Abstract

The linearized perturbation method for fringe pattern analysis and its extension to multifringe analysis have been recently introduced [Hilaire, Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1993)]. For isolating the error component that is due to information processing—as opposed to image-acquisition errors—experimental calibration experiments were conducted by use of computer-generated fringe patterns. The effects of noise, fringe completeness, image resolution, illumination, quantization, and displacement magnitude are tested and discussed in evaluating the software’s performance and accuracy.

© 1997 Optical Society of America

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References

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  1. D. W. Robinson, G. T. Reid, eds, Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).
  2. G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986–1987).
    [CrossRef]
  3. C. Guo, “Computational metrology of parallelism and application to precision machinery,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1997).
  4. T. P. Hilaire, “Optical sensing and fringe pattern analysis of translational errors in machine carriages,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1993).
  5. T. P. Hilaire, J. Pegna, “Multi-fringe pattern analysis of circular zone plates,” J. Electron. Imag. (to be published).
  6. T. P. Hilaire, J. Pegna, “Two-dimensional sub-micron interferometric translation straightness measurement for machine elements,” in Proceedings of the Symposium on Mechatronics for Manufacturing Metrology and Error Analysis 1995 (American Society of Mechanical Engineers, Fairfield, N.J., 1995), pp. 495–506.
  7. J. Pegna, C. Guo, T. P. Hilaire, “Design of a nanometric position sensor based on computational metrology of the circle,” in Proceedings 95 Design Technical Engineering Conference: Advances in Design Automation (American Society of Mechanical Engineers, Fairfield, N.J., 1995).
  8. J. Pegna, C. Guo, T. P. Hilaire, “Algorithmic circularity measurement for fringe analysis and sub-micron position sensing,” in Computer Aided Tolerancing, F. Kimura, ed. (Chapman & Hall, London, 1996), pp. 283–297.
    [CrossRef]
  9. M. Hiraoglu, “Characterization and calibration of cameras for machine-vision metrology,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1992).
  10. W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).
  11. Q. Tian, M. N. Huhns, “Algorithms for subpixel registration,” Comput. Vision Graphics Image Process. 35, 220–233 (1986).
    [CrossRef]
  12. J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 1, 31–39 (1983).
    [CrossRef]
  13. P. Carré, “Installation et utilisation du comparateur photo-électrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]

1986

Q. Tian, M. N. Huhns, “Algorithms for subpixel registration,” Comput. Vision Graphics Image Process. 35, 220–233 (1986).
[CrossRef]

1983

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 1, 31–39 (1983).
[CrossRef]

1966

P. Carré, “Installation et utilisation du comparateur photo-électrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Carré, P.

P. Carré, “Installation et utilisation du comparateur photo-électrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Guo, C.

C. Guo, “Computational metrology of parallelism and application to precision machinery,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1997).

J. Pegna, C. Guo, T. P. Hilaire, “Design of a nanometric position sensor based on computational metrology of the circle,” in Proceedings 95 Design Technical Engineering Conference: Advances in Design Automation (American Society of Mechanical Engineers, Fairfield, N.J., 1995).

J. Pegna, C. Guo, T. P. Hilaire, “Algorithmic circularity measurement for fringe analysis and sub-micron position sensing,” in Computer Aided Tolerancing, F. Kimura, ed. (Chapman & Hall, London, 1996), pp. 283–297.
[CrossRef]

Hilaire, T. P.

J. Pegna, C. Guo, T. P. Hilaire, “Algorithmic circularity measurement for fringe analysis and sub-micron position sensing,” in Computer Aided Tolerancing, F. Kimura, ed. (Chapman & Hall, London, 1996), pp. 283–297.
[CrossRef]

T. P. Hilaire, J. Pegna, “Multi-fringe pattern analysis of circular zone plates,” J. Electron. Imag. (to be published).

T. P. Hilaire, “Optical sensing and fringe pattern analysis of translational errors in machine carriages,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1993).

J. Pegna, C. Guo, T. P. Hilaire, “Design of a nanometric position sensor based on computational metrology of the circle,” in Proceedings 95 Design Technical Engineering Conference: Advances in Design Automation (American Society of Mechanical Engineers, Fairfield, N.J., 1995).

T. P. Hilaire, J. Pegna, “Two-dimensional sub-micron interferometric translation straightness measurement for machine elements,” in Proceedings of the Symposium on Mechatronics for Manufacturing Metrology and Error Analysis 1995 (American Society of Mechanical Engineers, Fairfield, N.J., 1995), pp. 495–506.

Hiraoglu, M.

M. Hiraoglu, “Characterization and calibration of cameras for machine-vision metrology,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1992).

Huhns, M. N.

Q. Tian, M. N. Huhns, “Algorithms for subpixel registration,” Comput. Vision Graphics Image Process. 35, 220–233 (1986).
[CrossRef]

Kenyon, R. V.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 1, 31–39 (1983).
[CrossRef]

Parker, J. A.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 1, 31–39 (1983).
[CrossRef]

Pegna, J.

T. P. Hilaire, J. Pegna, “Multi-fringe pattern analysis of circular zone plates,” J. Electron. Imag. (to be published).

J. Pegna, C. Guo, T. P. Hilaire, “Algorithmic circularity measurement for fringe analysis and sub-micron position sensing,” in Computer Aided Tolerancing, F. Kimura, ed. (Chapman & Hall, London, 1996), pp. 283–297.
[CrossRef]

T. P. Hilaire, J. Pegna, “Two-dimensional sub-micron interferometric translation straightness measurement for machine elements,” in Proceedings of the Symposium on Mechatronics for Manufacturing Metrology and Error Analysis 1995 (American Society of Mechanical Engineers, Fairfield, N.J., 1995), pp. 495–506.

J. Pegna, C. Guo, T. P. Hilaire, “Design of a nanometric position sensor based on computational metrology of the circle,” in Proceedings 95 Design Technical Engineering Conference: Advances in Design Automation (American Society of Mechanical Engineers, Fairfield, N.J., 1995).

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

Reid, G. T.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986–1987).
[CrossRef]

Tian, Q.

Q. Tian, M. N. Huhns, “Algorithms for subpixel registration,” Comput. Vision Graphics Image Process. 35, 220–233 (1986).
[CrossRef]

Troxel, D. E.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 1, 31–39 (1983).
[CrossRef]

Comput. Vision Graphics Image Process.

Q. Tian, M. N. Huhns, “Algorithms for subpixel registration,” Comput. Vision Graphics Image Process. 35, 220–233 (1986).
[CrossRef]

IEEE Trans. Med. Imaging

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging 1, 31–39 (1983).
[CrossRef]

Metrologia

P. Carré, “Installation et utilisation du comparateur photo-électrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Lasers Eng.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986–1987).
[CrossRef]

Other

C. Guo, “Computational metrology of parallelism and application to precision machinery,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1997).

T. P. Hilaire, “Optical sensing and fringe pattern analysis of translational errors in machine carriages,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1993).

T. P. Hilaire, J. Pegna, “Multi-fringe pattern analysis of circular zone plates,” J. Electron. Imag. (to be published).

T. P. Hilaire, J. Pegna, “Two-dimensional sub-micron interferometric translation straightness measurement for machine elements,” in Proceedings of the Symposium on Mechatronics for Manufacturing Metrology and Error Analysis 1995 (American Society of Mechanical Engineers, Fairfield, N.J., 1995), pp. 495–506.

J. Pegna, C. Guo, T. P. Hilaire, “Design of a nanometric position sensor based on computational metrology of the circle,” in Proceedings 95 Design Technical Engineering Conference: Advances in Design Automation (American Society of Mechanical Engineers, Fairfield, N.J., 1995).

J. Pegna, C. Guo, T. P. Hilaire, “Algorithmic circularity measurement for fringe analysis and sub-micron position sensing,” in Computer Aided Tolerancing, F. Kimura, ed. (Chapman & Hall, London, 1996), pp. 283–297.
[CrossRef]

M. Hiraoglu, “Characterization and calibration of cameras for machine-vision metrology,” Ph.D. dissertation (Rensselaer Polytechnic Institute, Troy, N.Y., 1992).

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

D. W. Robinson, G. T. Reid, eds, Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).

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Figures (22)

Fig. 1
Fig. 1

Interferometer for translational error measurements.

Fig. 2
Fig. 2

Sample circular zone plate pattern produced by the interferometer shown in Fig. 1. The two-dimensional position of the center C is the measurement parameter.

Fig. 3
Fig. 3

Optimization by use of perturbations of a circle: (a) Three points defining a circle called the base circle. (b) Optimization of the base circle by variation of the center location and radius.

Fig. 4
Fig. 4

Perturbation on four concentric fringes. This brings the successive radii in compliance with Eq. (3) and minimizes the total least-squares sum of normal errors.

Fig. 5
Fig. 5

Procedure for center-location determination: (a) Computer-generated image. (b) Peak extraction. (c) Fringe labeling. (d) Fringe skeletonizing. (e) Approximate center computation on the fringe with the maximum number of points. (f) Final optimization with four consecutive fringes.

Fig. 6
Fig. 6

Diagram showing the movement of the pattern from its original position along a polar angle θ by a radial amount ρ.

Fig. 7
Fig. 7

Radial displacement ρ of the center of ρ = 0.1 pixel: K = 0.02, φ = 0, test image of 640 × 480 pixels, noise-free center at (320, 240).

Fig. 8
Fig. 8

Radial displacement of the center of ρ = 0.01 pixel: K = 0.02, φ = 0, test image of 640 × 480 pixels, noise-free center at (320, 240).

Fig. 9
Fig. 9

Radial displacement of the center of ρ = 0.1 pixel. The pattern is nonsymmetrical, and the center of the pattern is located within the image. K = 0.02, φ = 0, test image of 640 × 480 pixels, noise-free center at (270, 190).

Fig. 10
Fig. 10

Radial displacement of the center of ρ = 0.1 pixel. The pattern is centered outside the image. K = 0.02, φ = 0, test image of 640 × 480 pixels, noise-free center at (320, -240).

Fig. 11
Fig. 11

Radial displacement of the center of ρ = 0.5 pixel. The pattern is centered outside the image. K = 0.02, φ = 0, test image of 640 × 480 pixels, center at (-320, -240), minimum radius r = 600 pixels.

Fig. 12
Fig. 12

Radial displacement of the center of ρ = 0.1 pixel. The centered pattern is corrupted with Gaussian noise. K = 0.02, φ = 0, test image of 640 × 480 pixels, Gaussian noise σ = 5, center at (320, 240).

Fig. 13
Fig. 13

Radial displacement of the center of ρ = 0.1 pixel. The off-center pattern is corrupted with Gaussian noise. K = 0.02, φ = 0, test image of 640 × 480 pixels, Gaussian noise σ = 5, center at (270, 190).

Fig. 14
Fig. 14

Error as a function of the number of complete fringes used in the optimization when the image is corrupted with Gaussian noise. φ = 0, test image of 640 × 480 pixels, Gaussian noise σ = 5, center at (320, 240), true center at ρ = 0.1 pixel.

Fig. 15
Fig. 15

Radial displacement of the center of ρ = 0.1 pixel. The pattern is corrupted with Gaussian noise. K = 0.02, φ = 0, test image of 640 × 480 pixels, Gaussian noise σ = 5, center at (320, -240), minimum radius r = 150 pixels.

Fig. 16
Fig. 16

Radial displacement of the center of ρ = 0.5 pixel. The pattern is corrupted with Gaussian noise. K = 0.02, φ = 0, test image of 640 × 480 pixels, noise-free center at (-320, -240), minimum radius r = 600 pixels.

Fig. 17
Fig. 17

Error of the center location for three different image resolutions (measured in pixels of the original image). True center at ρ = 0.1 pixel.

Fig. 18
Fig. 18

(a) Illumination function across the fringe pattern. (b) Error on the center location when different levels of illumination are applied to the image. φ = 0, test image of 640 × 480 pixels, noise-free center at (320, 240), true center at ρ = 0.1 pixel.

Fig. 19
Fig. 19

Error on the center location when different levels of quantization are applied to the image. K = 0.03, φ = 0, test image of 640 × 480 pixels, noise-free center at (320, 240), true center at ρ = 0.1 pixel.

Fig. 20
Fig. 20

Simulation of the effect on the captured fringe pattern when angular defects occur between the reference-mirror plane and the camera plane: (a) No angular defect—the pattern is circular concentric. (b) Angular defect of 45° about the vertical axis—the pattern is no longer circular.

Fig. 21
Fig. 21

Center location along the principal directions for an angular defect about the vertical direction equal to 0.05 rad and the direction of displacement rotated by 0.1 rad about the horizontal direction for a computer-generated image.

Fig. 22
Fig. 22

Simulation of a theoretical image of the effect of detector-plane angular defects on the center location for different thresholds T.

Equations (7)

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

JΔxC, ΔyC, Δr=j=1NΔxCcj+ΔyCsj+Δr-ξj2,
k=ii+3j=1Nkξjk-ΔxCcji+ΔyCsji+Δri2,
ri+32-3ri+22+3ri+12-ri2=0.
JΔxC=JΔyC=JΔri=JΔri+1=JΔri+2=JΔri+3=0, ri+3Δri+3-3ri+2Δri+2+3ri+1Δri+1-riΔri=0.
JΔxC=JΔyC=JΔri=JΔri+1=JΔri+2=0, ri+2Δri+2-2ri+1Δri+1+riΔri=0.
Ix,y=2551+cosKx-xc2+y-yc2+φ.
XO=XC+ρ cos θ, YO=YC+ρ sin θ.

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