Regular Grid Data Research Articles

High order numerical differentiation and approximation of Laplace fields using regular grid data

Specialized numerical schemes for calculation of high order field derivatives are considered. The input data are a regular rectangular mesh with potential values at its nodes, the output data are high order derivative values at grid nodes and associated expressions which enable the calculation of potentials, field values or high order derivatives inside a grid cell with high accuracy. An important feature is that differentiation and approximation techniques are based on a priori knowledge about the Laplace equation which allow the production of numerical schemes with higher order properties.

Improvement of particle-based volume rendering for visualizing irregular volume data sets

We present a technique for previewing large-scale irregular volume datasets using an improved particle-based volume rendering (PBVR) technique. Volume rendering of irregular grid volume data is considerably more complicated than that of regular grid data, since the sampling and compositing processes, which must be done in visibility order, are not straightforward. In our original PBVR, rendering involves generating and projecting sub-pixel-size, opaque, and emissive particles without visibility ordering. To make it easier to preview large-scale irregular volume datasets, we improve our original PBVR technique in two respects. The first is that we exploit its scalability to develop a cell-by-cell particle generation technique. The second is that we reduce the memory cost of the frame buffer using a pixel-by-pixel superimposing technique. To measure the effectiveness of our proposed method, we apply it to huge irregular volume datasets composed of 71 mega hexahedral cells or 1 giga tetrahedral cells.

Dynamic hybrid terrain representation based on convexity limits identification

Hybrid digital terrain models combine terrain data with different topologies and resolutions. Cartographic digital terrain models are typically composed of regular grid data that can be locally refined by adding a Triangulated Irregular Network (TIN) that represents morphologically complex terrain parts. Direct rendering of both datasets to visualize the digital terrain model generates discontinuities, as the meshes are disconnected. The utilization of complete/partial precomputed tessellation solutions solves the problem of quality, but limits the applicability of the representation to models with a fixed relative position between datasets. In this paper, we present a new scheme for hybrid terrain representation that permits the dynamic generation of the adaptive tessellation required to join the grid and TIN models. Our proposal permits the dynamic modification of the relative position between datasets. This increases the representation capabilities for those applications where this property is interesting as, for example, urban and landscape planning applications. The algorithm we propose is based on the identification of convex areas on the TIN and the efficient generation of triangles to join the models based on this convex structure. As a result, high quality models without discontinuities are obtained, increasing the flexibility of previous solutions based on fixed precomputations.

Unified Hybrid Terrain Representation Based on Local Convexifications

Hybrid digital terrain models represent an effective framework to combine and integrate terrain data with different topology and resolution. Cartographic digital terrain models typically are constituted by regular grid data and can be refined by adding locally TINs that represent morphologically complex terrain parts. Direct rendering of both data sets to visualize the digital terrain model would generate geometric discontinuities as the meshes are disconnected. In this paper we present a new meshing scheme for hybrid terrain representations. High quality models without discontinuities are generated as the different representations are softly joined through an adaptive tessellation procedure. Due to the complexity of the algorithms involved in the tessellation procedure, we propose a mixed strategy where part of the information is pre-computed and efficiently encoded. This way, for rendering the model, the tessellation information has to be decoded and only additional simple operations have to be performed.

Soil Carbon Maps

The quality of soil property maps may be improved and spatial sampling intensities reduced by incorporating secondary data to enhance spatial estimates. The purpose of this study was to evaluate how scale of sampling and secondary spatial information (terrain attributes) affected the quality of spatial estimates of soil C. A field in Central Michigan was sampled using 30.5- and 100-m regular grids and the samples were analyzed for total C. Extracting a 61-m grid from the 30.5-m regular grid (G30) data set created an additional data set. Total C maps were created at each scale using ordinary kriging, kriging with a trend model, cokriging, kriging with an external drift, and multiple regression. Each resulting map was compared with an independent validation data set (n = 24) to evaluate map quality. At the 30-m grid scale, there were modest differences between maps created with ordinary kriging (root mean squared error [RMSE] = 2.9 g kg−1, isotropic model; RMSE = 3.0 g kg−1, anisotropic model), kriging with a trend model (RMSE= 2.8 g kg−1), cokriging (RMSE = 2.9 g kg−1), kriging with an external drift (RMSE = 2.6 g kg−1), and multivariate stepwise regression (RMSE = 3.2 g kg−1). Prediction errors were generally larger at the 61-m grid scale and procedures that utilized secondary the terrain data (cokriging, RMSE = 3.8 g kg−1; kriging with an external drift, RMSE = 3.8 g kg−1; stepwise multiple regression, RMSE = 3.0 g kg−1) outperformed procedures that did not (isotropic ordinary kriging, RMSE = 4.3 g kg−1; kriging with a trend model, RMSE = 4.3 g kg−1). At the 100-m grid (G100) scale, geostatistical procedures were not appropriate because of the small sample size (n = 12) yet multiple regression performed well (RMSE = 3.8 g kg−1). Maps of soil C created with regression most resembled the soil color patterns evident in an aerial photograph of the field.

Soil Carbon Maps

The quality of soil property maps may be improved and spatial sampling intensities reduced by incorporating secondary data to enhance spatial estimates. The purpose of this study was to evaluate how scale of sampling and secondary spatial information (terrain attributes) affected the quality of spatial estimates of soil C. A field in Central Michigan was sampled using 30.5‐ and 100‐m regular grids and the samples were analyzed for total C. Extracting a 61‐m grid from the 30.5‐m regular grid (G30) data set created an additional data set. Total C maps were created at each scale using ordinary kriging, kriging with a trend model, cokriging, kriging with an external drift, and multiple regression. Each resulting map was compared with an independent validation data set (n= 24) to evaluate map quality. At the 30‐m grid scale, there were modest differences between maps created with ordinary kriging (root mean squared error [RMSE] = 2.9 g kg−1, isotropic model; RMSE = 3.0 g kg−1, anisotropic model), kriging with a trend model (RMSE= 2.8 g kg−1), cokriging (RMSE = 2.9 g kg−1), kriging with an external drift (RMSE = 2.6 g kg−1), and multivariate stepwise regression (RMSE = 3.2 g kg−1). Prediction errors were generally larger at the 61‐m grid scale and procedures that utilized secondary the terrain data (cokriging, RMSE = 3.8 g kg−1; kriging with an external drift, RMSE = 3.8 g kg−1; stepwise multiple regression, RMSE = 3.0 g kg−1) outperformed procedures that did not (isotropic ordinary kriging, RMSE = 4.3 g kg−1; kriging with a trend model, RMSE = 4.3 g kg−1). At the 100‐m grid (G100) scale, geostatistical procedures were not appropriate because of the small sample size (n= 12) yet multiple regression performed well (RMSE = 3.8 g kg−1). Maps of soil C created with regression most resembled the soil color patterns evident in an aerial photograph of the field.

Resel filtering to aid visualisation within an exploratory data analysis system

The work leading to this paper has been designed around the fusion of two concepts: Exploratory Data Analysis (EDA) and resel processing. Socio-economic data are collected in irregular spatial units that are here described as resels. An EDA system, RESEL, has been developed to demonstrate and apply methods that have been adapted from filtering processes on regular gridded data to an irregularly tessellated surface. Problems both in adapting filter algorithms and applying resel geometry were found and the spatial science developed is discussed at length. An EDA system was used so that patterns in enumeration district census data could be visually enhanced and, at the same time, the process of resel filtering itself explored. The results from resel filtering are described and ways in which they may provide a new methodology for visualisation and spatial data analysis are discussed.

Fast variogram computation with FFT

Two programs are presented to compute direct- and cross-variograms, direct and cross-covariograms, and pseudo-cross-variograms. The programs are written in MATLAB and are based on the Fast Fourier Transform algorithm (FFT). The programs accept complete, or incomplete, regular grid data. The FFT appoach is shown to be faster than the spatial approach for this type of data. It gives exactly the same numerical variogram values as programs operating in the spatial domain. These programs could be most useful in image analysis, where images are usually 256 × 256 pixels, 512 × 512 pixels, or larger. For such large images, FFT is many orders of magnitude faster than the spatial approach.

Feature-based derivation of drainage networks

Abstract An approach to the automatic derivation of drainage networks from digital elevation models (DEMs) using regular-gridded data is described. Features such as depressions and plateaus, on which drainage directions are undefined, are handled as entities in their own right, and are identified using operations on connected regions. Such an approach is a departure from the traditional pixel-based (local) approaches and facilitates the construction of a consistent drainage network. A single or multiple flow drainage model can be implemented. Accumulation and watershed delineation of the drainage network are performed by recursively climbing ascending paths from sink points. The algorithms are significantly faster than those of other approaches and operate within an efficient buffering scheme which facilitates handling of large DEMs and multiple data sources.

A cell‐transverse display algorithm for regular and rectilinear 3D grid data

AbstractThis paper proposes an efficient direct imaging algorithm for constructing iso‐surfaces from regular and rectilinear 3D grid data in scientific and engineering visualization. The basic idea is to generate and draw polygons simultaneously by processing the cells spanned by grids in decreasing order of distance from the current viewpoint. Iso‐surfaces are generated in five or six tetrahedrons into which the cells are subdivided, and are sent to a graphics device or drawn into a frame buffer on the fly. The execution order of each of the tetrahedrons is identical and is determined by the current viewpoint. Since the algorithm does not need to store intermediate polyhedral data and does not require a depth buffer memory for hidden surface removal, it is applicable to a large quantity of data on a 3D grid, such as computed tomography (CT) data. It is also particularly powerful for semi‐transparent display, because transparency calculation can be reduced to image compositing operations if the polygons are drawn in order of their z‐depth from the current viewpoint.