Perception of spatiotemporal random fractals: an extension of colorimetric methods to the study of dynamic texture Vincent A. Billock, Douglas W. Cunningham, and Paul R. Havig Logicon, Inc., U. S. Air Force Research Laboratory, P. O. Box 317258, Dayton, Ohio 45437-7258 Brian H. Tsou U. S. Air Force Research Laboratory, Building 146, Wright-Patterson Air Force Base, Ohio 45433 Received October 23, 2000; accepted April 4, 2001; revised manuscript received April 16, 2001 Recent work establishes that static and dynamic natural images have fractal-like 1/ faspatiotemporal spectra. Artiﬁcal textures, with randomized phase spectra, and 1/ faamplitude spectra are also used in studies of tex-ture and noise perception. Inﬂuenced by colorimetric principles and motivated by the ubiquity of 1/ faspatial and temporal image spectra, we treat the spatial and temporal frequency exponents as the dimensions char-acterizing a dynamic texture space, and we characterize two key attributes of this space, the spatiotemporalappearance map and the spatiotemporal discrimination function (a map of Mac Adam-like just-noticeable-difference contours). © 2001 Optical Society of America OCIS codes: 330. 1730, 330. 5510, 330. 6100, 330. 6110, 330. 6790, 100. 6740. 1. GENERAL INTRODUCTION Early color science struggled with problems of quantify-ing, representing, and predicting color appearance. Thedevelopment of colorimetry solved many of these prob-lems with the creation of color spaces that quantify colors,relate them in appearance, and make predictions aboutdiscriminability. 1,2Texture perception suffers from simi-lar difﬁculties and seems in need of similar solutions. Ina pioneering study, Richards showed that colorimetrictexture and ﬂicker matches can be made to arbitrary spa-tial and temporal stimuli by mixing several spatial ortemporal ''primaries. '' 3,4This approach ﬁts well with both the multiscale structure of natural images and withthe multichannel visual processing of spatial and tempo-ral information, but the dimensionality of a spatiotempo-ral match would be cumbersomely large. This dimen-sionality can be reduced in some situations where there isa lawful relationship between the contributions at variousscales. Such an approach is suggested by fractal math-ematics and by a plethora of recent studies on fractal-likeproperties of natural images. Fractional Brownian tex-tures (favorite objects of study in recent textureexperiments 5-7), natural images,8-11and ''fractal forger-ies'' of natural images12-14all share 1/ fbspatial frequency spectra. Similarly, temporal sequences of naturalimages 15-18and the electronics used in imaging devices19,20produce 1/ fatemporal spectra. This suggests that a reasonable model for the amplitude spectra ofmany spatiotemporal textures is A ~fs,ft!5Kfs2bft2a. (1) where fsand ftare spatial and temporal frequency, re-spectively. Throughout this paper we will refer to dy-namic textures—with randomized phase spectra and power law [Eq. (1)] Fourier amplitude spectra—as spa-tiotemporal random fractals. 21We mapped the percep-tual space of such dynamic fractal textures, quantifyingappearance and discriminability. Speciﬁcally, we soughtto create an appearance map of the a,bdynamic texture space, analogous to a CIE space with labeled color bound-aries. We also sought to characterize discriminability ofdynamic textures, by measuring just noticable differences(JNDs) for the four cardinal directions in this space,analogous to the measurement of Mac Adam's ellipses incolorimetry. 2. EXPERIMENT 1: QUANTIFICATION OF APPEARANCE SPACE A. Introduction Every color researcher has a fairly clear notion of the ap-pearance of a speciﬁed aperture color based on its CIE co-ordinates. This expectation is based on a general aware-ness of the approximate CIE coordinates of many spectraland desaturated colors. Most CIE charts plot the spec-tral locus and the blackbody locus. Some reference CIEcharts are even helpfully colored over their entire surface. One can consider such a chart an ''appearance map'' of thecolorimetric space (see Fig. 1 for an example). 22-24We set out to produce an appearance map for spatiotemporalrandom-fractal space by varying the aandbparameters of Eq. (1) and asking several observers to characterize theappearance of these dynamic fractals. B. Methods 1. Participants The four observers (PH, DC, SF, and VB) were all myopes wearing their refractive prescriptions, yielding correc-2404 J. Opt. Soc. Am. A/Vol. 18, No. 10/October 2001 Billock et al. 0740-3232/2001/102404-10$15. 00 © 2001 Optical Society of America