Adaptive aperture defocused digital speckle photography

Adaptive aperture defocused digital speckle photography

Jose M. DiazdelacruzDepartment of Applied Physics, Faculty for Industrial Engineering, Polytechnic University of Madrid, Jose Gutierrez

Abascal 2, 28006 Madrid, Spain

[email protected]

Received 12 April 2007; revised 21 June 2007; accepted 25 June 2007;posted 26 June 2007 (Doc. ID 82020); published 14 August 2007

Speckle photography can be used to monitor deformations of solid surfaces. Its measuring characteristics,such as range or lateral resolution, depend heavily on the optical recording and illumination setup. I showhow, by the addition of two suitably perforated masks, the effective optical aperture of the system mayvary from point to point of the surface, accordingly adapting the range and resolution to local require-ments. Furthermore, by illuminating narrow areas, speckle size can be chosen independently from theoptical aperture, thus lifting an important constraint on the choice of the latter. The technique, which Ibelieve to be new, is described within the framework of digital defocused speckle photography undernormal collimated illumination. Mutually limiting relations between the range of measurement and thespatial frequency resolution turn up both locally and when the whole surface under study is considered.They are deduced and discussed in detail. Finally, experimental results are presented. © 2007 OpticalSociety of America

OCIS codes: 030.6140, 120.4290.

1. Introduction

When a solid object undergoes a load change, its de-formation field exhibits different behaviors over thesurface. For instance, when a vertical force is appliedat the free end of a horizontal cantilever beam, slopechanges are bigger and more uniform in the vicinityof the load than near the supported end [1]. In otherwords, the ranges for spatial frequencies and magni-tudes of the slope variation fields are not evenly dis-tributed over the object surface.

On the other hand, when optical methods are usedto measure the surface deformation, they are gener-ally tuned to provide adequate characteristics for thewhole surface, so that there is one range of measure-ment and one lateral resolution. This may lead tocompromised solutions where the high values ex-pected in some areas reduce the lateral resolutionavailable even in points with anticipated lower val-ues.

Defocused speckle photography has long beenused to measure the distribution of out-of-plane ro-tations over a surface under load changes. However,its measuring ranges for rotation and spatial fre-

quency are the same over the whole area understudy. Moreover, when digital recording systemsare used, these values are strongly conditioned bythe camera resolution. I describe what I believe tobe a new enhancement of the system that makes itscapabilities more adaptable both by allowing differ-ent measuring characteristics over the surface andby untying its relation to camera resolution.

When a visible laser beam is scattered by a roughsurface, the reflected light intensity exhibits a grainydistribution, called a speckle pattern. The origin ofthis phenomenon is the interference of the light com-ing from all the points of the surface. If noncoherentlight is used, the interference patterns vary soquickly that only the average intensity is observedand therefore speckles appear only under coherentillumination.

The intensity pattern of the light scattered from arough surface can be collected by an optical systemand recorded on a plane. Each setup determines theway in which the light from different points in thesurface interfere and thus the amplitudes, spatialfrequencies, average speckle size, and other charac-teristics of the intensity pattern at the recordingplane [2]. For a given setup, the speckle distributionrepresents a unique signature of the surface underobservation. When the surface undergoes a mechan-

0003-6935/07/246105-08$15.00/0© 2007 Optical Society of America

20 August 2007 � Vol. 46, No. 24 � APPLIED OPTICS 6105

ical transformation the interference combinations atevery point in the recording plane are altered, butsometimes they can be partially reconstructed in adifferent point at the recording plane. In this case,the speckle pattern is said to be shifted (displaced)and decorrelated (slightly modified).

Speckle photographic techniques explore the pos-sibilities of determining the object transformationfrom the speckle shift that takes place at the record-ing plane. When it is at the back focal plane of theoptical setup, the method is called defocused specklephotography (DSP). If a digital detector (such as aCCD camera) is used to record the speckle pattern,then the system is said to be a defocused digitalspeckle photographic (DDSP) system. Under normalcollimated illumination, this technique is sensitive toout-of-plane rotations (or tilts) of the surface underobservation. The speckles are displaced in the backfocal plane of the lens by a distance that is propor-tional to the rotation angle (provided it is small) [3].It will be assumed that the investigated area is partof a rough planar surface.

The recorded speckle pattern (or specklegram) isstored in a computer. Then a load change is appliedto the object and a second specklegram is obtained.Once the two specklegrams are available, computeralgorithms are applied to find the speckle displace-ment with subpixel accuracy. Finally, the distribu-tion of the tilt throughout the area is evaluated.

When digital recording is used, speckle size consid-erations may play an important role in the design ofthe optical system. Speckles should not be smallerthan the detector cell, because the speckle patternwould be spatially averaged in the recorded speckle-gram. This leads to a reduction in the speckle con-trast and eventually to a total loss of the pattern. Onthe other hand, if speckles are much wider than thedetector cells, many pixels are necessary to computethe speckle displacement, leading to poorer lateralresolution. Moreover, random errors in the results ofdigital processing depend on speckle size, so that itshould be kept as small as possible, yet taking intoaccount the previous considerations. Some studieshave been published in order to set the optimalspeckle size in digital speckle photography, so that itsoptimum value s* lies close to

s* � 2w, (1)

where w is the pitch of the cells in the sensor array[4–6].

Speckle size in the recording plane of a defocusedspeckle photographic system is determined by thenarrowest aperture am of the light arriving at a pointof the detector:

s �1.22�f

am, (2)

and, therefore, it should be assured that

am �1.22�f

s* . (3)

As already stated, DSP is used to assess the distri-bution of slope variations over a surface when a loadchange is produced. The aperture of the optical sys-tem determines both the range and the lateral reso-lution of measurement. It will be shown that bigapertures allow larger ranges and narrower resolu-tions, and small apertures work in the opposite way.

The core of the method described in this paper isthe use of a pair of coupled masks to illuminate andcollect light from the surface into the digital camera.The masks should be suitably perforated so as toprovide the speckle size, effective aperture, lateralresolution, and measuring range from a distributedsystem approach. Here the mathematical relationssatisfied by the main parameters and their mutuallimitations are analyzed.

Previously described implementations of the methodexhibit two limiting characteristics:

(a) The optical system has one aperture, and thus,the measuring range and lateral resolution are sharedby all the points in the surface under observation.

(b) The aperture of the system sets the alreadymentioned measuring characteristics and the specklesize, so that it is not possible to tune them indepen-dently.

The system presented here relies on a slightly mod-ified implementation that improves on the aforemen-tioned problems. First, the optical setup is describedand then its main features are analyzed.

In short, as already stated, lateral resolution andmaximum range depend on the effective aperture ofthe optical system, so that in order to relieve theaperture determination from speckle size consider-ations, we use an illumination mask that produces anillumination pattern made of a discrete set of narrowcircles of diameter a, so that the speckle size is pri-marily determined by a. In addition, a second mask isadded to assign a different entrance pupil for everyilluminated area. Therefore, it is possible to havedifferent speckle sizes, lateral resolutions, and mea-suring ranges over the illuminated area.

Recording areas and their individual cells are oftenrectangular, although in this work, for the purposeof simplicity, they are assumed to be square. In thefollowing sections, the sides of the sensing area andthe individual cell squares will be assumed to be b, w,respectively. In addition, a focal length f, an object tolens optical distance d, an L � L square observedarea, and a lens aperture diameter D are assumed.Further, to use the maximum recording area, thefollowing relation,

bf �

Ld , (4)

is supposed to hold.

2. Antecedents

The first paper describing a defocused two-exposuremethod to measure out-of-plane rotations was writ-

6106 APPLIED OPTICS � Vol. 46, No. 24 � 20 August 2007

ten by Tiziani [7] and was later extended for vi-bration analysis [8]. If normal illumination andobservation are used, the speckle shift at the record-ing plane is given by [9]

dx � 2f�, (5)

dy � �2f�, (6)

where �, � are the (small) rotation angles around thex, y axis of a Cartesian system placed on the meanplane of the object surface, and f is the focal length ofthe recording system. Lateral displacements do notappreciably alter these values.

Gregory considered divergent illumination andshowed that when the optical system is focused on theplane that contains the image of the point sourceconsidering the object surface as a mirror, the speckleshift depends only on out-of-plane tilts [10–12].Chiang and Juang described a method to measure thechange in slope by defocused systems [13]. A greatnumber of more recent papers document the use ofdefocused speckle photography to measure in-planeand out-of-plane rotations and strains [14–17]. TodayCCD cameras store the specklegrams taken beforeand after the mechanical transformation in a digitalcomputer; adequate algorithms reveal the speckleshift distribution with subpixel accuracy [18–21].

Figure 1 represents a typical setup for measuringslope changes in a solid surface. A laser source (LS)emits a light beam, half of which goes through thebeam splitter (BS) and reaches the rough surface (S).Part of the light scattered from S is reflected by theBS and recorded at the back focal plane (R) of lens L1.Henceforth, this system will be referred to as a directillumination digital defocused speckle photography(DIDDSP) setup.

For a defocused recording system, the speckle sizeis given by [22]

s �1.22�f

D . (7)

When a part of the object surface undergoes anout-of-plane rotation �, the light scattered from itexperiences a rotation 2� and completely falls offthe aperture of the system when

2� �Dd . (8)

Thus the maximum measurable rotation is

� �D2d . (9)

Neglecting diffraction effects, the diameter of thearea that reflects light to the same point at the de-tector plane is D. Consequently, the lateral resolution� of the measurements is equal to D. The relation

�

�

12d (10)

represents the mutual limitation on range and spa-tial frequency, which holds for any aperture.

With regard to the requirements posed by the useof a digital system, it follows that, if the optimumspeckle size is s*, diameter D should be chosen ac-cording to

D �1.22�f

s* , (11)

and hence

� �1.22f�2ds* , (12)

�1.22�f

s* . (13)

3. Adaptive Aperture Model

In Fig. 2 the new system is depicted. A beam from theLS is expanded and spatially filtered to obtain a col-limated beam at least as wide as the area understudy. The beam is split by the perforated mask (IM)into a set of narrow beams of diameter ai, provid-ed that the wavelength � is much smaller thanai �� ai�. A BS lets half of the radiation arrive atthe diffuse surface �. The light reflected by � reachesthe BS again; half of its intensity goes through asecond perforated mask (AM) and is finally recordedon the back focal plane of lens L1. The circular holes

Fig. 1. DIDDSP setup for recording defocused specklegrams of arough surface.


in the AM are aligned with the rays coming from thecenters of the holes in the IM to the center of L1. Werefer to this setup as an adaptive aperture defocuseddigital speckle photographic (AADDSP) system. Anequivalent uniaxial system is depicted in Fig. 3. Fig-ure 4 represents an IM (a), a corresponding AM (b)and the resulting specklegram (c).

Each collimated beam emerging from an IM illu-minates a circle i of diameter ai in the object surface�. At each point in i a light cone originates limitedby a corresponding circle ei of diameter ci in the AM.The aperture diameter D of L1 is to be computed so asto exclude the light that pasess through any othercircle �ej, j � i� in the AM, yet allowing all the radi-ation from the corresponding circle �ei� to reach therecording plane, where it forms a speckled circle ri ofdiameter gi.

The speckle size at ri depends mainly on the nar-rowest aperture encountered by the cone (which iseither ai or ci), so that by assuring the condition

ci � 4ai, (14)

the speckle size is made practically independent fromci and can be approximated by

si �1.22�f

ai. (15)

Condition (14) also limits the decorrelation and in-tensity fading that occurs toward the border of thecircle.

The optimum speckle size s* depends mainly on thecamera resolution, so that the diameters ai should allbe equal to

a �1.22�f

s* . (16)

Next, it has to be assured that the circles ei on theAM do not overlap, provided that the circles ri in therecording plane do not. Taking into account that ev-ery circle ri of diameter gi inside the sensor areacorresponds to a circle ei of diameter ci in the aperturemask and that a lateral separation b at the recordingplane corresponds to the AM width p (see Fig. 3), it isnecessary that

pci

�bgi

, (17)

where gi is given by

gi � fci

m, (18)

p � nbf , (19)

Fig. 2. AADDSP setup, showing two suitably perforated masks toenhance measuring possibilities.

Fig. 3. AADDSP uniaxial equivalent to make geometry simpler.

Fig. 4. (a) Illumination mask (IM); (b) aperture mask (AM); (c)speckle circles at the recording plane.


with m, n as the distances from the IM to the objectand the lens surfaces, respectively, so that

m n. (20)

On the other hand, spatial considerations limit thepossible values for m:

m � L, (21)

so that

L m d2 �

fL2b, (22)

that entails

f � 2b, (23)

which is easily satisfied by current popular CCD cam-eras.

As yet another restriction, it is necessary to keeplasers in an illuminated spot from getting throughthe aperture of another spot and reaching the en-trance of the system. If the separation between thecenters of two neighboring holes at the IM is �, thedistance between the centers of the correspondingcircles at the AM is

�c �n�

d , (24)

that determines an angle for the deviated ray fromthe chief one

�c �c2

m , (25)

which has to be stopped, thus

D2 d

�c �c2

m �nm � �

d2m c. (26)

Nevertheless, the aperture must let the laser beamsfrom the corresponding circles at the IM go throughthe AM, so that

D � cdm . (27)

Taking into account that the measuring range � forthe tilts is given by

� �c

2m, (28)

we arrive at

�d D2

nm � � �d, (29)

so that the maximum measuring range for all thesurface is

� �D2d, (30)

which implies a minimum lateral resolution � at thepoints where the maximum range is allowed, so that

�2md�

n , (31)

which is further limited by the nonoverlapping con-dition for the circles at the detector plane

� 2d�, (32)

which, taking into account condition (20), is morerestrictive, so that

� 2d�. (33)

In order to make condition (14) easier to fulfill, andconsidering Eq. (28), the setup will be arranged sothat

m � n �d2. (34)

Substitution for D in Eq. (33) yields

L

�b

2f� . (35)

The term b�f in popular cameras is of the order of0.2. If a maximum measuring range of 1 � 10�2 radis desired then a 10 � 10 (or bigger) matrix can beobtained.

Condition (14) translates into

4a � 2�m, (36)

which for � � 1 � 10�2 rad and d � 100 cm yields

a � 2.5 mm, (37)

which is satisfied for typical detector cells whosewidth w is of the order of 10 �m and requires valuesof a

a �1.22f�

2w � 0.5 mm, (38)

assuming � � 0.5 �m, f � 20 mm, w � 12.2 �m.Finally, condition (14) would yield

� �2.44�f

wd . (39)


4. Discussion

As stated earlier, is the maximum measuring rangeand � is its associated minimum lateral resolution, sothat the sampled points on the surface may havebetter lateral resolutions �� , although at the costof lower ranges �� . Taking into account Eqs. (35)and (4), it follows that the possible values of �, forAADDSP are mutually limited by Eq. (10), exactly asfor DIDDSP. Once , � are set in AADDSP, the mea-suring range � and the lateral resolution � for eachsampled point in the surface can be chosen takinginto account their mutually limiting relation

� �

d � �, (40)

so that all possible pairs ��, �� are those contained inthe hatched area in Fig. 5. The maximum resolvablespatial frequency corresponds to a lateral resolution�2, with the further restriction arising from condi-tion (39). The possible values for �DI, DI using con-ventional DIDDSP are those placed on the dashedline in Fig. 5. The main advantages of AADDSP arethe possibility of having different measuring charac-teristics throughout the surface and the tuning of Dwith attention only to the measuring characteristics,because the speckle size is independently set by thechoice of a.

For � � 0.5 �m, f � 20 mm, w � 12.2 �m,d � 100 cm, DIDDSP would require values of D �0.5 mm and �DI � 0.5 � 10�3 rad, DI � 0.5 mm.With AADDSP, there are ample choices for . If� � 1 � 10�2, then a minimum resolution � can varyfrom 5 to 10 mm.

Assuming a maximum measuring range , thevalue of D is established by the equation

D � 2d�. (41)

Accordingly, the possible values of �, � lie in the seg-ment joining the points �, ��, ��2, 0� excludingthose that do not satisfy condition (39). This condition

may be rewritten as

� � 8�DI, (42)

which entails

� 8DI, (43)

so that for the same optical and recording system, theworst lateral resolution in AADDSP is always at leasteight times poorer than in DIDDSP. However, thisdrawback may be outweighed by the advantages thatwill be mentioned in Section 6.

If abstraction is made of speckle and pixel sizes,(for instance, by considering different optic elements)the DIDDSP technique may be tuned so that DI, �DI

may lay on any point of segment AB in Fig. 5. ForAADDSP, it is a segment (not just a point) that can bechosen, and it is any segment whose slope is twicethat of AB and whose right-top end lies on AB.

5. Experimental Results

The experimental test of the AADSP technique in-volved the preparation of illumination and aperturemasks. Both had five circular holes, although whileall the diameters were nearly equal �� 0.5 mm� inthe former, they ranged from 3 to 7 mm in the latter.Thin 10 � 10 cm square tin plates were perforated bypin tips in order to make the illumination and aper-ture masks. They were rigidly held in place by grip-ping supports fastened to the optical table on whichthe rest of the elements were fixed. The average di-ameter of each hole was measured by means of aprofile projector. An 8-bit monochrome CCD camerawith an f � 20 mm focal length, 752 � 582 effectiveoutput pixels, 11.6 � 11.2 �m cell size, and 8.8� 6.6 mm sensing area was connected through animage capture board to a 1 GByte RAM and 1.66 GHzpersonal computer. A red He–Ne laser source(632.8 nm wavelength) was used. Digitized imageswere divided into square subimages, each one con-sisting of a speckled circle on a dark background. Arough thin rectangular metal plate was held alongone side and a load was applied on the opposite so asto obtain the theoretical small flexural deflectiongiven by

z�x� � �� x3

2L3 �3x2

2L2�, (44)

where is the deflection at x � L, assuming that theplate is held at x � 0. Derivation of Eq. (44) withrespect to x yields

z��x� � ��3x2

2L3 �3x

L2�, (45)

so that the tilt distribution ��x� when is changed to� � � is theoretically given by

Fig. 5. Measuring range versus lateral resolution for DIDDSPand AADDSP.


��x� � ��3x2

2L3 �3x

L2�, (46)

and represented in Fig. 6. In the experiments carriedout, � was set to 0.4 mm by pushing down the sideat x � L (L was 10 cm).

A 5 cm long five-hole aperture mask was used tomeasure the tilt of the plate at different zones alongits span. The positions on the plate of the centers ofthe illuminated circles are given in the second columnof Table 1. The illumination mask holes were all thesame diameter, because their role was to set thespeckle size. The holes in the aperture mask werelarger near the supported side for the first zone, be-cause bigger tilts were expected there. The hole sep-aration was larger near the pushed plate side,because the theoretical model predicted less unifor-mity there. Their sizes ci are given in the third col-umn of Table 1. On the other hand, holes weresmaller near the supported side, because the greatertilts happened near the other end of the plate.

One photograph was taken before loading the plate�� 0�, and a second one after deflecting it. Thedistance d from the lens to the rough surface was 1 m.Tilts, according to Eq. (46), depend on the deflectionwhen loaded. By pushing down the free edge of theplate, the tilts appearing in column 4 of Table 1 areexpected. The next columns represent the experimen-tal results and the deviation between theoretical and

measured values, respectively. Four runs were car-ried out without significant differences betweenthem. The table reflects the results of one of theseruns. It should be noted that the DIDDSP systemcould not measure the tilts tested in the experiment,because, as stated earlier, its range would have notexceeded 0.5 � 10�3 rad.

6. Conclusion

One advantage of this new method is its greaterrange of measurement, which is at least eight timeslarger than in DIDDSP, assuming the same digitalcamera. A second advantage is the possibility of dif-ferent sensitivities in measuring tilts at differentpoints of the surface. The DIDDSP method has onesingle value for measuring range and lateral resolu-tion, which can be matched by AADDSP in the max-imum range points, although this technique allowsfiner lateral resolutions in other points at the cost ofsmaller ranges, according to the relation representedby segment AB in Fig. 5.

The third advantage of AADDSP is that the re-quirements imposed by the recording system (specklesize) can be met without restraining the choice pos-sibilities for range or lateral resolution. This is be-cause the speckle size is set by properly choosing thediameter a of the holes in the illumination mask. InDIDDSP it was the speckle size that set the apertureand thus the measuring characteristics. Moreover,in many cases (such as plate bending analysis), thepractical values of slope change allowed by the size ofthe recording cells in DIDDSP fall too short andAADDSP finds its primary applications. Finally, thistechnique is particularly interesting for assessing de-viations from standard behaviors. The illuminationand aperture masks are designed to provide an ade-quate measuring range at every point utilizing areference model for the normal tilt distribution (ob-tained either from mathematical equations or previ-ous experience). Then, if some abnormality is presentat the object, the system can detect and measure itsdifferential effect at any sample point covered by thecamera.

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Adaptive aperture defocused digital speckle photography