Calculation of area change and volume displacement frommoire topograms of deformed membranes

Joris J. J. Dirckx and Willem F. Decraemer

A method is proposed to calculate the area change and the volume displacement of a deformed membranefrom the measurement of its displacement amplitude on a regular grid of observation points. The method isdemonstrated on a circular latex membrane deformed by air pressure. The displacement data are obtainedby phase shift moire interferometry. Area changes down to 0.5% and volume changes down to 16.1 mm3 aremeasured, and error flags are determined. Key words: Metrology, moire.

1. Introduction

When a membrane is loaded by a static or dynamicpressure it is deformed. Metrological techniques suchas holographic interferometry and moire interferome-try are generally used to measure the displacementamplitudes of such deformations.Y13 The deformationof the membrane causes an overall volume displace-ment and a change in the membrane surface area. Weshow how these two quantities can be determinedwhen displacement amplitudes are measured over aregular grid of observation points. The method isdemonstrated using shape data recorded with a phaseshift shadow moire interferometer.

II. Method

A. Area Calculation

Suppose that, with a not further specified metrologi-cal technique, the height z(xy) of a membrane hasbeen measured on a regular grid of points (xy) for themembrane in its rest position and for the membranedeformed by pressure and that the obtained data arestored as a matrix in a computer.

Before area change or volume displacement calcula-tions can be started the exact stationary boundary ofthe surface has to be determined. This can be done bymaking the difference between the shape measure-ment of the observed membrane in its rest position andthat of the deformed membrane. This matrix of dif-

The authors are with University of Antwerp-RUCA, Laboratoryof Biomedical Physics, B-2020 Antwerp, Belgium.

Received 5 June 1990.0003-6935/91/192757-04$05.00/0.© 1991 Optical Society of America.

ference values Az(xy) is then displayed on a monitor ina gray scale representation (variable intensity to repre-sent different Az values), and contours of equal differ-ence are superimposed on the picture (a standard fea-ture of most image processing or graphic softwarepackages). The boundary of the membrane is thelocus of points where the displacement is zero and is,therefore, precisely determined by the zeroth-orderdisplacement contour. The position of this contour inthe matrix is measured at a sufficiently high number ofpoints to obtain a smooth border line. In our proce-dure this is done by a cursor on the monitor screenwhich displays the contour map. The cross hair can bepositioned with a mouse or cursor control keypad, andthe coordinates are read into a file. Finally, a smallprogram interpolates between the indicated points sothat the closed boundary is finally defined at all rowsand columns crossing the membrane surface.

To explain the surface calculation procedure, let usconsider Fig. 1. Every four neighboring points in thematrix of measured z-values define a surface element.Each element may be subdivided into two triangles(shaded and dotted) as indicated in the figure. For asufficiently dense grid these flat triangles may be con-sidered as coinciding with the actual surface. Thearea of such a triangle can readily be calculated fromthe (xy,z) coordinates of its cornerpoints. For theshaded triangle in Fig. 1 the area is, e.g., given by

/2 2 + (Z+lj+l- j+1)2. * [dy + (Zij-Zij+1)2] (1)

where dx and dy are the distances between two subse-quent matrix columns and matrix rows, respectively.The elementary surface areas are now calculated for allmatrix points that lie within the boundary, and theirsum is a finite element approximation of the area of thetotal surface.

1 July 1991 / Vol. 30, No. 19 / APPLIED OPTICS 2757

Fig. 1. Height z of a curved surface measured on a regular grid ofpoints (X,Y). The area of the surface is calculated by adding thearea of the shaded and dotted triangles as indicated for all matrixpoints covering the surface. The area of each triangle is obtainedfrom the coordinates of its cornerpoints. The volume between thesurface and the X-Y plane is the sum of the prisms indicated, again

for all matrix points on the surface.

Finally, a small refinement of the algorithm makes itpossible to reduce the quantization error. At the be-ginning and at the end of a given row k the interpolatedboundary coordinate x will seldom coincide with thelast matrix node xk which lies within the boundary. Ifthe condition

I(x - X) + (X'+l - Xk+0) > d (2)

is met, half of the surface of the square containing xkand x'+, is added to the area comprised between row kand row k + 1.

B. Volume Change Calculation

In an analog way the volume between the surfaceand an arbitrary chosen reference plane, e.g., the X- Yplane in Fig. 1, can be calculated. Every triplet ofneighboring matrix points defines an elementaryprism with a triangular base on the X- Yplane and topon the object surface. The volume of a prism may becalculated in good approximation as the mean height(or z) of the top vertices times the surface of the flattriangle on the reference plane (which is the same forall matrix points). For the prism in Fig. 1 with thebase defined by [(xi,yj),(xi,y+1),(xi+i,yj+i)] the volumeis given by

[d. dy (zij + zij+l + zi+lj+,)]/6. (3)

The total volume between the reference plane and thesurface is again determined by the sum of all the prismvolumes defined by the matrix points which lie withinthe boundary.

The choice of the z-location of the reference plane isfully arbitrary, as the value of interest is not the calcu-lated volume but the volume change. This change isobtained as the difference in volume between twoshape measurements (using the same reference plane).

Ill. Demonstration Measurement

A. Apparatus

In the following experiment the shape of the investi-gated membrane is measured using phase shift moiretopography, a noncontacting optical technique whichallows automated full field shape measurement of dif-fusely reflecting objects. The moire method, the ap-paratus used, and the calibration procedure have beendescribed extensively elsewhere.46 Our apparatusworks with a 512- X 512-pixel matrix. For the givenexperimental geometry the depth measuring resolu-tion was 20 ,um and the spacing between two subse-quent pixels 46 Am in the x-axis direction and 32 gm inthe y-axis direction.

B. Membrane Mounting

To show the possibilities of the presented technique,we demonstrate it on a circular, flat, stretched, elasticmembrane. We used a piece of pure latex rubber, 0.16mm thick, cut out of a Lawton surgical glove. Themembrane is deformed by applying air pressure at oneside. To obtain axial symmetric behavior it is impor-tant that the tension is uniformly distributed through-out the membrane. Therefore, we devised a specialholder which is schematically depicted in Fig. 2. First,the circular piece of membrane is deposited on theinterior rim of part A. Then the O-ring is put on top ofthe membrane, and part B of the holder is pressed on.By doing this the O-ring is stretched by the inclinatedsides of the rims of parts A and B, and the membrane isuniformly stretched together with the O-ring. Final-ly, part C is screwed onto part A until the membrane istightened between the flat clamp surfaces of parts Aand B. The front side of part A has been chamfered sothat the illumination system of the moire setup doesnot cast a shadow of the holder onto the membranesurface. The diameter of the opening of the holder is7.7 mm, so that the actual surface of the flat membraneis 46.6 mm2 .

With this method of mounting we may reasonablyexpect the outward and inward movements of themembrane under, respectively, over pressures and un-derpressures to be fairly symmetric and hence findequal area and volume change for positive and negativepressures of the same magnitude.

C. Error Estimation

When measuring the z-coordinate with the moireinterferometer the obtained data will inevitably con-tain some noise. Due to this noise the calculated areaof the surface enclosed by the defined boundary willalways be slightly larger than the actual area. In the 1-D analog we can readily understand this as the lengthof a smooth curve from one point to another always

2758 APPLIED OPTICS / Vol. 30, No. 19 / 1 July 1991

Si(mm2)

47.0 . I

46.5 -

46.0

-4

Fig. 2. Membrane holder: the membraneMis uniformly stretchedby the O-ring when pressing part B of the holder onto part A; part C

is screwed onto part A until the stretched membrane is tightenedbetween the flat clamp surfaces.

:4

-2 'I 2 'AP(kPa)

AS

(%)

3

2

0

Fig. 3. Membrane surface area as a function of applied pressure.The scale on the right gives the relative area change. Within theerror flags the area change is equal for overpressures and underpres-

sures of the same magnitude.

AVi(mm 3 )

being shorter than the length of the drunkman's walkalong this curve. The effect of the noise can be keptsmall and can be reduced by smoothing the obtaineddata. The amount of smoothing is limited, however,because by too much smoothing all surface shape de-tail is lost.

As we have shown in a previous paper 6 the amount ofnoise on the z-coordinate of an object measured withthe moire interferometer depends directly on the z-position of the point. The calculated area will, there-fore, vary slightly with the overall position of the objectsurface along the z-axis. When the membrane is pres-surized it will change its z-position and thus the noiseat the various surface points. This will induce an errorin the calculated area change superimposed on theactual area change due to stretching of the membrane.To determine the upper limit of this artificial effect wehave positioned the unpressurized flat membrane atten different z-positions chosen in the region in whichthe surface will move during pressurization. Ideallythe area should remain unchanged for all of thesepositions, but due to the varying noise effect we founda variation of 0.35% between the largest and smallestarea value. This maximal difference gives the upperlimit of the artificial area change and can, therefore, beused as an error flag on the obtained area data.

In a similar way an upper limit for the error on thevolume change measurement was determined. Whenthe flat membrane is positioned at ten subsequentequidistant z-positions, a linear increment of the vol-ume between the membrane and the reference plane isto be expected. The maximal deviation from thislinearity gives the upper limit for the error on thevolume change measurement. For a measured mem-brane area of 45.77 mm 2 we found an error of 1.2 mm 3

or 0.026 mm 3/mm 2 of the surface area.The calculated area is not exactly equal to the area of

10-

10-

-4

I-2 0 'AP

(kPa)

Fig. 4. Overall volume displacement of the membrane as a functionof applied pressure. For the highest pressure values there is a

deviation from linearity.

the membrane but rather gives the area of the surfacewhich lies within the indicated boundary. As theboundary is found by interpolation between a limitednumber of indicated points, small portions of themembrane surface will be lost at the circumference.For the flat membrane we measured an area of 45.77mm2 or 1.8% less than is theoretically to be expected.On the surface change of the deforming membrane thiseffect cancels out; as for all measurements the sameboundary is used, so that there we may use the noise-induced error of 0.35%.

Finally, we also repeated the area measurements ofthe unpressurized flat membrane for different z-posi-tions. The reproducibility proved to be very good, asthe largest difference between corresponding measure-ments did not exceed 0.05%. For two successive mea-surements of the area of the membrane deformed un-der a 3-kPa pressure load, the difference was <0.1%.

1 July 1991 / Vol. 30, No. 19 / APPLIED OPTICS 2759

Z Z a

IAVI(m m3)

16

1 2

8 1

4

0 0 1 2 3 IA PI

(kPa)Fig. 5. Absolute value of overall volume displacement vs the abso-lute value of applied pressure. The data corresponding to overpres-sures are indicated by triangles. Within the error flags the value ofIAVI is the same for overpressures and underpressures of the same

magnitude.

D. Results

The shape of the membrane was measured for therest position and for the membrane deformed by ap-plying 1-, 2-, and 3-kPa over pressure and underpres-sure. Figure 3 shows the area of the membrane as afunction of pressure. For corresponding over pres-sures and underpressures we see that, within the errorflags, the area change is the same, as expected. For +3kPa the area increase is (2.6 0.35%), and for -3 kPathe area increases with (3.08 + 0.35)%, both relative tothe area of the undeformed membrane. The figureclearly demonstrates that our method indeed offersthe possibility of measuring small area changes, downto 0.5%, induced by applying small pressures to a mem-

brane. In Fig. 4 we see the volume displacement of themembrane as a function of applied pressure. For +3kPa the volume between the membrane and the refer-ence plane decreases with (16.1 ± 1.2) mm3, and for thecorresponding underpressure of -3 kPa the volumeincrease is (17.2 + 1.2) mm3. For the highest pressurelevels we notice a deviation from linearity, because theelasticity modulus of the latex rubber and the tensionin the membrane do not remain constant with increas-ing stretch. In Fig. 5 we plotted the absolute value ofthe volume change vs the absolute value of the appliedpressure. The values of JAVI obtained for overpres-sures are indicated by triangles; those obtained forunderpressures are indicated by circles. From Fig. 5 itis seen that also for the volume displacement the ex-pected symmetry between inward and outward motionis present within the error flags.

References1. C. M. Vest, Holographic Interferometry (Wiley, New York,

1979).2. 0. D. D. Soares, Optical Metrology (Martinus Nijhoff, Dordrecht,

1987).3. J. Der Hovanesian and Y. Y. Hung, "Moire Contour-Sum Con-

tour-Difference, and Vibration Analysis of Arbitrary Objects,"Appl. Opt. 10, 2734-2738 (1971).

4. J. J. J. Dirckx, W. F. Decraemer, and G. Dielis, "Phase ShiftMethod Based on Object Translation for Full Field Automatic 3-D Surface Reconstruction from Moire Topograms," Appl. Opt.27, 1164-1172 (1988).

5. J. J. J. Dirckx and W. F. Decraemer, "Phase Shift Moire Appara-tus for Automatic 3-D Surface Measurement," Rev. Sci. Instrum.60, 3698-3701 (1989).

6. J. J. J. Dirckx and W. F. Decraemer, "Automatic CalibrationMethod for Phase Shift Shadow Moire Interferometry," Appl.Opt. 29, 1474-1476 (1990).

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