Optical measurement of marble and limestone surface polish
Francisco Gasc6n and Miguel Balbas
An optical method for measuring marble and limestone surface finish is described. This method is capable ofordering the different varieties of rocks subjected to the same industrial finishing process consistent with anarrangement made by an average human observer. The measure of finish is defined as the ratio between thespecularly reflected luminous flux from a sample and the total diffused light.
1. Introduction
Construction rock mining represents a very higheconomic volume. One of the industrial processesnecessary for its ornamental use is polishing. There-fore, it is important to measure the degree of polishobtained to compare machining methods and times, toidentify the kinds of rock easiest to work with and toprovide an objective polish expertise in cases of litiga-tion.
The degree of rock polish is a subjective and complexconcept, and it has more than one definition.
Optical properties are often the first to be noticed bythe user. If the surface lacks inhomogeneity, as on aperfect mirror, only a specularly reflected image exists.If the surface is heterogeneous, the diffused light is sostrong that the image is no longer perceived but ratheris seen as a bright surface; this type of surface is termedperfectly diffusing and the intensity distribution of thediffused light follows Lambert's law. Thus the polishis perfect when the surface behaves as a perfect mirrorand totally imperfect when it behaves as a perfectdiffuser.
Optical measurements of the polish of ornamentalrocks have been carried out by Asuni et al.1 There areother measures recommended for the characterizationof surface appearance such as gloss, haze, and distinct-ness of image; see the following standards: ISO 2813and ASTM D523, E430 and D4039; see also Christie'swork2 on metallic appearance, also referred to inASTM E430-78.
The authors are with Escuela Tecnica Superior de Ingenieros deMinas, Departamento de Fisica, 28003-Madrid, Spain.
Our purpose is to find a method that gives an objec-tive answer to the question: Is this marble or lime-stone well polished? It is necessary for an objectivemeasure of the polish to be comparable to the subjec-tive appraisal, simple to carry out, repeatable, andstandardizable.
An experimental method will now be discussed thatmay contribute toward improving the procedures forevaluating polished rock surfaces. The present re-search was carried out with ten different varieties ofmarble and limestone, all subjected to industrial fin-ishing processes. We had a sample of each variety.
First, we ordered the samples by a subjective meth-od. Then we measured the polish by an objectivemethod; this method is an improvement on the one weused for granite surface polish.3 Finally, we compareboth methods.
II. Subjective Method
Let i be one sample and j another. Each pair wascompared according to the following specification (Fig.1): A pattern P was made by a quadricle whose fre-quency varies from one side to the other from 1/13 to1/9 lines/mm with a width of dark trace variable from 4to 2 mm, respectively. Pattern P is set up on a diffuserglass and its upper side illuminated by three fluores-cent lamps. The mean incidence angle was 450, thedistance from pattern P to incidence point I on thesample was 0.12 m and the distance from I to observero was 3.5 m. Samples i and j were set together besideeach other for best comparison.
Thirty observers looked at the pair of samples i and jand if judging sample i to be better polished thansample j, wrote +; if the first is worse than the latter,they wrote -, and if they were equal, they wrote 0. Sothe number of comparisons was 45 X 30 = 1350.
Table I summarizes the results. The matrix ele-ment aij is the number of observers judging i betterpolished than j. The element aji is the number ofobservers thinking the i worse than j, and the differ-ence 30 - (aij + aji) is the number of ties.
We establish the following superclassification crite-rion: A superclassification relation is sure and weassert that the sample i is better polished than thesample j if and only if
aij> aij-aji>1 30-(aij + aji) 1aji ' 30 3 ' aij 2
Table I and these conditions lead to the superclassi-fication relations shown in Table II. The physical(objective) method must not contradict these rela-tions.
To obtain a complete ordering, we have applied Al-way's method,4 calculating for each i sample the coeffi-cient
30
Ai = Eaij,j=1
or the relative coefficient or mean rank
ri = - Ai/30.
Table III shows the coefficients r, Ai, and the ordernumber n, n = 1 being the worst polish.
111. Basis of the Objective Measures
A sample illuminated at 450 incidence by a cylindri-cal-circular beam of area A" may be studied by a pho-toelectric detector of area A (A > A") placed a distanceD from the center of the elliptically illuminated sur-face. The angle under which the emerging flux F ismeasured, when it reaches the detector, is and thearea illuminated on the sample A' = A"/cos450 =A"'A/2 (Fig. 2).
A'
Fig. 2. Incident beam and emerging light.
If the sample is a perfect mirror, the measured fluxwill have a sharp maximum for 0 = 450. If the sampleis a perfect diffuser, its luminance is independent ofthe observation angle, and the flux arriving at zone dAof the detector, proceeding from zone dA' of the sample(Fig. 3), will be
where dI = intensity,dQ = solid angle that encloses dA and has
the vertex in dA',L = luminance, andV = angle between the line from dA' to
dA and the detector normal.The totally diffused flux by a perfect diffuser sample
to the hemispace should be
FTD = J dA' JL cosed Q = rLA'. (1) Fig. 4. Variables in the geometric factor.
The diffused flux arriving at the detector proceedingfrom a perfect diffuser should be
FD J J L osa cosedAdA'
LA' 1 ff coso cosodAdA'AJA' A 7rr2
Since the geometrical factor of A seen from A' isdefined as
1 A J cosa cosodAdA'
we obtain FD = rLA'O.The angle in Fig. 3 is the semiangle under which the
detector is seen from the center of the illuminatedellipse.
Let us make two measurements:(1) With a large , L, the geometrical factor is X0L
and the diffused flux arriving at the detector is
(2)FDL = 7rLA'XL-
(2) With a small 6, 6s, we obtain ks and
FDS = 7rLA'0S, (3)
having got the small 5 by translating the detector to theleft in the direction of segment D.
From Eqs. (1) and (2),
FTD = - (4)AL
and from Eqs. (2) and (3),
FDS - FDL (5)OL/(S
If the sample is a badly polished mirror, we assume,as a hypothesis, that its surface is a mixture of smallzones of perfect mirror and others of perfect diffuser.If the sample is partly absorbent, the flux will be small-er. If the pattern proposed is illuminated, it will pro-duce a specular beam of luminous flux Fsp and sectionA" and a scattered beam FTD which propagates in alldirections of the hemispace.
It is reasonable to define the polish coefficient p asFSp.
Fsp + FTD
Thus it will be p = 1 for the specular surface and p = 0for the perfect diffuser.
If the detector is placed under L, it measures FL,both the specular and diffuse fluxes arriving at it,
FL = Fsp + FDL, (6)
and placing the detector under 6s, it measures Fs,
Fs = Fsp + FDS-
According to Eq. (5),
Fs = Fsp + FDL , (7)
and from Eqs. (6) and (7),
ILFS- OSFLF =
O ¢L ' OS(8)
FDL = (FL - Fs).OL S
Thus from Eqs. (4) and (9) we obtain
FDL 1 -(L-F)FTD = = O(FLs)
Therefore with Eqs. (8) and (10),OLFS- OSFL
(10)
LFS SFL + FL FS
IV. Calculation of the Geometric Factor
The geometric factor has been calculated for a 45°angle of incidence as shown in Fig. 2. For the meaningof symbols and geometric concepts, also see Figs. 3 and4. We have
The detector area A and the incident beam area A"must be large enough to allow great precision in theirmeasurement. A must be larger than A" to receive notonly the specular beam (area A") but also a fraction ofthe scattered light. The ratio A/A" = 2 is a compro-mise solution.
The maximum value for 6 is 450, but in practice S willbe <440.
For the numerical integration of the geometricalfactor we have taken I intervals for #', a', and 'y and I/2intervals for Al. The number I was chosen to have anerror <1/1000. The geometric factor results are givenin Table IV.
V. Mean Value and Uncertainty
To calculate the average value for p, several mea-surements are taken on various parts of the sample,and FSi and FLi are found. With a single measurementwhose beam should contain all the mentioned parts asa whole, we would obtain
Fs itP 2;Fs + ;FTDi
, 2 FS- - FL-FS - IFL, + --FLi - 2F'S OL PS OL S
OLFS -oSFL
OLFS - SFL + FL - Fs
Fs and FL being the mean values of Fsi anf FLi, respec-tively.
On the other hand, the luminous flux is dF = EdA,where E is the illumination on a point of the detector'ssurface. If E now represents the mean illumination ofthe detector,
F [ FL=EL A,
ae i f A Fs = Es b A,
and the expression for p becomes
O=LES - OSEL
LES - OSEL + EL - ES
This formula gives the mean value of p as a function ofthe geometric factors and the mean illumination of thedetector.
To calculate the systematic uncertainty, we considerthe following:
(1) The radius R of the detector and the radius R" ofthe diaphragm limiting the incident beam can be mea-sured and reproduced with a precision of 0.01 cm.Then
d (A)= R d + 2R dR1 .A"/ R 2 R"3
With nominal values R = 2.00 cm, A/Al = 2, dR = dR"= 0.01 cm, the above equation gives
d ()=0.048.(All)
(2) The distance D is measured with a scale with dD= 0.1-cm divisions. The semiangle is measured bythe ratio between the detector radius R and the dis-tance D. Then
5 arctan -)d =, d6 dR + R dDh D)= 1+ (R/D)2 (D D2
So, for 5 = = 340 DL = 3.0 cm d6L = 1.00 and for5 = s= 2 =*Ds = 573mm=*d5s = 0.010.
(3) The geometric factor may be obtained by nu-merical computation with a precision as large as welike. But 0 is a function of A/A" and , so
(4) The detector meter is calibrated with a lamp of2042 K color temperature which has been calibrated byanother standard lamp. The calibration curve gives E=E(V). Then
dE= E dV,av
where dV = 1 mVand E/OVis computable for VL andVs.
(6) The random uncertainty of p is calculated withthe experimental arrangement unaltered. Then FLand Os are fixed but not the individual values of illumi-nation. Here t being the Student factor for a degree ofconfidence of 95%, n the number of measurementsmade, Esi one of the measurements, and ES the meanmeasurements value, the random uncertainty R U(ES)of Es is
RU(Es) = t n(n -1)
and analogously for EL. Therefore the random uncer-tainty of p is
The experimental arrangement is shown in Fig. 5with the following nomenclature:
Lis an incandescent lamp with a small helical fila-ment calibrated to be an A standard illuminantwith input voltage VA. In our lamp VA = 6.0 V.The lamp's voltage was stabilized to ±0.01 V.
Ci is the collimating lens.Dlis a small diameter diaphragm (in our case 0.6
mm).C2is the objective.D2is the diaphragm to get the light beam diameter =
2.83 ± 0.1 mm.Mis the sample.
PDis the photometric detector (luxmeter).D3is the detector's diaphgram (40.0 + 0.1-mm diam-
eter).
The elements L, C1, D1, C2, and D2 were mounted onan arm of the optical bench and DF with D3 on theperpendicular arm.
Once L, C1, D1 , C2, and D2 were lined up and focused,the light beam was projected on a screen from 34 maway, and the spot diameter was measured to be 12 cm.Then the beam semidivergence A/2 is
PD
D3
I D2
o0
Fig. 5. Experimental arrangement.
L8 7 6 5S-
9 10 11 12
16 j1514 131
Fig. 6. Points of measurement.
12 2.83A 2 2 Atan =3 0 = 0.1.2 3400 2
The incident angle was 450, as mentioned earlier. Thechosen angles were AL = 340 and s = 2. Thesamples were 30 X 30 cm. The measurements weremade in the 8 odd zones of each sample (Fig. 6).
VII. Objective Results
The objective results of each variety: polish coeffi-cient p, random uncertainty RU, and systematic SUare shown in Table V. The order of the varietiesaccording to the increasing value of p is 10, 1, 6, 8, 9, 5,3,7,2,4.
VIII. Comparison Among Methods
The verification remains that the objective methoddoes not contradict the results of the statistical analy-sis carried out on the subjective visual classification.Comparing the physical results shown in Table V with
the subjective ones in Table II, we see that all therelations agree except that 9 is better than 5 because pq= 0.1079 and p5 = 0.1112.
The unsatisfied relation is only an apparent contra-diction, because we know that the true value of a mag-nitude is, with a 95% probability, between the intervaldefined by the mean value plus or minus the totaluncertainty. Then
p = 0.1079 + 0.0391 = 0°1470,10.0688,
p5 = 0.1112 + 0.0478 = 01010.0634.
So the p values do not distinguish whether sample 5 or9 is polished better.
The resulting ordering given by the mean ranks rimay be compared with those obtained from the objec-tive p values through Spearman's coefficient.5 If di isthe place difference of sample orderings, Spearman'scoefficient S is
S = 1-6 (x d2)/[10(102 - 1)];
in this case S = 0.83.As a third criterion we try to represent the p coeffi-
cients shown in Table V as a function of the meansubjective ranks shown in Table III. We obtain theregression line y = A + Bx and the linear correlationcoefficient r. The results are y = -0.01 + 0.04x and r= 0.72.
IX. Discussion
As we know the sample surfaces of rocks are hetero-geneous because of their mineralogical constitution.Furthermore the industrial polishing process givesplace to differentiated sharp zones with a poor finish.When an observer classifies a sample with bad zones,he penalizes the sample although its greater part ishighly polished. Thus the observer seems to take intoaccount not only the mean value but also the disper-sion of the measurement.
When the polish coefficient is measured in the opti-cal bench, we obtain the mean value p and the randomuncertainty RU. The last one is a measure of thedispersion. Then the subjective penalty may be afunction of the random uncertainty. Let us assumeboth quantities are proportional.
Table VI shows, in three decimal figures, p, p - RUandp-2RUfor each variety. If we take as the polishcoefficient the magnitude p - RU, we have for Spear-man's coefficient S = 0.88; and for p - 2RU we alsohave S = 0.94.
The statistics tables6 for two noncorrelated order-ings of ten samples give a probability << 1% for S = 0.88or S = 0.94.
For p - RU the regression line and the correlationcoefficient are y = -0.016 + 0.033x and r = 0.75; if wetake p- 2RU the linear correlation coefficient be-comes r = 0.65.
Therefore the best linear correlation coefficient cor-responds to p-RU. Moreover, the values of p-RUdo not contradict any subjective visual classifications.From the statistics tables we deduce that the probabil-ity of getting r 2 0.75 from two uncorrelated variablesis only 1% if the number of measurements is 10. So, astrong evidence exists for correlation between p - RUand the mean subjective ranks; the correlation may becalled highly significant.
The last two columns of Table V show the meanilluminations ES and EL of each variety. If we assumethe ES values are proportional to the glossmeter mea-surements, the order from the lowest gloss to the high-est is 10, 5, 6, 1, 3, 2, 4, 9, 7, 8, comparing this arrange-ment with the corresponding subjective order, we havefor Spearman's coefficient S = 0.81. Arranging ac-cording to the EL values, the order is 2,4,5,3,7,9,6, 10,8, 1, and the coefficient S = -0.54.
Comparing both orderings and their Spearman coef-ficients, we conclude that the glossmeter measure-ments depend on the distance from the sample to thephotometer. Therefore we do not recommend the useof the glossmeter for the measurement of marble andlimestone surface polish.
The authors wish to express their gratitude to theInstituto Geol6gico y Minero de Espaha for its finan-cial support and its authorization of the publication ofthis paper and to L. Garcia, J. Martin, and M. A. Diaz,assistants in the Department, for their professionalcontributions.
References
1. L. Asuni, G. Rossi, and I. Uras, "Analisi. . .Lucidat," paper pre-sented at First Convegno Internationale Sulla Cultivacione diPietre e Minerali Litoidi, Torino (1974), p. 13.
2. J. S. Christie, "An Instrument for the Geometric Attributes ofMetallic Appearance," Appl. Opt. 8,1777 (1969).
3. F. Gasc6n and M. Balbas, "Optical Measurement of GraniteSurface Polish," J. Test. Eval. 13, 367 (1985).
4. J. F. Marcotorchino and P. Michand, Optimisation en analyseordinate des donnees (Masson, Paris, 1979), p. 160.
5. J. R. Taylor, An Introduction to Error Analysis (UniversityScience Books, Mill Valley, CA, 1982), pp. 180-185.
6. E. Chacon and F. Miguez, Estadistica Aplicada (Rugarte S.L.,Madrid, 1980), pp. 444 and 445.
