fractal modeling of natural terrain - analysis and surface reconstruction with range data

7/28/2019 Fractal Modeling of Natural Terrain - Analysis and Surface Reconstruction With Range Data

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GRAPHICAL MODELS AND I MAGE PROCESSING

Vol. 58, No. 5, September, pp. 413-436. 1996ARTICLE NO. 0035

r

Fractal Modeling ofNatural Terrain: Analysis and Surface

Reconstruction with Range DataKENICHI ARAKAWA'AND ERI C KROTKOV

School ofComputer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania 1.5213Received J uly 10, 1995;revised J une3, 1996;accepted J une 17. 1996

In this paper we address two issues in modeling naturalterrain using fractal geometry: estimation of fractal dimension,and fractal surface reconstruction. For estimation of fractaldimension, weextend the fractal Brownian function approachto accommodate irregularly sampled data, and we developmethods for segmenting sets of points exhibiting self-similarity

over only certain scales.For fractal surface reconstruction, weextend Szeliski's regularization with fractal priors method touse a temperature parameter that depends on fractal dimen-sion. We demonstrate both estimation and reconstruction withnoisy range imagery of natural terrain. D 1% Academicpress,Inc.

1. INTRODUCTION

Many problems in the analysisofnatural surface shapesand the constructionofterrain maps to model them remainunsolved. One reason is that the familiar Euclidean geome-try of regular shapes, such as surfaces of revolution, doesnot capture well the irregular and less structured shapes

found in nature, such as a boulder field or surf washingonto a beach.Mandelbrot [20-221 proposed fractals as a family of

mathematical functions to describe natural phenomenasuch as coastlines, mountains, branching patterns oftreesand rivers, clouds, and earthquakes. Since Mandelbrot in-troduced them, fractal sets and functions have been foundto describe many other environmental properties [7], andhave received a great deal of attention from scientists,

This paper addresses two issues in fractal modelingI artists, and others.

(Fig. 1): .5 1. Estimating the fractal dimension ofnatural surfaces

given range data. Such estimates are descriptors ofnaturalterrain that express its roughness. These estimates are valu-able for such purposes as mobile robot path planningaround rough terrain, compression ofterrain maps, andgeological analysis ofterrain morphology.

' Current address: N?T Human Interface Labs, 3-9-11 M idori-cho,Musashino-shi,Tokyo 180J apan. E-mail: [email protected].

2. Reconstructing natural surfaces, given sparse rangdata and the fractal dimension. Surface reconstruction provides knowledge of surface geometry that is valuable fosuch purposes as mobile robot obstacle avoidance, decompression ofterrain data, and realistic terrain visualization

The first issue has been explored by a number ofauthors

who have given algorithms for estimating fractal dimension. In this paper, we extend that work to the case onatural terrain patterns acquired by a laser rangefindeThis extension requires handling fairly noisy depth datand handling points that are not spaced regularly ovethe terrain.

The need tohandle irregularly spaced data derives fromlaser rangefinders and camera-based computer vision sytems, which typically acquire depth data in a sensor-centered spherical coordinate system. As one would expecregularly spaced samples in the spherical system map ontirregularly spaced samples in a Cartesian system. Figure illustrates how, in a Cartesian system, a sensor (eithe

camera or rangefinder) acquires denser range data fromcloser objects, and sparser range data from farther objectsdespite a regular sampling in the image.

The second issue has not received much attention. Ithis paper, we define the natural surface reconstructioproblemofconstructing dense elevation maps ofnaturasurfaces, given sparse and irregularly spaced depth data

This problem differs from the traditional surface reconstruction problem in requiring that the reconstructed surface realistically reflect the rough, original surface. In contrast to approaches to surface reconstruction that impossmoothness constraints, our approach to natural surfacreconstruction imposes roughness constraints.A problem related to the second issue isto compute th

uncertainty on the reconstructed surface. We have developed and demonstrated a Monte Carlo algorithm for computing this uncertainty [2], but will not present it in thipaper since it does not directly concern fractal modeling

This paper is organized as follows. In Section 2, wpresent an extension toexisting fractal dimension estimation algorithms, enabling them to cope with irregularly

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Copyright 0 1YY6 by Academic Press, IncAll rights ofreproduction in any formresewed


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414 ARAK AWA AND KROTKOV

Range FractalDtmenszon r - - - - - - - 7ISurface

Images 7 - - - - -fractal I- I,~imensi on lReconstruction7I - - _ - -Estimation: !usingFractal I

I I ElevationMaps

I L - - _ - - - - d

FIG. 1. Approach to modeling natural terrain based on fractal ge-ometry.

spaced data. In Section3,we present a surface reconstruc-tion technique that computes elevation maps at arbitraryresolution, yet preserves the roughnessofthe original pat-tern. In the final section, we summarize the findings andidentify directions for future research.

2. ESTIMATING FRACTAL DIMENSION

In this section, we first review related research. Next,we define formally some terms related to self-similarityand self-affinity. With these definitions in hand, we thendescribe the fractal Brownian function approach, and pres-

ent results ofexperiments with synthetic patterns. Finally,we extend the approach to handle irregularly spaced data,and present results for range images.

2.1. Related Research

Several classes of techniques for estimating fractal di-mension have been reported in the literature: box-count-ing,&-blanket,spectral analysis, and fractal Brownian func-tion approaches.

The box-counting approach counts the number of boxesof various sizes which cover a fractal pattern [36]. Oneshortcoming of this approach was identified (and circum-vented) by Keller et al. [15], who report that the fractaldimension estimated by such methods saturates as it ap-proaches 3.0.

The&-blanketmethod was proposed by Peleg et al. [25]as a variation on the box-counting approach, and theyapplied the method to classification of natural textures.

The method has been explored by Dubuc [lo], and Dubucet al. [ l l ] for estimating the fractal dimension of bothcurves and surfaces.

As suggested by Mandelbrot [21], if the target patternis assumed to be generated under the fractional Brownianmotion model, spectral analysis methods [16, 261 can beapplied to estimate the fractal dimension. These methodsestimate the fractal dimension by linear regression on thelog ofthe observed power spectrum as a function of thefrequency. Wornell [37] proposed a representation of 1fprocesses using orthogonal wavelet bases, and applied it tovarious one-dimensional signal processing tasks, includingestimating fractal dimension with the maximum likeli-hood approach.

Other approaches that do not fit comfortably in theabove categories have been developed. For example,Mar-agos and Sun [23]use morphological operations with vary-ing structuring elements to evaluate the fractal dimension,and develop an iterative optimization method that con-verges to the true fractal dimension.

FIG. 2. Typical sampling pattern for rangefinder or camera. This figure shows the elevations in a Cartesian system when a scanning rangefinderobserves a horizontal plane. The sensor acquires denser range data fromcloser objects, and sparser range data from farther objects.


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Traces of fractional Brownian motion are one class offractal patterns.

Pentland [26] defined afractal Brownian functionas anextension ofstatistical self-affinity that characterizes self-affine processes, including fractional Brownian motion. Arandom function f ( t ) is a fractal Brownian function if, forall t and At. there exists H in

which is independent of At, where g(x) is a cumula-tive distribution function. In this definition, AfAr =f ( t +At) - ( t ) is statistically self-affine, and H is a self-affinity parameter, related to the fractal dimension D off ( t ) by D =2 - H . Ifg(x) is a zero-mean Gaussian withunit variance, thenf( t)represents fractional Brownian mo-tion BH(t).If, in addition, H =1/2, then f(t) representsclassical Brownian motion B(t).

Interpreting t as a vector quantity t extends this defini-tion to higher topological dimensions. In this case, the A?appearing in the denominator of(2) must be rewritten asthe norm IlAtll. Iff(t) is a pattern in D-dimensional Euclid-ean space, then D =E +1- H [36]. For instance, if weanalyze fractal dimension of natural terrain, we can expressthe terrain as an elevation mapf(t) ona horizontal planet =( x, y) and the fractal dimension can be estimated by

Pentland [26]proves that under certain conditions (con-stant illumination, constant albedo, and a Lambertian sur-face reflectance function), a three-dimensional surface witha spatially isotropic fractal Brownian shape produces animage (i) whose intensity surface is fractal Brownian, and(ii) whose fractal dimension is identical to that of the com-ponents of the surface normal. He also shows that thedefinition of a fractal Brownian functiononintensity I(t)-instead off ( t ) in (2)-can be rewritten as

D = 3 - H .

where E(AZliAtll)is the expected value of the change inintensity Z(t) over distance IlAtll. Note that AIllAtl1is al-ways positive.To evaluate the suitability of this fractal model for im-

agesofnatural surfaces, he observed the empirical distribu-tions of intensity differences AZllAtl1for different distancesIIAtll. He observed the distributions to be approximatelyGaussian. Moreover, he computed the standard deviationS(AZllAtll)of each distribution, and found the points (log((At(/,log S(AIllAtll))to lie on a line. From this line in log-log space, he estimated the slope H, which is

(4)

Given H , the fractal dimension of the two-dimensionalpattern isD =3 - H.

Y okoya [38] also assumed that intensity in imagesis distributed by a fractal Brownian function, and thatg(x) - N(0, v). He developed a method for estimatingfractal dimension similar to Pentlands. Instead of the stan-dard deviation in (4), he used the expected valueE(AZl1311):

Both methods are reasonably robust against noisy data,because they use statistics computed from a large numberof data points. Yokoyas method, in particular, tolerateszero-mean normally distributed sensor noise, because themethod implicitly performs an averaging operation.

The computational complexity of both methods isO(iV~Atsearc,,~~)for regularly sampled data, where N is thenumber of data points, and IIAtsearchl(is the maximum searchsize. Because Pentlands method computes the standarddeviation, it requires slightly more computation than Yo-koyas method, which computes the first moment.

2.4. Estimation from Irregularly SampledElevation Data

In previous sections, we have discussed fractal dimensionestimation from image intensity Z(t); here, we considerelevation z(d) (with d =( x, y)) instead of[(t).The proce-dure for estimating fractal dimension from a set of irregu-larly sampled elevations z(d) is stated in the followingthree steps.

1. Compute statistics ofIz,,-

tx+li x.y+ri J .In the sensor frame, consider two points on thexy plane:( x,y) and ( x +dx, y +dy).The Euclidean distance be-

tween them is Ad =ddx +dy. We are interested instatistical variations in the absolute valueofthe differencein elevation between these two points: AzAd= Iz,,? -~ ~ + ~ , ~ + ~ , , l .Pentlands method requires the standard devia-tion ofthe distribution ofelevation differences; Yokoyasmethod requires the expected value ofthe distribution ofelevation differences.

Because the data points are distributed irregularly onthe xy plane in the sensor frame, we must extend the

original methods, which assume the data is distributed reg-ularly, or equivalently, that the sampling interval is con-stant. For i =0, 1,. . .,m, andAdk


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FRACTAL MODELING

sarea

FIG. 3. Accommodating irregular sampling intervals. Because ofir-regular sampling, we cannot detect a pair ofpoints located a certaindistanceAd, away on thexy plane. As a permissible area, we set a circlewhose width is 2.saround a point ( x . y)with z. I fthe point ( x +dx , y +dy) exists in the area, its elevation z is used to computeA, and B,.

permissible area including a circle of radius Adj (Fig. 3).Suppose there is a data point at ( x + d x , y +dy) withelevation z. If [Adj- Ad( is less than E , then the pointlies in the permissible area, and we update the countersA,, B j ,and C, asAj +Aj +) Z - 1 , Bj +Bj + ( Z - z)*, Cj+C,+1.After considering all pairs of data points, we ensure thatC,is larger than a threshold number ofpairs. IfCi is small,then we question whether the number ofsamples was suf-ficient to compute reliable statistics, and discard this data.Otherwise, we compute the sample standard deviation forPentlands method by

and the sample mean for Yokoyas method by

2. Plot the points in log-log space and identify linearsegments.

For Pentlands method, the point coordinates are (logAdi,logShd,).For Y okoyas method, the point coordinates are

Because most natural patterns exhibit self-similarity onlyover certain scales, and not over all scales, it is necessaryto segment sets ofpoints that are linear. We investigatedtwo approaches to this segmentation problem.

The first approach is polyline fitting using the minimaxmethod, as proposed by Kurozumi [17]. In the field ofdocument image processing, this technique is frequentlyused to detect line segments. The technique segments thegiven points into several sets ofpoints which distribute

(1% Ad,, og EAd,).

3F NATURAL TERRAIN 41within narrow rectangles, i.e., nearly along lines. The widtof the rectangle must be specified asathreshold. The cardnality of the sets is a natural criterion for determininwhich should be used to estimate the fractal dimensionHowever, the cardinality is fairly sensitive to the rectanglwidth, thus making it difficult to select the proper threshol

for this Segmentation technique.The second approach employs iterative least-square linefitting [l].Using this technique, we can construct a set opoints that lie within a specified distance of the line. Thitechnique is not sensitivetochanges of the threshold. However, the technique selects only the first linear part satisfying the criterion. In the case where several fractal patterns exist, each with a different fractal dimension, multipllinear segments will appear in the log-log plot. Simplselecting the first will preclude consideration of the othersIn these experiments, we use one or the other method;future topic of research is to combine them to identify aofthe fractal patterns.

3. Estimate fractal dimension from the slope of lineasegments.

When the points lie on a line in the log-log space, wcan estimate the fractal dimension ofthe pattern by thedifference between the Euclidean dimension of the patternand the slope ofthe line formed by the points.

2.5. Experiments with Synthetic Patterns

We implemented the modified method based on thefractal Brownian function approach, and applied it tosparse synthetic elevation data. We created the data bydiscarding95%ofthe data points, selected randomly from

synthetic fractal images (similar to the tops in Figs. 10-

1generated by the Successive Random Addition (SRAmethod [28]).Figure4plots actual fractal dimension versuthe fractal dimension estimated using the modified Yokoyamethod. The estimated values increases monotonicallywith the actual fractal dimension, andnosaturation occurs

The method tends to overestimate for D 2.55.The results are similar, although not identical, to those from dense synthetic images

To determine the robustness of the methods againstGaussian noise, we repeated these trials on subsampledsynthetic elevations with additive noise distributed N(0a&),

added independently to x , y and z(d).The signal-to-noise ratio (SNR) is computed as20log,, osloN[dB],wherethe (+ is the maximum among the variances ofx, y andz(d)signals. Figure 4 illustrates the results. Table 1tabu-lates the statisticsofthe fractal dimensions estimated from100setsofelevations with various Gaussian random gener-ators. Figure 4 and Table 1indicate that for SNR of 25dB or greater, the method computes reasonable estimates

(see 111).


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estimatedFD2.95

2.90

2.85

2.80

21 5

2.70

2.65

2.60

2.55

2.502.45

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2.35230

2252.20

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ARAKAWA AND KROTKOV

2.10 2.20 2.30 2.40 250 260 270 2.80FIG. 4. Errors with Y okoyas method with irregularly sampled synthesized data. The graph on the left-hand side plots the real fractal dimension

against the dimension estimated by the modified Yokoyas method. The solid line illustrates the ideal relation. The graph on the right-hand sideillustrates the effect of Gaussian noise. The dotted and dashed lines represent estimates from data without noise and from data with noise withSNR =30, 25, 20, 15,and 10dB.

ofthe fractal dimension, because significant degradationcannot be seen compared to the estimates from data with-out noise.

2.6. Experiments with Range Images

In this section, we apply our method to range imagesacquired with a scanning laser rangefinder manufactured

by Perceptron [181.The sensor acquires range images withrespect to a spherical

-

polar coordinate system. Equal sam-pling intervals in this coordinate system become unequaland irregular when mapped into a Cartesian system. Thus,the data points are not equally spaced when expressed inCartesian coordinates.

We selected eight patterns from range images acquired

TABLE 1Estimation ErrorsofFractal Dimensions from Irregularly Sampled Elevations with Different Gaussian Noise

SNR =30 dB SNR =25 dB SNR =20 dBTrue No noise

FD error[%] Mean[%] Std. dev. Worst[%] Mean[%] Std. Dev. Worst[%] Mean[%] Std. Dev. Worst[%]

2.1 8.00

2.2 5.182.3 3.172.4 1.542.5 0.282.6 0.652.7 1.482.8 2.29

8.15 1.30x 10-3 8.315.38 1.26x 10-3 5.513.20 1.25x 10-3 3.331.52 1.25x 10-3 1.660.27 1.23x 10-3 0.410.68 1.23x 10-3 0.791.47 1.19x 10-3 1.562.28 1.17 x 10-3 2.36

9.16 2.05 X 9.355.96 1.94X 6.163.50 1.84 x 10-3 3.681.67 1.71x 10-3 1.830.35 1.58 x 10-3 0.490.63 1.50 x 10-3 0.781.42 1.41 X IO- 1.562.24 1.34 x 10-3 2.35

13.59 3.70 X lo- 14.079.02 3.26 x IO- 9.385.42 2.90 x lo- 5.672.75 2.55 x IO- 2.950.88 2.20 x lo- 1.060.38 1.89x 10- 0.601.31 1.65x I O -% 1.522.17 1.52 X 10 2.37


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FRACTAL MODEL lNG OF NATURAL TERRAIN 41with the rangefinder. Figures 5-7 illustrate the images,which are presented in order of increasing roughness, asdetermined subjectively by the authors. We estimated thefractal dimension of the regions indicated by white rect-angles.

Before applying the procedures to estimate fractal di-mension, we checked whether the data acquired by the

rangefinder satisfies the conditions on fractal Brownianfunctions stated in (2). Figure8histograms z, ,~- x t d x , y t d yfor Pattern 2, with Ad =0.4, 0.6, and 0.8 m. From thefigure it is clear that the condition in (2), that g ( x ) be acumulative distribution function, is satisfied. We concludethat the patterns are fractal Brownian functions.

A further condition on the distributions, imposed byYokoyas method, is that they are normal. We conductedx2tests for Gaussianity and observed negative results, i.e.,that the probabilitiesofthe data being normally distributedwere quite low. This suggests that it is not probable thatthe points were created by a fractional Brownian motionprocess (which is a special caseofa fractal Brownian func-

tion). However, to the extent that the distributions aresymmetric, and exhibit a central tendency, there is some

justification in proceeding to apply Yokoyas method, de-spite the negative2 test results.

Figure9 shows the result ofapplying Yokoyas methodto the range image regions: the points (log Ad, log EAd)segmented by iterative least-square fitting. The points dis-tribute linearly, therefore, we observe self-similarity in allof these natural terrain patterns. On some patterns, plottedpoints distribute in several setsoflinear parts. If we intendto estimate all fractal dimension in such distribution, seg-mentation by polyline fitting is appropriate. However, wemust set a parameter ofallowable error for fitting, because

the segmentation results are highly sensitive to the pa-rameter.Table 2 lists the fractal dimensions estimated by Yo-

koyas method, with both segmentation techniques. Someofthe results for segmentation by polyline fitting requiredcareful selection of the amountofallowable error. We alsoillustrate in parentheses the fitting error normalized by logE(AzAd).All errors are small enough to determine thatthe patterns are fractal. Moreover, the rows are orderedby roughness, as perceived by the authors. The order ofestimated fractal dimension correlate strongly to the intu-itive order (the last three patterns-sandy flat floors-arealmost identical). These results suggest that the fractaldimension estimated can be utilized as a measure ofroughness ofnatural terrain.

The computational complexity of the method utilizedhere is O(W),where N is the number ofpixels, becauseit is necessary to calculate distances between all pairs ofpixels in order to determine which pairs lie in the permissi-ble area. On a Sun4/40 with 24 MB ofphysical memory,estimating the fractal dimension for Pattern 2 (10,000pix-

els) requires 1.2 X 10 s, and Pattern 5 (19,600pixelsrequires 5.5 x lo3s.

3. FRACTAL RECONSTRUCTION OFNATURAL SURFACES

The surface reconstruction problem can be formulateas follows. Given a scattered set of surface elevation measurements, produceacomplete surface representation saisfying three conditions:

It must take the form of a dense array of inferred

It must pass approximately through the original dat

It must be smooth where new points are inferred.

measurements with regular spacing.

points.

The surface reconstruction problem may be called fitting problem by computer graphics researchers, and anapproximation problem by others. It is closely related tothe surface interpolation problem, for which the secondcondition requires the surface to pass exactly through theoriginal data points.

The smoothness constraint in the third condition is inappropriate for natural surfaces, which as a rule exhibiroughness over a wide range ofscales [7].Thus, for thenatural surface reconstruction problem, the third conditionabove becomes the following:

It must be realistically rough where new points are in-ferred.

This revised condition imposes a requirement that theroughness of the original pattern be known. In turn, thisimposes a requirement that the surface reconstructiontechniquesadapt in a nonuniform manner tothe roughnessofthe original pattern.

In this section, we use the estimated fractal dimensionto control the roughness of the reconstructed surfaces.Specifically, we estimate the fractal dimension at a coarsescale (given by the spacing of the sensed range data) anduse it at a finer scale (between range data samples). Thisapproach relies on the property that as scale changes, thefractal dimension does not.

3.1. Related Research

The surface reconstruction problem has been formulated

as an optimization problem, and solutions have been ob-tained through relaxation methods. For example, Grimson[13] suggested that given a set of scattered depth con-straints, the surface that best fits the constraints passesthrough the known points exactly and minimizes the qua-dratic variation of the surface. He employed a gradientdescent method to find such a surface. Extending this ap-proach to use multiresolution computation, Terzopoulos


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420 ARAK AWA A ND KROTKOV

Pattern1

Pattern 2

Pattern 3

FIG. 5. Rangefinder images (1).The figure illustrates image pairs: processed range (left) and raw reflectance (right). Only the range images areused to estimate fractal dimension.


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FRACTAL MODELING OF NATURAL TERRAIN 42

-Pattern 5

--

Pattern 6

FIG. 6. Rangefinder images(2).


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Pattern 7

Pattern8

FIG. 7. Rangefinder images (3).

[33] proposed a method minimizing the discrete potentialenergy functional associated with the surface. In this for-mulation, known depth and/or orientation constraints con-tribute as spring potential energy terms. Poggie et al. [27]reformulated these approaches in the context ofregular-ization.

Discontinuities in the visible surface have been a centralconcern in the approaches taken by Marroquin with Mar-kov random fields [24],by Blake and Zisserman with weakcontinuity constraints [4],and by Terzopoulos with contin-uation methods [35].

Burt [8] developed a method that relies onlocally fittingpolynomial surfaces to the data. The method achieves com-putational efficiency through computation by parts, wherethe value computed at a given position is based on pre-viously computed values at nearby positions.

Boult [5] developed surface reconstruction methodsbased on minimization with semireproducing kernelsplines, and with quotient reproducing kernel splines. He

compared the time and space complexity of these and othermethods for a number of different cases.

Stevenson and Delp [29] presented a two-stage algo-rithm for reconstructing a surface from sparse constraints.The first stage forms a piecewise planar approximation tothe surface, and the second stage performs regularizationusing a stabilizer based on invariant surface characteristics.By virtue of the selection ofstabilizer, the algorithm isapproximately invariant to rigid 3D motion of the surface.

The natural surface reconstruction problem has receivedless attention than the surface reconstruction problem. Inthe fieldofapproximation, Barnsley [3] introduced iteratedfunction systems with attractors that are graphs of a contin-uous function f that interpolate a given data set {(xi,y;)}so that f ( x i ) =y; . It appears that these functions are wellsuited for approximating fractal functions. Barnsley con-centratedonexistence proofs and moment theory for thesefunctions, and there does not appear to be a firm connec-tion to the issues at hand.


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Z(d+D)-Z(d)Histogram ofPattern 2 with Ad =0 4 m

z(d+D)-z(d)Histogram ofPattern 2 with Ad =0.6m

Z(d+O)-Z(d)Histogram ofPattern 2 with Ad=0.8m

FIG.8. Empirical distribution functions. These graphs illustrate histo-grams ofz ~ . ~- zr4,,,,.,,,with Ad =d d x 2+dy2on Pattern 2.

Y okoya et al. [39]present a technique for interpolatingshapes described by a fractional Brownian function. The

technique follows a random midpoint displacement ap-proach [28]. At each level of recursion, the midpoint isdetermined as a Gaussian random variable whose ex-pected value is the mean of its four nearest neighbors.Next, the technique displaces this midpoint by an amountthat depends on the fractal dimension and the standarddeviation ofthe fractional Brownian function. Thus, theirtechnique is both stochastic and adaptive. However, thereare key limitations:

1.The technique requires an equal spacing betweensamples of the original pattern.

2. The technique cannot generate stationary randomfractals. This is a result ofa compromise between compu-tational expense and generality.

3.The technique, like a midpoint displacementmethod, cannot generate stationary random fractal incre-ments [21].

I Some modified techniques 128, 30) have been proposed to solve thenonstationary increment problem.

I I I3.0 -2.5 -2.0 -1.5 -1.0 0. 5 0.0 0.5 1.0Result on Pattern 2 (slope=0.455) IOQ d

log dResult on Pattern 4 (slopr=0.766)

1 14 0 -2.5 -2.0 -1.5 -1.0 0.5 0.0 0.5 l!Olog d

Result on Pattern 6 (slope=0.889)FIG. 9. Results with Y okoyas method on rangefinder images. These

graphs illustrate the segmented points and the lines fitted to them. Thelabel (logd, logE ( d ) )corresponds to (log Ad, logEA,,)in the text.

Szeliski [32] showed that regularization based on thethin-plate model and weak-membrane model generatesfractal surfaces whose fractal dimensions are 2 and 3.

TABLE 2Fractal Dimensions Estimated with Different

Linear Segmentations

Pattern D (error) LSF D (error) Polyline1 2.691 (0.012) 2.661 (0.025)

2.545 (0.021) 2.512 (0.016)2

3 2.498 (0.016) 2.486 (0.014)2.234 (0.024) 2.239 (0.025)4

2.210 (0.009) 2.196 (0.010)5

6 2.111 (0.018) 2.118 (0.021)7 2.090 (0.009) 2.102 (0.014)8 2.038 (0.018) 2.010 (0.022)


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424 ARAKAWA AND KROTKOV

FIG. 10. Reconstruction ofsynthetic surface (D =2.3) using Szeliskis method. (top to bottom) Synthetic original, subsampled. and recon-structed surfaces.

respectively. He then developed a probabilistic methodfor visual surface reconstruction using Maximum A Poste-riori (MAP) estimation based on the fractal prior (seeSection3.2).The method generates surfaces whose fractaldimension lies between 2 and 3. Szeliskis approach range data from natural terrain.

provides the central inspiration for this work. Our contri-bution is to extend his approach, amending a numberof technical details concerning the temperature parame-ter, and applying the extended approach to nonsynthetic


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FRACTAL M ODELING OF NATURAL TERRAIN4

,5 1

FIG. 11. Reconstruction ofsynthetic surface (D =2.5) using Szeliski's method. (top to bottom) Synthetic original, subsampled, and recon-structed surfaces.


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FIG. 12. Reconstruction ofsynthetic surface (D =2.7) using Szeliski's method. (top to bottom) Synthetic original, subsampled. and re(structed surfaces. :on-


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FIG. 14. Surfaces reconstructed from synthetic data using nonzero temperatures. From top to bottom, the fractal dimensions are set to 2.3. 2.5,and 2.7.The original synthesized and subsampled elevations are the same as for Figs. 10-12.


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429FRACTAL MODELING OF NATURAL TERRAIN

l og(mean[z( t+d)- z(~ l )1.000.50

0.00

-0.50

-1.00

- 1S O-2.00

-2.50-3.00

-3.50

4.OO

-4.50

-5.00

-5.50

-6.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00FIG. 15. Scaling behavior ofreconstructions using nonzero temperatures.

Using (7), Szeliski found empirically that minimizing theenergy of only the finest layer, the prior model behavesas a fractal whose dimension isD in the vicinity of fre-quency o. Equation (8) changes the frequency, thus vary-ing the fractal dimension ofdifferent resolutions in themultiresolution decomposition.

Szeliski applied Gauss-Seidel relaxation for energy min-imization.The energy can be rewritten in the quadraticform

(9)

(10)

1

2

1

2

E ( u )=-uTAu- U% +c=-(U -u * ) ~(u- * ) +k,

withA =A,/ T,+Ad,b=A d,and the optimal elevationsthat minimize the energy u* =A-b. Because the energyterm is quadratic, the relaxation method reaches the mini-mum energy, and the optimal elevations are computedwith T, =0.

which is a Gaussian with mean u+and variance Tplaij(alsocalled a Gibbs or Boltzmann distribution). Thus, settingTpto a nonzero value changes the variance (noise) ofthe reconstructed surface.

3.3. EffectofTemperatureon Surface Reconstruction

The temperature parameter Tpcontrols the diffusion ofthe local energy distribution [32].Dothe fractal character-istics ofthe reconstructed surface depend on T,?

TABLE 4Temperatures Determined by NewFormalization for SyntheticData

D D* T,,, (D* - D(2.2 2.236 4.4x 10-6 6.0x lo-2.3 2.308 1.5x 10. 4.4 x 10-22.4 2.390 4.5 x lo-< 2.6 x 10

1.8x lo-2.5 2.476 1.4 x1.7x 10 -22.6 2.561 4.1 x 10-4

The probability distribution correspondingtothe energy 2.7 2.643 1.3x 10. 3.5 x 10-22.8 2.716 3.8 x 10-3 5.6x lo-?function is


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FIG. 16. Surfaces reconstructed using temperatures computed by new formalization. From top to bottom. the fractal dimensions are set to 2.3.2.5, and 2.7. The original synthesized and subsampled elevations are the same as for Figs. 10-12.


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FRACTAL MODELING OF NATURAL TERRAIN

l og(mean[z(t +d)-z(r ) J )

43

To answer this question, we synthesized eight elevationmaps with fractal dimensions varying from 2.1to 2.8. Wesubsampled these elevation maps, set Tpto zero, and thenreconstructed the subsamples. Figures 10

-

12 depict thereconstructed surfaces for fractal dimensions 2.3, 2.5, and2.7. In each case, the reconstructed result is too smooth,as compared to the original synthetic patterns.

Figure 13plots estimates, using Yokoyas method, ofthe fractal dimension ofthe eight reconstructed surfaces.More precisely, the figure illustrates the scaling characteris-tics of the underlying data: the abscissa represents logarith-mic scale(e.g., over what size neighborhood isthe estimatecomputed), and the ordinate represents logarithmic spatialvariation (e.g., the amount ofvariation in surface eleva-tion). If the underlying data possesses fractal characteris-tics, then the curve in the log-log plot will be linear overa wide range ofscales, and the slope of the line will varyinversely with the fractal dimension (the greater the slope,the smaller the fractal dimension).

Results for the eight original synthetic data sets appearon the left-hand side ofFig. 13.The curves exhibit linearbehavior over most scales; the departure from linearity atlarger scales is an artifact of the technique for estimatingthe fractal dimension.

Results for the reconstructions appear on the right-handside of Fig. 13.To zeroth order, the curves are parallelimplying (incorrectly) that the surfaces have the same frac

tal dimension.Tofirst order, analysis reveals that the slopein each of the plots is too steep at higher frequencie(smaller scales); i.e., the fractal dimension of the reconstructed surfaces is too low. This is also apparent, qualitatively, in the reconstructions shown in Figs. 10-12.Thesresults demonstrate that surface reconstruction using temperature of zero produces overly smooth surfaces, aleast at higher frequencies. Thus, the answer is affirmativeto the questionofthe dependence of fractal characteristicson T p.

Since setting Tpto zero produces unsatisfactory recon-structions, what is the proper Tpfora given fractal dimen

TABLE5

Temperatures Determined by NewFormalization for Range Data

2.12 1.0x 10-7 1.0 x 10 2.23 1.8 X 10 4.5 x 10-22.51 1.0 x 10-4 1.6 x 10


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FIG. 18.and bottom).

Surfaces reconstructed from range data. The range data was acquired for rocky terrain (top) and two different sandy terrains (middle

sion D?To answer this question empirically, let Syn ( )synthesize a dense fractal pattern offractal dimension D,Sub( ) subsample a pattern, Rec( ) reconstruct a patternusing regularization with temperature T p,and Est( ) esti-mate the fractal dimension ofa pattern (again, using Y o-koyas method). Let D be given by

Est(Rec(Sub(Syn(D)), T,)) =b. (11)The proper temperature TpforD is that which minimizes

the difference between D and D. When searching for theminimum difference, we use the estimated fractal dimen-sionD* =Est(Syn(D)) insteadofD,because the estimated

fractal dimension is apt to besmaller than the real fractaldimension, especially forD>2.5.

Table 3records seven empirically determined tempera-tures, and the differences between the fractal dimensionofthe original patterns and ofthe reconstructed results.All the differences are small, lending credence to the con-

clusion that setting Tpappropriately permits the methodto preserve the roughness of the original patterns, evenon reconstructed surfaces,

Figure 14shows surface reconstructions using three ofthe empirically determined temperatures. The recon-structed surfaces are reasonably rough compared to theoriginal synthetic patterns.

Figure 15plots the estimated fractal dimension ofthe


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l og(mean[z(t +d)-z(t )l )-1.00 , ___I-#2, b2.51I

0.00

-=-.80 --A- ___-5.00 __-6.00 -4.00 -2.00

............................,....#4.W-2.24#6. J k 2 . I2__ __ _-_-_- _-_- - - - --

FIG. 19. Scaling behavior of reconstructions from range data with new temperatures

reconstructed results. The curves display fairly linear be-havior over most scales, and unlike the right-hand side ofFig. 13, they no longer appear parallel or exhibit steepslopes at higher frequencies.

3.4. Temperature asa Function ofFractal Dimension

The results from the previous section demonstrate con-trol over the fractal characteristics ofthe reconstructedsurface by setting appropriate nonzero temperatures.However, the temperatures do not appear to have anymeaning regarding fractal dimension; the temperaturessimply control the amount oflocal diffusion [32]. In thissection, we formalize the temperatures as a function offractal dimension using an analogy to the SRA method.

TheSRA method synthesizes fractional Brownian mo-tion. It adds normally distributed random values to eleva-tions in the multigrid representation. The variances are

controlled according to the resolution ofeach layer I by

whereD is the fractal dimension of the pattern to be syn-thesized.

Szeliskis method adds a normally distributed randomvaluetothe elevationsofeach layer in the multigrid repre-sentation by setting a nonzero temperatureT,,.The tem-perature is proportional to the variance of the Gaussian,

and controls the amountofdiffusion in the high-frequencydomain. His method uses the same temperature (samevariance) for all layers.

In order to synthesize patterns that preserve fractalnessat higher frequencies, we set the temperature TI,/at eachlayer I by analogy with the SRA method

where T,,(D) is the temperature for the finest-resolutionlayer, and u0is the standard deviation ofelevation valuessampled at the finest resolution.

The two unknowns areq,andk. Pentlands method forfractal dimension estimation [26] directly computes theparameter 0,.To computek, it suffices to know one tem-perature Tpo(D),and then to follow the iterative methodtaken in the previous section to determine the proper tem-peratures.To test this formalization oftemperature as a function


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FIG. 20. Surface reconstructed from digital terrain data for Mt. Erebus. Original elevation data(top)and reconstructed surface when the fractaldimension is set to 2.65 (bottom).

of fractal dimension, we applied it to three different typesofdata: synthetic data, range data from a scanning laserrangefinder, and digital terrain map data.

For synthetic data, we first determined Tpo(D)for D =2.4. From this known temperature, it follows thatk =4.7Xlo-. Using this value ofk, Eqs. (12) and (13) computetemperatures for different D. Table 4 records the com-puted temperatures, as well as the difference between thefractal dimensionD* =Est(Syn(D))and the fractal dimen-sion D given by (11).The differences between D* andbare negligible.

Figure 16illustrates the surfaces reconstructed usingthese new temperatures. The surfaces appear appropri-ately rough and highly realistic. Figure 17 plots the esti-mated fractal dimension ofthe reconstructed surfaces. Itshows that the reconstructed surfaces maintain linearityover a wide range of scales.

We acquired range data from laser rangefinder images

ofa test area with sand on the ground, and some meter-scale rocks. We selected a relatively smooth area oftheterrain consisting mainly ofsand. We computed the propertemperature for this pattern and calculated k as 4.0 Xlo-*.Table 5shows the temperatures computed using thisvalueofk.The differences betweenD* andbare compara-ble to those observed for synthetic data (Table 4).

Figure 18shows three reconstructed surfaces. The topsurface is reconstructed from data corresponding to arougher area of the terrain consisting mainly ofrocks, and

the other surfaces are reconstructed from data correspond-ing to smoother areas consisting mainly ofsand. Using thesame parameters (k, go,Tpo),the method adapts to theroughnessofthe original surface, reconstructing the rocky

The parameter k isconstant for patternsofany fractal dimension. solong as they are generated by the same process. For this new data. adifferent generating process acts. Therefore, we must recompute k.


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area rather roughly, and reconstructing the sandy arearather smoothly.

Figure19plots the scaling behavior ofthe reconstructedelevation maps. These are not as linear as for the syntheticdata. However, they do exhibit enough of a linear tendencyto demonstrate scale-invariance.

We acquired digital terrain data from an aerial cartogra-phy databaseofMount Erebus, an active volcano in Ant-arctica. We estimated the fractal dimension D* to be ap-proximately 2.3, with a. =68.0. We used the value ofkdetermined above for the range data, and reconstructedthe sparse data. Figure20shows that the method producesresults that are dense and fairly realistic, including eventhe shape of craters.

Our approach requires large amounts of computation.Reconstruction ofa 256 X 256 elevation map typicallyconsumes around 2 h on a Sun4/75 with 24 MB physicalmemory. The surface reconstruction approach has beenimplemented on a massively parallel machine, reducingsubstantially the computation time (a speedup of100for

a 64K M asPar) [40].4. DISCUSSION

In this paper, we addressed two issues in fractal modelingof natural terrain: estimation of fractal dimension, andfractal surface reconstruction.

With respect to the first issue, we investigated the fractalBrownian function approach to the problem ofestimatingfractal dimension. We extended published algorithms toaccommodate irregularly sampled data supplied by a scan-ning laser rangefinder, and applied the extended methodsto noisy range imagery ofnatural terrain (sand and rocks).

The resulting estimates of fractal dimension correlatedclosely to the human perception ofthe roughness of theterrain. We conclude that it is reasonable and practical tomodel natural terrain as a fractal pattern, and that thefractal dimension is a reasonable measure of roughnessof terrain.

Problems remaining to be addressed include determin-ing the region in which to conduct fractal analysis, identi-fying which linear parts of the log-log curves are mostsignificant, and segmenting multifractal patterns. We mayneed further simulations using different noise models (e.g.,non-Gaussian additive noise, quantization, and truncation)in order to study the sensitivity of methods to estimatefractal dimension, and to determine how appropriate the

estimation is.With respect to the second issue, we described an ap-

proach to fractal surface reconstruction that producesdense elevation maps from sparse inputs. The reconstruc-tion stochastically performs energy minimization using reg-ularization, in which the prior knowledge terms includeroughness constraints, and in which the temperature term

depends on the fractal dimension. We demonstrated themethod using sparse elevation data from rugged, naturalterrain, and showed that it adaptively reconstructs surfacesdepending on the roughness of the original data. The re-constructed elevations are realistic and natural.

As future work, we consider two topics. First,oursurface

reconstruction approach does not take discontinuities intoaccount. Natural terrain contains many discontinuities,such as step edges around stones. Our method does notproduce realistic results reconstructing sparse depth datawith discontinuties. Many researchers have considered thisproblem and have derived methods that we expect will fitwell with our approach.

In nature and more frequently in artificial scenes, manypatterns are not truly self-similar, but are anisotropic and/or the fractal dimension changes with scale or changesspatially. Several methods for handling such patterns havebeen reported in the literature. Bruton and Bartley [6] haveshown a method to generate fractal images with spatially-variant characteristics. Kaplan and Kuo [14] proposed amethod to synthesize scale-variant fractal textures usingthe extended self-similar (ESS) model. Our approach can-not be applied to such patterns directly. However, theMAP-based method can interpolate the surfaces with spa-tially-variant roughness [32]. Further, our approach canchange the roughness at a specific scale by controlling thetemperature at each resolution.

Another future work should identify the sensitivity ofour fractal interpolation method, determining the changein fractal dimension caused by small variations in the tem-perature settings. This work will be important for applica-tions requiring small surface roughness tolerances.

In conclusion, fractal dimension is a powerful descriptor

ofnatural terrain, a descriptor related to the ambiguousproperty of roughness. We expect that ourcontributions-analysis and surface reconstruction using fractal dimen-sion-will advance fractal modeling of natural terrain be-yond mathematical curiosity, and closer to practicalapplications.

ACKNOWLEDGMENT

The authors thank Dr. Richard Szeliski and Dr. Hiroshi K aneko foruseful discussions. This research was partially supportedbyNippon Tele-graph andTelephoneCo.,and also by NA SA under GrantNAGW-1175.

REFERENCES

1. K . Arakawa and E. Krotkov. Estimating Fractal Dimrnsion fromRange ImagesofNatural Terrain,Technical Report CMU-CS-91-156,Schoolof Computer Science. Carnegie Mellon University. Pittsburgh.

J uly 1991.

2. K. Arakawa and E. Krotkov. Fractal Stcrfacr Rrconstriiction withUncertainty Estimation: Modeling Natuml Terrain.Technical Report


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fractal modeling of natural terrain - analysis and surface reconstruction with range data

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