Line integral convolution

In scientific visualization, line integral convolution (LIC) is a technique proposed by Brian Cabral and Leith Leedom to visualize fluid motion, such as the wind movement in a tornado. Compared with simpler integration-like techniques, where one follows the flow vector at each point to produce a line, it has the advantage of producing a whole image at every step. It is a method from the texture advection family.

Intuitively, the flow of a vector field is visualized by adding a random static pattern of dark and light paint sources. As the flow passes by the sources, each parcel of fluid picks up some of the source color, averaging it with the color it has already acquired in a manner similar to throwing paint in a river. The result is a random striped texture where points along the same streamline tend to have similar color.

Algorithmically, the technique generates a random gray level image at the desired output resolution. Then, for every pixel in the image, the forward and backward streamline of a fixed arc length is calculated. The convolution of a suitable convolution kernel with the gray levels of all the pixels that lie in this streamline is the value assigned to the current pixel in the output image.

Mathematically: let ${\displaystyle \mathbf {u} }$ be the vector field. Then a streamline parametrized by arc length can be defined as ${\displaystyle {\frac {d{\boldsymbol {\sigma }}(s)}{ds}}={\frac {\mathbf {u} ({\boldsymbol {\sigma }}(s))}{|\mathbf {u} ({\boldsymbol {\sigma }}(s))|}}}$. Let ${\displaystyle {\boldsymbol {\sigma }}_{\mathbf {r} }(s)}$ be the streamline that passes through the point ${\displaystyle \mathbf {r} }$ for ${\displaystyle s=0}$. Then the image color at ${\displaystyle \mathbf {r} }$ can be set to

...
Wikipedia