Nonlinear Error Compensation Method for Phase-Shifted Fringe Using 1/6-Period Lookup Table
Objective Structured light technology has been widely used in industrial inspection,cultural relic protection,biomedical,and other fields due to its non-contact and full-field imaging advantages.As one of the mainstream three-dimensional(3D)imaging methods,phase-shifting profilometry(PSP)measures the target surface shape information by projecting multiple phase-shifted patterns and capturing the corresponding target patterns.Since the 3D shape information is closely related to the phase distribution,it is crucial to achieve high-accuracy measurements.However,due to the gamma nonlinearity in both the projector and camera,the ideal sinusoidal fringe patterns are distorted,thereby introducing errors.This kind of non-sinusoidal fringe pattern results in phase errors,which are a major error source affecting the three-dimensional reconstruction accuracy and further degrading the measurement precision.Although the large-step PSP can reduce the nonlinear error,it takes a long time.Therefore,it is very meaningful to develop a phase error correction algorithm with both high accuracy and fast speed.Methods The N-step PSP has the advantages of fast measurement speed,high accuracy,and a non-contact nature,making it widely used in the phase measurement field.Since the three-step PSP is most easily affected by gamma nonlinearity,we take the three-step PSP as an example to illustrate the principles.According to the calculation formula of nonlinear error in the three-step PSP,it can be deduced that the phase error has periodicity and symmetry within 2π period.Given the periodic characteristic of phase error,we first propose a 1/3 period lookup table(LUT)method and then a 1/6 period lookup table(sLUT)method considering the symmetry of phase error.A standard whiteboard is imaged to calculate the actual phase values using the three-step PSP and theoretical phase values using the twelve-step PSP(Fig.2).The number of elements for the constructed full-period LUT is 360.According to the periodicity of phase error,a 1/3 period LUT is constructed with 120 elements.Considering the symmetry of phase error,the sLUT with only 60 elements is constructed.The sLUT is simulated and tested on real objects to compare the error correction effects and correction times of the whole-period LUT,1/3 period LUT,and sLUT methods.Results and Discussions The proposed sLUT method is rigorously evaluated and compared against the conventional full-period LUT approach through simulation testing and experimental validation.The simulation is tested on a standard sphere and peaked surface.The standard deviations(STDs)of the results corrected by whole-period LUT,1/3 period LUT,and sLUT are calculated.The results show that compared with whole-period LUT,sLUT achieves equivalent performance in terms of error correction.A fringe projection system based on an industrial camera(model:Basler a2A1920-160ucBAS)and digital projector(model:DLP Light-crafter 4500)is used.The camera resolution is 1920 pixel×1200 pixel and the projector resolution is 912 pixelX1140 pixel.Experimental validation is performed on a processing device with CPU(AMD Ryzen 5 5600H),GPU(NVIDIA GeForce RTX 3050Ti Laptop),and 16 GB memory system.The test results show that the maximum difference of STDs between the two methods is only 0.002 rad,indicating that the sLUT achieves equivalent error correction performance as whole-period LUT.A particularly notable aspect is that the sLUT achieved these results using only 60 table elements,representing an 83％reduction over the 360 elements comprising the whole-period LUT.This parameter efficiency allows for faster computation while still enabling high-fidelity nonlinear error modeling.Quantitative analysis shows the average error correction time is reduced from 0.97 s for the whole-period LUT to just 0.12 s for the sLUT(Table 1),showing an approximately 8-fold speed enhancement.In summary,both simulation and physical experimentation provide a strong validation that the proposed sLUT methodology offers correction accuracy on par with whole-period LUT while significantly improving computational efficiency and highlighting its significant advantages for practical phase metrology applications.Conclusions We propose a new method for addressing nonlinear phase errors in PSPs,the error correction method based on the sLUT.This method takes into account both the periodicity and symmetry of the phase errors and constructs a lookup table focusing only on the phase errors within a 1/6-period range.Experimental results demonstrate that,while reducing the parameter size of the sLUT by 83％,the same error correction performance as the traditional LUT can be achieved.Additionally,the computation time for error compensation is reduced by a factor of 1/8.The experimental results also indicate that the 1/3-period LUT achieves the highest accuracy in phase correction.This may be attributed to the 1/3-period LUT's ability to more accurately capture the periodic characteristics of the phase errors during the correction process.Compared to the full-period LUT and sLUT,the parameter values may not fully match the periodicity of the phase errors,resulting in inferior correction performance.However,the optimal parameter size for the sLUT may vary with changing experimental conditions.Therefore,further research is needed to achieve a balance between accuracy and speed by determining the most suitable parameter size for the sLUT.
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