3D Point Spread Function Research Articles

The high-numerical approach for the 3D sampling theorem

Point-spread-function engineering constitutes an interesting tool for the improvement of the performance of 3D optical imaging devices like high-resolution optical scanning microscopes. Apart from the gains in resolution, which provide an easy method for the estimation of the superresolving abilities of pupil filters, other analytical tools are necessary for the fast computation of the 3D point spread function. Such tools should take into account the non-paraxial nature of microscopy imaging. In this sense, in this paper we report the high numerical-aperture version of the 3D sampling theorem.

Spectro-Perfectionism: An Algorithmic Framework for Photon Noise-Limited Extraction of Optical Fiber Spectroscopy

ABSTRACTWe describe a new algorithm for the “perfect” extraction of one-dimensional (1D) spectra from two-dimensional (2D) digital images of optical fiber spectrographs, based on accurate 2D forward modeling of the raw pixel data. The algorithm is correct for arbitrarily complicated 2D point-spread functions (PSFs), as compared to the traditional optimal extraction algorithm, which is only correct for a limited class of separable PSFs. The algorithm results in statistically independent extracted samples in the 1D spectrum, and preserves the full native resolution of the 2D spectrograph without degradation. Both the statistical errors and the 1D resolution of the extracted spectrum are accurately determined, allowing a correct χ2 comparison of any model spectrum with the data. Using a model PSF similar to that found in the red channel of the Sloan Digital Sky Survey spectrograph, we compare the performance of our algorithm to that of cross-section based optimal extraction, and also demonstrate that our method allows coaddition and foreground estimation to be carried out as an integral part of the extraction step. This work demonstrates the feasibility of current and next-generation multifiber spectrographs for faint-galaxy surveys even in the presence of strong night-sky foregrounds. We describe the handling of subtleties arising from fiber-to-fiber cross talk, discuss some of the likely challenges in deploying our method to the analysis of a full-scale survey, and note that our algorithm could be generalized into an optimal method for the rectification and combination of astronomical imaging data.

Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system

Accurate estimation of the three-dimensional (3D) position of particles is critical in applications like biological imaging, atom/particle-trapping, and nanomanufacturing. While it is well-known that localization accuracy better than the Rayleigh resolution limit is possible, it was recently shown that, for photon-limited cases, 3D point spread functions (PSFs) can be shaped to increase accuracies over a 3D volume [Pavani and Piestun, Opt. Express 16, 22048 (2008)]. Here, we show that in the detector-limited regime, the gain in accuracy occurs in all three dimensions throughout the axial range of interest. The PSF is shaped as a double helix, resulting in a system with fundamentally better 3D localization accuracies than standard PSF systems, capable of achieving single-image subnanometer accuracies.

A Postprocessing Method for Compensation of Scatter and Collimator Blurring in SPECT: A Proof-of-Concept Study

Attenuation, scatter, and blurring are 3 major contributors to SPECT image degradation. Image reconstruction without compensation for these degradations results in reduced contrast and reduced quantitative accuracy. In this proof-of-concept study, we present an efficient postprocessing method to compensate for the scatter and blurring effect in SPECT. A raw image is first reconstructed with attenuation correction only. Then, a 2-dimensional (2D) point spread function (PSF) in the image domain is estimated to model the scatter and blurring. This spatially variant 2D PSF is fitted with an asymmetric gaussian function. The accuracy of the estimated 2D PSF is compared with that estimated from the Monte Carlo simulations and the scatter response functions in the projection domain. A further-blurring-and-deconvolution method is used to restore images with the spatially variant 2D PSF. The method is tested using computer simulations and a phantom experiment. The preliminary results demonstrate an improvement in image quality, with increased image contrast and quantitative accuracy, and the feasibility of this postprocessing method. We present a proof-of-concept study for a postprocessing method to compensate for scatter and blurring. Our results indicate that the method is a promising alternative to the state-of-the-art compensation methods thanks to its easy and fast implementation.

Image artifacts in digital breast tomosynthesis: Investigation of the effects of system geometry and reconstruction parameters using a linear system approach

Digital breast tomosynthesis (DBT) is a three-dimensional (3D) x-ray imaging modality that reconstructs image slices parallel to the detector plane. Image acquisition is performed using a limited angular range (less than 50 degrees) and a limited number of projection views (less than 50 views). Due to incomplete data sampling, image artifacts are unavoidable in DBT. In this preliminary study, the image artifacts in DBT were investigated systematically using a linear system approximation. A cascaded linear system model of DBT was developed to calculate the 3D presampling modulation transfer function (MTF) with different image acquisition geometries and reconstruction filters using a filtered backprojection (FBP) algorithm. A thin, slanted tungsten (W) wire was used to measure the presampling MTF of the DBT system in the cross-sectional plane defined by the thickness (z-) and tube travel (x-) directions. The measurement was in excellent agreement with the calculation using the model. A small steel bead was used to calculate the artifact spread function (ASF) of the DBT system. The ASF was correlated with the convolution of the two-dimensional (2D) point spread function (PSF) of the system and the object function of the bead. The results showed that the cascaded linear system model can be used to predict the magnitude of image artifacts of small, high-contrast objects with different image acquisition geometry and reconstruction filters.

Spatial resolution properties in cone beam CT: A simulation study

This work is intended to investigate the spatial resolution properties in cone beam CT by estimating the point spread functions (PSFs) in the reconstructed 3D images through simulation. The point objects were modeled as 3D delta functions. Their projections onto the detector plane were analytically derived and blurred with 2D PSFs estimated and used to represent the detector and focal spot blurring effects. The 2D PSF for detector blurring was computed from the line spread function measured for a typical a-Si/CsI flat panel detector used for general radiography. The focal spot blurring effect was simulated for an x-ray source with a nominal focal spot size of 0.6 mm and 1.33 x magnification at the rotating center. Projection images were computed and sampled with an interval significantly smaller than the detector pixel size to avoid aliasing. Images were reconstructed using the Feldkamp algorithm with the five different filter functions. Reconstructed PSFs were plotted and analyzed to investigate the effects of detector blurring alone, focal spot blurring alone, or a combination of the two on the PSFs and their variations with the radial distance and z-level. Effects of binning and reconstruction filters were also studied. Our results show that the PSFs due to detector blurring are largely symmetric and vary little with the locations of the point objects. With focal spot blurring only or added to detector blurring, the PSFs along the rotation axis were largely symmetric but became increasingly asymmetric as the point objects were moved away from the rotation axis. The PSFs were found to become wider in the axial (anode to cathode) direction as the objects were moved toward the cathode side. The 3D PSFs may be approximated by an ellipsoid with three different axial lengths. They were found to point upright along the rotating axis but tilt toward the rotating axis as the point object was moved away from the axis.

A combined nonlinear contrast image reconstruction algorithm under inexact point-spread function

The problem of image reconstruction is considered for the case when the right-hand side of the 2D integral equation and the point-spread function (the integral equation kernel) are given with random errors. A stable image reconstruction algorithm is proposed. It is a combination of a regularizing algorithm for solving an integral equation (frequency filtering) and a local nonlinear filter (spatial filtering). Characteristics of the 2D point-spread function of the regularizing algorithm are introduced. The regularization parameter is chosen according to the required regularizing algorithm resolution. For eliminating the random reconstruction error, the regularized solution is subjected to nonlinear local filtering that preserves high-frequency information components of the image.

Imaging of small spherical structures in CT: simulation study using measured point spread function

Size and density measurements of objects undertaken using computed tomography (CT) are clinically significant for diagnosis. To evaluate the accuracy of these quantifications, we simulated three-dimensional (3D) CT image blurring; this involved the calculation of the convolution of the 3D object function with the measured 3D point spread function (PSF). We initially validated the simulation technique by performing a phantom experiment. Blurred computed images showed good 3D agreement with measured images of the phantom. We used this technique to compute the 3D blurred images from the object functions, in which functions are determined to have the shape of an ideal sphere of varying diameter and assume solitary pulmonary nodules with a uniform density. The accuracy of diameter and density measurements was determined. We conclude that the proposed simulation technique enables us to estimate the image blurring precisely of any 3D structure and to analyze clinical images quantitatively.

A 3D point-kernel multiple scatter model for parallel-beam SPECT based on a gamma-ray buildup factor

A three-dimensional (3D) point-kernel multiple scatter model for point spread function (PSF) determination in parallel-beam single-photon emission computed tomography (SPECT), based on a dose gamma-ray buildup factor, is proposed. This model embraces nonuniform attenuation in a voxelized object of imaging (patient body) and multiple scattering that is treated as in the point-kernel integration gamma-ray shielding problems. First-order Compton scattering is done by means of the Klein–Nishina formula, but the multiple scattering is accounted for by making use of a dose buildup factor. An asset of the present model is the possibility of generating a complete two-dimensional (2D) PSF that can be used for 3D SPECT reconstruction by means of iterative algorithms. The proposed model is convenient in those situations where more exact techniques are not economical. For the proposed model's testing purpose calculations (for the point source in a nonuniform scattering object for parallel beam collimator geometry), the multiple-order scatter PSF generated by means of the proposed model matched well with those using Monte Carlo (MC) simulations. Discrepancies are observed only at the exponential tails mostly due to the high statistic uncertainty of MC simulations in this area, but not because of the inappropriateness of the model.

Adaptive optics-optical coherence tomography: optimizing visualization of microscopic retinal structures in three dimensions

Adaptive optics-optical coherence tomography (AO-OCT) permits improved imaging of microscopic retinal structures by combining the high lateral resolution of AO with the high axial resolution of OCT, resulting in the narrowest three-dimensional (3D) point-spread function (PSF) of all in vivo retinal imaging techniques. Owing to the high volumetric resolution of AO-OCT systems, it is now possible, for the first time, to acquire images of 3D cellular structures in the living retina. Thus, with AO-OCT, those retinal structures that are not visible with AO or OCT alone (e.g., bundles of retinal nerve fiber layers, 3D mosaic of photoreceptors, 3D structure of microvasculature, and detailed structure of retinal disruptions) can be visualized. Our current AO-OCT instrumentation uses spectrometer-based Fourier-domain OCT technology and two-deformable-mirror-based AO wavefront correction. We describe image processing methods that help to remove motion artifacts observed in volumetric data, followed by innovative data visualization techniques [including two-dimensional (2D) and 3D representations]. Finally, examples of microscopic retinal structures that are acquired with the University of California Davis AO-OCT system are presented.

True three-dimensional proximity effect correction in electron-beam lithography

Proximity effect in e-beam lithography is mainly due to the “nonideal” distribution of exposure (energy deposited in the resist). The proximity effect correction schemes developed so far employ a two-dimensional (2D) model, i.e., exposure variation along the resist depth dimension is not considered. The exposure distribution estimated by the 2D model can be significantly different from the actual exposure distribution, especially for the nanoscale patterns. In this article, a three-dimensional (3D) correction method which uses a 3D point spread function in controlling e-beam dose distribution within each circuit feature in order to achieve a certain desired 3D remaining resist profile after development is described. The dose to be given to each region of a feature is determined based on the estimated remaining resist profile (with the emphasis on the sidewall shape) through iterations. Simulation results demonstrating the potential improvements by the 3D correction are provided.

The point spread function of spiral CT

Computed tomography (CT) scanners are usually described by their in-plane resolution and slice-sensitivity profile (SSP). Other imaging systems are characterized by their point spread function (PSF). The PSF is an excellent basis for the analysis, design and enhancement of imaging systems. The 3D PSF of CT systems has rarely been considered, and has usually been approximated by a 3D Gaussian. We present mathematical analysis of the PSF of single-slice and multi-slice fan-beam and cone-beam CT, for major reconstruction algorithms. We show that the PSF has a complicated, non-separable 3D shape. It is anisotropic in the xy plane and twisted in the z direction. Furthermore, the PSF is space variant in all three axes. In particular, it rotates as the input impulse function moves in the z direction. The PSF may also have effective discontinuities that can lead to streaking artefacts. Indirect measurements of the PSF can be misleading. We support the theoretical results by direct experimental measurements of the PSF.

Analytic derivation of the longitudinal component of the three-dimensional point-spread function in coded-aperture laminography

Near-field coded-aperture data from a single view contain information useful for three-dimensional (3D) reconstruction. A common approach is to reconstruct the 3D image one plane at a time. An analytic expression is derived for the 3D point-spread function of coded-aperture laminography. Comparison with computer simulations and experiments for apertures with different size, pattern, and pattern family shows good agreement in all cases considered. The expression is discussed in the context of the completeness conditions for projection data and is applied to explain an example of nonlinear behavior inherent in 3D laminographic imaging.

Supergridded cone-beam reconstruction and its application to point-spread function calculation

In cone-beam computed tomography (CBCT), the volumetric reconstruction may in principle assume an arbitrarily fine grid. The supergridded cone-beam reconstruction refers to reconstructing the object domain or a subvolume thereof with a grid that is finer than the proper computed tomography sampling grid (as determined by gantry geometry and detector discreteness). This technique can naturally reduce the voxelization effect, thereby retaining more details for object reproduction. The grid refinement is usually limited to two or three refinement levels because the detail pursuit is eventually limited by the detector discreteness. The volume reconstruction is usually targeted to a local volume of interest due to the cubic growth in a three-dimensional (3D) array size. As an application, we used this technique for 3D point-spread function (PSF) measurement of a CBCT system by reconstructing edge spread profiles in a refined grid. Through an experiment with a Teflon ball on a CBCT system, we demonstrated the supergridded volume reconstruction (based on a Feldcamp algorithm) and the CBCT PSF measurement (based on an edge-blurring technique). In comparison with a postreconstruction image refinement technique (upsampling and interpolation), the supergridded reconstruction could produce better PSFs (in terms of a smaller FWHM and PSF fitting error).

Design of Two-Dimensional Ultrasonic Phased Array Transducers

Two-dimensional (2D) phased arrays have the potential to significantly change the way in which engineering components in safety critical industries are inspected. In addition to enabling a three-dimensional (3D) volume of a component to be inspected from a single location, they could also be used in a C-scan configuration. The latter would enable any point in a component to be interrogated over a range of solid angles, allowing more accurate defect characterization and sizing. This paper describes the simulation and evaluation of grid, cross and circular 2D phased array element configurations. The aim of the cross and circle configurations is to increase the effective aperture for a given number of elements. Due to the multitude of possible array element configurations a model, based on Huygens’ principle, has been developed to allow analysis and comparison of candidate array designs. In addition to the element configuration, key issues such as element size, spacing, and frequency are discussed and quantitatively compared using the volume of the 3D point spread function (PSF) as a measurand. The results of this modeling indicate that, for a given number of elements, a circular array performs best and that the element spacing should be less than half a wavelength to avoid grating lobes. A prototype circular array has been built and initial results are presented. These show that a flat bottomed hole, half a wavelength in diameter, can be imaged. Furthermore, it is shown that the volume of the 3D reflection obtained experimentally from the end of the hole compares well with the volume of the 3D PSF predicted for the array at that point.

Phase-retrieved pupil functions in wide-field fluorescence microscopy.

Pupil functions are compact and modifiable descriptions of the three-dimensional (3D) imaging properties of wide-field optical systems. The pupil function of a microscope can be computationally estimated from the measured point spread function (PSF) using phase retrieval algorithms. The compaction of a 3D PSF into a 2D pupil function suppresses artefacts and measurement noise without resorting to rotational averaging. We show here that such 'phase-retrieved' pupil functions can reproduce features in the optical path, both near the sample and in the microscope. Unlike the PSF, the pupil function can be easily modified to include known aberrations, such as those induced by index-mismatched mounting media, simply by multiplying the pupil function by a calculated aberration function. PSFs calculated from such a modified pupil function closely match the corresponding measured PSFs collected under the aberrated imaging conditions. When used for image deconvolution of simulated objects, these phase-retrieved, calculated PSFs perform similarly to directly measured PSFs.

SEMI-AUTOMATIC REMOVAL OF FOREGROUND STARS FROM IMAGES OF GALAXIES

ABSTRACT A new procedure, designed to remove foreground stars from galaxy proviles is presented here. Although several programs exist for stellar and faint object photometry, none of them treat star removal from the images very carefully. I present my attempt to develop such a system, and briefly compare the performance of my software to one of the well-known stellar photometry packages, DAOPhot (Stetson 1987). Major steps in my procedure are: (1) automatic construction of an empirical 2D point spread function from well separated stars that are situated off the galaxy; (2) automatic identification of those peaks that are likely to be foreground stars, scaling the PSF and removing these stars, and patching residuals (in the automatically determined smallest possible area where residuals are truly significant); and (3) cosmetic fix of remaining degradations in the image. The algorithm and software presented here is significantly better for automatic removal of foreground stars from images of galaxies than DAOPhot or similar packages, since: (a) the most suitable stars are selected automatically from the image for the PSF fit; (b) after star-removal an intelligent and automatic procedure removes any possible residuals; (c) unlimited number of images can be cleaned in one run without any user interaction whatsoever.

Temporal bone volumetric image deblurring in spiral computed tomography scanning

We developed a method for volumetric image deblurring in spiral (helical) computed tomography (CT) scanning with a three-dimensional (3D) Gaussian point spread function (PSF) to improve the quality of temporal bone spiral CT images for assessing the position of cochlear implants electrodes. A patient was scanned after cochlear implantation, and the temporal bone was reconstructed into a volume with 128 voxels per dimension, 0.1 mm per voxel side, and x 10 gray-scale expansion. The 3D PSF in spiral CT imaging was assumed to be Gaussian separable transversely and longitudinally. Standard deviations of the PSF were derived and subjectively adjusted. The image was then deconvolved using Wiener filtering and maximum-likelihood deconvolution methods. Image quality was assessed both visually and quantitatively using cross-sectional area at half of the maximum (CAHM) of the implanted array as the figure of merit. Substantial image deblurring was achieved via deconvolution. Subjectively, anatomic structures were more clearly shown. Deconvolution reduced the CAHM by approximately one third, on average. Three-dimensional deconvolution had better image quality than two-dimensional deconvolution. The maximum-likelihood method produced superior image quality but took longer to process relative to Wiener filtering. Volumetric image deblurring is practical with a Gaussian PSF. The maximum-likelihood method is preferred if time permits. Deconvolution facilitates the study of fine details of the temporal bone and cochlear implant.

Effects of a Finite-sized Pinhole on 3D Image Formation in Confocal Two-photon Fluorescence Microscopy

Abstract We investigate three-dimensional (3D) imaging properties in confocal two-photon (2-p) fluorescence microscopy with a finite-sized detector. The 3D intensity point spread function, the 3D optical transfer function (OTF), the strength of optical sectioning and the signal levels for point, planar and volume objects are calculated in order to reveal their relationships to the detector size. It is shown that confocal 2-p fluorescence imaging with a point detector gives super-resolution, compared with conventional 2-p imaging without a confocal pinhole. Although optical sectioning can be attained in the latter case, its strength is inferior to that in the confocal case. In particular, a finite-sized detector can result in negative values in the 3D OTF and reduce the effective cut-off spatial frequencies, thus degrading the capability of confocal 2-p fluorescence microscopy for 3D imaging. Unlike the single-photon fluorescence microscope, there is no missing cone of spatial frequencies in the 3D OTF for...