Papers_We_Love
February 28, 2017
960

# 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.

## Papers_We_Love

February 28, 2017

## Transcript

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

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

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

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

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

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