## 1 Introduction  
The past several decades have seen revolutionary developments in three-dimensional (3D) optical shape measurement techniques [1; 2], which are now widely used across many different fields, including reverse engineering, 3D games, digitization of cultural relics, clothing, biometrics and other biological fields [3; 4; 5; 6; 7; 8; 9]. In the area of structured light techniques, phase-calculation-based fringe pattern projection has made rapid progress so far, because it has the advantages of full-field measurement, non-contact operation, automatic data processing, and fast data acquisition [10; 11; 12; 13]. Nevertheless, there are a large number of specular objects that need to be measured [14]. Current fringe pattern projection techniques cannot deal with this demand, especially in the measurement of specular objects that have discontinuous and/or isolated surfaces. With the increasing demands of industrial development, much relative research has been done to solve the problem of specular object measurement. Shape reconstruction from gradient data [15; 16; 17; 18] based on phase measuring deflectometry (PMD, also called fringe reflection profimetry) [19; 20; 21; 22; 23; 24; 25; 26; 27; 28] has been widely studied because of its high dynamic range, full-field measurement, fast data acquisition, and automatic data processing. Cross path integration techniques, Fourier transform integration, and area wavefront reconstruction methods have been demonstrated [29]. However, existing PMD methods have the drawback of local error accumulation and are incapable of measuring discontinuous specular objects while reconstructing 3D shapes by integrating gradient data. Many improved algorithms have been proposed to avoid this integration procedure. [PERSON] et al. [30] presented an iterative method that uses discrete cosine transforms to deal with the integration problem, with an incomplete gradient dataset in a Southwell configuration. [PERSON] et al. [31] presented a method of measuring the absolute height and slope of a specular surface with an incident ray, camera pinhole, and dummy assistant surfaces. [PERSON] et al. [32] selected a binary shifting strip as a structured light pattern for 3D reconstruction of the specular surface. Recently, [PERSON] et al. [33] have proposed a model phase measuring deflectometry technique that uses a mathematical method to reconstruct 3D shapes of specular objects. In all these PMD methods, calibration of the system parameters plays a key role in the measurement of the specular surface because it directly influences the accuracy of the measurement system. [PERSON] et al. [34] presented an iterative optimization algorithm to calibrate stereo deflectometry. It includes two steps: a stepwise method and a self-calibration method; however, errors accumulated via the iterative principle is a potential problem. [PERSON] et al. [35] introduced a calibration method that uses a markerless flat mirror. This method requires a two-dimensional cosine fringe pattern to be reflected three times. It is easy for the images to become out of focus, which influences the accuracy of extraction of the feature points on the screen. System calibration is thus still a challenging problem in measuring 3D shapes by PMD. A novel DPMD (direct PMD) method that can obtain the full-field 3D shape of specular objects with discontinuous and/or isolated surfaces has been developed to solve the problem of measuring complex specular surfaces [36; 37; 38]. This method builds a direct relationship between the absolute phase and the depth information. The DPMD method can calculate depth data from one directional fringe patterns at two display screens, instead of two orthogonal fringe patterns. Moreover, 3D shape data of specular objects could be obtained from two unwrapped phase maps directly, without the procedure of integrating gradient data.