Shape and erosion of pebbles Douglas Durian, H. Bideaud, P. Duringer, André Schröder, Carlos Marques
To cite this version:
Douglas Durian, H. Bideaud, P. Duringer, André Schröder, Carlos Marques. Shape and erosion of pebbles. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2007, 75 (2), pp.021301. 10.1103/PhysRevE.75.021301. hal-00144239
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D.J. Durian,1,2 H. Bideaud,2 P. Duringer,3 A. Schr¨oder,2 C.M. Marques2 1Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA 2LDFC-CNRS UMR 7506, 3 rue de l’Universit´e,67084 Strasbourg Cedex, France and 3CGS-CNRS UMR 7517, Institut de G´eologie, 1 rue Blessig, 67084 Strasbourg Cedex, France (Dated: November 28, 2006) The shapes of flat pebbles may be characterized in terms of the statistical distribution of curvatures measured along their contours. We illustrate this new method for clay pebbles eroded in a controlled laboratory apparatus, and also for naturally-occurring rip-up clasts formed and eroded in the Mont St.-Michel bay. We find that the curvature distribution allows finer discrimination than traditional measures of aspect ratios. Furthermore, it connects to the microscopic action of erosion processes that are typically faster at protruding regions of high curvature. We discuss in detail how the curvature may be reliable deduced from digital photographs.
PACS numbers: 45.70.-n,83.80.Nb,91.60.-x,02.60.Jh
I. INTRODUCTION forms due to weight, hardness or anisotropy differences. In spite of the diversity of factors at play in shape mod- The roundness of pebbles on a beach has long been a ification, the complete evolution of the pebble shape is source of wonder and astonishment for scientists in many fully determined by (i) the initial form described by some fields [1, 2]. Explanations for the pebble shapes were born number of faces, edges and vertices and (ii) the position, from the simple pleasure of understanding nature but also size and orientation of the successive ablations. from the hope that a pebble, or a collection of pebbles, might carry lithographically imprinted the signature of Given that the erosion process evolves by a succession their erosion history. Reading that imprint would then, of localized events on the pebble surface, it is surprising for instance, reveal if a pebble was eroded on a beach, a that the majority of the precedent attempts to charac- river or a glacier, or if it traveled a long distance down terize the pebble shapes were restricted to the determi- a stream. It even perhaps would reveal for how long nation of global quantities such as the pebble mass or the erosion forces have been at work on that object. Of the lengths of its three main axes [3]. Clearly, in order obvious interest in Geology [3], a physical understand- to capture both the local nature of the erosion process ing of the formation of erosion shapes would also allow and the statistical character of the successive elementary for a better control of many industrial processes lead- cuts, one needs to build a new detailed description of the ing to rounded objects such as gem stone or clay bead pebble shapes based on quantities that are more micro- grinding in tumblers or fruit and vegetable peeling in sev- scopic and more closely connected to evolution processes. eral mechanical devices. Diverse mathematical tools have In Ref. [8] we proposed curvature as a key microscopic been developed for geometrical shape analysis of crys- variable, since, intuitively, protruding regions with large talites, cell membranes, and other far from equilibrium curvature erode faster than flatter regions of small cur- systems [4–7]; however, these do not seem applicable to vature. We then proposed the distribution of curvature pebbles. around a flat, two-dimensional, pebble as a new statisti- The evolution of a pebble shape under erosion can ar- cal tool for shape description. And finally we illustrated guably be viewed as a succession of elementary cuts that and tested these ideas by measuring and modeling the act at the surface of the body to remove a given amount erosion of clay pebbles in a controlled laboratory appa- of material. This converts young, polyhedral-like shapes ratus. with a relatively small number of large sides and sharp vertices into more mature shapes with a high number of small sides and smooth vertices. The size and the shapes In this paper we elaborate on our initial Letter [8], of each of these successive ablations, as well as the surface and we apply our methods to naturally-occurring rip- sites where the cutting happens, are determined both by up clasts found in the tidal flats of the Mont St.-Michel the conditions under which erosion takes place and by bay. Section II begins with a survey of shape quantifi- the nature of the material being eroded. Exposure of a cation for two-dimensional objects, in general, and reca- young, polyhedral-like shape to the rough tumbling of a pitulates our new curvature-based method. Section III steep stream slope will result in relatively large cuts of provides further details of the laboratory experiments on the angular sections, while exposure to the gentle erosion clay pebbles. Section IV presents a new field study of of wind or water is more likely to lead to small cuts al- the Mont St.-Michel rip-up clasts. And finally, following most parallel to the existing flat sides. Also, the same the conclusion, two methods are presented in the Appen- sequence of external forces acting on two identical origi- dix for reliably extracting the local curvature from digital nal shapes of different materials will result into distinct photographs. 2
II. 2D SHAPE QUANTIFICATION scription. Also, there is no a priori guaranty that the variables are Gaussian distributed, and one needs a di- The issue of rock shape is of long-standing interest in rect space method to better assess the importance of non- the field of sedimentology [9–17]. Two basic methods linear phenomena. Non-linearity is, we suspect, intrinsi- have become sufficiently well established as to be dis- cally embedded in the erosion mechanisms of pebbles. If cussed in introductory textbooks [3]. The simplest is a one considers for instance a shape represented by a single visual chart for comparing a given rock against a stan- harmonic in the r(θ) representation, it is clear that the dard sequence of rocks that vary in their sphericity and peaks will wear more rapidly than the valleys. Therefore angularity. A rock has high “sphericity” if its three di- the erosion rate cannot be a function of the harmonic mensions are nearly equal. And it is “very angular”, inde- number only; it must either be a non-linear function of pendent of its sphericity, if the surface has cusps or sharp the amplitude itself or a function coupling many harmon- ridges; the opposite of “very angular” is “well rounded”. ics. While useful for exposition, such verbal distinctions are Our aim is to provide an alternative measure of pebble subjective and irreproducible. The second method is to shape that is well-defined, simple, and connects naturally form dimensionless shape indices based on the lengths of to local properties involved in the evolution process. We three orthogonal axes. From the ratios, and the ratios restrict our attention to flat pebbles, where an obvious of differences, of the long to intermediate to short axes, shape index is the aspect ratio of long to short axes. one can readily distinguish rods from discs from spheres. Since erosion processes generally act most strongly on And a given rock may be represented by a point on a the rough, pointed portions of a rock, we will focus on the local curvature of the pebble contour. Technically, triangular diagram according to the values of three such K T indices, with rod / disc / sphere attained at the corners. curvature is a vector given by = d /ds, the deriva- This practice is nearly half a century old [18]. Neverthe- tive of the unit tangent vector with respect to arclength less, there is still much debate about which of the infinite along the contour [30]. More intuitively, the magnitude number of possible shape indices are most useful [19–23]. of the curvature is the reciprocal of the radius of a circle In any case, such indices cannot capture fine distinctions that mimics the local behavior of the contour. Here we in shape, let alone the verbal distinctions of angular vs shall adopt the sign convention K > 0 where the contour rounded. Furthermore, they provide no natural connec- is convex (as at the tip of a bump) and K < 0 where tion to the underlying physical process by which the rock the contour is concave (as where a chip or bite has been was formed. removed from an otherwise round pebble). In the Ap- pendix, we describe two means by which the curvature More recent methods of shape analysis employ may be reliably measured at each point along the peb- Fourier [24–28], or even wavelet [29], transforms of the ble contour. Note that the average curvature is simply contour. This applies naturally to flat pebbles or grains, related to the perimeter of the contour: but also to flat images of three-dimensional objects. The advantage of Fourier analysis over shape indices is that, P = 2π/ K , (1) with enough terms in the series, the exact pebble con- h i tour can be reproduced. For simple shapes, the contour which is obviously correct when the shape is a circle. may be described in polar coordinates by radius (eg dis- To describe the shape of a pebble, a very natural quan- tance from center of mass) vs angle, r(θ), and the corre- tity is the distribution of curvatures, ρ(K), defined such sponding transform. However, this representation is not that ρ(K)dK is the probability that the curvature at single-valued for complex shapes with pits or overhangs. some point along the contour lies between K and K +dK Generally, the contour may be described by Cartesian co- [8]. In order to distinguish different distributions, as a ordinates vs arclength, x(s), y(s) , and the correspond- { } practical matter, it is more reliable [31] to use the cumu- ing transforms. In any case, the relative amplitudes of lative distribution of curvatures different harmonics give an indication of shape in terms K of roughness at different length scales. In a different ′ ′ f(K) = ρ(K )dK (2) area of science, Fourier representations have proven es- Z0 pecially useful for analysis of fluctuations and instabili- ties of liquid interfaces, membranes, etc. [4–6]. In prac- Literally, f(K) is the fraction of the perimeter with cur- tice, for these systems, shape fluctuations are sampled vature less than K. Note that f(K) increases from 0 during some time interval and then the average Fourier to 1 as K : 0 ; the minimum curvature is where amplitudes extracted by averaging over many different f(K) first rises above→ ∞ 0, the maximum curvature is where realizations of the shape. Also, because these phenom- f(K) first reaches 1, and the median curvature is where ena are linear, each Fourier component grows or shrinks f(K) = 1/2. Unlike for ρ(K), it is not necessary to bin at some amplitude-independent rate and the evolution is the curvature data in order to deduce f(K). Instead, just fully determined by a dispersion relation. Unfortunately sort the curvature data from smallest to largest and keep these features do not hold for the erosion of pebbles. Be- a running sum of the arclength segments, normalized by cause each pebble shape only provides one configuration, perimeter. Finally, so that the shapes of pebbles of dif- average quantities need to be built from a different pre- ferent sizes may be compared, it is useful to remove the 3
1.6 (a) have the same aspect ratio, a = 3/2, their shapes are 1.5 obviously different. This emphasizes how a single num- 1.4 ber is insufficient to quantify shape. The shape differ- 1.3 r ences do, however, show up nicely in the forms of f(K). 1.2 1.1 The ellipse is closest to a circle, with a distribution of 1 curvatures that is most narrowly distributed around the 0.9 average and hence with an f(K) that is most like a step (b) superellipse function. The superellipse is farthest from a circle, with 3 ellipse oval four long nearly-flat sections and four high-curvature cor- circle 2 ners; its curvature distribution is broadest. The oval is intermediate. K/
0 0π 0.5π 1π θ 1.5π 2π III. LABORATORY EXPERIMENT 0 (c) The examples given in Fig. 1 correspond to regular, 0.2 highly symmetric shapes of two dimensional convex “peb- bles”. In practice, natural or artificial erosion processes 0.4 lead to curvature functions with an important statistical
f(K) 0.6 component. In this section we elaborate on the labora- tory experiments of Ref. [8], designed to study both the 0.8 statistical nature of the curvature distribution and the in- fluence of the original shapes of the pebbles on the final 1 output of a controlled erosion process. 0 0.5 1 1.5 2 2.5 3 Laboratory pebbles were formed from “chamotte” clay, K/
FIG. 3: Shape evolution of a 5 × 5 cm square pebble eroded in the laboratory, by the method explained in the text. The images were taken at 5 min intervals, and are shown at the same magnification.
FIG. 2: The wearing apparatus used for the laboratory ex- 3 periments. The rotating metal tray is 30 × 30 × 7 cm . distributions f(K) are given in Fig. 4, where the inset shows the extracted contours. Here the initial shape is square, with four long nearly-flat regions and four it hits one of the walls in the bottom part of the container, short high-curvature regions. Thus the initial f(K) rises and then rolls down along that wall as the basin keeps steeply around K = 0 and extends with relatively little its rotation. After a short stop at a container corner the weight out to K K . At first, the action of ero- pebble starts a new cycle again. We performed prelimi- sion is most rapid≫ at h thei high-curvature corners, with nary tests in order to determine both ideal basin orien- the flat regions in between relatively unaffected. Thus tation and ideal rotation frequency for our experiments. the high-K tail of f(K) at first is suppressed, and weight As expected, above some maximum rotation frequency builds up across (0.5 2) K . Next the rounded cor- the pebble becomes immobilized in the basin: centrifu- ners erode further and− graduallyh i extend across the flat gal forces maintain the pebble on a given position against sections. Thus weight in f(K) is gradually concentrated the wall. Also, under some minimum inclination angle, more and more toward K . After about 15-20 minutes, no fall of the pebble is observed, while a high inclination when the flat sections areh nearlyi gone, the form of f(K) doesn’t allow the pebble to reach it’s maximum altitude. fluctuates slightly but ceases to change in any system- Altogether, we found it suitable to operate at a basin atic manner. In otherwords, the shape of the pebble has ◦ angle of 45 and a rotation frequency of one cycle per reached a final limiting form. Further erosion will affect second. Using the latter experimental conditions, we ob- pebble size, but not pebble shape! served that the in-plane dimension of a pebble decreased To test the universality of the final shape, we repeat by around a factor 2 after 30 minutes. Thus, a signifi- the same experiment both for other squares as well as cant wearing of a pebble could be observed after a few for a variety of other initial shapes such as rectangles, minutes rotation. In practice, each pebble was eroded triangles, and circles. A number of different examples under the described conditions during 30 minutes, while showing both the initial and final shapes are shown in a picture of the pebble was taken after each 5 minutes Fig. 5. In all cases, the cumulative curvature distrib- wearing. Hence, for each of the pebbles studied, we ob- ution f(K) shows a systematic evolution at short times tained about 7 pictures, corresponding to the initial peb- and slight fluctuations about some average shape at later ble and to six following states of the wearing pebble. For times, just as in Fig. 4. The more angular or oblong the some of the pebbles 8 or 9 pictures at 5 minutes interval initial shape, the more erosion is needed to reach a sta- were taken. The images were then analyzed following the tionary final shape. The average final f(K) is shown for method described in the appendix. the various initial shapes in Fig. 6. Evidently, these all An example for the shape evolution produced by this display the same quantitative form independent of the method is given in Fig. 3, with photographs shown every initial shape. Even f(K) for an initially-circular pebble five minutes. The corresponding cumulative curvature broadens from a step function to the same form as all the 5
1
0.8 200
100 0.6
f(K) 0 0.4
-100 0.2
-200 -200 -100 0 100 200 0 -2 0 2 4 6 8 K/
FIG. 4: (Color online) Cumulative curvature distribution, f(K), for the evolving pebble depicted in the inset (and pic- tured in Fig. 3). As the pebble becomes progressively rounder, the curvature distribution narrows and approaches a final shape. The time interval between successive contours is 5 minutes; the early contours of evolving shapes are shown as different blue dashes, while the later contours of stationary shape are shown as solid red. The same color and curve types are used in the main plot. For the inset, the axes are given in pixel units, equal to 0.132 mm. For contrast, a circle is shown by points. others. The final f(K) for all initial shapes can thus be av- FIG. 5: The initial and final forms of different shapes eroded eraged together for a more accurate description of the in our experiment, all shown with the same magnification. stationary shape produced by the laboratory erosion ma- The erosion times for the final shapes are 40 min for square, chine. The result is shown by the open circles in the same 30 min for rectangle, 30 min for triangle, 35 min for polygon, plot, Fig. 6. Differentiating, we obtain the actual curva- 35 min for circle, and 30 min for lozenge. Note that the final ture distribution, ρ(K), shown on the right axis. It is shapes are all roughly circular. fairly broad, with a full-width at half-maximum equal to about 1.6 K . The actual shape is not quite symmetrical, skewed towardh i higher curvatures. The closest simple ana- shown in Fig. 7. Though the dynamic range is not lytic form would be a Gaussian, exp[ (K K )2/(2σ2)]. great, the data are consistent with an exponential de- − −h i crease, P (t) = P (t0) exp[ (t t0)/τ]. The best fit to The actual distribution is slightly skewed toward higher − − curvatures, but the best fit gives a standard deviation of this form is shown by a solid curve; it gives a decay con- σ = 0.70 K , as shown in Fig. 6. It is easy to imagine stant of τ = 44 min. Exponential erosion is, in fact, that theh widthi of this distribution could be set by the observed in field and laboratory studies [12]. It is to be strength of the erosion process. For example, if the angle expected whenever the strength of the erosion is propor- of the rotating pan were lowered, then the erosion would tional to the pebble size, as in our lab experiments where be more gradual and more like polishing; in which case the impulse upon collision is proportional to the pebble’s a rounder stationary shape may be attained with a nar- weight. rower distribution of curvatures. The form of f(K), as well as its width, could also be affected. These types of questions can be addressed, both in laboratory and field IV. FIELD STUDY studies, now that we have an incisive tool like f(K) for quantifying shape. As a first field test of our method of analysis, we col- To further study the erosion produced by our labo- lected mud pebbles in the Mont St.-Michel bay, France. ratory apparatus, we now consider how the perimeter The littoral environment located at the inner part of the of the pebble decreases with time, P (t). Since the ini- Norman-Breton Gulf is characterized by a macro-tidal tial behavior depends on the specific initial shape, we dynamics. This location exhibits the second largest tide focus on subsequent erosion once the (universal) station- in the world after the Bay of Fundy, Canada. During ary shape is achieved. If the final stationary shape of the spring tide periods the upper part of the tidal flats the curvature distribution is reached at time t0, then collects a muddy sediment. This mud dries up during the quantity of interest is really P (t)/P (t0) vs t t0. the following neap tide period where the sediments are The results, averaged over all laboratory pebbles,− are exposed to the air. In certain areas, between the large 6
1 0.6 Laboratory pebbles 0.5 0.8 squares (4) rectangles (5) pentagons (5) 0.4
0.6 ρ
triangle (1) (K) polygon (1) 0.3 f(K) 0.4 lozenge (1) 0.2 circle (1) 0.2 ALL 0.1
0 0 -2 0 2 4 6 8 K/
FIG. 6: (Color online) Integrated curvature distribution, f(K), for the final shapes of laboratory pebbles with vari- ous initial shapes, as labeled. These are indistinguishable to within measurement accuracy; their average is shown by the open circles. The corresponding average curvature distribu- FIG. 8: Typical shapes of Mont St.-Michel rip-up clasts. The tion, ρ(K) is obtained by differentiation and is shown on the immature pebbles in the top row were collected close to their right axis along with a fit to a Gaussian shape. origin; the sub-mature pebbles in the middle row were col- lected further downstream; the smooth, mature pebbles in 1 the bottom row were collected on a nearby sandbar. As these
lab pebble data pebbles eroded, their shapes became rounder, an effect quan- 0.8 tified in the next figure. The bars indicate, in each row, a Exp[-(t-t )/(44 min)] o length of 2 cm. 1-(t-to )/(44 min)
o 0.6
0.4 sub-mature pebbles found further “downstream”. As a P(t)/P(t ) result of erosion, these pebbles are smaller and smoother 0.2 than the cobble. Thirty five samples were examined; for these, the average perimeter is 180 mm. The third class 0 0 10 20 30 40 50 is rounded, mature pebbles found on a nearby sand bar. t-to The relation of these pebbles to the other two classes is not clear. Seventeen samples were examined; for these, FIG. 7: (Color online) Perimeter vs time, where t0 is when the average perimeter is 220 mm. the stationary shape has been reached. Typical photographs for each of these classes are shown in Fig. 8. The average of the cumulative curvature distri- bution for all samples in each class is shown in Fig. 9. The equinoxial neap tides, the exposure of mud sediments angularity of the large cobble is reflected in the breadth to air may last for several months. During this period, of the curvature distribution. Roughly a quarter of the this mud layer will develop a vast network of desiccation perimeter has negative curvature, and roughly a tenth cracks. This network then leads to fragmented plates of a has curvature five times greater than the average. For polygonal shape with 20 to 40 cm size. During the next the other two classes, the curvature distribution is pro- spring tide period, the plates in the erosional area can gressively more narrow. The relative steepness of the be eroded by tide currents, thus re-incorporated into the f(K)’s shows that all of these shapes are less round than sedimentary cycle. During the tide process these clasts the final pebbles produced by the laboratory erosion ma- are progressively eroded over many months. Thus the chine. mud cohesion allows enough observation time for the life The width of the curvature distribution can be speci- of a clast to be observed within a distance of order of fied quantitatively by the standard deviation, σ. Results one hectometer, from the original erosional area down to are normalized by the average curvature, and are shown the latest stages of abrasion. Our approach is similar in for the field and laboratory pebbles in Table I. The peb- spirit to field studies of river pebbles, where downstream bles with steeper f(K) indeed have smaller widths. For distance serves as a surrogate for time. The mud clasts example, the width for the immature field cobble is about here have the advantage of remaining in a smaller area 3-4 times that of the average laboratory pebble. While and of eroding on a human timescale. the dimensionless width of the distribution, σ/ K , is a We have analyzed the shapes of three classes of rip- useful number for comparisons, it does not distinguishh i up clast photographed at three distinct locations on the between curvature distributions of different shape. The tidal flats near Mont St.-Michel. The first is large sub- actual functional form of the curvature distribution can angular cobble found near the site of formation. Ten be specified to some extent by comparing its moments samples were examined; for these immature pebbles, the with that of a Gaussian. In particular, the “skewness” average perimeter is 550 mm. The second class is medium and “kurtosis” are dimensionless numbers defined by the 7
1 a stationary form. When this stationary state is reached, the pebble contour still changes but, within small fluc- 0.8 tuations, its curvature distribution remains the same, provided that the curvature is normalized by its average 0.6 value. Secondly, we have found that the final stationary
f(K) form of the distribution is independent of the original 0.4 immature sub-mature pebble shapes. This not only shows that the curvature mature distribution is a property of the erosion process itself, but 0.2 laboratory average it also opens the interesting possibility of establishing a classification of different erosion processes according to 1 -2 0 2 4 6 8 the type of curvature distribution they generate. K/
APPENDIX: CURVATURE ANALYSIS V. CONCLUSION The goal of this section is to provide a detailed, prac- We have studied the formation of two-dimensional peb- tical description of two means to measure the local cur- ble shapes. As in Ref. [8] we introduced a local descrip- vature at each point along the pebble contour. In both tion of the erosion process, based on the distribution cases the starting point is an image of the pebble. For function of the curvature, measured along the pebble con- our work we use a digital camera Canon Power Shot G1 tours. This description captures both the local character with a resolution of 1024 768 pixels. It should be just of the erosion events, and the statistical nature of the as effective to scan conventional× photographs, or even to erosion process. scan pebbles themselves. To determine the x,y coor- For pebbles generated in the laboratory, we have shown dinates of the contour, we import the images{ into} NIH that the curvature distribution has two important prop- Image [32], which includes routines for finding edges and erties. First, the erosion drives the distribution towards for skeletonizing the result. An example of the digitized 8 pebble contour, and the smooth reconstructions to be dis- 200 cussed below, is given in Fig. 10. We show the contour in pixel units, since the actual calibration is needed only 150 to determine size, not shape. Note from the inset that 100 the contour points are indeed pixelized and skeletonized, -110 with each point having only two neighbors located at ei- 50 ther 1 or 0 units away in the x- and y-directions. If the -120 digitization± process is faithful, then the uncertainty in each pixel is about 0.5 units in each direction. This is 0 -130 ± not small compared to the distance between neighboring -140 pixels, so a smoothing or fitting routine is necessary to -50 reconstruct the actual contour and thereby extract the -150 -100 curvature distribution. -160 100 110 120 130 140 To illustrate the difficulties of extracting curvature, let -150 us begin with two methods that are, in fact, unsatis- factory. It is perhaps tempting to simply smooth the -200 -200 -150 -100 -50 0 50 100 150 200 data, replacing each point with a weighted average of neighbors lying within some window. Weights could be FIG. 10: (Color online) Reconstruction of a smooth pebble cleverly chosen to de-emphasize points at the edge of the contour from the pixelized digital representation. The solid window, for example. This fails, however, since it’s far curve is based on fitting to cubic polynomials at each point. from obvious how to choose a suitable window size. For The dashed curve is based on the iteration scheme of Fig. 11. instance, the pixelized representation of the straight sec- tion given by y = 0.1x for 0 < x < 10 is a step function y = 0(1) for x < (>)5. A large window would be needed value and derivatives of the cubic polynomial by to even approximately reconstruct the original line. How- 2 2 ever, such a sufficiently large smoothing window would r + 2rθ rrθθ K = 2 2−3/2 , (A.1) erase fine features if applied elsewhere along the contour. (r + rθ ) Since smoothing filters provide no feedback on quality, vi- 2 2 sual inspection of the result would be necessary to choose where rθ = dr/dθ and rθθ = d r/dθ [30]. an optimal window size at each point along the contour. Two tricks seem necessary to achieve satisfactory re- This is not only subjective, but rather impractical. As sults. The first is to weight the data most heavily near an alternative, it is perhaps tempting to implement an the center of the window. We use a Gaussian weighting automated version of Wentworth’s curvature gauge [9]. function with a standard deviation equal to 1/4 of the This is a device with circular notches of various diam- width of the window. This ensures that points at the eters into which portions of a pebble may be pressed. edges have essentially no influence. Therefore, the fitting The computational analogue would be to find the best results do not vary rapidly as the window is slid along the nonlinear-least-squares fit to a circle at each point along contour. This guarantees that the reconstructed curve the pebble contour. As with smoothing, one difficulty is and its first two derivatives are continuous, which is a to find the optimal window over which to do the fitting. crucial requirement for measuring the curvature. A compounding difficulty is that essentially nowhere is The second trick is to choose the window size appro- the pebble exactly circular, so even with ‘the’ optimal priately. This is actually the most difficult and subtle window there is substantial disagreement between data aspect of the whole problem. If the window is too small, and fit. A spectacular example of this problem is at an then the fit will reproduce the bumps and wiggles of the inflection point, where the curvature changes from posi- pixelization process; usually the curvature will be over- tive to negative. estimated. If the window is too large, then the fit will significantly deviate from the data; usually the curvature To overcome such difficulties, we propose to fit digi- will be underestimated. And while the curvature tends tized contour data to a cubic polynomial at each point to decrease systematically with window size, there is in along the contour. A cubic is the lowest order polynomial general, unfortunately, no plateau between these two ex- needed in order to avoid systematic error when the cur- tremes where the curvature is relatively independent of vature varies gradually across the fitting window, which window size and hence clearly represents the true value. is the usual case. In order to avoid having to rotate To pick the window appropriately requires careful un- the coordinate system to ensure that the contour y(x) derstanding of the numerical fitting procedure and the is a single-valued function, we instead convert to polar feedback it provides. Since the fitting function is a poly- coordinates. Thus we define the origin by the center- nomial, the minimization of the χ2 total square deviation of-mass of the contour and perform fits to r(θ) where from the data reduces to solving a set of linear equations. r = x2 + y2 and θ = tan−1(y/x). Once a satisfactory This in turn reduces to inverting a matrix. If the window fit isp achieved, the curvature may be deduced from the is too small then the fit will be ‘ambiguous’ in the sense 9 that χ2 is small but the error in the fitting parameters is of the actual contour with two provisos. First, at each large. Mathematically, the matrix to be inverted is essen- step, we locally smooth the trial pixelization by replacing tially singular. A good strategy is therefore to start with each point by its average with its two immediate neigh- a small window and increase its size until the matrix is bors. Second, we keep only every fourth or fifth point no longer singular. This can be accomplished using a lin- in the original pixelized data and perform all operations ear least-squares fitting routine based on singular-value on this subset. When done, we compute the curvature decomposition [31]. However, the uncertainty in fitting literally by the change of slope with respect to arclength parameters for the first suitable window is generally too using the straight segments between adjacent points. large (nearly 100%). So we increase the window size two The cumulative curvature distribution given by this it- pixels at a time until the error in curvature has been re- eration scheme is shown in Fig. 11 as a function of the duced by a factor of ten. This defines the largest suitable number of points kept. When too many are kept, the re- window, beyond which systematic errors due to incorrect constructed curve follows the bumps and wiggles of the functional form begin to appear. We have not been able original pixelization too closely; the curvature distribu- to define the largest suitable window based on the value tion is too broad. When too few are kept, the recon- 2 of χ . For the final result, we take a weighted average of structed curve incorrectly smooths over small-scale fea- the cubic fit parameters over all suitable windows, where 1 the weights are set by the uncertainties in fitting para- meters as returned by the fitting routine. An example 0.8 of a reconstructed contour from this procedure is shown by the solid curve in Fig. 10. The recontruction is sat- 0.6 number of points kept every 2nd isfactorily smooth; also, it clearly avoids the pixel noise f(K) every 3rd 0.4 without smoothing over significant small-scale features in every 4th every 5th the contour. 0.2 every 6th Since the cubic fitting method is rather involved, and every 7th cubic fit since the choice of window sizes is still slightly subjective, 0 we have developed an alternative method. The starting -0.5 0 0.5 1 1.5 2 2.5 3 3.5 point is the fact that the actual digital representation K/
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