A visualizing application of line integral convolution techniques

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Abstract

The line integral convolution (LIC) technique, a texture synthesis technique, has served as a useful method for visualizing vector data. However, a number of shortcomings have been identified in the LIC technique, not the least of which are that the technique is computationally expensive and adopts a more or less brute force approach. This paper presents a modification of the original LIC method which addresses these shortcomings. Our method, although used for visualizing atmospheric vortical flows, is also applicable to other atmospheric phenomena. In our method we employ a particular colouring scheme to unambiguously identify the nature of the vortical flows irrespective of the hemisphere in which they occur.

Introduction

In just over a decade or so many methods for artificially generating textures have been suggested. These methods cover a variety of applications. In the field of scientific visualization, texture-based methods are of special interest and are gaining increasing favour because they allow the display of vector fields in an unrivaled spatial resolution (Stalling and Hege, 1995). Traditionally, small arrows or other symbols indicating vector magnitude and direction have been used to represent vector data. This approach is restricted to a rather coarse spatial resolution. More sophisticated methods include the display of streamlines (Helman and Hesselink, 1991), stream surfaces (Hultquist, 1992), flow volumes (Max et al., 1993), as well as various particle tracing techniques (Hin and Post, 1993; Ma and Smith, 1992; van Wijk, 1993). These methods are well suited for revealing characteristic features of vector fields. However, they depend strongly on the proper choice of seed points.

Numerous schemes have been developed to ensure a reasonably accurate analysis of streamlines (Hultquist, 1992; Jobard and Lefer, 1997a; Turk and Banks, 1996; Ueng et al., 1996). These schemes have all involved the method of placing seed points within the vector field and computing curves that are tangential everywhere to the flow direction of the wind. However, as Shen and Kao (1997) pointed out, in order to capture the subtleties of the flow-field, a large number of streamlines had to be computed, resulting in the cluttering of the display.

The technique of image convolution is a fairly common one in image processing. Given an input image, each pixel value in the output image is determined by computing the weighted-average of a small region of pixel values from the input. The weighting variable used to multiply the input pixel value is defined by a function called the convolution kernel. Based on the dimension of the small region covered by the convolution kernel for each pixel, the operation can then be classified as one-dimensional, two-dimensional, or three-dimensional convolution. One-dimensional image convolution can be used to create textures of arbitrarily oriented lines.

For two-dimensional vector fields, texture-based methods have become an attractive option for visualization. These methods depict all parts of the vector field and thus are not susceptible to missing characteristic data features. van Wijk (1991) has presented an interesting approach towards a global visualization of flows by utilizing a spot noise technique. He used a random texture which is convolved along a straight-line segment, and oriented parallel to the local vector direction. Cabral and Leedom (1993) presented line integral convolution (LIC) by modifying van Wijk's method. LIC is a texture synthesis technique that uses an elegant algorithm for visualizing vector fields.

Many authors have embraced the LIC method and have either improved on the original technique or created applications for its direct implementation. Forsell (1994) extended LIC to work on the curvilinear type of grid and showed that for the resulting curvilinear-gridded field, the location of any point in the domain can be represented by a physical co-ordinate system (in physical space) or by a computational co-ordinate system (in computational space). He proposed a solution that stretches the convolution kernel length in computational space for pixels that are in regions with higher grid density, so that its corresponding length in physical space becomes the same everywhere.

Although the use of LIC to visualize vector fields has proven to be quite effective, for large or dense vector fields it is not fast enough to perform real-time data exploration. An extension proposed by Stalling and Hege (1995) has gone a long way towards addressing this problem. Their approach is based on two key observations. First, a streamline starting from any point in the domain actually passes through many pixels. These pixels can share this streamline when computing the convolution, so redundant numerical integrations can be avoided. The second observation is that adjacent pixels in the same streamline use very similar pixel sets for the convolution. Therefore, the LIC value computed for one pixel can be reused by its neighbours, with small modifications, to accelerate the convolutions.

In order to satisfy the convolution process, Stalling and Hege used a box filter, which is a constant function k. Using the box filter, the convolution result for pixel x0 can be expressed as:I(x0)=k∑i=−LLT(xi)k=1(2L+1),where xi is the pixel at the ith step of the streamline integration

T(xi) is the input image (texture) value at xi

L is the convolution length.

Once I(xm) is known (m=0 in this case), I(xm+1) and I(xm−1) can be computed as:I(xm+1)=I(xm)+k[T(xm+1+L)−T(xm−L)],I(xm−1)=I(xm)+k[T(xm+1−L)−T(xm+L)].

The results for I(xm+1) and I(xm−1) are stored into pixel xm+1 and xm−1, respectively. The convolution then continues one step further in both streamline directions using the above formula.

In this paper we provide a modification of the LIC technique, different from that provided by other authors (Charlery, 2000; Interrante and Grosch, 1997; Jobard and Lefer, 1997b; Shen et al., 1996; Wegenkittl and Groller, 1997; Wegenkittl et al., 1997). We opted to use LIC as our underlying technique because the technique is well known, the literature on LIC is extensive, and LIC is able to visualize large and detailed vector fields in a reasonable display area. Also the LIC technique exhibits the desirable properties of accuracy, locality of calculation, simplicity, controllability, and generality.

Section snippets

Atmospheric vortical flow

The most important parameter in analyzing the atmosphere is the wind field. The signatures of all the weather systems such as high and low pressure systems, troughs, ridges and fronts, are all expressed within the information content of the wind field. To accurately represent the behaviour of the atmosphere, therefore, it is imperative that we find a way to visualize the vector field.

The atmosphere is driven by its general circulation. Within this circulation are embedded a number of smaller

Controlling LIC

The primary deficiency in the original LIC method, as applied to visualizing atmospheric flow, is in the algorithm's method of handling the circulation centres. All the available published research adopt one of two approaches. In (Cabral and Leedom, 1993) the convolution equation is applied to one pixel at a time or, in (Stalling and Hege, 1995) the convolution equation is applied to the entire length of the streamline connecting the starting pixel. In either case, the centre of a circulation

Algorithms

In this paper we have utilized a vector field with data points given at discrete locations on a uniform grid. Vector values at intermediate locations have been computed by bilinear interpolation. For streamline integration we used a Runge–Kutta integrator, which requires four evaluations to proceed from some point x to some other point x′ located a step size h ahead on the same streamline.

Our implementation of the algorithm (hereinafter referred to as MetLIC), involves a modification of the

Example

The methodology employed by the algorithm can be illustrated using the following example. Consider the vector data field as indicated by Fig. 1. For simplicity the magnitude of the vectors in all the grid cells, apart from those labeled A, B, C, D, E, F, G, and H, are equal and non-zero. In the labeled cells the vectors are all zero. The data area is expressed in two-dimensional Cartesian coordinates and divided equally between the Northern and Southern hemispheres. The x-axis and y-axis

Conclusion

The primary objective of this paper was to present a visualization scheme which portrays the vortical flow within the atmosphere by using LIC techniques. This was achieved by implementing a modification to the LIC methods, called MetLIC, using vector field data. Another objective was to provide a mechanism for identifying the nature of the vortical systems and how they impact the atmospheric flow within the visualization scheme. We achieved this objective by using a colour scheme to identify

References (25)

  • H. Theisel

    Geometric conditions for G3 continuity of surfaces

    Computer Aided Geometric Design

    (1997)
  • S. Axler

    Linear Algebra Done Right

    (1997)
  • B. Cabral et al.

    Imaging vector fields using line integral convolution

  • Charlery, J., 2000. Visualizing Atmospheric Vortical Flow Using Line Integral Convolution Techniques, Ph.D....
  • L.K. Forsell

    Visualizing flow over curvilinear grid surfaces using line integral convolution

  • J.L. Helman et al.

    Visualizing vector field topology in fluid flows

    IEEE Computer Graphics and Applications

    (1991)
  • A.J.S. Hin et al.

    Visualization of turbulent flow with particles

  • J.P.M. Hultquist

    Constructing stream surfaces in steady 3-D vector fields

  • Interrante, V., Grosch, C., 1997. Strategies for effectively visualizing 3D flow with volume LIC. In: Proceedings of...
  • B. Jobard et al.

    Creating evenly-spaced streamlines of arbitrary density

    Visualization in Scientific Computing

    (1997)
  • Jobard, B., Lefer, W., 1997. The motion map: efficient computation of steady flow animations. In: Yagel, R., Hagen, H....
  • Jobard, B., Erlebacher, G., Hussaini, M., 2000. Hardware-accelerated advection for unsteady flow visualization. In:...
  • Cited by (0)

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    Present address: Department of Computer Science, Mathematics and Physics, University of the West Indies, Cave Hill, Barbados.

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