three-dimensional angle measurement based on propagation...

Three-dimensional angle measurement based onpropagation vector analysis of digital holography

Lingfeng Yu,1,* Giancarlo Pedrini,1 Wolfgang Osten,1 and Myung K. Kim2

1Institut für Technische Optik, Universität Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany2Department of Physics, University of South Florida, Tampa, Florida 33620, USA

*Corresponding author: [email protected]

Received 16 January 2007; revised 18 March 2007; accepted 20 March 2007;posted 21 March 2007 (Doc. ID 78933); published 18 May 2007

We propose what we believe to be a novel method for highly accurate three-dimensional (3D) anglemeasurement based on propagation vector analysis of digital holography. Three-dimensional rotations inspace can be achieved by use of a CCD camera and a multifacet object, which reflects an incident waveinto different directions. The propagation vectors of the reflected waves from the object can be extractedby analyzing the object spectrum of the recorded hologram. Any small rotation of the object will inducea change in the propagation vectors in space, which can then be used for 3D angle measurement.Experimental results are presented to verify the idea. © 2007 Optical Society of America

OCIS codes: 120.0120, 090.1760.

1. Introduction

Techniques to measure small rotation angles of anobject are widely used in both automotive and indus-trial applications. For example, optical clinometersare used for calibration, alignment, machine setting,boresighting, fire control alignment, and inspectionwhen the tightest tolerances are expected and re-quired [1]. During the past decade, many differentangle measurement techniques have been developed.Among them, the most widely used are those basedon autocollimators [2,3], interferometers [4–6], andtotal internal reflection [7–9]. An autocollimator is anoptical apparatus that projects an illuminated reticleto infinity and receives the reticle image after reflec-tion from a mirror stuck onto a measured object. Angleinformation is obtained by comparing the positionsbetween the reflected image of the collimator reticleand the eyepiece reticle. Autocollimators are simplein design; however, the size of the system will be verylarge if high accuracy is required. Angle-sensitiveinterferometers are widely used because of their highaccuracy and ease of use. Their angle measurement isbased on changes in optical path length, which has anonlinear sinusoidal relationship with the rotation

angle; thus sinusoidal error may occur. Furthermore,it is difficult to reduce the size of the system since itsresolution is determined by the length of the sinearm. The method based on total internal reflectionuses two equal-intensity beams, which are incidentupon two separate prisms and reflected in the vicinityof the critical angle. The small rotation angle of thetwo prisms can be estimated by measuring the reflec-tance of the beams from the two prisms, but it is quitesensitive to variations in the intensity of the lightsource or stray light. There are also some other opti-cal techniques based on fringe analysis [10], specklecorrelation [11], or orthogonal parallel interferencepatterns [12].

However, all the above-mentioned techniques weredeveloped and applied for one-dimensional (1D) ortwo-dimensional (2D) angle measurement. Motivatedby the increasing demands on the precision of micro-systems, compact three-dimensional (3D) angle mea-surement systems with high accuracy are needed.However, a simple combination of multiple 1D or 2Dangle measurement techniques in one system is usu-ally not the best solution for 3D angle measurement,and it will make the whole system rather complicatedin structure. Recently, C. H. Liu et al. [13] proposed a3D angle measurement method that uses two position-sensitive detectors to measure the spot positions ofdiffracted waves from a grating. This method is

0003-6935/07/173539-07$15.00/0© 2007 Optical Society of America

10 June 2007 � Vol. 46, No. 17 � APPLIED OPTICS 3539

based on the measurement of the spot movementson different sensors, and its measurement resolu-tions around three axes are limited to 0.1–0.2 minwithin a measurement range of 1–2 degrees.

This paper proposes what we believe to be a novelmethod for 3D angle measurement based on propa-gation vector analysis of digital holography, where asingle CCD camera is used and high accuracy isachieved. A multifacet object (or more precisely, anobject that can reflect or diffract an incident wave todifferent directions) is particularly used as an indi-cator of rotation. Any small 3D rotations of the objectwill induce a change in the propagation vectors inspace, which will be manifested by the spatial spec-trum of the object. Thus, by evaluating the change ofthe object spectrum in holograms, change of the wavepropagation vectors can be traced and 3D angle mea-surement is possible. To the best of the authors’knowledge, this paper presents the first 3D anglemeasurement method based on a single detector.

2. Principle

Suppose the multifacet object is observed by a digitalholographic system based on a Michelson interferom-eter as shown in Fig. 1. The laser beam is split atbeam splitter BS into reference and object beams,and each part is focused by lens L1 onto the focalpoint F1 or F2. Point F2 is also the front focus ofobjective L2, so the object is illuminated with a col-limated beam. The plane S is imaged to the CCDcamera by lens L2. In the reference arm the beam isalso collimated by lens L3, which results in a magni-fied image at the CCD camera of an interferencepattern that would exist at S if the object wave weresuperposed with a plane wave there. Thus an off-axishologram can be recorded by slightly tilting the ref-erence mirror REF. The object angular spectrum canbe easily separated by a bandpass filter if the off-axisangle of the two beams is properly adjusted [14–17],and the object complex amplitude at different plane z,E�x, y, z�, can be reconstructed. From Fourier optics,E�x, y, z� can be expressed as

E�x, y, z� ��S��, �; z�exp�i2��x � �y��d�d�, (1)

where S��, �; z� is the corresponding angular spec-trum of the object wave and � and � are the spatialfrequencies of x and y, respectively. The complex-exponential function exp�i2��x � �y�� may be re-garded as a plane wave propagating with spatialfrequencies � � cos ��, � � cos ��, and � cos��, where cos �, cos �, and cos are the directioncosines of the normal propagation vector n of theplane-wave component, and � ��2 � �2 � �2 is thespatial frequency along the z axis. From above,the field E�x, y, z� can be viewed as a composition ofmany plane-wave components propagating with dif-ferent angles in space and with the complex ampli-tude of each component equal to S��, �; z�. Each point��, �� on S��, �; z� presents a plane-wave componentpropagating with a unit normal vector

n � �cos �, cos �, cos �T � ��, �, �T, (2)

where the superscript T represents the transpose of avector.

For simplicity, if a two-facet object is considered,the normal propagation vectors of the two reflectedplane-wave components from the object are labeled asn1 and n2 in an absolute x–y–z coordinate system.Figure 2(a) shows the top view of the two-facet objectin the x–y–z coordinate system. We also use frame {A}or xA–yA–zA to represent this absolute frame in Figs.2(b)–2(d). To fulfill 3D angle measurement, anotherlocal coordinate system xB–yB–zB or frame {B} is in-troduced to attach to the object. Thus, as the object isrotating, local frame {B} will also change its orienta-tion in space. Thus, by tracing the relative orienta-tion or rotation of frame {B} to absolute frame {A}, a3D angle can be measured. Note that we can assumethat frames {A} and {B} share the same origin, sinceonly 3D rotations are of interest, and translationswill not affect the angle measurement.

It is interesting to note that, except for a con-stant (�) scale difference, the xA–yA projections of thetwo normal vectors n1 and n2 are exactly the same asthe spatial frequency distribution ��i, �i� �i � 1, 2� ofthe object spectrum, as indicated by Fig. 2(b). Thus,by extracting the object spectrum in digital hologra-phy and analyzing the direction cosines of the twoplane-wave components, the propagation vectors n1

� ��1, �1, 1�T and n2 � ��2, �2, 2�T can be preciselyobtained. In practice, the spatial frequencies ��i, �i�can be obtained by either fitting the reconstructedphase information of each object facet to a form�i�x, y� � 2��ix � �iy� by a least-squares method orsimply evaluating the peak position of the object spec-trum.

Note that, although the object is composed of twomirrors, the surface is not a diffuse one; however, thesplit or the connecting edges between the two mirrorsare diffuse, and by observing which we can tellwhether the recorded or reconstructed object is in focusor not. By reconstructing the object in focus by thedigital holographic method, diffraction of the edges orFig. 1. (Color online) Apparatus for 3D angle measurement.

3540 APPLIED OPTICS � Vol. 46, No. 17 � 10 June 2007

mixing between the two reflected object waves can beavoided. Furthermore, as will be discussed later, it isalso possible to use the digital holography method toreconstruct wavefields and analyze propagation vec-tors of more complex objects at different reconstructionpositions, where a straightforward fast Fourier trans-form (FFT) fringe analysis [10] for the interferencepattern (interferogram) would be inadequate.

The obtained phase information represents thephase difference of the two-facet object relative tothe reference mirror. Thus it is reasonable to de-fine the xA–yA plane parallel to the reference mirrorand the zA axis coincident with its normal direction.Note that the phase information ��x, y� of each objectfacet only gives 2D angle information. In other words,neither digital holography nor the FFT fringe analy-sis method itself provides 3D angle information. Wepropose a novel algorithm to achieve 3D angle infor-mation by combining the (two) spatial frequencypairs ��i, �i� of the object. Vectors n1 and n2 define aplane P in space, as shown in Fig. 2(c). To evaluatethe 3D orientation, another local frame {B} is attachedto plane P. The three axes of {B} can be specified inframe {A} as

AzB � �n1 � n2�� n1 � n2 ,

AxB � �n1 � n2�� n1 � n2 ,

AyB � AzB � AxB, (3)

where the symbol � means the cross product of twovectors and || represents the modulus of a vector.Actually, frame {B} can also be defined as any otherforms by choosing different local axes, but not limitedto Eq. (3). The rotation matrix of frame {B} rela-tive to frame {A} can be calculated by describing thecoordinate system {B} in the system {A} as B

AR ��AxB

AyBAzB�. Once the rotation matrix is known, we

can decompose the object orientation as three rota-tions taking place about an axis in the fixed referenceframe {A}. For example, starting from a position co-incident with frame {A}, frame {B} can be obtained byrotating about zA by an angle �1, then rotating aboutyA by an angle �2, and finally rotating about xA by anangle �3; thus the total transform matrix B

AR can beexpressed as

BAR � RxA��3�RyA��2�RzA��1�, (4)

where Ri�� represents the transfer matrix of a rota-tion about an axis i by an amount of �. Expanding theabove matrix multiplication and comparing it withthe known 3 � 3 rotation matrix B

AR � tij, (i, j � 1,2, 3) from Eq. (3), the fixed rotation angles can finallybe calculated as

�2 � arctan�t13, �t232 � t33

2�,�3 � arctan��t23�cos �2, t33�cos �2�,

�1 � arctan��t12�cos �2, t11�cos �2�, (5)

Fig. 2. (Color online) (a) Top view of the object in an absolute x–y–z frame; (b) propagation vectors of the two-facet object; (c) local frame{B} attached to the object; (d) shift of the propagation vectors due to 3D rotations in space.


where arctan(y, x) is a two-argument arctangentfunction. Thus the 3D orientations of the object canbe measured in frame {A}, which is equivalent to anabsolute angle measurement. This is referred to as aninverse problem to calculate the rotation angles whilethe rotation matrix is already known. Relative anglemeasurement is also possible by tracing the changesof these angles themselves. For example, after athree-dimensional rotation of the object, the twopropagation vectors will change to n…1 � ��1, ��1,

…

1�and n…2 � ��2, ��2,

…

2�, as shown in Fig. 2(d). Similarlyas above, the attached local frame {B} now changes to�B�, whose rotation matrix relative to frame {A} canbe calculated as

B…AR. The change of rotation angles �1,

�2, and �3 can be easily obtained by following the sameangle decomposition process as above and comparingthe angles with their initial values. Rotation anglesin a different rotation sequence or in any other coor-dinate systems can also be obtained.

3. Experiments

Our first experiment is aimed at demonstrating thatthe wave propagation information from a multiple-facet object can be recorded by a single hologram. AUSAF-1951 resolution target with an area of 2.62mm � 2.62 mm and 354 � 354 pixels is illuminatedwith a green 532 nm Nd:YAG laser and is positioned2 mm away from the S plane as shown in Fig. 1. Theresolution target is intentionally split into two parts,which reflect an incident wave into two propagationdirections. Figure 3(a) shows the recorded hologramand Fig. 3(b) shows its spectrum. One can clearly seethat the two propagation components are separatedin spectrum domain and can be easily extracted byuse of a filter. The spectrum of the right-down part ofthe resolution target is filtered and shown in Fig. 3(c),based on which the corresponding amplitude andphase information are reconstructed as Figs. 3(d) and3(e), respectively. Thus the spatial frequency pair��1, �1� can be obtained by unwrapping and fitting(a portion of) the phase map in Fig. 3(e) with a simpleleast-squares method [10], and the propagation vec-tors can be achieved with high precision. Since the 3Dangle measurement is based on the correct acquisi-tion of the propagation vectors, it is important toapply a good phase-fitting method. Some other meth-ods [18–20] might also be used for phase fitting here.

Phase-unwrapping errors might occur if phase un-wrapping is directly implemented based on the rawphase map, which has dense phase jumps and phasenoises. To diminish these possible phase errors, theobject spectrum can first be shifted Ns pixels in the �(horizontal) direction and Ms in the � (vertical) direc-tion to near the center of the spectrum, as shown inFig. 3(f), and the residual fringes on the recon-structed phase map will have lower frequency, asshown in the right-down portion of Fig. 3(g), whichcan then be unwrapped and fitted to get a residualspatial frequency pair ��1r, �1r�. The total spatial fre-quency ��1, �1� can be obtained as

�1 � X�1Ns � �1r, �1 � Y�1Ms � �1r, (6)

where X (or Y) is the size of the squared image areaand Ns and Ms are the shifted pixel numbers of thespectrum component in the � and � directions, respec-tively. Similarly, Figs. 3(h) and 3(i) show the recon-struction of the up-left portion of the resolutiontarget. The above clearly show that both the ampli-tude and phase information of a two-facet object canbe obtained by a single hologram. Note that phaseinformation is of more interest here, since it indicatesthe propagation directions of the reflected waves fromthe object, although the amplitude information canalso be reconstructed in digital holography.

For 3D angle measurement, a similar two-facet ob-ject is accomplished by sticking two plane mirrorstogether. The relative angle � between the two planemirrors is about 0.5°. First of all, rotation around thex axis is studied as shown in Fig. 1. The maximummeasurement angle around the x (or y) axis of theproposed method is given as

�x,max � sin�1 �M2�x�, (7)

which is determined by the resolution of the CCDcamera �x, the wavelength �, and the magnification

Fig. 3. (Color online) (a) Hologram of a split resolution target;(b) spectrum of the hologram; (c) filtered spectrum component ofthe right-down part of the resolution target; (d), (e) reconstructedamplitude and phase images, respectively, from (c); (f) obtained byshifting (c) to near the center of the spectrum domain; (g) re-constructed phase map from (f) containing fringes with lowerfrequency; (h), (i) reconstructed amplitude and phase images, re-spectively, of the up-left part of the target.


M of the optical system. In our experiment, if thesystem is simplified so that M � 1, the 7.4 �m reso-lution of the CCD camera and the 532 nm wave-length determine a 2.06° measurement range aroundthe x (or y) axis.

Figure 4 shows another example to use the pro-posed method for 3D angle measurement. The ho-logram (or interferogram) of the two-facet object isshown in Fig. 4(a), and Fig. 4(b) shows its spectrum.In the left part of Fig. 4(b) one can clearly see twoseparated real spectrum components, which repre-sent the waves propagating in two different direc-tions. Figure 4(c) shows the object spectrum afterrotation mostly around the x axis. Figures 4(d)–(f)show the spectrum, phase map, and unwrappedphase map of the right facet, respectively. Thus thespatial frequency pair ��1, �1� can be obtained by fit-ting the phase map in Fig. 4(f) to a form �1�x, y� �2��1x � �1y� by a least-squares method, and finallythe normalized wave propagation vectors n1 can becalculated as n1 � �0.015274 0.010582 0.99983�T

from Eq. (2). Similarly, Figs. 4(g)–4(i) are the spec-trum and the wrapped and unwrapped phase mapof the left facet of the object. The normalized wavepropagation vectors n2 are calculated as n2 ��0.014721 0.0009231 0.99989�T. Here we have de-fined the xA–yA plane parallel to the reference mirroras shown in Fig. 4, and the zA axis is coincident with itsnormal direction. Following from Eqs. (3)–(5), the ab-solute angle orientations of frame {B} are calculated as�1 � 86.724°, �2 � 0.8593°, and �3 � �0.3296°. Sim-ilarly, the absolute angle orientations from Fig. 4(c)are also calculated as �1 � 86.722°, �2 � 0.86156°,and �3 � �0.25274° in frame {A}. Of course, the rel-ative angle rotations can also be measured by com-

paring the above three angles around different axesfor each pose.

To study the performance of the proposed method,the measurement system is first calibrated by anautocollimator. The system is used to measure therelative rotations of the object around the x axis.Figure 5(a) shows the calibration result in a measure-ment range of 0–670 arc sec, which is mainly limitedby the measurement limit of the autocollimator. Themaximum deviation from the linearity of the mea-surement (or the calibration error) is about �1 arcsec. Since the object rotates only along one (x) axis,the fringe analysis method [10] can also be used forangle measurement in this case. The comparison(or calibration) of the angle measurement results be-tween the proposed method and the fringe analysismethod is shown in Fig. 5(b), which shows a muchsmaller calibration error of �0.005 arc sec and clearlyshows the agreement between the proposed methodand the fringe analysis method.

The measurement range around the z axis could beas large as 360°, which is not possible in any othercurrently available techniques. Figure 6 shows thecalibration of the measured angle with the readout ofa rotation stage within a 10° measurement range.Since the used rotation stage has only a limited res-olution of 0.5°, the autocollimator is used for calibra-tion as well by observing the object along the x axis inFig. 1. Because of the range limit of the autocollima-tor, we only calibrate the system within a range of

Fig. 4. (a) Hologram or interferogram of the two-facet object;(b) spectrum of (a); (c) spectrum after rotation; (d), (e), (f) spectrum,phase map, and unwrapped phase map, respectively, of the rightfacet; (g), (h), (i) those of the left facet.

Fig. 5. (Color online) Angle measurement around the x axis:(a) calibration test of the proposed method with an autocollimator;(b) comparison with the fringe analysis method [10].


0.23° (or 828 arc sec), which is highlighted with asmall circle in the figure. Note that the fringe anal-ysis method cannot be used for measurement in thiscase since it is insensitive to rotations around the zaxis. The up-left small figure shows that the calibra-tion error to the autocollimator is about �0.005°,which is not as good as that around the x (or y) axisin Fig. 5(a). This is mainly because of the small anglebetween the two facets of the object �� 0.5°� in ourcurrent experiment prototype, which makes the sys-tem relatively sensitive to the phase map noise of thesystem. The phase noise will result in the deviation ofthe calculated propagation vectors from their truevalue; this will affect the accuracy of the measure-ment. The system performance could be improved byusing a bigger � �� 2�x,max� between the two facets,reducing the phase noise, controlling vibration, im-proving the coherent length of the light source, orincreasing the pixel number in the CCD camera. Inpractice, it is also possible to design a multiple-facetobject so that more propagation vectors may be avail-able, either to improve the measurement results byaverage and compensation or to guarantee a maxi-mum measurement range around different axes bychoosing different propagation vectors for analysis.

Although only mirrorlike objects are consideredabove, it is still possible to extend the proposedmethod for more complex and rough objects withspeckle noise. Note that the FFT fringe analysismethod does not work in this case since it requires arelatively clean phase map. However, the digital ho-lography method can be used to reconstruct the objectwave and extract the object spectrum at any specificplane. In this case the Fourier spectrum of the objectwill appear as an extended area with speckle noise,but not a well-defined point as in the case of a mir-rorlike object. It is possible to get the propagationvector change information by directly evaluating theobject spectrum in the frequency domain, but not thephase map. For example, the shift of the object spec-trum can be measured by evaluating the cross covari-ance of the spectrum in the Fourier plane of theobject. Although this method might be limited by thepixel resolution of the imaging sensor, a resolution of

0.3 mdeg (or 5 �rad) was reported in [11]. For a roughobject, the light from the object will be scattered todifferent directions, so there are no absolute propa-gation directions of the object wave. If necessary,however, some featured points (such as the weightcenter or maximum cross-covariance point) of the ob-ject spectrum might still be used as the absolute prop-agation directions as in a mirrorlike object. Furtherwork regarding this aspect is currently under way,and corresponding results will be reported in a forth-coming paper.

4. Conclusion

In conclusion, a new method for 3D angle measure-ment has been proposed based on the principle ofpropagation vector analysis. A multiple-facet object isused as an indicator of 3D rotations of a destinationobject, and the wavefields reflected from the multiplefacets are recorded in a single hologram. Any small3D rotation of the object will induce a change in thepropagation vectors of the reflected waves in space,which can be extracted by analyzing the object spec-trum of the recorded hologram and evaluating thephase maps of each wave component. Finally, 3Dangle measurement is achieved by use of the pro-posed propagation vector analysis algorithm. Al-though the system used in our experiments is notfully optimized, the experiments presented abovehave clearly demonstrated the effectiveness of themethod. The method may also be extended for com-plex and rough objects, but is not limited to mirror-like objects.

This work was supported in part by the NationalScience Foundation. L. Yu gratefully acknowledgesthe financial support of the German Alexander vonHumboldt Foundation.

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