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16.4 Line Integral Convolution (LIC) with Texture

Line integral convolution is texture based technique for visualizing vector
fields and has the advantage of being able to visualize large and
detailed vector fields in a reasonable display area.

Line integral convolution involves selectively blurring a reference image
as a function of the vector field to be displayed. The reference image
can be anything, but to make the results clearer, is usually an
spatially uncorrelated image (e.g., a noise image). The resulting image
appears stretched and squished along the directions of the distorting
vector field streamlines, visualizing the flow with a minimum of display
resolution. Vortices, sources, sinks and other discontinuities are
clear shown in the resulting image, and the viewer can get an immediate
grasp of the flow fields ``big picture''.

In each case, you start with a vector field, sampled as a discrete grid of
normalized vectors. You also need an image that is non-uniform and
spatially uncorrelated, so correlations you apply to it will be more obvious.
The goal is to process the image with the vector field, using line integral
convolution, so you can visualize it. Note that in this technique, you will
concentrate on the direction of the flow field, not its velocity; this is
why the vector values at each gridpoint are normalized.

The processed image can be calculated directly using a special convolution
technique. A representative set of vector values on the vector grid are
chosen. Special convolution kernels are created shaped like the local
stream line at that vector by tracing local field flow forwards and
backwards some user-defined distance. The resulting curve is used as a
convolution kernel to convolve the underlying image. This process is repeated
over the entire image using a sampling of the vectors in the vector field.

Mathematically, for each location in the input vector field,
a parametric curve is generated which passes through the
location and follows the vector field for some distance in either
direction. To create an output pixel , a weighted sum of the
values of the input image along the curve is computed. The
weighting function is . Thus the continuous form of the
equation is:

To discretize the equation, use values along the
curve :

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2001-01-10