Gaussian Random Spheres Research Articles

Can particle shape information be retrieved from light-scattering observations using spheroidal model particles?

We address the question if and how observations of scattered intensity and polarisation can be employed for retrieving particle shape information beyond a simple classification into spherical and nonspherical particles. To this end, we perform several numerical experiments, in which we attempt to retrieve shape information of complex particles with a simple nonspherical particle model based on homogeneous spheroids. The discrete dipole approximation is used to compute reference phase matrices for a cube, a Gaussian random sphere, and a porous oblate and prolate spheroid as a function of size parameter. Phase matrices for the model particles, homogeneous spheroids, are computed with the T-matrix method. By assuming that the refractive index and the size distribution is known, an optimal shape distribution of model particles is sought that best matches the reference phase matrix. Both the goodness of fit and the optimal shape distribution are analysed. It is found that the phase matrices of cubes and Gaussian random spheres are well reproduced by the spheroidal particle model, while the porous spheroids prove to be challenging. The “retrieved” shape distributions, however, do not correlate well with the shape of the target particle even when the phase matrix is closely reproduced. Rather, they tend to exaggerate the aspect ratio and always include multiple spheroids. A most likely explanation why spheroids succeed in mimicking phase matrices of more irregularly shaped particles, even if their shape distributions display little similarity to those of the target particles, is that by varying the spheroids’ aspect ratio one covers a large range of different phase matrices. This often makes it possible to find a shape distribution of spheroids that matches the phase matrix of more complex particles.

Dependence of the single-scattering properties of small ice crystals on idealized shape models

Abstract. The projections of small ice crystals (with maximum dimension <50 μm) appear quasi-circular when imaged by probes on aircraft flying through cloud. Therefore, idealized models constructed to calculate their single-scattering properties have included quasi-spherical models such as Chebyshev particles, Gaussian random spheres, and droxtals. Recently, an ice analogue grown from sodium fluorosilicate solution on a glass substrate, with several columns emanating from a common center of mass, was shown to be quasi-circular when imaged by state-of-the-art cloud probes. In this study, a new idealized model, called the budding Bucky ball (3B) that resembles the shape of the small ice analogue is developed. The corresponding single-scattering properties (scattering phase function P11 and asymmetry parameter g) are computed by a ray-tracing code. Compared with previously used models, 3B scatters less light in the forward and more light in the lateral and backward directions. The Chebyshev particles and Gaussian random spheres show smooth and featureless P11, whereas droxtals and 3Bs, which have a faceted structure, show several peaks in P11 associated with angles of minimum deviation. Overall, the difference in the forward (lateral; backward) scattering between models are up to 22% (994%; 132%), 20% (510%; 101%), and 16% (146%; 156%) for small ice crystals with respective area ratios of 0.85, 0.77, and 0.69. The g for different models varies by up to 25%, 23%, and 19% for particles with area ratios of 0.85, 0.77, and 0.69, respectively. Because the single-scattering properties of small ice crystals depend both on the choice of the idealized model and the area ratios used to characterize the small ice crystals, higher resolution observations of small ice crystals or direct observations of their single-scattering properties are required.

Light scattering by Gaussian random ellipsoid particles: First results with discrete-dipole approximation

We introduce the stochastic geometry of a Gaussian random ellipsoid (GE) and, with the discrete-dipole approximation, carry out preliminary computations for light scattering by wavelength-scale GE particles. In the GE geometry, we describe the base ellipsoid by the three semiaxes a ⩾ b ⩾ c . The axial ratios b: a and c: a appear as two shape parameters additional to those of the Gaussian random sphere geometry (GS). We compare the scattering characteristics of GE particles to those of ellipsoids. Introducing irregularities on ellipsoids smoothens the angular scattering characteristics, in a way analogous to the smoothening of spherical particle characteristics in the case of GS particles.

Model of light scattering by dust particles in the solar system: Applications to cometary comae and planetary regoliths

We analyze both the intensity and linear polarization of cosmic dust particles by using the physically exact superposition T-matrix method in a fixed orientation for various aggregates of spheres and DDA for the aggregates of Gaussian random spheres. We study both the spherical geometry (in cometary comae) and cylindrical slabs (for regoliths) up to 2000 monomers with size parameters less than ∼3. It is straightforward to produce the observed linear polarization in both geometries while the typically convex and strong opposition spike seems to require wide regolith geometries. The dependence of various parameters on light scattering has also been studied in a rather detailed form. In applications to the cometary polarization we can fit the data in six colors from UV to the J band at a very good accuracy. We, however, emphasize that we do not claim our model to be unique. The most important parameters here are the refractive index and the size distribution of submicron particles. Rest of the parameters has only a minor role. We also found that it is critically important to use several realizations from any assumed particle geometry model because corresponding scattering characteristics can vary quite a lot.

Evaluation of Features and Classifiers for Classification of Early-stage Breast Cancer

Microwave Imaging (MI) has been extensively investigated for a number of years to develop a technique to detect breast cancer at the earliest stages of development. MI for this application is based on exploiting the dielectric contrast between normal breast tissue and cancerous tissue at microwave frequencies. Recent findings have reported overlapping dielectric properties between malignant and benign tumour tissue. Therefore, differentiating between benign and malignant tumours based on the dielectric properties alone is potentially problematic. Alternative methods for the classification of breast tumours have to be developed to improve the sensitivity and specificity of MI. In this paper, the Radar Target Signatures (RTS) of tumours are used to classify tumours, since the RTS contain valuable information about the shape and surface texture of tumours. The most important characteristics of any classification system are the feature extraction method used, along with the type of classifier itself. In this paper, the performance and robustness of three different feature extraction methods in combination with three classifiers are examined. A database of 352 tumours, comprising tumours of four shapes and sizes is created using Gaussian Random Spheres to evaluate the feature extraction methods. The three feature extraction methods — Principal Component Analysis (PCA), Independent Component Analysis (ICA), and Discrete Wavelet Transform (DWT) — are compared using three classification methods — Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA) and Support Vector Machines (SVM). Significantly, the performance of the feature extraction methods is evaluated using tumours in both a dielectrically homogeneous and heterogeneous breast models.

Introduction to light scattering by Gaussian random particles

Background, current status, and future prospects are offered for “Light scattering by Gaussian random particles: Ray-optics approximation” [1]. The stochastic geometry of the random particle is called the Gaussian random sphere. The radial distance of the Gaussian sphere is lognormally distributed. Two logarithmic radial distances at a given great-circle angle apart relate to one another according to the covariance function. Sample Gaussian particles can be conveniently generated using a Legendre polynomial expansion for the covariance function and a spherical harmonics expansion for the logarithmic radial distance. The ray-optics approximation consists of the geometric-optics and forward-diffraction parts fully accounting for polarization. It is valid for particles much larger than the wavelength of incident light and with central phase differences much larger than unity. The numerical ray-tracing algorithms are general and, in principle, applicable computationally to arbitrarily shaped non-spherical particles.

INVESTIGATION OF CLASSIFIERS FOR EARLY-STAGE BREAST CANCER BASED ON RADAR TARGET SIGNATURES

Ultra Wideband (UWB) radar has been extensively investigated as a means of detecting early-stage breast cancer. The basis for this imaging modality is the dielectric contrast between normal and cancerous breast tissue at microwave frequencies. However, based on the dielectric similarities between a malignant and a benign tumour within the breast, difierentiating between these types of tissues in microwave images may be problematic. Therefore, it is important to investigate alternative methods to analyse and classify dielectric scatterers within the breast, taking into account other tumour characteristics such as shape and surface texture of tumours. Benign tumours tend to have smooth surfaces and oval shapes whereas malignant tumours tend to have rough and complex surfaces with spicules or microlobules. Consequently, one classiflcation approach is to classify scatterers based on their Radar Target Signature (RTS), which carries important information about scatterer size and shape. In this paper, Gaussian Random Spheres (GRS) are used to model the shape and size of benign and malignant tumours. Principal Components Analysis (PCA) is used to extract information from the RTS of the tumours, while eight difierent combinations of tumour classiflers are analysed in terms of performance and are compared in terms of two possible approaches: Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA).

Laboratory-based grain-shape models for simulating dust infrared spectra

Analysis of thermal dust emission spectra for dust mineralogy and physical grain properties depends on laboratory-measured or calculated comparison spectra. Often, the agreement between these two kinds of spectra is not satisfactory because of the strong influence of the grain morphology on the spectra. We investigate the ability of the statistical light-scattering model with a distribution of form factors (DFF model) to reproduce experimentally measured infrared extinction spectra for particles that are small compared to the wavelength. We take advantage of new experimental spectra measured for free particles dispersed in air with accompanying information on the grain morphology. For the calculations, we used DFFs that were derived for aggregates of spherical grains, as well as for compact grain shapes corresponding to Gaussian random spheres. Irregular particle shapes require a DFF similar to that of a Gaussian random sphere with sigma=0.3, whereas roundish grain shapes are best fitted with that of a fractal aggregate of a fractal dimension 2.4-1.8. In addition we used a fitting algorithm to obtain the best-fit DFFs for the various laboratory samples. In this way we can independently derive information on the shape of the grains from their infrared spectra. For anisotropic materials, different DFFs are needed for the different crystallographic axes. This is due to a theoretical problem, which is inherent to all models that are simply averaging the contributions of the crystallographic directions.

Breast Tumor Characterization Based on Ultrawideband Microwave Backscatter

Characterization of architectural tissue features such as the shape, margin, and size of a suspicious lesion is commonly performed in conjunction with medical imaging to provide clues about the nature of an abnormality. In this paper, we numerically investigate the feasibility of using multichannel microwave backscatter in the 1-11 GHz band to classify the salient features of a dielectric target. We consider targets with three shape characteristics: smooth, microlobulated, and spiculated; and four size categories ranging from 0.5 to 2 cm in diameter. The numerical target constructs are based on Gaussian random spheres allowing for moderate shape irregularities. We perform shape and size classification for a range of signal-to-noise ratios (SNRs) to demonstrate the potential for tumor characterization based on ultrawideband (UWB) microwave backscatter. We approach classification with two basis selection methods from the literature: local discriminant bases and principal component analysis. Using these methods, we construct linear classifiers where a subset of the bases expansion vectors are the input features and we evaluate the average rate of correct classification as a performance measure. We demonstrate that for 10 dB SNR, the target size is very reliably classified with over 97% accuracy averaged over 360 targets; target shape is classified with over 70% accuracy. The relationship between the SNR of the test data and classifier performance is also explored. The results of this study are very encouraging and suggest that both shape and size characteristics of a dielectric target can be classified directly from its UWB backscatter. Hence, characterization can easily be performed in conjunction with UWB radar-based breast cancer detection without requiring any special hardware or additional data collection.

The 10 $\boldsymbol{\mu}$m amorphous silicate feature of fractal aggregates and compact particles with complex shapes

We model the m absorption spectra of nonspherical particles composed of amorphous silicate. We consider two classes of particles, compact ones and fractal aggregates composed of homogeneous spheres. For the compact particles we consider Gaussian random spheres with various degrees of non-sphericity. For the fractal aggregates we compute the absorption spectra for various fractal dimensions. The 10m spectra are computed for ensembles of these particles in random orientation using the well-known Discrete Dipole Approximation. We compare our results to spectra obtained when using volume equivalent homogeneous spheres and to those computed using a porous sphere approximation. We conclude that, in general, nonspherical particles show a spectral signature that is similar to that of homogeneous spheres with a smaller material volume. This effect is overestimated when approximating the particles by porous spheres with the same volume filling fraction. For aggregates with fractal dimensions typically predicted for cosmic dust, we show that the spectral signature characteristic of very small homogeneous spheres (with a volume equivalent radius m) can be detected even in very large particles. We conclude that particle sizes are underestimated when using homogeneous spheres to model the emission spectra of astronomical sources. In contrast, the particle sizes are severely overestimated when using equivalent porous spheres to fit observations of 10m silicate emission.

Light scattering by small feldspar particles simulated using the Gaussian random sphere geometry

The single-scattering properties of Gaussian random spheres are calculated using the discrete dipole approximation. The ensemble of model particles is assumed to be representative for a feldspar dust sample that is characteristic for weakly absorbing irregularly shaped mineral aerosol. The morphology of Gaussian random spheres is modeled based on a statistical shape analysis using microscope images of the dust sample. The size distribution of the dust sample is based on a particle sizing experiment. The refractive index of feldspar is estimated using literature values. All input parameters used in the light scattering simulations are thus obtained in an objective way based on the true properties of the mineral sample. The orientation-averaged and ensemble-averaged scattering matrices and cross sections of the Gaussian random spheres are compared with light scattering simulations using spheroidal shape models which have been shown to be applicable to the feldspar sample. The Gaussian random sphere model and the spheroidal shape model are assessed using the measured scattering matrix of the feldspar dust sample as a reference. Generally, the spheroidal model with strongly elongated prolate and strongly flattened oblate shapes agrees better with the measurement than the Gaussian random sphere model. In contrast, some features that are characteristic for light scattering by truly irregular mineral dust particles are rendered best by the Gaussian random sphere model; these features include the flat shape of the phase function and a minimum in the scattering matrix element F 22 / F 11 as a function of the scattering angle.

Spectral estimation of Gaussian random circles and spheres

A maximum a posteriori estimator is presented for estimating the parameters of a lognormal random field defined on the unit circle S 1. The measurement data are star-like curves, sample functions that are insufficiently known as equivalent classes of plane curves, where a curve is determined except translations and rotations in R 2 . The estimation problem is also considered in the case of a lognormal random field defined on the unit sphere S 2, where the measurements are sample function surfaces, determined except translations and rotations in R 3 .

Microwave backscattering by nonspherical ice particles at [formula omitted] using second-order perturbation series

Scattering of microwaves by an ensemble of nonspherical ice particles is studied using a scattering model based on a second-order perturbation series at 5.6 GHz (C-band). Particle shapes are defined using a Gaussian random sphere geometry. Particle inhomogeneity is taken into account using three different effective-medium approximations: Maxwell–Garnett, Bruggeman, and Coherent Potential mixing rules. By systematically varying particle size, liquid water content, Gaussian shape parameters, and internal structure, it is found that liquid water content is the most important factor for the co-polarized backscattering; the shape is relatively unimportant. For depolarized backscattering, the shape is of fundamental importance, although the other factors are significant too. Surprisingly, the type of nonsphericity is found to be important for depolarization even for scatterers that are in the Rayleigh region: elongated targets depolarize clearly stronger than more irregular shapes. This finding seems not to be strongly size dependent, at least for size parameters from 0.0059 to 0.47, and indicates that the accurate modeling of shape is important for polarization quantities even in the Rayleigh region.

Ray optics regime for Gaussian random spheres

We examine the validity regime of the ray optics approximation for Gaussian random spheres. The Gaussian sphere is fully characterized by the mean and the covariance function of the radius. We calculate the first two moments of the total, Gaussian curvature on such particles, and utilize the second moment in establishing the ray optics regime. As an intermediate result, we obtain the 6 × 6 covariance matrix for the logarithmic radius and its first and second partial derivatives with respect to the spherical angles. The results are useful when analysing the applicability of our earlier ray optics computations for ensembles of Gaussian spheres.