Multi-DimensionalFunctions

cs5984: Information Visualization

Chris North

Multi-Dimensional Functions

• y = f(x1, x2, x3, …, xn)

• Continuous: y = x13 + 2x2

2 - 9x3

• Discrete: xi are sampled in a bounded region• Xi = [0,1,2,…,100]

• How is this different from Multi-Dimensional Data?

• Huge scale: 6D with 10 samples/D = 1,000,000 data points

• Data values at every point in the space

Multi-D Data Visualizations…

• Don’t work well for multi-D functions

• Example:

5d func sampled on 1-9 for all inputs

Parallel coords

1-D: Easy

• y = f(x)

x

y

2-D: Easy

• y = f(x1, x2)

• Height field:

x2

x1

y

2-D: Easy

• Color map: y color

3-D: Hard• y = f(x1, x2, x3)

• Color cube: y color

• What’s inside?

x1

x2

x3

4D: Really Hard

• y = f(x1, x2, x3, x4, …, xn)

• What does a 5D space look like?!?

• Approaches:• Hierarchical axes (Mihalisin)

• Nested coordinate frames (Worlds within Worlds)

• Slicing (HyperSlice)

• Our spiffy new approach: Radial Focus+Context (Sanjini)

Hierarchical Axes• 1D view of 3D function: (Mihalisin et al.)

f(x1, x2, x3)

x3

x2

x1

Hierarchical Axes• 2D view of 4D function (using color maps)

• y = f(x1, x2, x3, x4, …, xn)

• Discrete: xi = [0,1,2,3,4]

x1

x2

x3

x4

y = f(x1,x2,0,0) as color

Hierarchical Axes• Scale?

• 6d = 3 levels in the 2d approach

• 10 samples/d = 1,000,000 data points = 1 screen

• For more dimensions, zoom in on “blocks”

• For alternate 2d color maps, reorder dimensions

Nested Coordinate Frames

• Feiner, “Worlds within Worlds”• Sandip, Ben

Slicing

• Van Wijk, “HyperSlice”• Kumar, Kunal

Radial Focus+Context

• Sanjini

• infovis.cs.vt.edu

Comparison

• Hierarchical axes (Mihalisin): • < 6d by 10 samples, ALL slices, view only 2d at a time

• Nested coordinate frames (Worlds within Worlds)• < 5-8d, continuous, no overview, 3d hardware

• Slicing (HyperSlice): • < 10d by 100 samples, 2d slices

• *Radial Focus+Context (Sanjini)• < 10d by 1000 samples, overview, all d uniform, rays

• Way to go, sanjini!

Next Week

• Book chapter 7

• Tues: 1-D• Plaisant, “Lifelines”

» mahesh, jon

• Eick, “SeeSoft”» jeevak, alex

• Thurs: 1-D• Mackinlay, “Perspective Wall”

» ahmed, ganesh

• Hibino, “MMVIS”» atul, dananjan

Project

• Proposal due today

• Literature review: due Sept 27• Goal: become world’s expert on your topic area

• A few pages

• What have other people done on the topic?

• How will yours be new & different?

• Will be the “Related Work” section of your final paper

• Resources: ACM & IEEE DL, follow refs, people