Visualizing Densities

By

Edward J. Wegman

and

Qiang Luo

ABSTRACT

This paper focuses on visualizing densities. We first give a small generalization of kernel density estimators which is appropriate for smoothing general point masses including statistical data, but also more general data forms. We give a heuristic discussion to show that our smoother has some desirable approximation properties. We also show that for this class of kernel smoother the 1- 2- or 3-dimensional marginal densities of a high-dimensional kernel density approximator have the same formula as the 1- 2- or 3-dimensional kernel density approximator. We conclude that, for visualization purposes, it is unnecessary to compute kernel density approximators in higher than three dimensions. We develop the relationship between the isopleths, the gradient and the surface normals for a density with two-dimensional support. We show that these form a trihedron. We also develop the algorithm for computing the surface normal for the isopleths of a density with three-dimensional support. With this information in hand, we discuss rendering and lighting models, contouring algorithms, stereoscopic display algorithms, and visual design considerations. We conclude with some examples and a discussion of our experiences in using rendering and lighting, transparency, stereoscopy, dynamic rotation and dynamic thresholding techniques to visualize densities.

Keywords: High Interaction Graphics, Density Visualization, Lighting Models, Transparency, Stereoscopic, Density Approximation, Visual Design

PAPER

This web site comes in two parts:

• the version of the paper submitted for publication including color graphics in pdf format, and
  
  
• the full resolution graphics. In order to see the stereo graphics, you will need a pair of anaglyph (red-blue) glasses. Please send email to Ed Wegman and I will send you a pair of glasses by regular mail. Please be sure to include your postal address.
  
  
  • Figure 1.1 An example of a failed wireframe density representation. The grid is dense to represent fine structure, but because of the density of the grid lines, the fine structure is lost in black ink. See Figures 8.1a, b and c for comparison.
    
    
  • Figure 1.2 Failed anaglyph stereo wireframe. Multilevel surfaces obscure any structure visible. The 3-D effect is essentially useless because of the severe overplotting. Compare with Figure 8.2.
    
    
  • Figure 3.1a Pseudo confidence bands based on the gradient of a 2-dimensional density. The density is based on points randomly scattered around a C-shaped curve. This is a representation of the magnitude of the gradient of the density surface. The "trough" in this image represents the estimate of the C-shaped curve while the ridges represent a confidence band. This is an example of a surface with lighting and specular reflection but no transparency or stereoscopic effects.
    
    
  • Figure 3.1b Pseudo confidence bands based on gradient of a 3-dimensional density. The basic data is randomly scattered off of a helical tube in three dimensions. The surfaces shown are inner and outer confidence surfaces based on slicing the magnitude of the gradient. This is a 3-D analogue to Figure 3.1a. This is an example of a full-color anaglyph stereoscopic image with transparency and lighting effects. In this anaglyph image, red should cover the right eye.
    
    
  • Figure 8.1a View of 2-D density surface based on pattern recognition application. This is a joint density plot for an F statistics and a y-intercept arising from a regression analysis related to a calculation for fractal dimension. The density surface is given a metallic feature so that there is a relatively high specular reflection. There are four lights: 1) pink to the upper right, 2) blue to the left, 3) green to the upper left and 4) white high, slightly right, and in front.
    
    
  • Figure 8.1b Rotated view of same density surface as in Figure 8.1a, but a somewhat different perspective. Again there are four lights. This image is a stereo pairs plot. This is a straight ahead version, left-eye view is on left and right-eye view on right. Interchanging them yields the cross-eyed view. These pairs can be put into a stereopticon to aid three dimensional visualization. When not using transparency as in this Figure, a white background is effective.
    
    
  • Figure 8.1c Same perspective of density surface as shown in Figure 8.1b, but here rendered with full color anaglyph stereo and using transparency algorithms. Use of red-green or red-blue glasses will assist in stereo viewing. Red should cover right eye for proper viewing. Black backgound is essential when using transparency.
    
    
  • Figure 8.1d Same perspective as in Figure 8.1c. Here we give a pure anaglyph stereo view, rather than a full-color anaglyph view. Again, red should cover right eye.
    
    
  • Figure 8.2a View of 3-D density contours based on mass distribution of supergalactic clusters. Here there are four lights, two whites, a pink and a blue light. This image portrays an equal density contour and so literally represents the shape of space. This is a full-color stereo anaglyph image with lighting, rendering and transparency. Red should cover the right eye.
    
    
  • Figure 8.2b This is a pure anaglyph stereo version of the contours of the mass distribution of supergalactic clusters. Red should cover right eye.
    
    
  • Figure 8.3a This is a rendered, lit iso-density contour constructed from MRI data. Frontal view of the iso-density contours of the MRI data. In this image we are using transparency, but not stereoscopic displays. Prominent in this view are the eyeballs, the near linear features just inside the eyeball sockets, the brain stem, and the ear canals.
    
    
  • Figure 8.3b Isodensity contours based on the same data as in Figure 8.3a, again using the transparency algorithms to view internal structure. Features of interest include the cartilage in the tip of the nose, the skin, subdural lining and brain, the eyeballs and the shape in the middle of the brain which looks like an eyeglass earpiece.
    
    
  • Figure 8.3c Isodensity contours as in Figure 8.3b, but here rendered as pure anaglyph stereo with transparency. In this stereo view, red should cover right eye.