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Article Excerpt
INTRODUCTION
The perception of size is a fundamental process in vision: perceiving and navigating through the natural environment involves the continuous assessment of the sizes of objects and spaces (Kosslyn, Flynn, Amsterdam, & Wang, 1990). In engineered environments, the size of an object is often used to communicate critical information (Wickens, & Hollands, 2000) to pilots, drivers, operators of heavy or dangerous equipment, and many others who rely on displays, graphs, and gauges. The magnitudes of 2-D representations of objects are often used as surrogates for speed, distance, temperature, and other variables that are intrinsically 1-D. Objects are also commonly used to represent number in statistical graphs and displays--for example, in scientific visualization and industrial instrumentation. Although displayed on a 2-D surface (a screen, a sheet of paper, or a gauge), the representation of an object may have an apparent dimensionality of 1-D, 2-D, or 3-D, depending on whether a line, an area, or a volume is used. It is natural and accurate to represent 1-D and 2-D objects on a surface, but 3-D objects cannot be faithfully portrayed in the plane. In the last instance, the illusion of depth must depend on the use of 2-D cues such as occlusion, relative height, shading, linear perspective, texture gradient, and motion parallax.
In a real-time display, the numbers displayed may vary continuously over time, an apparent 3-D object may move, or the viewing position of the observer may shift. For example, if a box varying in height is viewed from a fixed position, the shape of the top of the box may be varied in perspective to assist in creating the illusion of three dimensionality. In addition to the normal 2-D cues, motion parallax may be used to enhance the illusion of depth--motion parallax is one of the most powerful enhancers of the illusion of depth (Rogers & Graham, 1979). In elaborate data analytic applications that enhance the illusion of depth (see examples at the following Web sites: Visualize, Inc., http://www.visualizetech.com/; Miner3D, http://miner3d.com/: ADVIZOR Solutions, Inc., http://www.advizorsolutions.com/), the observer's view of several objects may be altered simultaneously.
Data mining software applications often allow observer-controlled movement (e.g., by rotation, oscillation, or varying the height and/or azimuth of the viewpoint), thus giving a more convincing illusion of depth (Chittaro, Combi, & Trapasso, 2003; Keim, 2002).
Psychophysics
Recent fashion has embraced the use of objects that have two or three apparent dimensions (Carswell, Frankenberger, & Bernhard, 1991). The practice is so widespread that it is now unusual to see simple line graphs or 2-D bar charts, except in the scientific literature. However, although the psychophysical relation between perceived size (P) and physical size ([PI]) is known to be approximately linear for lines, it is usually nonlinear for areas and volumes, for which the exponents of power functions fitted to the data are less than unity (see Figure 1). These exponents are often referred to as Stevens's law exponents and have typically been found to vary with the dimensionality of the object--exponents for length are near 1.0, those for area are near 0.8, and those for volume are near 0.6 (Baird, 1970; Stevens & Galanter, 1957; Teghtsoonian, 1965).
[FIGURE 1 OMITTED]
In a study designed to assess how people judge objects used in graphs and displays, Spence (1990) found that the Stevens's exponent for size did not always reflect the apparent dimensionality. Some of Spence's stimuli (rectangles, boxes, and cylinders represented in two dimensions) had two or three apparent dimensions, and yet the estimated exponents were in the region of unity. Spence speculated that effective dimensionality rather than apparent dimensionality was responsible for the observed variation in exponents. He defined effective dimensionality as the number of dimensions showing variation, independent of what the dimensionality appeared to be. Thus a set of boxes with identical bases, but varying in height, would have an effective dimensionality of one but an apparent dimensionality of three. Similarly, a set of bars in a bar chart would have an effective dimensionality of one but an apparent dimensionality of two.
However, Spence's experiment was not explicitly designed to manipulate the effective dimensionality of the objects used, and his speculation was post hoc. Also, his 3-D stimuli were static--no cues other than perspective were used to enhance the illusion of three dimensionality.
Accuracy of Processing
It is always possible to render 1-D data in either a 2-D or a 3-D format. However, since the earliest work on the subject (e.g., Croxton & Stein, 1932), there have been disagreements regarding the propriety, accuracy, efficiency,...
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