Error compensation in two-step triangular-pattern phase-shifting profilometry

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Abstract

This paper presents an analysis of error compensation using a newly modified two-step triangular-pattern phase-shifting measurement method, developed to reduce periodic measurement errors due to gamma nonlinearity and defocus of the projector and camera. Experimental analysis revealed that a trade-off is necessary in choosing a higher minimum projector input intensity to use the more linear region of input-to-output intensity mapping, and a lower minimum input intensity for greater dynamic range of input intensity. The modified two-step triangular-pattern phase-shifting method performs two-step triangular-pattern phase-shifting twice, the second time with an initial phase offset of one-eighth of the pitch, and generates the three-dimensional object height distribution by averaging the two obtained object-height distributions. The modified two-step triangular-pattern phase-shifting method consistently had higher measurement accuracy than the unmodified method. Errors were reduced by 23.4% at the midrange of depth using an input intensity value of 40, which yielded the highest measurement accuracy and up to 64% and 54% at small and large depths, respectively.

Introduction

Fringe projection is one of the most widely used techniques for three-dimensional (3-D) object-shape measurement. The technique is based on projecting periodic fringe or grating patterns onto an object surface and then viewing the patterns, which are distorted according to the topography of the object, from another direction. The image intensity distribution of the deformed fringe pattern or grating is captured by a CCD camera, and then sampled and processed to retrieve the phase distribution through phase extraction techniques. Coordinates of the 3-D object surface are then determined by triangulation. Phase-measuring interferometry techniques [1], [2], [3] can be implemented to extract phase information and increase measurement resolution. Among these techniques, phase-shifting methods [4], [5] are the most widely used.

Based on digital fringe-projection [6], intensity ratio [7], [8], [9], and phase-shifting techniques [4], [5], a new 3-D shape measurement method, called the two-step triangular-pattern phase-shifting method [10], has been recently developed. Compared with the traditional sinusoidal, trapezoidal [11], and previous linear-coded triangular [12], [13] phase-shifting methods, this method involves less processing due to the combination of simple computation of the intensity ratio and fewer images required to measure the 3-D object. It also has a better depth resolution compared to the traditional intensity ratio [7], [8] based methods and less ambiguity when the triangular pattern is repeated [9] to reduce sensitivity to measurement noise. However, the measurement accuracy of the two-step triangular-pattern phase-shifting method is limited by gamma nonlinearity and defocus of the projector and camera. This paper presents an analysis of error compensation using a newly modified two-step triangular-pattern phase-shifting method, developed to reduce errors due to gamma nonlinearity and defocus of the projector and camera.

Section 2 briefly introduces the 3-D shape measurement system using two-step triangular-pattern phase shifting, and presents an experimental analysis of the gamma nonlinearity and defocus errors, and a method of compensating the errors by modified two-step triangular-pattern phase shifting. Section 3 describes error-compensation experiments and compares the results of the unmodified and modified two-step triangular-pattern phase-shifting methods. A discussion of the proposed method is given in Section 4, followed by conclusions in Section 5.

Section snippets

Two-step triangular-pattern phase shifting

A schematic diagram of the experimental setup for the triangular-pattern phase-shifting fringe-projection measurement system is shown in Fig. 1. It consists of a computer (P4 3.04 GHz with 1 GB memory), a digital light processing (DLP) projector (In Focus LP600, 8 bit), a CCD camera (Sony XCHR50, 8 bit), and a flat plate for calibration. The projector is used to project the triangular fringe pattern, which is generated digitally by computer, onto the object surface. The projected triangular

Error-compensation experiments

To determine the performance of the modified two-step triangular-pattern phase-shifting method in compensating errors, four tests were carried out. All measurement errors reported are RMS values of errors in depth with respect to ground truth based on all pixels of the 648×494 resolution image. In the first test, two sets of measurements were carried out, one using the basic two-step triangular-pattern phase-shifting method, and the second using two-step phase-shifting with an initial phase

Comparison of modified two-step and four-step triangular-pattern phase shifting

In the modified two-step triangular-pattern phase-shifting method, the shifts in the second set of measurements were specifically selected to compensate the errors in the first set of measurements. Both sets of measurements had a shift of one-half of the pitch between steps, and a one eighth of pitch shift was made between the two sets of measurements. This resulted in the following shifts: 0, T/2, −T/8, 3T/8. This error compensation approach was different from simply repeating measurements to

Conclusion

Experimental error analysis of the two-step triangular-pattern phase-shifting method indicates that the minimum intensity of the computer-generated pattern input to the projector has an important effect on the measurement accuracy because of the gamma nonlinearity. A trade-off is required in choosing a higher minimum input intensity value to use the more linear region of input-to-output intensity mapping, and a lower minimum input intensity to achieve a greater range of input intensity values.

Acknowledgments

This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada (research grant and Industrial Postgraduate Scholarship), Neptec Design Group Ltd., and Communications and Information Technology Ontario (CITO)—Ontario Centres of Excellence (OCE).

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