Imaging Vector Fields Using Line Integral Convolution Erich Ess

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.

Where do Vector Fields Arise • Fluid Dynamics • Climate information • Physics including modelling electromagnetic fields, gravity, and particle motion

Prior Techniques • Several techniques with specific drawbacks which Line Integral Convolution attempts to solve

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

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

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

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

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

DDA Convolution Example

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

DDA Convolution Well Suited Poorly Suited

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

LIC: The Sampling Process Visual

Convolution

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

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

Convolution • With k(w)=1 the kernel looks like this k(w)

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

Convolution with an advanced kernel k(w)

Sample Kernels • Hanning Filter Functions

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