2404 J. Opt. Soc. Am. A/Vol. 18, No. 10 /October 2001 Billock et al.

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/f a spatiotemporal spectra.Artifical textures, with randomized phase spectra, and 1/f a amplitude spectra are also used in studies of tex-ture and noise perception. Influenced by colorimetric principles and motivated by the ubiquity of 1/f a spatialand 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 MacAdam-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 INTRODUCTIONEarly 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.1,2 Texture 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.’’ 3,4 This approach fits well withboth 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 textureexperiments5–7), natural images,8–11 and ‘‘fractal forger-ies’’ of natural images12–14 all share 1/f b spatial frequencyspectra. Similarly, temporal sequences of naturalimages15–18 and the electronics used in imagingdevices19,20 produce 1/f a temporal spectra. This suggeststhat a reasonable model for the amplitude spectra ofmany spatiotemporal textures is

A~ fs , ft! 5 Kfs2bft

2a. (1)

where fs and ft are spatial and temporal frequency, re-spectively. Throughout this paper we will refer to dy-

0740-3232/2001/102404-10$15.00 ©

namic textures—with randomized phase spectra andpower law [Eq. (1)] Fourier amplitude spectra—as spa-tiotemporal random fractals.21 We mapped the percep-tual space of such dynamic fractal textures, quantifyingappearance and discriminability. Specifically, we soughtto create an appearance map of the a, b dynamic texturespace, 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 MacAdam’s ellipses incolorimetry.

2. EXPERIMENT 1: QUANTIFICATION OFAPPEARANCE SPACEA. IntroductionEvery 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).22–24 Weset out to produce an appearance map for spatiotemporalrandom-fractal space by varying the a and b parametersof Eq. (1) and asking several observers to characterize theappearance of these dynamic fractals.

B. Methods

1. ParticipantsThe four observers (PH, DC, SF, and VB) were all myopeswearing their refractive prescriptions, yielding correc-

2001 Optical Society of America

Billock et al. Vol. 18, No. 10 /October 2001 /J. Opt. Soc. Am. A 2405

Fig. 1. Appearance map of CIE 1931 color space. Reproduced from Ref. 22 by permission of the Optical Society of America.

tions to at least 6/6 binocular acuity. All participants arepublished psychophysicists and highly experienced ob-servers, with prior work in the psychophysics of ‘‘white’’and ‘‘colored’’ (spectrally nonuniform) spatial noise.Three of the observers were also authors of this study; oneadditional observer (SF) was naı̈ve to the purpose of theexperiment.

2. ApparatusAll stimuli were generated and presented on a SiliconGraphics O2 at 30 Hz. The lookup table was modified to

linearize the display. All stimuli were viewed binocu-larly with natural pupils in a well-lit room (the luminanceof a diffuse reflector in the plane of the monitor was 3.5cd/m2). Subjects were comfortably fixed in place by achin rest at a viewing distance of 40 cm from the monitor.

3. StimuliThe stimuli were 70 spatiotemporal, gray-scale, random-phase fractal-like image sequences. For each stimulusimage, the average luminance was constant at 8.57 cd/m2

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(not including the diffuse room illumination). Owing tocomputational constraints, each fractal was limited to64 3 64 pixels (18 3 18 mm) in size and 64 frames in du-ration. Thus each stimulus subtended 2.58 arc deg em-bedded in a 43.9 (H) by 36.4 (V) deg dark surround andlasted 2.133 s. Each 64-frame stimulus sequence wascreated by first producing a series of random white noiseimages. Gray-scale values are computed digitally withuse of 15-bit floating-point numbers and are displayed

with 8 bits (256-level gray scale). Each image in the se-ries was Fourier transformed, and the amplitudes of allspatial frequencies were equalized to ensure that thenoise was uniformly white. The resulting frequency dis-tributions were then filtered so that the amplitude spec-trum varied over time following the power law relation-ship 1/ft

a and over space (isotrophically over bothCartesian coordinates) following the power law relation-ship 1/fs

b @s 5 (x2 1 y2)1/2#. Ten spatial exponents (b

Fig. 2. Appearance map of spatiotemporal fractal space. The appearance map is based on the responses of four observers. To helprender the chart more readable to a diverse audience, the real-world descriptions provided by the subjects (which were highly individu-alistic and culturally based) have been replaced where possible with stimulus-based descriptions provided by the other subjects.

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5 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, and 2.2) werefactorially combined with seven temporal exponents(a 5 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4) to create the 70dynamic image sequences we employed. The resultingfiltered spectra were inverse Fourier transformed to gray-scale levels for display. For noise stimuli the traditionalmeasures of contrast have proved inadequate. A usefulmetric of noise contrast is root mean square (RMS)contrast,25 which we express here in percentage terms asthe standard deviation of the gray levels in the display di-vided by the mean. The standard deviation and mean ofthe gray-scale distribution in each stimulus were mea-sured. The gray levels in the final image were thenrenormalized to a standard deviation of 28 and a meanlevel of 127, corresponding to a RMS contrast of 22%.

4. ProcedureObservers were asked to provide a written description ofthe phenomenal appearance of each fractal sequence andwere free to use any terms they chose. They were givenan unlimited amount of time to respond. Each fractal se-quence was shown in a continuous loop until the observerindicated readiness to proceed to the next display. The

Fig. 3. Static, random phase, fractal-like textures produced by1/f b spatial-frequency filtering of random white noise. Staticsnapshots of five of the ten fractal exponents (for one seed valueof the random number generator) are shown here. From top tobottom, the values of the exponent b are 0.4, 0.8, 1.2, 1.6, 2.0.Note that as the spatial exponent increases, the apparent coarse-ness of the texture also increases.

fractals were presented one at a time in order, startingwith the lowest spatial and temporal exponents. All tem-poral exponents were shown for a given spatial exponentbefore the spatial exponent was increased.

C. Results and DiscussionThere was considerable regularity across the four observ-ers’ descriptions, so the responses were combined to pro-duce a consensus map of the appearance of the spatiotem-poral texture space (see Fig. 2). Very idiosyncratic termsprovided by some of the subjects were replaced by moregeneral terms consistent with that description. As ex-pected from previous work,5–8,26,27 the primary effect ofincreasing the spatial exponent b was to increase the ap-parent coarseness of the fractal texture (see Fig. 3 for anillustration of the effect of b on static fractal textures).The primary effect of increasing the temporal exponent awas to decrease the apparent speed and jitter of the dy-namic textures. Perceptually, for a exponents of 0.2 and0.4, the motion appeared very jittery; e.g., motion to theright was more often than not followed by motion to theleft. In the theory of fractional (biased) Brownian mo-tions, such behavior is called antipersistent and is differ-entiated from random (Brownian) motions and persistentmotions.28–31 Interestingly, when the spatial and tempo-ral exponents were both within the range of natural im-ages, the subjects tended to provide real-world descrip-tions of the stimuli.

There were strong interactions between the spatial andtemporal dimensions for the appearance of dynamic tex-ture. There are three primary examples of this interac-tion. First, the emergent texture regions were oftenmore apparent, larger, and more diffuse at higher tempo-ral exponents (slower speeds). Second, when the tempo-ral exponent was between 0.6 and 1.0, the texture oftenappeared to rotate, but only for smaller spatial exponents.Finally, when the temporal exponent was between 0.4 and0.8 and the spatial exponent was small, there was a ten-dency to see two transparent textures, with one moving infront of the other.

3. EXPERIMENT 2: DISCRIMINATION OFDYNAMIC TEXTURESA. IntroductionIn addition to being able to anticipate the appearance ofan aperture color, given its CIE coordinates, color re-searchers are also able to gauge how tolerant subjectswould be to small perturbations in the CIE coordinates.Several studies, especially Wright’s and MacAdam’s pio-neering work,32–35 have measured the just noticeable dif-ferences for perturbations of CIE coordinates. Becausethe CIE space is usually depicted as a plane, the locus ofJNDs around a single point is usually an ellipse. A mapof such ‘‘MacAdam ellipses’’ in the CIE space (Fig. 4) is auseful tool, which we wished to characterize for the spa-tiotemporal random-fractal space. So far as we areaware, prior fractal discrimination studies have been lim-ited to spatial discrimination.26,27

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Fig. 4. MacAdam’s ellipses.33 Discrimination thresholds (JNDs) for various directions in CIE 1931 color space. Ellipses are for Mac-Adam’s observer PGN and are shown at ten times their actual size. Reproduced by permission of the Optical Society of America.

B. Methods

1. Participants and ApparatusThese were the same as in Experiment 1.

2. StimuliThe stimuli were identical to those used in Experiment 1,with the sole exception that static fractals were also em-ployed.

3. ProcedureThe smallest detectable increase (Above JND) and de-crease (Below JND) in the spatial exponent b and in thetemporal exponent a were separately measured with anadaptive staircase procedure. For each of the four typesof discrimination, a reference and a comparison imagewere presented side by side (1.1 mm apart) for 2.133 s oruntil the observer responded, whichever came first. Ineach case, the subject was given instructions that wereequivalent to identifying the reference stimulus. The lo-cation of the reference stimulus (left versus right) wasrandomized across trials. Immediately after the ob-server responded (using a computer keyboard), feedbackwas provided on the accuracy of this response. If the ob-server did not respond within the display time, the screenwas blanked and the observer was prompted for an an-swer. If the observer accurately identified the reference

stimulus three times in a row (79% correct criteria), thedifference between the exponents of the comparison andreference images was decreased by 10%. If the observerresponded incorrectly, the difference between the expo-nents was increased by 10%. Each staircase continuedfor eight reversals, and the mean of the last six reversalswas used as a measure of the threshold.

When spatial JND’s were measured, the reference andthe comparison fractals were assigned the same temporalexponent and the spatial exponent of the comparisonstimulus was adaptively varied. The observers wereasked to identify the reference image. That is, they wereto find the more ‘‘fine-grained’’ texture for the Above dis-criminations and the less fine-grained texture for the Be-low discriminations.

When temporal JND’s were measured, the spatial ex-ponent was held constant while the temporal exponentwas adaptively varied. The observers were again askedto identify the reference image. In other words, theywere to find the faster and more jittery texture for theAbove discriminations and the slower and less jittery tex-ture for Below discriminations.

For all subjects, the 80 Above spatial discriminationswere presented first, followed by the 80 Below spatial dis-criminations, then the 70 Above temporal discrimina-tions, and finally the 70 Below temporal discriminations.

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Within each type of discrimination, the order of presenta-tion of the fractal exponents was randomized, with eachobserver receiving a different random order. We alsorandomized the ‘‘seed’’ values of the random number gen-erator used to generate the white noise before filtering.The same seed values were used within each staircase butnot for subsequent staircases. Thus the data in Figs.5–9, which are collapsed across subjects or conditions,represent discrimination for an ensemble of randomphase spectra rather than any particular pattern of ran-dom phases. Each subject required roughly 50 h to com-plete the measurements.

C. Results and AnalysisFigure 5 plots just-noticeable-difference contours (analo-gous to MacAdam ellipses) for the dynamic fractals. It isinteresting to note that the ellipse size is fairly constantover much of the space. In color, much attention hasbeen devoted to obtaining colorimetric spaces in whichdiscrimination thresholds are relatively constant: the socalled Uniform Color Spaces (e.g., CIE 1976 UniformColor Space).1,2 The spatial and temporal discrimina-tions will be discussed separately in more detail below.

1. Spatial Just-Noticeable DifferencesOverall, spatial discriminations were easiest when b wasbetween 1.4 and 1.8. This is consistent with previous re-search on static fractal discrimination.26,27 Spatial

JND’s are plotted in Fig. 6 as a function of spatial expo-nent. The data for both Above and Below JND’s are re-markably similar across subjects.

2. Effect of Temporal Characteristics on SpatialDiscriminationsIf the spatial and temporal dimensions are independentin dynamic fractals, as is often assumed, similar discrimi-nation thresholds should be found for dynamic fractals re-gardless of temporal exponent. Figure 7 helps to illus-trate the point that the spatial JND’s did not vary muchas a function of the temporal nature of the stimuli. Al-though there is a slight suggestion of lower spatial JNDsfor the lowest temporal exponents, it seems that spatialdiscriminations are largely independent of the temporalcharacteristics in the stimuli.

3. Temporal Just-Noticeable DifferencesFigure 8 plots the temporal JND’s as a function of tempo-ral exponent. For each subject, temporal discriminationswere easiest when a was between 0.8 and 1.0, a typicalrange of exponents for natural stimuli.15–18 The patternof discriminations was remarkably consistent across sub-jects. Of particular interest is that the antipersistentstimuli were extremely difficult to discriminate. More-over, for three of the four subjects, the Below JND’s forthe antipersistent stimuli tended to be elevated forsmaller temporal exponents. Notice that for these three

Fig. 5. Wright–MacAdam-like contours for discrimination thresholds in four directions in a spatiotemporal fractal space. Average offour observers. The contours are simple quarter-circles fitted to the JNDs that span their quadrants. These contours are not theo-retically motivated and are used to reduce the confusion that occurs for overlapping JND crosses. Note that over a wide range of thespace the discrimination thresholds vary only slightly; e.g., the space approaches the ideal of a uniform colorimetric space (such as theCIE 1976 uniform chromaticity space).1,2

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observers, the JND at a 5 0.2 was greater than 0.2; i.e.,the exponent a 2 Da becomes negative, and the slope ofthe 1/f a2Da spectra for the comparison image is slightly

positive. This can occur only for the Below discrimina-tions and is the most likely explanation of the asymmetrybetween Above and Below thresholds.

Fig. 6. Spatial JND’s collapsed across temporal exponents. Here the spatial discriminations are plotted as a function of spatial expo-nent. Each subfigure is for an observer named by his initials.

Fig. 7. Spatial JND’s collapsed across spatial exponents. Here the spatial discriminations are plotted as a function of temporal expo-nent. The temporal exponent labeled ‘‘0’’ is a place holder for the static condition. Observers are identified by initials.

Billock et al. Vol. 18, No. 10 /October 2001 /J. Opt. Soc. Am. A 2411

Fig. 8. Temporal JND’s collapsed across spatial exponents. Here the temporal discriminations are plotted as a function of temporalexponent. Each subfigure is for an observer named by his initials.

Fig. 9. Temporal JND’s collapsed across temporal exponents. Here the temporal discriminations are plotted as a function of spatialexponent. Each subfigure is for an observer named by his initials.

4. Effect of Spatial Characteristics on TemporalDiscriminationsFigure 9 plots the temporal JND’s as a function of spatialexponent. For all four observers, thresholds increased as

the spatial exponent increased. In other words, discrimi-nating the temporal nature of the stimuli became moredifficult as the texture became coarser. This makes in-tuitive sense, as it would be easier to see a 1-mm object

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move 2 mm (200% of its size) than it would be to see a100-m object move the same distance (0.002% of its size).

4. GENERAL DISCUSSIONThe fractal space described here was chosen not only forits multiscale elegance but for its potential applicabilityto real-world images and image sequences. Fractals—mathematical entities that are self-similar over manyspatial or temporal scales—are ideal tools for represent-ing natural phenomena that take place on many scalesand for studying vision, which contains multiple channelstuned on the spatial and temporal dimensions. Randomfractal textures (sometimes called fractional Browniantextures) have been widely studied because their 1/f b

spectra simulate the spectra of natural images withoutbeing confounded by phase information. If b is in therange of a typical natural object (b 5 0.9–1.2),11 subjectswill often spontaneously report its appearance12: a psy-chophysical version of the venerable children’s game ofnaming the shapes in clouds. If phase is manipulated,fractal textures can take on the appearance of naturalscenes: Fractal forgeries of landscapes, coastlines andclouds are extremely compelling.13,14 In the temporal di-mension, sequences of events with 1/f a spectra are ubiq-uitous and fractal forgeries are especially striking; 1/facoustic sequences sound musical.36 In vision, 1/f a am-plitude spectra are found for dynamic images, where a isindicative of the character of the movement.18,28–31 Theuse of a common mathematical framework for character-izing both dynamic noise and dynamic images may alsofacilitate the study of masking of images by noise.

ACKNOWLEDGMENTSWe thank Jer Sen Chen, Steven Fullenkamp, and EricHeft for technical support.

Douglas Cunningham’s current address is Max PlanckInstitute for Biological Cybernetics, Spemannstrasse 38,Tubingen 72076, Germany.

REFERENCES AND NOTES1. D. B. Judd and G. Wyszecki, Color in Business, Science and

Industry, 2nd ed. (Wiley, New York, 1963).2. G. Wyzecki and W. S. Stiles, Color Science: Concepts and

Methods, Quantitative Data and Formulae, 2nd ed. (Wiley,New York, 1982).

3. W. Richards, ‘‘Quantifying sensory channels: generalizingcolorimetry to spatiotemporal texture, touch and tones,’’Sens. Processes 3, 207–229 (1979). In general, Richardsfound that arbitrary textures could be matched by mixturesof four separated texture primaries and that temporalmodulations could be matched by mixtures of three sepa-rated flicker primaries.

4. The analogy to colorimetry cannot be pushed too far here.Brill, in an interesting gedanken experiment, showed thatunder some circumstances Grassman’s additivity law is vio-lated for some textures: M. H. Brill, ‘‘Formalizing Grass-man’s laws in a generalized colorimetry,’’ Sens. Processes 3,370–372 (1979).

5. J. E. Cutting and J. J. Garvin, ‘‘Fractal curves and complex-ity,’’ Percept. Psychophys. 42, 365–370 (1987).

6. T. Kumar, P. Zhou, and D. A. Glaser, ‘‘Comparison of humanperformance with algorithms for estimating fractal dimen-

sion of fractional Brownian statistics,’’ J. Opt. Soc. Am. A10, 1136–1146 (1993).

7. A. Pentland, ‘‘Fractal-based description of surfaces,’’ inNatural Computation, W. Richards, ed. (MIT Press, Cam-bridge, Mass., 1988), pp. 279–299.

8. A. P. Pentland, ‘‘Fractal-based description of naturalscenes,’’ IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6,661–674 (1984).

9. D. J. Field, ‘‘Relations between the statistics of natural im-ages and the responses of cortical cells,’’ J. Opt. Soc. Am. A4, 2379–2394 (1987).

10. D. J. Tolhurst, Y. Tadmor, and T. Chou, ‘‘The amplitudespectrum of natural images,’’ Ophthalmic Physiol. Opt. 12,229–232 (1992).

11. V. A. Billock, ‘‘Neural acclimation to 1/f spatial frequencyspectra in natural images transduced by the human visualsystem,’’ Physica D 137, 379–391 (2000).

12. B. E. Rogowitz and R. F. Voss, ‘‘Shape perception and low-dimensional fractal boundaries,’’ Human Vision and Elec-tronic Imaging: Models, Methods, and Applications, J. P.Allebach and B. E. Rogowitz, eds., Proc. SPIE 1249, 387–394 (1990).

13. R. F. Voss, ‘‘Random fractal forgeries,’’ in Fundamental Al-gorithms for Computer Graphics, R. A. Earnshaw, ed.(Springer, Berlin, 1985), pp. 805–835.

14. B. B. Mandelbrot, The Fractal Geometry of Nature (Free-man, New York, 1983).

15. M. P. Eckert, G. Buchsbaum, and A. B. Watson, ‘‘Separabil-ity of spatiotemporal spectra of image sequences,’’ IEEETrans. Pattern Anal. Mach. Intell. 14, 1210–1213 (1992).

16. D. W. Dong and J. J. Atick, ‘‘Statistics of time varying im-ages,’’ Network Comput. Neural Syst. 6, 345–358 (1995).

17. J. H. van Hateren, ‘‘Processing of natural time series by theblowfly visual system,’’ Vision Res. 37, 3407–3416 (1997).

18. V. A. Billock, G. C. De Guzman, and J. A. S. Kelso, ‘‘Fractaltime and 1/f spectra in dynamic images and human vision,’’Physica D 148, 136–146 (2001).

19. M. Savilli, G. Lecoy, and J. P. Nougier, Noise in PhysicalSystems and 1/f Noise (Elsevier, New York, 1983).

20. M. S. Keshner, ‘‘1/f noise,’’ Proc. IEEE 70, 212–218 (1982).21. Technically, fractals have infinite spectral bandwidths.

Purists would designate textures physically obtainable ondisplays as prefractals or pseudofractals.

22. K. L. Kelly, ‘‘Color designations for lights,’’ J. Opt. Soc. Am.33, 627–632 (1943).

23. P. Keller, ‘‘1976-UCS chromaticity diagram with colorboundaries,’’ Proc. Soc. Inf. Disp. 24, 317–321 (1983).

24. Color appearance maps of Kelly’s CIE 1931 and Keller’s1976 spaces are available from Photo Research, 3000 N.Hollywood Way, Burbank, California 91505.

25. B. Moulden, F. Kingdom, and L. F. Gatley, ‘‘The standarddeviation of luminance as a metric for contrast in random-dot images,’’ Perception 19, 79–101 (1990).

26. D. C. Knill, D. Field, and D. Kersten, ‘‘Human discrimina-tion of fractal textures,’’ J. Opt. Soc. Am. A 7, 1113–1123(1990).

27. Y. Tadmor and D. J. Tolhurst, ‘‘Discrimination of changes inthe second-order statistics of natural and synthetic images.’’Vision Res. 34, 541–554 (1994).

28. In the theory of fractional (biased) Brownian motion, themotion bias is quantified by the distance that a biasedwalker moves from the origin in unit time. A Brownian(random) walker’s distance over unbroken ground is pro-portional to At and has an amplitude spectrum exponent of1.0. Greater distance than this is covered if the motion ispersistently biased (if bias is perfect, distance is propor-tional to time), less distance is covered if the bias is anti-persistent. B. B. Mandelbrot and J. W. van Ness, ‘‘Frac-tional Brownian motions, fractional noises andapplications,’’ SIAM Rev. 10, 422–437 (1968).

29. Over broken ground, a random walker covers less distancefrom the origin. The case closest to our stimuli that hasbeen studied in statistical physics is Brownian motion on afractal fractured surface. The temporal frequency spec-trum exponent a for this case is 0.5. W. Lehr, J. Machta,

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and M. Nelkin, ‘‘Current noise and long time tails in biaseddisordered random walks,’’ J. Stat. Phys. 36, 15–29 (1984).This is in accord with our finding that stimuli with expo-nents below 0.5 appear to move in a jittery fashion, whereasexponents above 0.5 move smoothly.

30. In an interesting experiment Snippe and Koenderink stud-ied human ability to perceive correlations between mem-bers of a row of light sources. Positive correlations be-tween two lights were detected as apparent motion, but thecorresponding anticorrelation between the same two lightswas undetectable as motion. H. P. Snipe and J. J. Koen-derink, ‘‘Detection of noise-like luminance functions,’’ Per-cept. Psychophys. 55, 28–41 (1994).

31. One oddity of random motion is that the visual system doesnot seem to be able to compensate for motion blur forstimuli undergoing Brownian motion. S. N. J. Watama-

niuk, ‘‘Visual persistence is reduced by fixed-trajectory mo-tion but not random motion,’’ Perception 21, 791–802(1992). No one has yet studied deblurring for anticorre-lated (antipersistent) motion.

32. W. D. Wright, ‘‘The graphical representation of small colordifferences,’’ J. Opt. Soc. Am. 33, 632–636 (1943).

33. D. L. MacAdam, ‘‘Visual sensitivity to color differences indaylight,’’ J. Opt. Soc. Am. 32, 247–274 (1942).

34. L. Silberstein and D. L. MacAdam, ‘‘The distribution ofcolor matchings around a color center,’’ J. Opt. Soc. Am. 35,32–39 (1945).

35. W. R. J. Brown and D. L. MacAdam, ‘‘Visual sensitivities tocombined chromaticity and luminance differences,’’ J. Opt.Soc. Am. 39, 808–834 (1949).

36. R. F. Voss and J. Clarke, ‘‘1/f noise in music: music from1/f noise,’’ J. Acoust. Soc. Am. 63, 258–263 (1978).