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Volume: 28 | Article ID: art00026
The pyramid of visibility
  DOI :  10.2352/ISSN.2470-1173.2016.16.HVEI-102  Published OnlineFebruary 2016

A fundamental limit to human vision is our ability to sense variations in light intensity over space and time. These limits were first described systematically a half century ago in a series of three seminal papers: de Lange [1], van Nes and Bouman [2], and Robson [3] measured the visibility of temporal, spatial, and joint spatio-temporal sinusoidal variations. Additionally, the first two papers described how sensitivity depended on the the light level from which the deviations occurred. Their results provided an enduring foundation for all subsequent studies of contrast sensitivity. We have recently reanalyzed these reports and discovered a fundamental and remarkable simplification. In brief, we have found that for photopic retinal illuminances at moderate to high frequencies the log of human contrast sensitivity is a linear function of spatial frequency, temporal frequency, and the log of adapting retinal illuminance. As a surface in the space defined by spatial and temporal frequency, sensitivity thus forms a rectangular pyramid. Elsewhere we have described the boundaries of this surface, where it intersects the plane defined by the maximum contrast limit of one, as the “window of visibility.” [4, 5] The new linear formulation allows us to describe the complete surface as the “pyramid of visibility.” The height of the pyramid rises linearly with the log of retinal illuminance. As a result, the window of visibility is always a diamond that grows and shrinks, linearly, with the log of retinal illuminance. Elsewhere we have shown that under typical conditions log retinal illuminance is a linear function of log luminance [6], in which case the pyramid model also applies for sensitivity as a function of luminance. Almost 40 years ago, analyzing some of his own data, Kulikowski also noted the dependence of log contrast sensitivity on linear spatial and temporal frequency, and on the log of luminance [7]. His result appears not to have been widely understood, nor its practical significance appreciated. This result has deep theoretical and practical significance. With respect to theory, the independence of spatial, temporal, and light level effects constrains models of processing mechanisms and strategies. The practical significance is that rendition of visual information for the human eye is ultimately governed by the pyramid of visibility. There is no need to render beyond these limits, and these limits determine the visibility of artifacts in rendered information [4]. Consequently this surface provides a critical guide to design of a wide variety of visual display technologies. In particular, these limits determine the ultimate number of pixels per degree and frames per second required in electronic displays of static or moving imagery.

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Andrew B Watson, Albert J Ahumada, "The pyramid of visibilityin Proc. IS&T Int’l. Symp. on Electronic Imaging: Human Vision and Electronic Imaging,  2016,

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