Existing phase-shifting measurement methods involve processing of three acquired images or computation of functions that require more complex processing than linear functions. This paper presents a novel two-step triangular-pattern phase-shifting method of 3-D object-shape measurement that combines advantages of earlier techniques. The method requires only two image-acquisition steps to capture two images, and involves projecting linear grayscale-intensity triangular patterns that require simpler computation of the intensity ratio than methods that use sinusoidal patterns. A triangular intensity-ratio distribution is computed from two captured phase-shifted triangular-pattern images. An intensity ratio-to-height conversion algorithm, based on traditional phase-to-height conversion in the sinusoidal-pattern phase-shifting method, is used to reconstruct the object 3-D surface geometry. A smaller pitch of the triangular pattern resulted in higher measurement accuracy; however, an optimal pitch was found, below which intensity-ratio unwrapping failure may occur. Measurement error varied cyclically with depth and may partly be due to projector gamma nonlinearity and image defocus. The use of only two linear triangular patterns in the proposed method has the advantage of less processing than current methods that process three images, or methods that process more complex functions than the intensity ratio. This would be useful for high speed or real-time 3-D object-shape measurement.
In phase-shifting-based fringe-projection surface-geometry measurement, phase unwrapping techniques produce a continuous phase distribution that contains the height information of the 3-D object surface. Mapping of the phase distribution to the height of the object has often involved complex derivations of the nonlinear relationship. In this paper, the phase-to-height mapping is formulated using both linear and nonlinear equations, the latter through a simple geometrical derivation. Furthermore, the measurement accuracies of the linear and nonlinear calibrations are compared using measurement simulations where noise is included at the calibration stage only, and where noise is introduced at both the calibration and measurement stages. Measurement accuracies for the linear and nonlinear calibration methods are also compared, based on real-system measurements. From the real-system measurements, the accuracy of the linear calibration was similar to the nonlinear calibration method at the lower range of depth. At the higher range of depth, however, the nonlinear calibration method had considerably higher accuracy. It seems that as the object approaches the projector and camera for the higher range of depth, the assumption of linearity based on small divergence of light from the projector becomes less valid.
Two-step triangular phase-shifting has recently been developed for 3-D surface-shape measurement. Compared with
previous phase-shifting methods, the method involves less processing and fewer images to reconstruct the 3-D object.
This paper presents novel extensions of the two-step triangular phase-shifting method to multiple-step algorithms to
increase measurement accuracy. The phase-shifting algorithms used to generate the intensity ratio, which is essential for
determination of the 3-D coordinates of the measured object, are developed for different multiple-step approaches. The
measurement accuracy is determined for different numbers of additional steps and values of pitch. Compared with the
traditional sinusoidal phase-shifting-based method with same number of phase shifting steps, the processing is expected
to be reduced with similar resolution. More phase steps generate higher accuracy in the 3-D shape reconstruction;
however, the digital fringe projection generates phase shifting error if the pitch of the pattern cannot be evenly divided
by the number of phase steps. The pitch in the projected pattern must therefore be selected appropriately according to the
number of phase-shifting steps used.
Two-step triangular phase-shifting is a recently developed method for 3-D shape measurement. In this method, two
triangular gray-level-coded patterns, which are phase-shifted by half of the pitch, are needed to reconstruct the 3-D
object. The measurement accuracy is limited by gamma non-linearity and defocus of the projector and camera. This
paper presents a repeated phase-offset two-step triangular-pattern phase-shifting method used to decrease the
measurement error caused by the gamma non-linearity and defocus in the previously developed two-step triangularpattern
phase-shifting 3-D object measurement method. Experimental analysis indicated that a sensitivity threshold
based on the gamma non-linearity curve should be used as the minimum intensity of the computer-generated pattern
input to the projector to reduce measurement error. In the repeated phase-offset method, two-step triangular phaseshifting
is repeated with an initial phase offset of one-eighth of the pitch, and the two obtained 3-D object height
distributions are averaged to generate the final 3-D object-height distribution. Experimental results demonstrated that the
repeated phase-offset measurement method substantially decreased measurement error compared to the two-step
triangular phase-shifting method.
In fringe-projection surface-geometry measurement, phase unwrapping techniques produce a continuous phase distribution that contains the height information of the 3-D object surface. To convert the phase distribution to the height of the 3-D object surface, a phase-height conversion algorithm is needed, essentially determined in the system calibration which depends on the system geometry. Both linear and non-linear approaches have been used to determine the mapping relationship between the phase distribution and the height of the object; however, often the latter has involved complex derivations. In this paper, the mapping relationship between the phase and the height of the object surface is formulated using linear mapping, and using non-linear equations developed through simplified geometrical derivation. A comparison is made between the two approaches. For both methods the system calibration is carried out using a least-squares approach and the accuracy of the calibration is determined both by simulation and experiment. The accuracy of measurement using linear calibration data was generally higher than using non-linear calibration data in most of the range of measurement depth.
Traditional sinusoidal phase-shifting algorithms involve the calculation of an arctangent function to obtain the phase, which results in slow measurement speed. This paper presents a novel high-speed two-step triangular phase-shifting approach for 3-D object measurement. In the proposed method, a triangular gray-level-coded pattern is used for the projection. Only two triangular patterns, which are phase-shifted by 180 degrees or half of the pitch, are needed to reconstruct the 3-D object. A triangular-shape intensity-ratio distribution is obtained by calculation of the two captured triangular patterns. Removing the triangular shape of the intensity ratio over each pattern pitch generates a wrapped intensity-ratio distribution. The unwrapped intensity-ratio distribution is obtained by removing the discontinuity of the wrapped image with a modified unwrapping method commonly used in the sinusoidal phase-shifting method. An intensity ratio-to-height conversion algorithm, which is based on the traditional phase-to-height conversion algorithm in the sinusoidal phase-shifting method, is used to reconstruct the 3-D surface coordinates of the object. Compared with the sinusoidal and trapezoidal phase shifting methods, the processing speed is faster with similar resolution. This method therefore has the potential for real-time 3-D object measurement. This has applications in inspection tasks, mobile-robot navigation and 3-D surface modeling.