Geometric analysis and new discoveries for phase-shifting algorithms based on the orthogonal resolution and resultant of forces

In fringe projection profilometry, the wrapped phase extraction is an essential process for absolute phase unwrapping and even the computation of object height information. Over the past few decades, tremendous efforts have been devoted to developing various techniques for computing wrapped phase. By contrast, the phase-shifting techniques process more advantages including higher accuracy, higher spatial resolution, and lower sensitivity to variations of background intensity and surface reflectivity. At present, a variety of phase-shifting algorithms show the comprehensive mathematical deduction and their theories are very clear. Analysis from the perspective of theoretical integrity, however, the phase-shifting techniques lack the exploration of geometric algebra. For that reason, inspired by the orthogonal resolution and resultant of forces in physics, we present a geometric analysis method. Furthermore, exploiting the proposed method to explore the double three-step algorithm, four-step algorithm and extended averaging technique, we obtain three new discoveries. Simulations and experiments have been carried out to verify the performance of these new discoveries. In addition, these results also reflect the necessity of the geometric analysis method for phase-shifting techniques.