Nonlinear Color Scales for Interactive Exploration
Introduction
Color is an important dimension in visualizations. A well-chosen
encoding can make features and patterns in data easy to find and
analyze. There are many possible ways of coloring the values of a
single continuous variable, but they all involve the mapping of a set
of values to a color scale, an ordered arrangement of colors. Color
scales are also called color ramps. Some common
color scales are (1) a gradient from black to white (a greyscale
ramp) and (2) a rainbow-like scale that passes through blue, cyan,
green, yellow, and red.
Choosing a good color scale is an important step in crafting an
effective visualization. Users typically apply a predefined color
scale or create their own with a color scale editor. Users of the IBM
Data Explorer
visualization tool, for example, can create a new color scale using
a flexible color scale editor (see Figure 1),
or choose a predefined color scale with the guidance of the
PRAVDAColor module, which offers suggestions based on the
characteristics of their data and their expressed visualization goals.
Although there may be differences in detail, such as the colors
employed, the overall design of color scales for displaying a single
continuous variable is predictable. It usually consists of some
linear progression of colors, scaled such that the first color
corresponds to the minimum value and the last color to the maximum
value in the range of the variable. Although sufficient for most
cases, it is not too hard to imagine real data sets for which a single
color scale of this basic design would be inadequate. For example, it
would be a poor match for a data set in which most values are
clustered in a narrow range even though the minimum and the maximum
are widely separated (by, say, many orders of a magnitude). Because
the available colors must represent the entire range of values, few
colors would be devoted to the relatively narrow cluster of values,
and thus detail would be lost. If such detail is important, then
something must be done to bring it out. Obvious ways are by filtering
the data and by narrowing the view upon the data. These methods are
straightforward to implement and often very effective. This paper
proposes an additional method which may be useful in similar
situations. The basic idea is to distort the color scale such that
more colors are applied to the region of interest and fewer elsewhere.
Nonlinear Color Scales
Nonlinear color scales are a direct application of
nonlinear magnification techniques. Like other Focus+Context
techniques, the main goal of nonlinear color scales is to make the
most of a limited resource--the set of distinguishable shades of
colors in this case. By alloting more colors to areas of interest and
fewer colors to all other areas, color can become a tool for
interactively exploring fine detail. There are other well-established
and complementary methods for exploring data, such as filtering and
aggregating, but a technique based on color has the advantage of
making use of the power and immediacy of our visual system.
There are two equivalent formulations of this technique. The
technique can be implemented by distorting either the color space or
the value space. Figures 2 and 3 show how these formulations differ conceptually.
Their difference in appearance is only superficial, however, as the
effect is the same. In each figure, the left column shows distortion
in the color space, and the right shows distortion in the value space.
The vertical lines mark off equal intervals of values, and each column
shows five snapshots of a color scale undergoing some change. In Figure 2, the successive rows show an increase in
magnification, with the bottom row showing the highest level. In Figure 3, the rows show a left-to-right movement
of the focus of magnification.
Although these figures use the
hot-to-cold color ramp, the technique itself is applicable to any
color scheme, though some will be better suited than others. In
actual practice, the color scheme will most likely involve a change in
luminance or saturation, rather than hue, as a change in hue is
considered ill-suited to representing continuous variables by
some.
A nonlinear color scale is no more than a particular arrangement of
colors. Thus, in principle, it requires no more supporting machinery
than a color scale editor. For example, one could imagine adapting
the powerful color
gradient editor (see Figure 4) in GIMP for this purpose, as it already
supports nonlinear interpolation between colors. Even a tool as
powerful as this, however, would soon become tedious to use during
interactive exploration, if the user has to manually piece together
each of possibly many color scales. For this technique to be
maximally useful, there must be direct high-level support for creating
nonlinear color scales. The ideal mechanism would automatically
generate them based on a few intuitive parameters supplied by the
user. We present one possible implementation, in which the parameters
are magnification level and focus location.
Our approach is to map values to colors using a continuous nonlinear
function. We describe two such functions, but their choice is largely
arbitrary, as the technique of nonlinear color scales is not about the
use of any particular function. Any strictly-increasing function with
parameters for adjusting the magnification and focus could be used
instead. For the purposes of discussion, we assume that the value
space has been mapped into the closed interval [0, 1]. We further
assume that there is a mapping between the closed interval [0, 1] to
some user-chosen color scheme (some sequence of colors). Thus, the
central task is to map [0, 1] onto itself with nonlinear distortion.
Leung
and Apperley introduced the distinction between magnification and
transformation functions. The former is the derivative of the latter,
and expresses the desired degree of magnification at each point in the
domain. Peaks correspond to areas of high magnification, and troughs
to areas of low magnification. This function corresponds more closely
to our intuitive idea of the distortion process. Nevertheless, it is
the transformation function that actually performs the distortion.
Figures 5 and 6 show the
common form of our distortion functions. Figure
5 shows how changing the level of magnification affects the shape
of the magnification and transformation functions. Figure 6 shows the effect of moving the focus
location. In both figures, green and red lines represent the
functions under the starting and ending values, respectively, of the
parameter. The blue lines represent intermediate stages.
(a)
(b)
Figure 5. The (a) magnification and (b) transformation
functions corresponding to an increase in the magnification
at the central focus.
(a)
(b)
Figure 6. The (a) magnification and (b) transformation
functions corresponding to a left-to-right movement of
the focus at a constant level of magnification.
Given a transformation function T(x, alpha, beta), where
x (in [0,1]) is the coordinate to transform, alpha specifies
focus location, and beta specifies magnification level, we compute
y' (in [0,1]) as follows:
where
The following is a pair of magnification and transformation functions
which could be used for this purpose (alpha should vary between -1 and
1, and a good range for beta is [0.5, 5.5]):
Another pair, similar in shape to the first but with a gentler
drop-off in magnification, is the following (the above ranges for
alpha and beta also work here):
The above formulas are meant to be clear rather than maximally
efficient. One improvement would be the elimination of the scaling by
1/beta in the transformation functions, but doubtless there are
others, including the use of piecewise approximations.
Future Work
The above transformation functions suffer from a minor defect. Slope
directly corresponds to magnification level in the graphs of
transformation functions. However, Figure 6(b)
clearly shows that slope changes as the focus moves to either end of
the range with these transformation functions. This defect results
from our simplistic scaling of the range to [0,1]. We would like to
correct this in the future.
We would also like to try other transformation functions. Ones based
on raising x to a variable power seem promising. Another worth
trying is the transformation function of the
Graphical Fisheye View.
Resources
Nonlinear Magnification:
Nonlinear Magnification Home Page. A good starting point for
investigating this topic. There is a brief introduction and an
extensive collection of links to groups and publications.
T. Alan Keahey,
Nonlinear Magnification, Ph.D. Dissertation, Indiana University
Computer Science, December 1997. A thorough discussion of nonlinear
magnification.
Y.K. Leung and M.D. Apperley,
A review and taxonomy of distortion-oriented presentation
techniques, ACM Transactions on Computer-Human Interaction
1 (1994), no. 2, 126-160. Discusses some early techniques, and
presents a unified model for understanding them. The model is
based upon the concepts of magnification and transformation
functions.
M. Sarkar and M.H. Brown,
Graphical fisheye views of graphs, In Proceedings of
CHI '92, ACM, New York, pp 83-91. Discusses a technique,
analogous to a fisheye lens, for achieving a continuous nonlinear
distortion of 2D space (and not necessarily just of graphs, despite
the title).
Color scales:
Paul Bourke,
"Colour Ramping for Data Visualization". Describes some simple
color scales, including the greyscale and hot-to-cold ramps.
Lloyd Treinish, et al.,
"Why Should Engineers and Scientists Be Worried About Color?"
Discusses the limitations of the widely used "rainbow" color scale,
and offers suggestions for choosing good task-dependent alternatives.
Lloyd Treinish, et al.,
"A Rule-based Tool for Assisting Colormap Selection". Discusses
PRAVDAColor, a module to the IBM Data Explorer visualization tool,
which uses knowledge of human perception to guide users in selecting
effective color scales.
Lloyd Treinish, et al.,
"How NOT to Lie with Visualization". Discusses the ideas behind
PRAVDAColor in greater depth.
Color science:
Charles Poynton's Color FAQ. Discusses color spaces, color
management (maintaining color across media), and the fundamentals
of color.
David Bourgin's Color Spaces FAQ. Discusses the different color
spaces (such as RGB, HSV, HSL, CIE XYZ) in detail, and provides
equations for converting between them.
The Joy of Visual Perception. An online book about the
physiology and psychology of visual perception.
Colour in Computer Graphics. An introduction to color science,
discussing the physiology of vision, color models, and color
management. It provides some guidelines on the effective use of color.
Alex Byrne and David Hilbert,
A Glossary of Color Science.
Color Reference Library. A large collection of links
covering various color topics. There are also two related
collections,
Color Science/Theory and
Color & Computers.
Organizations:
Commission Internationale de l'Eclairage. The organization
responsible for the important CIE color models.
Young Hyun
Last modified: Fri Jul 13 15:45:51 PDT 2001
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