Hexagonal Sampling Grid Research Articles

Performance Analysis of Circular Antenna Array for Microwave Interferometric Radiometers

The microwave interferometric radiometer (MIR) is a passive sensor using synthetic aperture technique for microwave remote sensing. It uses a thinned antenna array to replace the traditional large real aperture antenna, so as to reduce the antenna size, weight, and manufacturing difficulty, and further increase the spatial resolution of the instrument. The antenna array configuration is the first concern for the MIR application. Various array configurations have been proposed. Among them the arrays with hexagonal sampling grid such as the Y-shaped and hexagonal arrays have drawn more attention because of the higher sampling efficiency. However, they do not have the best performance in radiometric sensitivity and radio frequency interference (RFI) mitigation. This paper presents the detailed analysis of the circular array performance and compares it with the Y-shaped and hexagonal arrays. Based on the early developed theory about the radiometric resolution, two new factors named as sampling sensitivity factor and uncertainty factor are first introduced and defined in this paper, which can be used to quantitatively characterize the array performance. The main beam efficiency of the MIR is also redefined to prevent it being uncomfortably greater than unity but still keep a physical significance. Numerical simulations with analysis are finally implemented to verify the theoretical results and also to demonstrate the superiority of the circular array in terms of radiometric sensitivity and RFI mitigation.

Performance Comparison of DCT Based Image Compression on Hexagonal and Rectangular Sampling Grid

The advantages of processing images on hexagonal lattice are higher degree of circular symmetry, uniform connectivity, greater angular resolution, and a reduced need of storage and computation in image processing operations. In this work a comparison of DCT based Image compression on hexagonal and rectangular sampling grid is performed. DCT based image compression is performed on both rectangular domain and hexagonal domain using alternate pixel suppressal method. Mean Square Error and Peak Signal to Noise Ratio is considered for the performance analysis. Compression on hexagonal domain gives better results compared to compression on rectangular domain 

Comparative Study of DCT based Image Compression on Hexagonal and Conventional Square Pixel Images

though, the storage space and transmission bandwidth can be reduced by compression, it will reduce the image fidelity, especially when the images are compressed at lower bit rates. Discrete Cosine Transform (DCT) based compression is an effective way of compression in which good compression ratio without losing too much of information can be obtained. In this work a comparison of DCT based Image compression on hexagonal and rectangular sampling grid is performed. Mean Square Error and Peak Signal to Noise Ratio is considered for the performance analysis. Compression on hexagonal domain gives better results compared to compression on rectangular domain

Fine‐scale Microhabitat Heterogeneity in a French Guianan Forest

ABSTRACTWe examined fine‐scale heterogeneity of environmental conditions in a primary rain forest in French Guiana to describe variation in microhabitats that plants may experience during establishment. We characterized both the range as well as the spatial structuring of 11 environmental factors important for seedling establishment in six hexagonal sampling grids, one each in gap and understory sites at three points representing the predominant geomorphic units in this primary forest. Each grid contained 37 sampling points separated by 31 cm–20 m. Monte‐Carlo tests of semivariograms against complete spatial randomness indicated that for many variables in all six sampling grids, spatial dependence did not exceed 1 m. A principal component analysis of all sampling points revealed a lack of spatial microhabitat structure, rather than homogeneous patches associated with canopy structure or geomorphology. Our results suggest that ample fine‐scale spatial heterogeneity exists to support the coexistence of plant species with differential abiotic requirements for regeneration.

The Petersen‐Middleton theorem and sampling of seismic data

ABSTRACTThe Petersen‐Middleton sampling theorem can lead to weaker sampling requirements than the Shannon sampling theorem and therefore proposes advantageous economic alternatives for the acquisition and processing of seismic data. In this paper we present a tutorial review on the Petersen‐Middleton sampling theorem and a unified description of its applications to seismic survey: the interpolation of spatially aliased seismic data, the dealiasing of seismic common midpoint and common offset gathers, the recording and processing of seismic data on a hexagonal sampling grid. The important aspects of this sampling theorem are highlighted and illustrated by synthetic examples.

Alpha stable modeling of human visual systems for digital halftoning in rectangular and hexagonal grids

Human visual system (HVS) modeling has become a critical component in the design of digital halftoning algorithms. Methods that exploit the characteristics of the HVS include the direct binary search (DBS) and optimized tone-dependent halftoning approaches. The spatial sensitivity of the HVS is low-pass in nature, reflecting the physiological characteristics of the eye. Several HVS models have been proposed in the literature, among them, the broadly used Näsänen’s exponential model, which was later shown to be constrained in shape. Richer models are needed to attain better halftone attributes and to control the appearance of undesired patterns. As an alternative, models based on the mixture of bivariate Gaussian density functions have been proposed. The mathematical characteristics of the HVS model thus play a key role in the synthesis of model-based halftoning. In this work, alpha stable functions, an elegant class of functions richer than mixed Gaussians, are exploited to design HVS models to be used in two different contexts: monochrome halftoning over rectangular and hexagonal sampling grids. In the two scenarios, alpha stable models prove to be more efficient than Gaussian mixtures, as they use less parameters to characterize the tails and bandwidth of the model. It is shown that a decrease in the model’s bandwidth leads to homogeneous halftone patterns, and conversely, models with heavier tails yield smoother textures. These characteristics, added to their simplicity, make alpha stable models a powerful tool for HVS characterization.

Blue-noise halftoning for hexagonal grids

In this paper, we closely scrutinize the spatial and spectral properties of aperiodic halftoning schemes on rectangular and hexagonal sampling grids. Traditionally, hexagonal sampling grids have been shunned due to their inability to preserve the high-frequency components of blue-noise dither patterns at gray-levels near one-half, but as will be shown, only through the introduction of diagonal correlations between dots can even rectangular sampling grids preserve these frequencies. And by allowing the sampling grid to constrain the placement of dots, a particular algorithm may introduce visual artifacts just as disturbing as excess energy below the principal frequency. If, instead, the algorithm maintains radial symmetry by introducing a minimum degree of clustering, then that algorithm can maintain its grid defiance illusion fundamental to the spirit of the blue-noise model. As such, this paper shows that hexagonal grids are preferrable because they can support gray-levels near one-half with less required clustering of minority pixels and a higher principal frequency. Along with a thorough Fourier analysis of blue-noise dither patterns on both rectangular and hexagonal sampling grids, this paper also demonstrates the construction of a blue-noise dither array for hexagonal grids.

Hexagonal sampling and hexagonal binning in 3D seismic data acquisition

The hexagonal sampling offers savings in data storage and processing of 3D poststack seismic data because the hexagonal sampling grid requires 13.4% fewer sample points compared to a square grid to represent data with the same degree of accuracy. Moreover, 3D poststack migration via McClellan transformations is better applied on a hexagonal grid than on a square one. Using a hexagonal grid of midpoints we can also reduce by 13.4% either the receiver or the source effort compared to a square grid of midpoints and yet still achieve the same resolution.

Landscape structure analysis of Kansas at three scales

Recent research in landscape ecology has sought to define the underlying structure of landscape pattern as quantified by landscape pattern metrics. One method used by researchers to address this question involves statistical data reduction techniques. In this study, principal components analysis (PCA) was performed on 27 landscape pattern metrics derived from a Kansas land cover data base at three spatial resolutions: 30 m, 100 m, and 1 km. A hexagonal sampling grid was used to subset the landscape and FRAGSTATS software was used to calculate landscape pattern metrics. The PCA reduced the number of variables from 27 to 5. A five-component PCA solution explained between 81 and 89% of the variation in the data set. The components were interpreted as overall landscape texture, patch shape and size, cropland and grassland class-specific metrics, patch interspersion, and a nearest neighbor attribute. These five dimensions were identified at each resolution level. The first component was stable throughout the resolution levels, whereas the order of importance changed for the latter four components. That most components consistently appeared at each resolution level supports the use of the same subset of pattern metrics for landscape monitoring in the region at different resolutions. The individual metrics emerging as most important were similar to those noted in other research and include the modified Simpson’s diversity index, the area-weighted mean patch fractal dimension, the interspersion and juxtaposition index, and the largest patch index for grasslands.

A hexagonal sampling grid for 3D recording and processing of 3D seismic data

3D seismic data are usually recorded and processed on rectangular grids, for which sampling requirements are generally derived from the usual 1D viewpoint. For a 3D data set, the band region (the region of the Fourier space in which the amplitude spectrum is not zero) can be approximated by a domain bounded by two cones. Considering the particular shape of this band region we can use the 3D sampling viewpoint, which leads to weaker sampling requirements than does the 1D viewpoint; i.e. fewer sample points are needed to represent data with the same degree of accuracy. The 3D sampling viewpoint considers regular nonrectangular sampling grids.The recording and processing of 3D seismic data on a hexagonal sampling grid is explored. The acquisition of 3D seismic data on a hexagonal sampling grid is an advantageous economic alternative because it requires 13.4% fewer sample points than a rectangular sampling grid. The hexagonal sampling offers savings in data storage and processing of 3D seismic data.A fast algorithm for 3D discrete spectrum evaluation and trace interpolation in the case of a 3D seismic data set sampled on a hexagonal grid is presented and illustrated by synthetic examples. It is shown that by using this algorithm the hexagonal sampling offers, approximately, the same advantage of saving 13.4% in data storage and computational time for 3D phase‐shift migration.

The processing of hexagonally sampled signals with standard rectangular techniques: application to 2-D large aperture synthesis interferometric radiometers

In Earth observation programs there is a need of passive low frequency (L-band) measurements to monitor soil moisture and ocean salinity with high spatial resolution 10-20 km, a radiometric resolution of 1 K and a revisit time of 1-3 days. Compared to total power radiometers aperture synthesis interferometric radiometers are technologically attractive because of their reduced mass and hardware requirements. In this field it should be mentioned the one-dimensional (1D) linear interferometer ESTAR developed by NASA and MIRAS a two-dimensional (2D) Y-shaped interferometer currently under study by European Space Agency (ESA). Interferometer radiometers measure the correlation between pairs of nondirective antennas. Each complex correlation is a sample of the "visibility" function which, in the ideal case, is the spatial Fourier transform of the brightness temperature distribution. Since most receiver phase and amplitude errors can be hardware calibrated, Fourier based iterative inversion methods will be useful when antenna errors are small, their radiation voltage patterns are not too different, and mutual coupling is small. In order to minimize on-board hardware requirements-antennas, receivers and correlators-the choice of the interferometer array shape is of great importance since it determines the (u,v) sampling strategy and the minimum number of visibility samples required for a determined aliasing level. In this sense, Y-shaped and triangular-shaped arrays with equally spaced antennas are optimal. The main contribution of this paper is a technique that allows the authors to process the visibility samples over the hexagonal sampling grids given by Y-shaped and triangular-shaped arrays with standard rectangular FFT routines. Since no interpolation processes are involved, the risk of induced artifacts in the recovered brightness temperature over the wide held of view required in Earth observation missions is minimized and signal to noise ratio (SNR) is preserved.

Sampling 3-D Seismic Data

A three-dimensional (3-D) seismic survey is usually achieved by recording a parallel profile network. The 3-D data thus obtained are sampled and processed in a cubic grid for which the sampling requirements generally are derived from the usual one-dimensional (1-D) viewpoint. The spectrum of 3-D seismic data has a band region (the region of the Fourier space in which the spectrum is not zero) that can be approximated by a domain bounded by two cones. Considering the particular shape of this support, we use the 3-D sampling theory to obtain results applicable to the recording and processing of 3-D seismic data. This naturally leads to weaker sampling requirements than the 1-D viewpoint. We define the hexagonal noncubic sampling grid and the triangular noncubic sampling grid and show that fewer sample points are needed to represent 3-D seismic data with the same degree of accuracy. Thus, using the hexagonal noncubic sampling grid, we point out that the maximum value of the spatial sampling interval along the profiles is larger by 15.6% that the one of the cubic sampling grid. We also point out that the triangular noncubic sampling grid requires a number of sample points equal to half the number requiredmore » by a cubic sampling grid.« less

3-D migration via McClellan transformations on hexagonal grids

Two‐dimensional (2-D) spatially varying convolutional filters for three‐dimensional (3-D) depth extrapolation are efficiently designed and implemented by using McClellan transformations. Although efficient, McClellan transformations only approximate circularly symmetric depth extrapolation filters. The accuracy of the extrapolation filters can be increased only at an increase in the computational cost of implementing those filters. By applying McClellan transformations on a hexagonal sampling grid, the accuracy of the extrapolation filters can be increased at a reduction in cost.

Sampling strategies for finding contaminated land

Sampling approaches aimed at finding contamination on and beneath the land surface are discussed, compared, and illustrated. Random sampling is the best choice when pre-sampling research shows that multiple and elongated ellipses of contamination will be found in a regular geometric configuration, and the objective of sampling is to minimize the chances of completely missing contamination. If missing small targets is not important, and if a simple layout of the sampling sites is important, then a hexagonal sampling grid should be used if contaminants are likely to be uniformly mixed with other materials or randomly located on the site.

Design of almost minimax FIR filters in one and two dimensions by WLS techniques

The design of Finite Impulse Response (FIR) filters in one or several dimensions can be performed with good computational efficiency using a Weighted Least Square (WLS) design. Minimax design, which is often preferred, is computationally burdensome, principally in two dimensions. This paper draws attention to the design of minimax filters using iterative WLS techniques for one-dimensional filters and extends the approach to two-dimensional filters. For two dimensions the techniques apply to both rectangular and hexagonal sampling grids. Examples demonstrate flexibility and good computational efficiency. The paper also illustrates a promising new approach to filter design which couples the very general WLS methodology to the less manageable but often preferred minimax performance criterion.

Computer graphics on a hexagonal grid

Aberrations in digital images can be attenuated by computing the image at higher-than-display resolution, convolving it with a two-dimensional filter kernel and decimating the filtered image back to display resolution for presentation. Traditionally, this has been performed using a rectangular sampling grid, which is parallel to the display scan direction. In this paper, the application of a hexagonal sampling grid is proposed. Rectangular and hexagonal processing are discussed, and it is shown that despite the differences in their mathematical bases, their algorithmic implementations are similar. A series of test patterns are presented and evaluated. It is concluded that hexagonal processing results in a greater reduction in aliasing in regions of vertical/near vertical features of images than is achieved by rectangular processing, without any degradation in other regions, and with negligible additional computational effort.