Abstract

We present a generalized method for reconstructing the shape of an object from measured gradient data. A certain class of optical sensors does not measure the shape of an object but rather its local slope. These sensors display several advantages, including high information efficiency, sensitivity, and robustness. For many applications, however, it is necessary to acquire the shape, which must be calculated from the slopes by numerical integration. Existing integration techniques show drawbacks that render them unusable in many cases. Our method is based on an approximation employing radial basis functions. It can be applied to irregularly sampled, noisy, and incomplete data, and it reconstructs surfaces both locally and globally with high accuracy.

© 2008 Optical Society of America

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References

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  2. C. Wagner and G. Häusler, “Information theoretical optimization for optical range sensors,” Appl. Opt. 42, 5418-5426(2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2005 (2)

2004 (2)

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” in Optical Metrology in Production Engineering, Proc. SPIE 5457, 366-376 (2004).
[CrossRef]

I. Horovitz and N. Kiryati, “Depth from gradient fields and control points: bias correction in photometric stereo,” Image Vis. Comput. 22, 681-694 (2004).
[CrossRef]

2003 (2)

F. J. Narcowich, J. D. Ward, and H. Wendland, “Refined error estimates for radial basis function interpolation,” Constructive Approx. 19, 541-564 (2003).
[CrossRef]

C. Wagner and G. Häusler, “Information theoretical optimization for optical range sensors,” Appl. Opt. 42, 5418-5426(2003).
[CrossRef] [PubMed]

2002 (1)

C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function f(x) when only df(x)/dx or d2f(x)/dx2 is known at discrete measurement points,” in X-Ray Mirrors, Crystals, and Multilayers II, Proc. SPIE 4782 (2002), 12-20.
[CrossRef]

2001 (1)

J. Villa, G. García, and G. Gómez, “Wavefront recovery in shearing interferometry with variable magnitude and direction shear,” Opt. Commun. 195, 85-91 (2001).
[CrossRef]

2000 (1)

1999 (2)

I. Weingärtner and M. Schulz, “Novel scanning technique for ultraprecise measurement of slope and topography of flats, aspheres, and complex surfaces,” in Optical Fabrication and Testing, Proc. SPIE 3739, 274-282 (1999).
[CrossRef]

N. Dyn, F. Narcowich, and J. Ward, “Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold,” Constructive Approx. 15, 175-208 (1999).
[CrossRef]

1998 (1)

1995 (1)

H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree,” Adv. Comput. Math. 4, 389-396 (1995).
[CrossRef]

1988 (2)

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[CrossRef]

G. Häusler and G. Schneider, “Testing optics by experimental ray tracing with a lateral effect photodiode,” Appl. Opt. 27, 5160-5164 (1988).
[CrossRef] [PubMed]

1980 (1)

Adv. Comput. Math. (1)

H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree,” Adv. Comput. Math. 4, 389-396 (1995).
[CrossRef]

Appl. Opt. (2)

Constructive Approx. (2)

F. J. Narcowich, J. D. Ward, and H. Wendland, “Refined error estimates for radial basis function interpolation,” Constructive Approx. 19, 541-564 (2003).
[CrossRef]

N. Dyn, F. Narcowich, and J. Ward, “Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold,” Constructive Approx. 15, 175-208 (1999).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 439-451 (1988).
[CrossRef]

Image Vis. Comput. (1)

I. Horovitz and N. Kiryati, “Depth from gradient fields and control points: bias correction in photometric stereo,” Image Vis. Comput. 22, 681-694 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

J. Villa, G. García, and G. Gómez, “Wavefront recovery in shearing interferometry with variable magnitude and direction shear,” Opt. Commun. 195, 85-91 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Proc. SPIE (3)

C. Elster and I. Weingärtner, “High-accuracy reconstruction of a function f(x) when only df(x)/dx or d2f(x)/dx2 is known at discrete measurement points,” in X-Ray Mirrors, Crystals, and Multilayers II, Proc. SPIE 4782 (2002), 12-20.
[CrossRef]

I. Weingärtner and M. Schulz, “Novel scanning technique for ultraprecise measurement of slope and topography of flats, aspheres, and complex surfaces,” in Optical Fabrication and Testing, Proc. SPIE 3739, 274-282 (1999).
[CrossRef]

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” in Optical Metrology in Production Engineering, Proc. SPIE 5457, 366-376 (2004).
[CrossRef]

Other (11)

J. Kaminski, S. Lowitzsch, M. C. Knauer, and G. Häusler, “Full-field shape measurement of specular surfaces,” in Fringe 2005, The 5th International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (Springer, 2005), pp. 372-379.

S. Lowitzsch, J. Kaminski, M. C. Knauer, and G. Häusler, “Vision and modeling of specular surfaces,” in Vision, Modeling, and Visualization 2005, G. Greiner, J. Hornegger, H. Niemann, and M. Stamminger, eds. (Akademische Verlagsgesellschaft Aka GmbH, 2005), pp. 479-486.

G. Häusler, C. Richter, K.-H. Leitz, and M. C. Knauer, “Microdeflectometry--a novel tool to acquire 3D microtopology with nanometer height resolution,” Opt. Lett. 33, 396-398 (2008).

A. Agrawal, R. Chellappa, and R. Raskar, “An algebraic approach to surface reconstruction from gradient fields,” in Proceedings of the IEEE International Conference on Computer Vision (ICCV), (IEEE, 2005), Vol. 1, pp. 174-181.

G. Häusler, “Verfahren und Vorrichtung zur Ermittlung der Form oder der Abbildungseigenschaften von spiegelnden oder transparenten Objekten,” DE Patent 19944354 (16 September 1999).

B.Girod, G.Greiner, and H.Niemann, eds., Principles of 3D Image Analysis and Synthesis (Kluwer, 2000).

B.Horn and M.Brooks, eds., Shape from Shading (MIT, 1989).

K. Schlüns and R. Klette, “Local and global integration of discrete vector fields,” in ,i>Advances in Computer Vision (Springer, 1996), pp. 149-158.

M. D. Buhmann, Radial Basis Functions (Cambridge U. Press, 2003).
[CrossRef]

S. Ettl, J. Kaminski, and G. Häusler, “Generalized hermite interpolation with radial basis functions considering only gradient data,” in Curve and Surface Fitting: Avignon 2006, A. Cohen, J. -L. Merrien, and L. L. Schumaker, eds. (Nashboro Press, 2007), pp. 141-149.

M. C. Knauer, T. Bothe, S. Lowitzsch, W. Jüptner, and G. Häusler, Höhe, Neigung oder Krümmung? in Proceedings of the 107th Annual DGaO Conference (Deutsche Gesellschaft für angewandte Optik, 2006), p. B30.

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Figures (5)

Fig. 1
Fig. 1

(a) Surface z of a typical smooth object and (b) its derivative d z / d x , together with (c) the power spectral density of the surface Φ z and (d) of its slope Φ d z / d x (all in arbitrary units). The power spectral density shows that differentiation reduces redundancy, contained in the low frequencies.

Fig. 2
Fig. 2

(a) Surface of a concave, spherical lens, reconstructed from local slope data. To be able to handle this data set consisting of 60 × 60 points, it has been split into a grid of 6 × 6 patches. (b) To illustrate this, the calculated fitting parameters (height constants and patch tilts) have been scaled before stitching the patches together. All units are in millimeters.

Fig. 3
Fig. 3

Surface reconstruction for simulated, noisy slope data of a sphere: (a) the reconstructed spherical surface and (b) a cross section of the absolute error of the reconstruction, both for realistic noise of 8 arcsec , and (c) the absolute error of the reconstruction for several noise levels. The absolute error increases only linearly.

Fig. 4
Fig. 4

Reconstruction of grooves on a spherical surface from simulated slope data, for realistic noise. The nominal height of the grooves range from 100 nm down to 1 nm , at constant width of 180 μm . After the reconstruction, the sphere was subtracted to make the grooves visible. The actual, reconstructed grooves are depicted in (a) full-field and in (b) cross section.

Fig. 5
Fig. 5

Height reconstruction of a part of a wafer from its local slope data. The data were acquired by a phase-measuring deflectometry sensor (measurement field 100 μm × 80 μm , height range 350 nm ). The diagonal groove (marked by the arrow) has a depth of 40 nm .

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

J ( z ) i = 1 N [ z x ( x i ) p ( x i ) ] 2 + [ z y ( x i ) q ( x i ) ] 2 .
2 z ( x ) = p x ( x ) + q y ( x ) ,
s ( x ) = i = 1 N α i Φ x ( x x i ) + i = 1 N β i Φ y ( x x i ) ,
{ s x ( x j ) = ! p ( x j ) s y ( x j ) = ! q ( x j ) for     1 j N .
( Φ x x ( x i x j ) Φ x y ( x i x j ) Φ x y ( x i x j ) Φ y y ( x i x j ) ) A M 2 N × 2 N ( α i β i ) α M 2 N × 1 = ( p ( x j ) q ( x j ) ) d M 2 N × 1 .
ϕ ( r ) = { 1 3 ( 1 r ) 6 ( 35 r 2 + 18 r + 3 ) for     r 1 ρ x 2 + y 2 1 ρ > 0 0 otherwise .
K ( f 2 ) x Ω 1 Ω 2 | s 1 ( x ) s 2 ( x ) f 2 ( x ) | 2 .
Δ z global tan ( σ α ) Δ x M ,

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