As an active optical three-dimensional (3D) measurement technique, fringe projection profilometry projects fringes onto the surface of the measured object, obtains phase information from the acquired deformed fringe image and performs 3D reconstruction. The traditional method employs standard sinusoidal fringes for measurement, which limits the projection speed of the digital projector and affects the measurement efficiency. Additionally, the Gamma value set by the projector makes the projected sinusoidal fringes lose their good sinusoidal properties and introduces nonlinear errors. Binary fringe has only two gray values to avoid nonlinear effect and improve projection efficiency. Therefore, binary fringe projection technologies have been widely concerned in the three-dimensional measurement of fringe projection. Among them, binary coding technology can solve the phase information loss caused by the decreased image quality during defocusing projection, and the error diffusion algorithm makes the fringe spacing no longer limited in practical applications. We propose a three-dimensional measurement method of binary coding combined with an error diffusion algorithm to optimize the sinusoidal quality of binary fringes and improve the measurement accuracy. On this basis, the serpentine path scanning method is adopted to further improve the sinusoidal fringes.

We put forward a fringe design method combining binary coding and an error diffusion algorithm. The continuous standard sinusoidal fringes of a period are regularly sampled, the sampling points are divided into equal intervals in the time domain, and the gray scale regions corresponding to each interval are processed by the error diffusion algorithm to generate corresponding binary fringes. The binary fringe is projected onto the surface of the object by a digital projector, and the obtained fringe image modulated by the measured object is superimposed to obtain the sinusoidal fringe containing the height information of the measured object. To further improve the phase quality of sinusoidal fringes, we modify the traditional error diffusion path and utilize the path scanning method of "odd lines spreading to the right and even lines to the left" to calculate the pixels. This method leverages four binary fringes instead of one sinusoidal fringe and then combines the four-step phase shift algorithm with the complementary gray code method to carry out phase unwrapping and complete the 3D measurement under the focusing state of the projector.

To verify the superiority of binary coding combined with the serpentine path scanning error diffusion algorithm, we conduct several sets of comparative experiments. In the phase quality comparison experiment, with the high-precision calibration board, the sine stripes generated by the serpentine path scanning have a better phase quality. The phase error results under each fringe period are approximate under the focusing and de-focusing states. When the generated sinusoidal fringe has a fringe period of 32, the phase root mean square error (RMSE) error of the solution is 0.0075 rad. Compared with the error before modification, the measurement phase error of the proposed algorithm is reduced by 24.19%, and it is lower than that of the traditional binary defocusing technique (Fig. 7). In the sinusoidal comparison experiment, the optimized scheme is compared with the non-optimized scheme and the traditional four-step phase shift algorithm, and the obtained sinusoidal fringe is sinusoidally fitted. When the fringe periods are 32 pixel and 96 pixel, the RMSE of the optimized scheme is 0.87457 and 1.0465, and the sum squared error (SSE) is 192.75 and 289.12 respectively (Fig. 9 and Table 1). In the precision comparison experiment, the standard precision ball with a diameter of 50.8140 mm is measured. The optimized scheme fits the diameter of the ball to 50.8178 mm, and the average distance between the point cloud data and the standard ball with the same center and diameter of 50.8140 mm is 0.006544 mm (Fig. 10 and Table 2). In the contrast experiment of deep objects, the plaster whose surface depth changes greatly is measured, and the optimized scheme reconstruction surface is smooth, which can avoid the influence of nonlinear errors.

By combining binary coding with an error diffusion algorithm, high-quality sinusoidal fringes are obtained under the action of low pass filter in the 3D measurement system. The measurement accuracy, phase error, sinusoidal mass, and actual measurement effect of sinusoidal fringe obtained by binary coding combined with serpentine path scanning and natural path scanning error diffusion are compared by simulation experiments. Simulation and experiments show that serpentine path scanning can further improve the quality of sinusoidal fringes based on the sound measurement results of the proposed algorithm. The proposed method employs the superposition of four binary fringes to generate a sinusoidal fringe, greatly reducing the fringe number compared with the existing binary coding technology. Therefore, the binary coding combined with the error diffusion algorithm proposed in our paper provides a new research approach in the three-dimensional measurement neighborhood of binary fringes.