Paraxial Image Research Articles

Separating edges from microstructure in X-ray dark-field imaging: evolving and devolving perspectives via the X-ray Fokker-Planck equation.

A key contribution to X-ray dark-field (XDF) contrast is the diffusion of X-rays by sample structures smaller than the imaging system's spatial resolution; this is related to position-dependent small-angle X-ray scattering. However, some experimental XDF techniques have reported that XDF contrast is also generated by resolvable sample edges. Speckle-based X-ray imaging (SBXI) extracts the XDF by analyzing sample-imposed changes to a reference speckle pattern's visibility. We present an algorithm for SBXI (a variant of our previously developed multimodal intrinsic speckle-tracking (MIST) algorithm) capable of separating these two physically different XDF contrast mechanisms. The algorithm uses what we call the devolving Fokker-Planck equation for paraxial X-ray imaging as its forward model and then solves the associated multimodal inverse problem to retrieve the attenuation, phase, and XDF properties of the sample. Previous MIST variants were based on the evolving Fokker-Planck equation, which considers how a reference-speckle image is modified by the introduction of a sample. The devolving perspective instead considers how the image collected in the presence of the sample and the speckle membrane optically flows in reverse to generate the reference-speckle image when the sample is removed from the system. We compare single- and multiple-exposure multimodal retrieval algorithms from the two Fokker-Planck perspectives. We demonstrate that the devolving perspective can distinguish between two physically different XDF contrast mechanisms, namely, unresolved microstructure- and sharp-edge-induced XDF. This was verified by applying the different retrieval algorithms to two experimental data sets - one phantom sample and one organic sample. We anticipate that this work will be useful in (1) yielding a pair of complementary XDF images that separate sharp-edge diffuse scatter from diffuse scatter due to spatially random unresolved microstructure, (2) XDF computed tomography, where the strong edge XDF signal can lead to strong contaminating streaking artefacts, and (3) sample preparation, as samples will not need to be embedded since the strong XDF edge signal seen between the sample and air can be separated out.

Fourier-optics imaging analysis with ABCD matrices: tutorial.

The use of ABCD (ray-transfer) matrices to analyze wave propagation through paraxial optical systems, with emphasis on imaging systems, is described. It is shown how to find the image, Fourier transform, and exit pupil planes. Different forms of the propagation integrals are given. The propagation integral for an imaging system, which includes an aperture stop, is derived. The relationships between the aperture stop and the exit pupil and the impulse response of a paraxial imaging system are derived.

General relations for optical design parameters under diffraction limited paraxial imaging

General relations for optical design parameters under diffraction limited paraxial imaging

A Phase Recovery Technique Using the Genetic Algorithm for Aberration Correction in a Coherent Imaging System.

For traditional imaging systems, high imaging quality and system miniaturization are often contradictory. In order to meet the requirements of high imaging quality and system miniaturization, this paper proposes a method to correct the aberration of coherent imaging optical systems. The method is based on the idea of phase recovery and the imaging principle of a coherent imaging system to recover the aberrations at the exit pupil of the system. According to the recovered aberrations, conjugate filters are constructed to correct the image quality in the frequency domain. The imaging quality of the system is improved without changing the original optical path, and the simplicity of the system is guaranteed. To solve the pupil frequency domain aberration more accurately, this paper adopts the dual competition and parallel recombination strategy based on the genetic algorithm and introduces the disaster model. The improved genetic algorithm can effectively restrain the appearance of the "precocity" phenomenon. Finally, the paraxial imaging optical path is simulated and verified by experiments. The results show that, after aberration correction, the image sharpness is improved and the edge information is richer, which verifies the feasibility of the coherent imaging system image quality enhancement method proposed in this paper.

X-Ray Dark-Field and Phase Retrieval Without Optics, via the Fokker-Planck Equation.

Emerging methods of x-ray imaging that capture phase and dark-field effects are equipping medicine with complementary sensitivity to conventional radiography. These methods are being applied over a wide range of scales, from virtual histology to clinical chest imaging, and typically require the introduction of optics such as gratings. Here, we consider extracting x-ray phase and dark-field signals from bright-field images collected using nothing more than a coherent x-ray source and a detector. Our approach is based on the Fokker-Planck equation for paraxial imaging, which is the diffusive generalization of the transport-of-intensity equation. Specifically, we utilize the Fokker-Planck equation in the context of propagation-based phase-contrast imaging, where we show that two intensity images are sufficient for successful retrieval of both the projected thickness and the dark-field signal associated with the sample. We show the results of our algorithm using both a simulated dataset and an experimental dataset. These demonstrate that the x-ray dark-field signal can be extracted from propagation-based images, and that sample thickness can be retrieved with better spatial resolution when dark-field effects are taken into account. We anticipate the proposed algorithm will be of benefit in biomedical imaging, industrial settings, and other non-invasive imaging applications.

Aided Imaging Phase Measuring Deflectometry Based on Concave Focusing Mirror

With the rapid development of aerospace, high-speed train, and automotive industries, the demand for the measurement of high-precision specular components is increasing. The acquisition of high-precision three-dimensional (3D) data is conducive to improving the performance of and extending the service life of these components. However, the existing 3D measurement methods of specular surfaces are affected by the inherent limitation of the depth of field (DOF) of camera lenses. Based on the principle of paraxial reflection imaging of a concave mirror, this paper introduces a concave mirror into a phase measuring deflectometry (PMD) system and proposes an aided imaging PMD (AIPMD) based on a concave focusing mirror. The proposed system realizes the clear imaging of the encoded patterns and the surface under test in the DOF of the camera lens, simultaneously. Meanwhile, the iterative coefficient specular reconstruction algorithm is studied based on this system. The feasibility and accuracy of the proposed method are verified by simulations and experiments.

Enhancing the design by optimizing MTF performance of MLA with overlapping aperture

In this paper, we introduced a novel microlens array (MLA) design approach with the use of overlapped apertures. The chief ray tracing and paraxial imaging optics are utilised in projection imaging analysis. This work makes MLA structure design requirements more flexible. The proposed design is small and compact. It provides homogeneous illumination over the whole field of view (FOV). In the demonstration, a comparison of the sample 3x3 MLA projector and the proposed 3x3 MLA projector with the same aperture size is illustrated with the help of overlapped apertures. Finally, the proposed MLA projection imaging simulations are accomplished. It substantially outperforms the basic design in terms of FOV and modulation transfer function (MTF). We performed comparative analysis for proposed design with the help of diffraction MTF analysis tool. Our research will aid in the development and exploration of more efficient and advanced compact MLA projection optical systems.

A Golay3 sparse aperture optical system of primary mirror with free-form surface

To image distant and dim objects, this paper designs a sparse aperture optical system with a small F number and a large filling factor. The design process is divided into three steps. To start with, the curvature radius and spacing of the primary and secondary mirrors are calculated under the guidance of the paraxial imaging theory. Then, based on the primary aberration theory, the conic coefficients of the primary and secondary mirrors are also calculated. Finally, the node aberration theory is used to increase the field of view and eliminate the residual aberrations. Through theoretical derivation and computer-aided methods, a Golay3 sparse aperture optical system with primary mirror used Zernike polynomial is designed. The full field of view is 0.3° and the F number is 5.

Boosting spatial resolution by incorporating periodic boundary conditions into single-distance hard-x-ray phase retrieval

A simple coherent-imaging method due to Paganin et al is widely employed for phase–amplitude reconstruction of samples using a single paraxial x-ray propagation-based phase-contrast image. The method assumes that the sample-to-detector distance is sufficiently small for the associated Fresnel number to be large compared to unity. The algorithm is particularly effective when employed in a tomographic setting, using a single propagation-based phase-contrast image for each projection. Here we develop a simple extension of the method, which improves the reconstructed contrast of very fine sample features. This provides first-principles motivation for boosting fine spatial detail associated with high Fourier frequencies, relative to the original method, and was inspired by several recent works employing empirically-obtained Fourier filters to a similar end.

Nonparaxial phasor-field propagation.

Growing interest in non-line-of-sight (NLoS) imaging, colloquially referred to as "seeing around corners", has led to the development of phasor-field (P-field) imaging, wherein the field envelope of amplitude-modulated spatially-incoherent light is manipulated like an optical wave to directly probe a space that is otherwise shielded from view by diffuse scattering. Recently, we have established a paraxial theory for P-field imaging in a transmissive geometry that is a proxy for three-bounce NLoS imaging [J. Dove and J. H. Shapiro, Opt. Express27(13) 18016 (2019)]. Our theory, which relies on the Fresnel diffraction integral, introduces the two-frequency spatial Wigner distribution (TFSWD) to efficiently account for specularities and occlusions that may be present in the hidden space and cannot be characterized with P-field formalism alone. However, because the paraxial assumption is likely violated in many, if not most, NLoS scenarios, in the present paper we overcome that limitation by deriving a nonparaxial propagation formula for the P field using the Rayleigh-Sommerfeld diffraction integral. We also propose a Rayleigh-Sommerfeld propagation formula for the TFSWD and provide a derivation that is valid under specific partial-coherence conditions. Finally, we report a pair of differential equations that characterize free-space TFSWD propagation without restriction.

角膜直接接触设备对角膜适配性的近轴光学成像分析

角膜直接接触设备对角膜适配性的近轴光学成像分析

X-ray Fokker\u2013Planck equation for paraxial imaging

The Fokker–Planck equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This forms a natural generalisation of the transport-of-intensity equation. The x-ray Fokker–Planck equation can simultaneously account for both propagation-based phase contrast, and the diffusive effects of sample-induced small-angle x-ray scattering, when forming an x-ray image of a thin sample. Two derivations are given for the Fokker–Planck equation associated with x-ray imaging, together with a Kramers–Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample micro-structure. These equations may be applied to the forward problem of modelling image formation in the presence of both coherent and diffusive energy transport. They may also be used to formulate associated inverse problems of retrieving the phase shifts due to a sample placed in an x-ray beam, together with the diffusive properties of the sample. The domain of applicability for the Fokker–Planck and Kramers–Moyal equations for paraxial imaging is at least as broad as that of the transport-of-intensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons.

Influence of astigmatism and field curvature on the correction zones of an optical imaging system.

By a detailed analysis of the dependence of aberrations on the numerical aperture and the field of view of the optical system, it is possible to find such values of the numerical aperture and the field of view, where the residual wave aberration is zero. These values can be called correction zones. Our work presents a theoretical analysis of the described problem, and general formulas are derived for the expression of wave aberration coefficients using correction zones for astigmatism and field curvature of the third and fifth order. Finally, an analysis of optimum values of correction zones and the optimum position of the image plane has been performed using derived equations with respect to maximization of the Strehl definition. The analysis of the correction zone and the position of the image plane is presented on two examples of the same optical system, where the image plane is located either in the paraxial image plane or in the optimum image plane.

Improvement of Scheimpflug systems with freeform surfaces.

It is well known that freeform surfaces are used to improve the resolution in systems without rotational symmetry. For Scheimpflug systems, the tilted object plane leads to variant magnification in the system imaging. Thus, the system suffers from non-rotationally symmetric aberrations, non-uniform resolution, and non-uniform intensity distribution. In this paper, the paraxial imaging condition of Scheimpflug systems is discussed. From the classical viewpoint, the aberration theory is used to understand, balance, and improve the system performance for variant object distance. For large object distance shift, it is necessary to apply freeform surfaces. With the initial system design method based on Gaussian brackets, the starting configuration of a Scheimpflug system with large object distance shift is obtained. Based on the extension of the Nodal aberration theory concerning the aberrations of freeform surfaces, the rules of selecting the freeform surface position in the system are introduced. By adding two freeform surfaces far away from the pupil, the aberrations are effectively corrected in the Scheimpflug system. The aberrations along the field are decomposed and represented using Zernike fringe polynomials to show the improvement of uniformity and resolution. This work provides insight into Scheimpflug system design with freeform surfaces.

Theory and analysis of a large field polarization imaging system with obliquely incident light.

Polarization imaging technology provides information about not only the irradiance of a target but also the polarization degree and angle of polarization, which indicates extensive application potential. However, polarization imaging theory is based on paraxial optics. When a beam of obliquely incident light passes an analyser, the direction of light propagation is not perpendicular to the surface of the analyser and the applicability of the traditional paraxial optical polarization imaging theory is challenged. This paper investigates a theoretical model of a polarization imaging system with obliquely incident light and establishes a polarization imaging transmission model with a large field of obliquely incident light. In an imaging experiment with an integrating sphere light source and rotatable polarizer, the polarization imaging transmission model is verified and analysed for two cases of natural light and linearly polarized light incidence. Although the results indicate that the theoretical model is consistent with the experimental results, the theoretical model distinctly differs from the traditional paraxial approximation model. The results prove the accuracy and necessity of the theoretical model and the theoretical guiding significance for theoretical and systematic research of large field polarization imaging.

Spherical aberration of an optical system and its influence on depth of focus.

This paper analyzes the influence of spherical aberration on the depth of focus of symmetrical optical systems for imaging of axial points. A calculation of a beam's caustics is discussed using ray equations in the image plane and considering longitudinal spherical aberration as well. Concurrently, the influence of aberration coefficients on extremes of such a curve is presented. Afterwards, conditions for aberration coefficients are derived if the Strehl definition should be the same in two symmetrically placed planes with respect to the paraxial image plane. Such conditions for optical systems with large aberrations are derived with the use of geometric-optical approximation where the gyration diameter of the beam in given planes of the optical system is evaluated. Therefore, one can calculate aberration coefficients in such a way that the optical system generates a beam of rays that has the gyration radius in a given interval smaller than the defined limit value. Moreover, one can calculate the maximal depth of focus of the optical system respecting the aforementioned conditions.

Accommodation and pupil behaviour of binocularly viewing early presbyopes.

To study accommodation behaviour of early presbyopes with the full suite of accommodative stimuli, and to monitor changes in spherical aberration, pupil size and image quality that accompany the accommodative response. Using a high resolution Shack-Hartmann aberrometer, we measured refractive state as a binocularly viewed 0.30 logMAR (6/12 or 20/40) letter E was moved from 2m to 20cm and simultaneously monitored pupil diameter and spherical aberration for 19 subjects (mean age: 42±7.18years). Refractive state was defined using three standard criteria: minRMS, paraxial, and optimum image quality using the Visual Strehl ratio metric (VSOTF). Because of changes in spherical aberration, accommodative gain measured with the paraxial criterion is generally greater (on average by 34%) than the minRMS gain in early presbyopes. The slope of stimulus/response curve is relatively stable up to age 41 and then declines progressively in spite of a full binocular set of accommodative cues. Reduced accommodative gain appears once accommodative amplitude has dropped to below 3D, and gain drops to <0.5 as amplitude drops to 1.5D. This decline happens at different ages in different presbyopes. Prior to the early 40s, changes in accommodation are restricted to a reduction in amplitude, but during the 40s the continued loss of accommodative amplitude is accompanied by a concurrent drop in accommodative gain. Therefore, reduced near image quality in early presbyopes is caused by lower accommodative amplitudes and gains, which may explain the apparent acceleration in symptoms and near add prescriptions during the mid to late 40s.

Qualitative analysis of the evaluation indicators and their related parameters of ametropic state

To investigate the theoretical basis and practical limitations of the existing calculation formulas in the evaluation of ametropic state. The evaluation indicators and their calculation parameters of ametropia were analyzed by using the reduced schematic model eye, the paraxial imaging principle, and the dimension laws. The existing formulas resulted from the reduced object vergence of object distance and image distance relation. Regarding the two measurement indicators of the existing formulas, diopter was misused for refractive power. "Ametropia degree" was a non-standard diction. Both of them were not suitable as the evaluation indicators. The outcomes of the existing formulas and their related plus or minus sign rules showed refractive corrections instead of refractive errors proper. For refractive errors, there was no suitable evaluation indicator. In the evaluation of ametropic state, there are fundamental problems in the existing formulas resulting from the reduced object vergence. The measurement indicators and their dimensional units are confused and misused. The calculation results refer to the refractive corrections only. The evaluation indicators for ametropia need to be further discussed.

Paraxial properties of three-element zoom system for laser beam expanders based on tunable-focus lenses.

The paper is focused on the problem of a theoretical analysis of paraxial imaging properties and initial optical design of the three-element zoom optical system for laser beam expanders using lenses with a tunable focal length. Equations which allow calculation of required optical powers of individual elements of the three-element zoom optical system for laser beam expander depending on the value of the axial position of the beam waist of the input Gaussian beam and the required magnification of the system are derived.

Computational method for correcting complex optical distortion based on FOV division.

We propose a computational method for correcting complex optical distortion in off-axis optical systems, such as the optical systems found in head-mounted and head-up displays. The proposed method divides the wide field of view (FOV) into subsections, thereby allowing the distortion to be calculated for each small FOV. Instead of applying the conventional distortion model, the distortion coefficients for each small FOV can be calculated using a simple linear polynomial. In addition, in contrast to the conventional distortion coefficients that refer to the deviation between the real and paraxial image, the distortion coefficients employed by this method directly characterize the relationship between the object and its image. Thus, using the polynomial in the reverse manner repeatedly for each small FOV with the corresponding distortion coefficients, a pixel lookup table is obtained, which can be used to accurately compensate for the distortion in the optical system. This method avoids complicated computations, and there are no requirements for intrinsic or extrinsic parameters. Our experiments verified the effectiveness of the method where the root-mean-square deviation of the projected distorted straight lines was corrected from 23 to 65 pixels to approximately 1 pixel.