Abstract

A new semispatial, robust, and accurate phase evaluation algorithm for spatial carrier fringe measuring systems is presented, which is a combination of a temporal algorithm and a spatial algorithm in order to use the advantages of both methods. Only two to four frames are needed for operation. A new interlaced detection mode is also presented. Thus the time of data acquisition can be reduced further. Limits are discussed and compared with conventional algorithms. An application of this algorithm with a fringe projection system is described and demonstrated by the measurement of objects with different optical properties. Using this algorithm it is possible to achieve short data-acquisition times of 80 ms in combination with a high vertical resolution. This fact is demonstrated by the measurement of living corneas.

© 1995 Optical Society of America

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References

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  1. K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis-Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, Pa., 1993), pp. 95–140.
  2. H. J. Tiziani, “Rechnergestützte Laser-Messtechnik,” Tech. Messen. 6, 221–230 (1987).
  3. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis-Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, Pa., 1993), pp. 141–193.
  4. B. Dörband, “Analyse Optischer Systeme,” Ph.D. dissertation (University of Stuttgart, Stuttgart, Germany, 1986), p. 39.
  5. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  6. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  7. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2505 (1987).
    [CrossRef] [PubMed]
  8. R. Windecker, H. J. Tiziani, “Topometry of technical and biological objects by fringe projection,” Appl. Opt. 34, 3644–3650 (1995).
    [CrossRef] [PubMed]

1995 (1)

1992 (1)

1987 (2)

1983 (1)

Burow, R.

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis-Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, Pa., 1993), pp. 95–140.

Dörband, B.

B. Dörband, “Analyse Optischer Systeme,” Ph.D. dissertation (University of Stuttgart, Stuttgart, Germany, 1986), p. 39.

Eiju, T.

Elssner, K.-E.

Grzanna, J.

Hariharan, P.

Kujawinska, M.

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis-Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, Pa., 1993), pp. 141–193.

Larkin, K. G.

Merkel, K.

Oreb, B. F.

Schwider, J.

Spolaczyk, R.

Tiziani, H. J.

Windecker, R.

Appl. Opt. (3)

J. Opt. Soc. Am. A (1)

Tech. Messen. (1)

H. J. Tiziani, “Rechnergestützte Laser-Messtechnik,” Tech. Messen. 6, 221–230 (1987).

Other (3)

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis-Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, Pa., 1993), pp. 141–193.

B. Dörband, “Analyse Optischer Systeme,” Ph.D. dissertation (University of Stuttgart, Stuttgart, Germany, 1986), p. 39.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis-Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics Publishing, Philadelphia, Pa., 1993), pp. 95–140.

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Figures (10)

Fig. 1
Fig. 1

Fourier analysis method with only one frame and with two frames with an additional π/2 phase shift introduced.

Fig. 2
Fig. 2

Examination of the influence of noisy input data on the phase calculation by computer simulation. For the simulated rms error of the phase, three-phase shift and modified Fourier analysis (FA) algorithms were used.

Fig. 3
Fig. 3

PV phase errors for (a) phase shift algorithms that are due to a linear phase shift error1 and (b) a modified Fourier analysis (FA) that is due to a linear phase shift error. The curves show the results for different surface angles.

Fig. 4
Fig. 4

(a) Relative phase error (standard deviation) that is due to a sinusoidal intensity modulation with a standard Fourier analysis. The diagram shows the simulation of three different modulation degrees for a 10° tilted plane. (b) The same diagram as (a) except that the plane is tilted by 30°.

Fig. 5
Fig. 5

Experimental setup of the fringe projection topometer.

Fig. 6
Fig. 6

Principle of the interlaced detection mode. In this case the time for data acquisition can be reduced by a factor of 2. Two frames have been mixed to get all four consecutive phase values into one picture.

Fig. 7
Fig. 7

(a) Portion of the topography of a German 1-pfenning coin obtained with the interlaced mode. The depicted data are not filtered. The data-acquisition time was 80 ms. (b) Cross section through the topography of (a) between the marks.

Fig. 8
Fig. 8

(a) Simulated intensity distribution on a tilted plane surface. (b) Phase errors caused by the intensity modulation of (a) after leveling.

Fig. 9
Fig. 9

(a) Difference in micrometers between an evaluation with and without contrast compensation of the optical rough surface of a German 50-pfennig coin. (b) The contrast distribution of the measurement of (a). The dark gray level signals a high fringe contrast. The units are arbitrary.

Fig. 10
Fig. 10

(a) In vivo measurement of a human cornea with fringe projection over a field of view of 8 × 12 mm2. The calculated mean radius of the best-fitting sphere is 7.9 mm. (b) Contour map representation of (a). The line spacing is 41.55 μm.

Equations (21)

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I ( x , y ) = I 0 ( x , y ) { 1 + m ( x , y ) F [ f x , x , Φ ( x , y ) ] } ,
I ( x , y ) = I 0 ( x , y ) { 1 + m ( x , y ) sin [ 2 π f x x + Φ ( x , y ) ] } .
I ( x , y ) = a ( x , y ) + b ( x , y ) sin [ 2 π f x x + Φ ( x , y ) ] .
I 1 ( x , y ) = a ( x , y ) + b ( x , y ) sin [ 2 π f x x + Φ ( x , y ) ] ,
I 2 ( x , y ) = a ( x , y ) + b ( x , y ) cos [ 2 π f x x + Φ ( x , y ) ] ,
I 3 ( x , y ) = a ( x , y ) - b ( x , y ) sin [ 2 π f x x + Φ ( x , y ) ] .
B 1 ( x , y ) = I 1 ( x , y ) - I 3 ( x , y ) 2 = b ( x , y ) sin [ 2 π f x x + Φ ( x , y ) ] ,
B 2 ( x , y ) = I 2 ( x , y ) - I 1 ( x , y ) + I 3 ( x , y ) 2 = b ( x , y ) cos [ 2 π f x x + Φ ( x , y ) ] .
C ( x , y ) = 1 2 B 1 2 + B 2 2 = b ( x , y )
B 1 ( x , y ) = sin [ 2 π f x x + Φ ( x , y ) ] ,
B 2 ( x , y ) = cos [ 2 π f x x + Φ ( x , y ) ] .
J s k ( x , y ) = i = x x + 1 / f x B k ( i , y ) sin ( 2 π f x i ) ,
J c k ( x , y ) = i = x x + 1 / f x B k ( i , y ) cos ( 2 π f x i ) .
Φ k ( x , y ) = arctan ( J s k J c k ) ,
Φ ( x , y ) = ½ ( Φ 1 + Φ 2 ) + π / 4.
Φ 1 ( x , y ) = arctan { i = x x + 1 / f x [ a ( i , y ) + b ( i , y ) sin ( 2 π f x i + c i ) ] sin ( 2 π f x i ) i = x x + 1 / f x [ a ( i , y ) + b ( i , y ) sin ( 2 π f x i + c i ) ] cos ( 2 π f x i ) } .
Φ 1 ( x , y ) = arctan { i = x x + 1 / f x [ a ( i , y ) + b ( i , y ) sin ( 2 π f ˜ x i ) ] sin ( 2 π f ˜ x i - c i ) i = x x + 1 / f x [ a ( i , y ) + b ( i , y ) sin ( 2 π f ˜ x i ) ] cos ( 2 π f ˜ x i - c i ) } .
I ( x , y ) = ¼ [ 1 - m sin ( 2 π f B x ) ] × { [ 1 × sin [ 2 π f x x + Φ ( x , y ) ] } ,
I 4 ( x , y ) = a ( x , y ) - b ( x , y ) cos [ 2 π f x x + Φ ( x , y ) ] .
B 1 ( x , y ) = I 1 ( x , y ) - I 3 ( x , y ) 2 = b ( x , y ) sin [ 2 π f x x + Φ ( x , y ) ] ,
B 2 ( x , y ) = I 1 ( x , y ) - I 4 ( x , y ) 2 = b ( x , y ) cos [ 2 π f x x + Φ ( x , y ) ] .

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