoding spherical aberration too via PSF rotation.

Validation of the blurring of a small object on CT images calculated on the basis of three-dimensional spatial resolution.

We determine three-dimensional (3D) blurring of a small object on computed tomography (CT) images calculated on the basis of 3D spatial resolution. The images were characterized by point spread function (PSF), line spread function (LSF) and slice sensitivity profile (SSP). In advance, we systematically arranged expressions in the model for the imaging system to calculate 3D images under various conditions of spatial resolution. As a small object, we made a blood vessel phantom in which the direction of the vessel was not parallel to either the xy scan-plane or the z-axis perpendicular to the scan-plane. Therefore, when scanning the phantom, non-sharpness must be induced in all axes of the image. To predict the image blurring of the phantom, 3D spatial resolution is essential. The LSF and SSP were measured on our scanner, and two-dimensional (2D) PSF in the scan-plane was derived from the LSF by solving an integral equation. We obtained 3D images by convoluting the 3D object-function of the phantom with both 2D PSF and SSP, corresponding to the 3D convolution. Calculated images showed good agreement with scanned images. Our technique of determining 3D blurring offers an accuracy advantage in 3D shape (size) and density measurements of small objects.

A three-dimensional model-based partial volume correction strategy for gated cardiac mouse PET imaging

Quantification in cardiac mouse positron emission tomography (PET) imaging is limited by the imaging spatial resolution. Spillover of left ventricle (LV) myocardial activity into adjacent organs results in partial volume (PV) losses leading to underestimation of myocardial activity. A PV correction method was developed to restore accuracy of the activity distribution for FDG mouse imaging. The PV correction model was based on convolving an LV image estimate with a 3D point spread function. The LV model was described regionally by a five-parameter profile including myocardial, background and blood activities which were separated into three compartments by the endocardial radius and myocardium wall thickness. The PV correction was tested with digital simulations and a physical 3D mouse LV phantom. In vivo cardiac FDG mouse PET imaging was also performed. Following imaging, the mice were sacrificed and the tracer biodistribution in the LV and liver tissue was measured using a gamma-counter. The PV correction algorithm improved recovery from 50% to within 5% of the truth for the simulated and measured phantom data and image uniformity by 5–13%. The PV correction algorithm improved the mean myocardial LV recovery from 0.56 (0.54) to 1.13 (1.10) without (with) scatter and attenuation corrections. The mean image uniformity was improved from 26% (26%) to 17% (16%) without (with) scatter and attenuation corrections applied. Scatter and attenuation corrections were not observed to significantly impact PV-corrected myocardial recovery or image uniformity. Image-based PV correction algorithm can increase the accuracy of PET image activity and improve the uniformity of the activity distribution in normal mice. The algorithm may be applied using different tracers, in transgenic models that affect myocardial uptake, or in different species provided there is sufficient image quality and similar contrast between the myocardium and surrounding structures.

Generation and Control of Wide-Field Three-Dimensional Structured Illumination for Advanced Microscopic Imaging

In a variety of practical microscopic imaging applications, many industries require not only lateral resolution improvement but also axial resolution improvement. The resolution in optical microscopy is limited by diffraction and determined by the wavelength of the incident light and the numerical aperture (NA) of the objective lens. The diffraction limit is mathematically described by a point spread function in the imaging system, and three-dimensional (3D) point spread functions describe both the lateral and axial resolutions. Thus, it is useful to focus on exceeding this limit and improving the resolution of optical imaging by the spatial control of structured illumination. Structured illumination microscopy is a familiar technique to improve resolution in fluorescent imaging, and it is expected to be applied to industrial applications. Microscopic imaging is convenient, non-destructive, and has a high-throughput performance and compatibility with a number of applications. However, the spatial resolution of conventional light microscopy is limited to wavelength scale and the depth of field is shallow; hence, it is difficult to obtain detailed 3D spatial data of the object to be measured. Here, we propose a new technique for generating and controlling wide-field 3D structured illumination. The technique, based on the 3D interference of multiple laser beams, provides lateral and axial resolution improvement, and a wide 3D field of view. The spatial configuration of the beams was theoretically examined and the optimal incident angle of the multiple beams was confirmed. Numerical simulations using the finite difference time domain (FDTD) method were carried out and confirmed the generation of 3D structured illumination and spatial control of the illumination by using the phase shift of incident beams.

Deconvolution of vibroacoustic images using a simulation model based on a three dimensional point spread function

Vibro-acoustography (VA) is a medical imaging method based on the difference-frequency generation produced by the mixture of two focused ultrasound beams. VA has been applied to different problems in medical imaging such as imaging bones, microcalcifications in the breast, mass lesions, and calcified arteries. The obtained images may have a resolution of 0.7–0.8mm. Current VA systems based on confocal or linear array transducers generate C-scan images at the beam focal plane. Images on the axial plane are also possible, however the system resolution along depth worsens when compared to the lateral one. Typical axial resolution is about 1.0cm. Furthermore, the elevation resolution of linear array systems is larger than that in lateral direction. This asymmetry degrades C-scan images obtained using linear arrays. The purpose of this article is to study VA image restoration based on a 3D point spread function (PSF) using classical deconvolution algorithms: Wiener, constrained least-squares (CLSs), and geometric mean filters. To assess the filters’ performance on the restored images, we use an image quality index that accounts for correlation loss, luminance and contrast distortion. Results for simulated VA images show that the quality index achieved with the Wiener filter is 0.9 (when the index is 1.0 this indicates perfect restoration). This filter yielded the best result in comparison with the other ones. Moreover, the deconvolution algorithms were applied to an experimental VA image of a phantom composed of three stretched 0.5mm wires. Experiments were performed using transducer driven at two frequencies, 3075kHz and 3125kHz, which resulted in the difference-frequency of 50kHz. Restorations with the theoretical line spread function (LSF) did not recover sufficient information to identify the wires in the images. However, using an estimated LSF the obtained results displayed enough information to spot the wires in the images. It is demonstrated that the phase of the theoretical and the experimental PSFs are dissimilar. This fact prevents VA image restoration with the current theoretical PSF. This study is a preliminary step towards understanding the restoration of VA images through the application of deconvolution filters.

A new linear transfer theory and characterization method for image detectors. Part II: Experiment

A novel generalized linear transfer theory describing the signal and noise transfer in image detectors has been developed in Part I (Niermann, this issue, [1]) of this paper. Similar to the existing notion of a point spread function (PSF) describing the transfer of the first statistical moment (the average), a noise spread function (NSF) was introduced to characterize the spatially resolved transfer of noise (central second moment, covariance). Following the theoretic results developed in Part I (Niermann, this issue, [1]), a new experimental method based on single spot illumination has been developed and applied to measure 2D point and 4D noise spread functions of CCD cameras used in TEM. A dedicated oversampling method has been used to suppress aliasing in the measured quantities. We analyze the 4D noise spread with respect to electronic and photonic noise contributions.

Accuracy of MRI wall shear stress estimation

Methods Three methods for WSS estimation were studied. These methods are based on 1) linear extrapolation (LE) of MRI velocity data, 2) MRI velocity data in combination with estimation of location of vessel wall, and 3) Fourier velocity encoding (FVE). Numerical velocity fields representing axisymmetric 2D velocity profiles were generated for WSS values ranging from 1-20 N/m. Based on the numerical velocity fields, phase-contrast MRI data voxels were simulated as follows: A jinc-function was used to model the 2D point spread function (PSF), and this PSF was used to obtain each voxel’s intravoxel velocity distribution. The phasecontrast MRI signal of each voxel was simulated by taking the Fourier transform of this distribution. To account for the fact that voxels cannot be positioned exactly at the wall in an MR-experiment, all simulations were carried out for ten different voxel positions uniformly distributed over one voxel length. In the LE method the spatial velocity derivative was estimated as the velocity difference between the two adjacent nearwall voxels divided by the distance between them. In the wall-based method, WSS was estimated by dividing the linear interpolated velocity at one voxel distance from the wall by the distance to the wall. Errors in segmentations of wall position were accounted for by modeling them as normally distributed with a standard deviation of 1/4 voxel size. In the FVE-based method, the WSS was obtained by first estimating the intravoxel velocity profile via a simulated FVE measurement and then computing the spatial velocity derivative near the wall. Note that the FVE-method uses larger voxels.

Three Dimensional Single Molecule Localization using Phase Retrieved Pupil Functions

Single particle tracking and single-molecule localization based super-resolution techniques rely on the precise and accurate localization of single fluorescent molecules. For two dimensional imaging, relatively simple models of a microscope point spread function (PSF), such as the two dimensional Gaussian, are adequate for most common imaging approaches. However, three dimensional localization is hampered by the fact that the image of a molecule near the focal plane contains little information about its axial position. In recent years, several three dimensional (3D) imaging techniques have been demonstrated that improve the localization of single fluorescent molecules along the axial direction, such as astigmatic imaging[1], a double helix point spread function (DH-PSF)[2], and dual focal plane methods[3]. Although these methods implement very different optical setups, they use either a simple Gaussian models, theoretical PSFs that do not account imaging system aberrations, or large, unwieldy, experimentally acquired 3D PSFs that can be prone to artifacts. Here we introduce a localization algorithm based on phase retrieved pupil functions. Pupil functions can contain information about specific aberrations present in the imaging system and can be used to calculate realistic 3D PSFs from a small set of Zernike polynomial coefficients that describe the pupil magnitude and phase. This compact representation of the 3D PSF allows the PSF to be efficiently calculated as needed in an iterative update method implemented on GPU hardware. We demonstrate the use of phase retrieved pupil functions for 3D localization in both the astigmatic and dual focal plane setups.Reference[1] Bo Huang, Science, 2008[2] Pavani, App.Phy.Sci, 2009[3] Manuel F Juette, NMETH, 2008

Optimal 3D single-molecule localization for superresolution microscopy with aberrations and engineered point spread functions

Photo-activation localization microscopy is a far-field superresolution imaging technique based on the localization of single molecules with subdiffraction limit precision. Known under acronyms such as PALM (photo-activated localization microscopy) or STORM (stochastic optical reconstruction microscopy), these techniques achieve superresolution by allowing only a sparse, random set of molecules to emit light at any given time and subsequently localizing each molecule with great precision. Recently, such techniques have been extended to three dimensions, opening up unprecedented possibilities to explore the structure and function of cells. Interestingly, proper engineering of the three-dimensional (3D) point spread function (PSF) through additional optics has been demonstrated to theoretically improve 3D position estimation and ultimately resolution. In this paper, an optimal 3D single-molecule localization estimator is presented in a general framework for noisy, aberrated and/or engineered PSF imaging. To find the position of each molecule, a phase-retrieval enabled maximum-likelihood estimator is implemented. This estimator is shown to be efficient, meaning it reaches the fundamental Cramer-Rao lower bound of x, y, and z localization precision. Experimental application of the phase-retrieval enabled maximum-likelihood estimator using a particular engineered PSF microscope demonstrates unmatched low-photon-count 3D wide-field single-molecule localization performance.

Marginal blind deconvolution of adaptive optics retinal images

Adaptive Optics corrected flood imaging of the retina has been in use for more than a decade and is now a well-developed technique. Nevertheless, raw AO flood images are usually of poor contrast because of the three-dimensional nature of the imaging, meaning that the image contains information coming from both the in-focus plane and the out-of-focus planes of the object, which also leads to a loss in resolution. Interpretation of such images is therefore difficult without an appropriate post-processing, which typically includes image deconvolution. The deconvolution of retina images is difficult because the point spread function (PSF) is not well known, a problem known as blind deconvolution. We present an image model for dealing with the problem of imaging a 3D object with a 2D conventional imager in which the recorded 2D image is a convolution of an invariant 2D object with a linear combination of 2D PSFs. The blind deconvolution problem boils down to estimating the coefficients of the PSF linear combination. We show that the conventional method of joint estimation fails even for a small number of coefficients. We derive a marginal estimation of the unknown parameters (PSF coefficients, object Power Spectral Density and noise level) followed by a MAP estimation of the object. We show that the marginal estimation has good statistical convergence properties and we present results on simulated and experimental data.

Point-spread function engineering to reduce the impact of spherical aberration on 3D computational fluorescence microscopy imaging

Wavefront encoding (WFE) with different cubic phase mask designs was investigated in engineering 3D point-spread functions (PSF) to reduce their sensitivity to depth-induced spherical aberration (SA) which affects computational complexity in 3D microscopy imaging. The sensitivity of WFE-PSFs to defocus and to SA was evaluated as a function of phase mask parameters using mean-square-error metrics to facilitate the selection of mask designs for extended-depth-of-field (EDOF) microscopy and for computational optical sectioning microscopy (COSM). Further studies on pupil phase contribution and simulated WFE-microscope images evaluated the engineered PSFs and demonstrated SA insensitivity over sample depths of 30 μm. Despite its low sensitivity to SA, the successful WFE design for COSM maintains a high sensitivity to defocus as it is desired for optical sectioning.

Point spread function and two-point resolution in Fresnel incoherent correlation holography

Fresnel Incoherent Correlation Holography (FINCH) allows digital reconstruction of incoherently illuminated objects from intensity records acquired by a Spatial Light Modulator (SLM). The article presents wave optics model of FINCH, which allows analytical calculation of the Point Spread Function (PSF) for both the optical and digital part of imaging and takes into account Gaussian aperture for a spatial bounding of light waves. The 3D PSF is used to determine diffraction limits of the lateral and longitudinal size of a point image created in the FINCH set-up. Lateral and longitudinal resolution is investigated both theoretically and experimentally using quantitative measures introduced for two-point imaging. Dependence of the resolving power on the system parameters is studied and optimal geometry of the set-up is designed with regard to the best lateral and longitudinal resolution. Theoretical results are confirmed by experiments in which the light emitting diode (LED) is used as a spatially incoherent source to create object holograms using the SLM.

TU‐G‐211‐06: Partial Volume Correction for Noninvasive PET Image‐Derived Simplified Kinetic Analysis

Purpose: To assess a partial volume correction (PVC) method for a non‐invasive PET image based simplified kinetic analysis.Methods and Materials: A phantom was designed with a cylinder tank and six fillable cylinders to simulate vessels with a range of inner diameters from 4.8 to 28.6mm. F‐18 in aqueous solution was injected into the vessels and the tank with an activity concentration ratio of 1.5 to 1. The PET FOV was 576mm and voxel size 4×4×4mm. CT images of this phantom were used to create a digital 3D PET phantom model. The inner and outer vessel cylinders and the tank were automatically contoured and the known true F‐18 activities were assigned to the vessels and tank volume. The CT‐based model was then convolved with a 3D Gaussian PET point spread function to create the simulated phantom. For both simulated and physical phantoms, PVC was applied to recover the F‐18 uptakes using a 3×3 geometric transfer matrix (GTM) method. Recovery coefficients of the vessels were calculated before and after PVC. Results: Severe underestimations in vessel uptake were observed for both simulated and physical phantoms before PVC. After PVC the vessel uptakes recovered closer to the true uptake values. For the simulated phantom, complete PVC recovery was achieved for all vessel sizes with an accuracy <0.2%. For the physical phantom, the PVC recovery was <5% for all vessel sizes except for the smallest vessel (4.8mm) where the vessel uptake was overestimated by 20%. Conclusions: Non‐invasive PET image based simplified kinetic analysis is feasible provided that partial volume corrections are applied to recover vessel tracer uptake. Blood activity may be corrected for partial volume error with better than 5% accuracy using the GTM method for vessels larger than 10mm in diameter. A patient study is needed to confirm these results in clinical settings.

Two-dimensional point spread matrix of layered metal–dielectric imaging elements

We describe the change of the spatial distribution of the state of polarization occurring during two-dimensional (2D) imaging through a multilayer and in particular through a layered metallic flat lens. Linear or circular polarization of incident light is not preserved due to the difference in the amplitude transfer functions for the TM and TE polarizations. In effect, the transfer function and the point spread function (PSF) that characterize 2D imaging through a multilayer both have a matrix form, and cross-polarization coupling is observed for spatially modulated beams with a linear or circular incident polarization. The PSF in a matrix form is used to characterize the resolution of the superlens for different polarization states. We demonstrate how the 2D PSF may be used to design a simple diffractive nanoelement consisting of two radial slits. The structure assures the separation of nondiffracting radial beams originating from two slits in the mask and exhibits an interesting property of a backward power flow in between the two rings.

Three-dimensional point spread function of multilayered flat lenses and its application to extreme subwavelength resolution

The three-dimensional (3D) point spread function (PSF) of multilayered flat lenses was proposed in order to characterize the diffractive behavior of these subwavelength image formers. We computed the polarization-dependent scalar 3D PSF for a wide range of slab widths and for different dissipative metamaterials. In terms similar to the Rayleigh criterion we determined unambiguously the limit of resolution featuring this type of image-forming device. We investigated the significant reduction of the limit of resolution by increasing the number of layers, which may drop nearly 1 order of magnitude. However, this super-resolving effect is obtained in detriment of reducing the depth of field. Limitations exist on the formation of 3D images.

Motion frozen 18F-FDG cardiac PET

BackgroundPET reconstruction incorporating spatially variant 3D Point Spread Function (PSF) improves contrast and image resolution. “Cardiac Motion Frozen” (CMF) processing eliminates the influence of cardiac motion in static summed images. We have evaluated the combined use of CMF- and PSF-based reconstruction for high-resolution cardiac PET. MethodsStatic and 16-bin ECG-gated images of 20 patients referred for 18F-FDG myocardial viability scans were obtained on a Siemens Biograph-64. CMF was applied to the gated images reconstructed with PSF. Myocardium to blood contrast, maximum left ventricle (LV) counts to defect contrast, contrast-to-noise (CNR) and wall thickness with standard reconstruction (2D-AWOSEM), PSF, ED-gated PSF, and CMF-PSF were compared. ResultsThe measured wall thickness was 18.9 ± 5.2 mm for 2D-AWOSEM, 16.6 ± 4.5 mm for PSF, and 13.8 ± 3.9 mm for CMF-PSF reconstructed images (all P < .05). The CMF-PSF myocardium to blood and maximum LV counts to defect contrasts (5.7 ± 2.7, 10.0 ± 5.7) were higher than for 2D-AWOSEM (3.5 ± 1.4, 6.5 ± 3.1) and for PSF (3.9 ± 1.7, 7.7 ± 3.7) (CMF vs all other, P < .05). The CNR for CMF-PSF (26.3 ± 17.5) was comparable to PSF (29.1 ± 18.3), but higher than for ED-gated dataset (13.7 ± 8.8, P < .05). ConclusionCombined CMF-PSF reconstruction increased myocardium to blood contrast, maximum LV counts to defect contrast and maintained equivalent noise when compared to static summed 2D-AWOSEM and PSF reconstruction.

Generation of a high depth of focus with constant transversal spot size using a phase-only pupil filter

A novel approach is proposed to obtain an extended depth of focus (DoF) in white light with constant transversal spot size within the DoF. It combines a phase-only pupil filter based on multiplexed radial zones with alternating quartic phase functions. The design is first tested via numerical simulations of the point spread function (PSF) based on the scalar diffraction theory. The results for a fourfold gain of the depth of focus are experimentally verified with a phase-only spatial light modulator liquid crystal device combined with a 3D PSF measurement system. A close conformity between the experimental and simulation results proves the effectiveness of the proposed approach.

First experimental demonstration of temporal hypertelescope operation with a laboratory prototype

In this paper, we report the first experimental demonstration of a Temporal HyperTelescope (THT). Our breadboard including 8 telescopes is firstly tested in a manual cophasing configuration on a 1D object. The Point Spread Function (PSF) is measured and exhibits a dynamics in the range of 300. A quantitative analysis of the potential biases demonstrates that this limitation is related to the residual phase fluctuation on each interferometric arm. Secondly, an unbalanced binary star is imaged demonstrating the imaging capability of THT. In addition, 2D PSF is recorded even if the telescope array is not optimized for this purpose.

Three-Dimensional Imaging in Aberration-Corrected Electron Microscopes

This article focuses on the development of a transparent and uniform understanding of possibilities for three-dimensional (3D) imaging in scanning transmission and confocal electron microscopes (STEMs and SCEMs), with an emphasis on the annular dark-field STEM (ADF-STEM), bright-field SCEM (BF-SCEM), and ADF-SCEM configurations. The incoherent imaging approximation and a 3D linear imaging model for ADF-STEM are reviewed. A 3D phase contrast model for coherent-SCEM as well as a pictorial way to find boundaries of information transfer in reciprocal space are reviewed and applied to both BF- and ADF-SCEM to study their 3D point spread functions and contrast transfer functions (CTFs). ADF-STEM is capable of detecting the depths of dopant atoms in amorphous materials but can fail for crystalline materials when channeling substantially modifies the electron propagation. For the imaging of extended (i.e., nonpointlike) features, ADF-STEM and BF-SCEM exhibit strong elongation artifacts due to the missing cone of information. ADF-SCEM shows an improvement over ADF-STEM/BF-SCEM due to its differential phase contrast eliminating slowly varying backgrounds, an effect that partially suppresses the elongation artifacts. However, the 3D CTF still has a cone of missing information that will result in some residual feature elongation as has been observed in A. Hashimoto et al., J Appl Phys 160(8), 086101 (2009).

A Parallel Product-Convolution approach for representing the depth varying Point Spread Functions in 3D widefield microscopy based on principal component analysis

We address the problem of computational representation of image formation in 3D widefield fluorescence microscopy with depth varying spherical aberrations. We first represent 3D depth-dependent point spread functions (PSFs) as a weighted sum of basis functions that are obtained by principal component analysis (PCA) of experimental data. This representation is then used to derive an approximating structure that compactly expresses the depth variant response as a sum of few depth invariant convolutions pre-multiplied by a set of 1D depth functions, where the convolving functions are the PCA-derived basis functions. The model offers an efficient and convenient trade-off between complexity and accuracy. For a given number of approximating PSFs, the proposed method results in a much better accuracy than the strata based approximation scheme that is currently used in the literature. In addition to yielding better accuracy, the proposed methods automatically eliminate the noise in the measured PSFs.