Geometrical Optics Model Research Articles

SGOT: A Simplified Geometric-Optical Model for Crown Scene Components Modeling over Rugged Terrain

Topography affects the fraction of scene components of the canopy and background, resulting in the observed reflectance distortion. Modeling the canopy reflectance over rugged terrain needs to account for topographic effects. For this purpose, the existing models greatly increased the mathematical complexity while improving description of terrain and crown structure, which dramatically decreased the computational efficiency so as to limit their universal application. In this study, we developed a simplified geometric-optical model (SGOT) for simulating the scene components over rugged terrain. The geotropism of tree growth was considered to make SGOT physically sound. The internal structure of crown was simplified to make SGOT mathematically simpler. Scene component observations derived from Persistence of Vision Ray-tracer (POV-Ray) on surfaces with different normal directions and simulations were made using Geometric-Optical and Mutual Shadowing Coupled with Topography Model (GOMST) and Geometric-Optical for Sloping Terrains Model GOST; models were combined to test the SGOT model. In addition, topographic factors and crown density effect on the scene components modeling were analyzed. The results indicated that SGOT has good accuracy (R2 for the areal proportions of sunlit crown (Kc), sunlit background (Kg), shaded crown (Kt), and shaded background (Kz) are 0.853, 0.857, 0.914, and 0.838, respectively) compared with POV-Ray simulation, and performs better than GOMST, especially in scenes with high crown density. Moreover, SGOT outperformed the compared models in computational efficiency (4% faster than GOMST and 29.5% faster than GOST). Finally, the simulations of the scene components distribution in different topographic factors and crown density were further discussed. SGOT and GOST can both capture scene component variations caused by terrain better than GOMST, but comparatively, SGOT provides a more efficient tool to simulate the crown scene components because of its physical soundness and mathematical simplicity, and consequently, it will facilitate the modeling of canopy reflectance over mountainous regions.

Utilization and efficient computation of polarization factor Q for fast, accurate BRDF modeling.

The Bidirectional Reflectance Distribution Function (BRDF) is of substantial use in remote sensing, scene generation, and computer graphics, to describe optical scatter off realistic surfaces. This paper begins by summarizing our prior work in relating wave optics and geometric optics models, culminating with the Modified Cook-Torrance (MCT) model. The MCT model is evaluated here against aluminum, Infragold, and silver paint at various wavelengths in the IR. In each case, the MCT model is shown to outperform a standard microfacet model. Then, this paper shows a non-trivial method of computing the primary new term, the polarization factor Q. This optimization requires manipulation of the polarization factor in the complex plane, and results in code that runs nearly 2 times faster when compared to the more straightforward implementation of Q. The code presented here is easily adapted to languages other than Matlab, as the code does not use complex variables and uses only cosines of relevant angles (which can trivially be computed by the dot product of unit vectors in scene rendering). It is anticipated that these results will lead to more widespread use of the polarization factor in scene rendering, to produce more accurate optical scatter results.

A new method for high resolution curvature measurement applied to stress monitoring in thin films

By combining the well-known grid reflection method with a digital image correlation algorithm and a geometrical optics model, a new method is proposed for measuring the change of curvature of a smooth reflecting substrate, a common reporter of stress state of deposited layers. This tool, called Pattern Reflection for Mapping of Curvature (PReMC), can be easily implemented for the analysis of the residual stress during deposition processes and is sufficiently accurate to follow the compressive-tensile-compressive stress transition during the sputtering growth of a Ag film on a Si substrate. Unprecedented resolution below 10−5 m−1 can be reached when measuring a homogeneous curvature. A comparison with the conventional laser-based tool is also provided in terms of dynamical range and resolution. In addition, the method is capable of mapping local variations in the case of a non-uniform curvature as illustrated by the case of a Mo film of non-uniform film thickness under high compressive stress. PReMC offers interesting perspectives for in situ accurate stress monitoring in the field of thin film growth.

An Improved Two-Scale Model for Sea Surface Scattering in the Transition Range of Incidence Angles

An improved two-scale model (I-TSM) for the incidence angle transition range (5-11°) is proposed by analyzing the contributions of both quasi-specular and Bragg scatterings to the total sea surface scattering. With normalized radar cross section (NRCS) products of surface wave investigation and monitoring (SWIM) instrument, the mean square slope (MSS) and reflection coefficient of the geometrical optics (GO) model are modified to simulate the quasi-specular scattering in I-TSM, and a Bragg scattering contribution factor adaptive to the incidence angle and wind speed is fitted to provide the Bragg scattering component for I-TSM. At different incidence angles and wind speeds, I-TSM combines the quasi-specular and Bragg scatterings in variable proportions by the fitting factor. I-TSM avoids the critical angle selection and separation problems of the traditional TSM in the incidence angle transition range. Compared with the GO model and traditional TSM, I-TSM has significantly higher accuracies with root mean square error (RMSE) of 0.309dB, relative error (RE) of 2.019% and scattering index (SI) of 0.03 when SWIM NRCS products are the standard data in the validation. I-TSM will provide high-precision NRCS simulation data in the incidence angle transition range for scattering characteristics analysis and echo simulation of SWIM.

Application of a Hypergeometric Model in Simulating Canopy Gap Fraction and BRF for Forest Plantations on Sloping Terrains

The influence of tree distribution and slope on canopy gap fraction (GF) and bidirectional reflectance factor (BRF) is shown here to be non-negligible. Trees are often assumed to be randomly distributed in natural forests due to random distribution of natural resources, but this assumption is not valid for forest plantations. A geometric optical model for forest plantations (GOFP) is a geometric optical model for forest plantations on horizontal surfaces based on the theory of exclusion distance among crowns. Sloping terrains change the exclusion distance among crowns, and inevitably affect the canopy GF and BRF. In this article, GOFP with a hypergeometric model (distances among trees are considered) on horizontal surfaces is modified as GOFP-T to simulate BRF for forest plantations on sloping terrains under two scenarios: (horizontal distances among crowns remain unchanged with slope) and (sloping distances among crowns remain unchanged with slope). Two three-dimensional (3-D) radiative transfer models (DART and LESS) and field measurements are used to evaluate and validate GOFP-T simulations. The results show that 1) the canopy GF, four component area ratios, and canopy BRF simulated by GOFP-T show high consistency with results from the two 3-D model: root-mean-square errors in GF, sunlit foliage, and sunlit ground are less than 0.02, 0.06, and 0.03, respectively; 2) forest coverage and canopy reflectance in GOFP-T are compared well with point cloud results from an airboard LiDAR system and Landsat 8 OLI surface reflectance products, respectively, indicating that GOFP-T has ability in simulating canopy reflectance for forest plantations on sloping terrains. GOFP-T with the hypergeometric model in this article is the first simple model to simulate canopy GF and BRF for forest plantations on sloping terrains.

Multi-Model Estimation of Forest Canopy Closure by Using Red Edge Bands Based on Sentinel-2 Images

In remote sensing, red edge bands are important indicators for monitoring vegetation growth. To examine the application potential of red edge bands in forest canopy closure estimation, three types of commonly used models—empirical statistical models (multiple stepwise regression (MSR)), machine learning models (back propagation neural network (BPNN)) and physical models (Li–Strahler geometric-optical (Li–Strahler GO) models)—were constructed and verified based on Sentinel-2 data, DEM data and measured data. In addition, we set up a comparative experiment without red edge bands. The relative error (ER) values of the BPNN model, MSR model, and Li–Strahler GO model with red edge bands were 16.97%, 20.76% and 24.83%, respectively. The validation accuracy measures of these models were higher than those of comparison models. For comparative experiments, the ER values of the MSR, Li–Strahler GO and BPNN models were increased by 13.07%, 4% and 1.22%, respectively. The experimental results demonstrate that red edge bands can effectively improve the accuracy of forest canopy closure estimation models to varying degrees. These findings provide a reference for modeling and estimating forest canopy closure using red edge bands based on Sentinel-2 images.

4D line-scan hyperspectral imaging

This paper proposes a 4D line-scan hyperspectral imager that combines 3D geometrical measurement and spectral detection with high spectral resolution and spatial accuracy. We investigated the geometrical optical model of a camera attaching with a spectrograph, theoretically explored the mathematical model for line-scan fringe projection profilometry, and established the 3D reconstruction and calibration methods under this proposed line-scan high-dimensional imaging system. The spectral resolution of the system is 2.8 nm, and the spatial root-mean-square-error is 0.0895 mm when measuring a standard sphere with a diameter of 40.234 mm. We measure a colored statue to showcase the intensity change along the dimension of wavelength. In addition, the quality and defect of the spinach leaves are inspected based on spectral data and depth data, which demonstrates the potential application of the system in the food industry.

2 × 2 Matrices: Manifolds, Realizations, Applications

Both geometric and wave optical models, as well as classical and quantum mechanics, realize linear transformations with matrices; for plane optics, these are 2×2 and of unit determinant. Students and some researchers could assume that the structure of this matrix group is fairly evident and hardly interesting. However, the properties and applications even of this lowest 2×2 case are already unexpectedly rich. While in mechanics they cover classical angular momentum, quantum spin, and represent ‘2+1’ relativity, in optical models they lead from the geometrical description of light propagation in the paraxial regime to wave optics via linear canonical transforms requiring a more penetrating view of their manifold structure and multiple covers. The purpose of this review article is to highlight the topological space of 2×2 matrices as it applies to classical versus quantum and wave models, to underline how the latter requires the double cover of the former, thus using 2×2 matrices as an alternative viewpoint of the quantization process, beside the traditional characterization by commutation and non-commutations of position and momentum.

Application and repeatability of ocular biometric measures for gaze position calibration in children.

The purpose of this study was to investigate the utility of two ocular biometric measurements to obtain Hirschberg ratios (HRs) in a binocularly normal paediatric population, and to assess the repeatability of this approach. Ocular biometry data from 80 participants (aged 5 to 14years) was obtained using the KM-1 LED manual keratometer and the Tomey Biometer AL-100 A-scan. HRs were calculated from corneal curvature and anterior chamber depth measurements in the horizontal and vertical meridians of each eye using a regression equation based on a geometric optics model. To assess intrasubject variability in the HRs obtained from biometry, measurements were repeated approximately 1h later. At the initial measurement, mean (SD, range) HRs were 10.77 (0.79, 9.14-12.73) and 11.02 (0.82, 9.48-13.32) °/mm for the horizontal and vertical meridians, respectively. There was a significant difference between the horizontal and vertical HRs (p<0.0001). Mean intrasubject variability of HR was 0.06 °/mm (95% Limit of Agreement [LOA]: -0.82 to 0.94 °/mm), and 0.05 °/mm (95% LOA: -1.05 to 1.15 °/mm) for the horizontal and vertical meridians, respectively. The results indicated that HRs obtained through ocular biometry in a binocularly normal paediatric population are consistent with previous studies in both strabismic children and adult cohorts. The HRs obtained with this technique were highly repeatable in this study population. This approach to gaze position calibration could be used in lieu of other empirical techniques in children.

Fine-scale leaf chlorophyll distribution across a deciduous forest through two-step model inversion from Sentinel-2 data

Leaf chlorophyll content (LCC) is a key physiological trait and is crucial for monitoring plant health and accurately modeling the terrestrial carbon cycle. However, spatially-continuous information on LCC variability at fine time-steps, and at fine spatial resolutions across regional spatial extents, is sparse. In this study, we improved a physically-based, two-step inversion approach by using an advanced canopy-to-leaf reflectance conversion model to estimate LCC at fine spatial resolution (20 m) from Sentinel-2 Multi-Spectral Instrument (MSI) data. The first step is to convert MSI canopy reflectance to leaf reflectance using look-up tables constructed from a geometric optical model (4-Scale). The second step is to estimate LCC from the modeled leaf reflectance using the PROSPECT-5 leaf optical model. Both leaf reflectance and LCC derived from MSI were validated against field measurements at a mixed temperate forest site in Canada to examine the accuracy of leaf area index (LAI) and LCC retrievals. The results demonstrate robust canopy-level inversions with strong relationships between measured and MSI-derived leaf reflectance (R2 = 0.995, p < 0.001, RMSE = 0.0143). The modeled LCC results were also strong when compared to measured LCC samples: R2 = 0.849, p < 0.001, and RMSE = 0.304 μg/cm2, respectively. The most important Sentinel-2 MSI band for LAI and LCC derivation was centered at 705 nm (Band 5). Importantly, this two-step radiative transfer inversion approach substantially improved upon the current LAI and LCC algorithms adopted by the Sentinel-2 Application Platform, which underestimated LAI by 52.93% and overestimated LCC by 44.45%. This work highlights the potential of the physically-based two-step inversion method for deriving leaf and canopy traits from Sentinel-2 at very fine spatial and temporal resolutions, for a wide range of terrestrial ecological applications.

Evaluation of GOFP over four forest plots using RAMI and UAV measurements

ABSTRACT Comparison and validation of canopy reflectance (CR) models are two important steps to ensure their reliability. Pure forest plantations are an ideal type of forest for validating CR models because of their simple background and the low variance in the crown structures which are usually assumed to be identical in most CR models. A Geometric Optical Model for Forest Plantations (GOFP) was compared using dataset in two radiation transfer model intercomparison exercise (RAMI) stands and validated using in situ dataset of detailed optical and structural data of two forest plantations in the Saihanba Forestry Center, China. The results show that (1) the tree distributions in stands described by the hypergeometric model in GOFP show good consistencies with the dataset in the two RAMI stands and measurements from the two Saihanba forest stands; and (2) the CRs simulated with GOFP are also compared well in the two RAMI stands and validated with measurements collected with unmanned aerial vehicles in the two Saihanba stands. GOFP shows a better consistency with the CR measurements than those from CR models for natual forestsbecause the tree distribution in forest plantations is described more reasonably in GOFP.

Ka-Band Radar Cross-Section of Breaking Wind Waves

The effective normalized radar cross section (NRCS) of breaking waves, σwb, is empirically derived based on joint synchronized Ka-band radar and video records of the sea surface from a research tower. The σwb is a key parameter that, along with the breaker footprint fraction, Q, defines the contribution of non-polarized backscattering, NP =σwbQ, to the total sea surface NRCS. Combined with the right representation of the regular Bragg and specular backscattering components, the NP component is fundamental to model and interpret sea surface radar measurements. As the first step, the difference between NRCS values for breaking and non-breaking conditions is scaled with the optically-observed Q and compared with the geometric optics model of breaker backscattering. Optically-derived Q might not be optimal to represent the effect of breaking waves on the radar measurements. Alternatively, we rely on the breaking crest length that is firmly detected by the video technique and the empirically estimated breaker decay (inverse wavelength) scale in the direction of breaking wave propagation. A simplified model of breaker NRCS is then proposed using the geometric optics approach. This semi-analytical model parameterizes the along-wave breaker decay from reported breaker roughness spectra, obtained in laboratory experiments with mechanically-generated breakers. These proposed empirical breaker NRCS estimates agree satisfactorily with observations.

Phase-contrast THz-CT for non-destructive testing.

A new approach for image reconstruction in THz computed tomography (THz-CT) is presented. Based on a geometrical optics model containing the THz signal amplitude and phase, a novel algorithm for extracting an average phase from the measured THz signals is derived. Applying the algorithm results in a phase-contrast sinogram, which is further used for image reconstruction. For experimental validation, a fast THz time-domain spectrometer (THz-TDS) in transmission geometry is employed, enabling CT measurements within several minutes. Quantitative evaluation of reconstructed 3D printed plastic profiles reveals the potential of our approach in non-destructive testing of plastic profiles.

Detection of Scots Pine Single Seed in Optoelectronic System of Mobile Grader: Mathematical Modeling

The development of mobile optoelectronic graders for separating viable seeds by spectrometric properties with high detection accuracy represents a very relevant direction of development for seed handling operations. Here, the main parameters of the radiation source and receiver for detecting a single seed in the diagnostic system of a mobile grader were modeled based on the principles of technical optics using Scots pine (Pinus sylvestris L.) seeds as a case study. Among the pine seeds in the seed batch, there are fossilized and empty seeds that are exactly the same in geometric and gravitational parameters as live seeds. For their separation from the seed batch, data from spectrometric studies in the near-infrared (980 nm) region can be used. To substantiate the parameters of the light source, a geometric optical model of optical beam formation was considered, while the energy model of optical beam formation was considered to substantiate the parameters of the light detector. The results of this study show that the signal value depended on the orientation of a single seed relative to the recording window. The beam angle from the radiation source should be within 45 degrees. The difference between the optical streams should be 50 microns, which made it possible to clearly detect the signal at a standard noise level of 15 microns and signal-to-noise detection accuracy ratio of 56.3 dB. This study expands theoretical knowledge in the field of the spectrometric properties of a single seed, considering the cases of its orientation relative to the optical beam, which affected the effective area of detection of the seed. The obtained data on the location of the main elements of the diagnostic system will speed up the design of mobile optoelectronic graders, and the development of a contemporary protocol for improving Scots pine seed quality.

DNN-FZA camera: a deep learning approach toward broadband FZA lensless imaging.

In mask-based lensless imaging, iterative reconstruction methods based on the geometric optics model produce artifacts and are computationally expensive. We present a prototype of a lensless camera that uses a deep neural network (DNN) to realize rapid reconstruction for Fresnel zone aperture (FZA) imaging. A deep back-projection network (DBPN) is connected behind a U-Net providing an error feedback mechanism, which realizes the self-correction of features to recover the image detail. A diffraction model generates the training data under conditions of broadband incoherent imaging. In the reconstructed results, blur caused by diffraction is shown to have been ameliorated, while the computing time is 2 orders of magnitude faster than the traditional iterative image reconstruction algorithms. This strategy could drastically reduce the design and assembly costs of cameras, paving the way for integration of portable sensors and systems.

A Modified Geometrical Optical Model of Row Crops Considering Multiple Scattering Frame

The canopy reflectance model is the physical basis of remote sensing inversion. In canopy reflectance modeling, the geometric optical (GO) approach is the most commonly used. However, it ignores the description of a multiple-scattering contribution, which causes an underestimation of the reflectance. Although researchers have tried to add a multiple-scattering contribution to the GO approach for forest modeling, different from forests, row crops have unique geometric characteristics. Therefore, the modeling approach originally applied to forests cannot be directly applied to row crops. In this study, we introduced the adding method and mathematical solution of integral radiative transfer equation into row modeling, and on the basis of improving the overlapping relationship of the gap probabilities involved in the single-scattering contribution, we derived multiple-scattering equations suitable for the GO approach. Based on these modifications, we established a row model that can accurately describe the single-scattering and multiple-scattering contributions in row crops. We validated the row model using computer simulations and in situ measurements and found that it can be used to simulate crop canopy reflectance at different growth stages. Moreover, the row model can be successfully used to simulate the distribution of reflectances (RMSEs &lt; 0.0404). During computer validation, the row model also maintained high accuracy (RMSEs &lt; 0.0062). Our results demonstrate that considering multiple scattering in GO-approach-based modeling can successfully address the underestimation of reflectance in the row crops.

Improvement of microwave emissivity parameterization of frozen Arctic soils using roughness measurements derived from photogrammetry

ABSTRACT Soil emissivity of Arctic regions is a key parameter for assessing surface properties from microwave brightness temperature (Tb) measurements. Particularly in winter, frozen soil permittivity and roughness are two poorly characterized unknowns that must be considered. Here, we show that after removing snow, the 3D soil roughness can be accurately inferred from in-situ photogrammetry using Structure from Motion (SfM). We focus on using SfM techniques to provide accurate roughness measurements and improve emissivity models parametrization of frozen arctic soil for microwave applications. Validation was performed from ground-based radiometric measurements at 19 and 37 GHz using three different soil emission models: the Wegmüller and Mätzler [1999, TGRS] model (Weg99), the Wang and Choudhury [1981, JGR] model (QNH), and a geometrical optics model (Geo Optics). Measured and simulated brightness temperatures over different tundra and rock sites in the Canadian High Arctic show that Weg99, parametrized with SfM-based roughness and optimized permittivity , yielded an RMSE of 3.1 K () for all frequencies and polarizations. Our SfM based approach allowed us to measure roughness with 0.1 mm accuracy at 55 locations of different land cover type using a digital camera and metal plates of know dimensions.

Understanding Ku-Band Ocean Radar Backscatter at Low Incidence Angles under Weak to Severe Wind Conditions by Comparison of Measurements and Models

The rain-free normalized radar cross-section (NRCS) measurements from the Ku-band precipitation radars (PRs) aboard the tropical rainfall measuring mission (TRMM) and the global precipitation measurement (GPM) mission, along with simultaneous sea surface wind truth from buoy observations, stepped-frequency microwave radiometer (SFMR) measurements, and H*Wind analyses, are used to investigate the abilities of the quasi-specular scattering models, i.e., the physical optics model (PO) and the classical and improved geometrical optics models (GO and GO4), to reproduce the Ku-band NRCS at low incidence angles of 0–18° over the wind speed range of 0–45 m/s. On this basis, the limitations of the quasi-specular scattering theory and the effects of wave breaking are discussed. The results show that the return caused by quasi-specular reflection is affected significantly by the presence of background swell waves at low winds. At moderate wind speeds of 5–15 m/s, the NRCS is still dominated by the quasi-specular reflection, and the wave breaking starts to work but its contribution is very small, thus, the models are found in excellent agreement with the measurements. With wind speed increasing, the impact of wave breaking increases, whereas the role of standard quasi-specular reflection decreases. The wave breaking impact on NRCS is first visible at incidence angles near 18° as wind speed exceeds about 20 m/s, then it becomes dominant when wind speed exceeds about 37 m/s where the NRCS is insensitive to wind speed and depends linearly on incidence angle, which cannot be explained by the standard quasi-specular scattering theory.

Evaluating Snow Bidirectional Reflectance of Models Using Multiangle Remote Sensing Data and Field Measurements

Because of the anisotropy of snow surface reflectance, it is essential to select a proper snow bidirectional reflectance model for extracting snow cover and inverting snow properties from remote sensing image, especially during the period of snow rapidly changed. In this letter, the ability of reproducing snow bidirectional reflectance by three semiempirical bidirectional reflectance distribution function (BRDF) models (Ross-Li, Roujean, and Raman-Pinty-Verstraete) and the asymptotic radiative transfer theory (ART) model was evaluated using the polarization and directionality of the earth reflectance (POLDER) data. In addition, the ART model was compared with the bicontinuous geometric optics (bic-GORT) model based on field measurements. The results indicated that the root mean square errors (RMSEs) are small and similar for all models during the stable-snow period. The physical models perform better than the semiempirical models in capturing the bidirectional signatures of snow during the periods of snow rapidly changed. The bic-GORT model achieves higher accuracy, while the ART model holds the advantages of simple and efficient to be used.

A modified geometrical optical model of row crops considering multiple scattering frame

A modified geometrical optical model of row crops considering multiple scattering frame