Erich Ess on Imaging Vector Fields Using Line Integral Convolution

Line Integral Convolution is one of the most intuitive data visualization techniques around. It's dropping paint in a river to see how the current is flowing: to visualize a vector field simply take an image and have the vector field smear the colors. The result is a powerful alternative to using arrows or stream lines. And while the intuition is very straightforward, the actual mathematics that power the technique are very complex.

Transcript

1. Imaging Vector Fields Using Line Integral Convolution Erich Ess
2. Vector Fields • A vector field is a function V with takes a location in space and returns a vector • Used to represent things like: fluid behavior, magnetic fields, gravity, etc.
3. Where do Vector Fields Arise • Fluid Dynamics • Climate information • Physics including modelling electromagnetic fields, gravity, and particle motion
4. Prior Techniques • Several techniques with specific drawbacks which Line Integral Convolution attempts to solve
5. Glyphs • Divide the space into a grid • At each point on the grid • Get the vector at that point • Draw a glyph (like an arrow) which points in the direction of the vector
6. Glyphs: Problems • Take up a lot of visual space • Limits how many glyphs you can display which means • Overly turbulent or large vector fields don’t render well with glyphs • It becomes hard to visualize or intuit how something would move through the vector field
7. Stream Lines • Stream Lines draw lines which start at a point and the follow the flow of the vector field • Pick a point in the space • Get the vector at that point and move a step in the direction of that vector • Draw a line connecting the previous point and the new point • Repeat
8. Stream Lines: Problems • Depend critically on where stream lines are placed • Complex information in turbulent vector fields can be missed entirely if the stream lines are not well placed
9. DDA Convolution • An intermediate step to Line Integral Convolution • DDA convolution takes a vector field and an input image (usually white noise), the image is mapped one to one to the vector field • For each vector a short line is followed and each pixel on the line is summed • The result is the value of the pixel at the vector point
10. DDA Convolution Example
11. DDA Convolution: Problems • Assumes that the local vector field can be modelled using a straight line • Works for where the local curvature is small • Does not work where the local curvature is high, such as turbulent fluids or small vortices
12. DDA Convolution Well Suited Poorly Suited
13. Line Integral Convolution • Combine DDA Convolution with Stream Lines • DDA Convolution: follow a straight line that’s parallel to the vector and sample each pixel in the input image at each step on this straight line • Line Integral Convolution: trace a stream line through the vector field and sample the input image at each step
14. LIC: The Sampling Process Visual
15. Convolution
16. Convolution • Each step on the curve maps to a pixel on the input image • The color of that pixel is sampled to get the color of the output pixel • At each point the convolution kernel is used to determine the weight that pixel color will have to the output
17. Convolution • The function k is a weighting function used to determine how much a specific pixel in the input image will contribute to the value of an output pixel • The simplest version of LIC just uses the unit function k(w)=1 for it’s kernel
18. Convolution • With k(w)=1 the kernel looks like this k(w)
19. Periodic Motion Filters • Using the basic kernel doesn’t show any information with regards to the direction of flow • Using more complex kernel functions allow us to use the direction of the vector field and the distance from the given point to impact the weight of a given pixel from the input image
20. Convolution with an advanced kernel k(w)
21. Sample Kernels • Hanning Filter Functions
22. Normalization • Playing with how normalization works can also be used to provide additional information • In the bottom image, normalization is used to highlight sources and sinks