Profilometry of three-dimensional discontinuous solids by combining two-steps temporal phase unwrapping, co-phased profilometry and phase-shifting interferometry

Highlights

• We describe how to combine co-phased profilometry with 2-step temporal phase-unwrapping to digitize discontinuous solids.

• This technique minimizes the problem of self-generated shadows in profilometry of discontinuous solids.

• This 3D profilometer of discontinuous solids uses 2-steps temporal phase unwrapping of two widely separated phase sensitivities.

Abstract

In this work we review and combine two techniques that have been recently published for three-dimensional (3D) fringe projection profilometry and phase unwrapping, namely: co-phased profilometry and 2-steps temporal phase-unwrapping. By combining these two methods we get a more accurate, higher signal-to-noise 3D profilometer for discontinuous industrial objects. In single-camera single-projector (standard) profilometry, the camera and the projector must form an angle between them. The phase-sensitivity of the profilometer depends on this angle, so it cannot be avoided. This angle produces regions with self-occluding shadows and glare from the solid as viewed from the camera׳s perspective, making impossible the demodulation of the fringe-pattern there. In other words, the phase data is undefined at those shadow regions. As published recently, this limitation can be solved by using several co-phased fringe-projectors and a single camera. These co-phased projectors are positioned at different directions towards the object, and as a consequence most shadows are compensated. In addition to this, most industrial objects are highly discontinuous, which precludes the use of spatial phase-unwrappers. One way to avoid spatial unwrapping is to decrease the phase-sensitivity to a point where the demodulated phase is bounded to one lambda, so the need for phase-unwrapping disappears. By doing this, however, the recovered non-wrapped phase contains too much harmonic distortion and noise. Using our recently proposed two-step temporal phase-unwrapping technique, the high-sensitivity phase is unwrapped using the low-frequency one as initial gross estimation. This two-step unwrapping technique solves the 3D object discontinuities while keeping the accuracy of the high-frequency profilometry data. In scientific research, new art are derived as logical and consistent result of previous efforts in the same direction. Here we present a new 3D-profilometer combining these two recently published methods: co-phased profilometry and two-steps temporal phase-unwrapping. By doing this, we obtain a new and more powerful 3D profilometry technique which overcomes the two main limitations of previous fringe-projection profilometers namely: high phase-sensitivity digitalization of discontinuous objects and solid׳s self-generated shadow minimization. This new 3D profilometer is demonstrated by an experiment digitizing a discontinuous 3D industrial-solid where the advantages of this new profilometer with respect to previous art are clearly shown.

Introduction

Since the 1980s, fringe-projection profilometry of three-dimensional (3D) objects has been a well-established and productive technique for digitizing solids [1], [2], [3]. Over the years, several improvements have been made to this fringe-projection profilometry technique [2], and this has been accompanied by important improvements to the theory of digital phase-demodulation methods applied to fringe-pattern analysis [3].

Digital fringe-projection for 3D profilometry continues to be an active research field as can be seen from the references on this paper [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. In 2010, Gorthi et al. reviewed the main techniques used in 3D shape measurement [2], and fringe analysis algorithms was in turn reviewed two years later [4]. Wang et al. [5] published practical considerations to take into account when measuring 3D shapes using fringe-projection profilometry. Su et al. [6], and Liu et al. [7] explored how to integrate N camera-projector systems that generate N fringe-patterns from different perspectives to obtain full 3D shape measurement. Classically, profilometers with non-coherent integration from N fringe-projections with N different perspectives must first estimate the fringe-boundaries of the N fringe-patterns [8]. This N-times fringe׳s boundary estimation is followed by N phase-demodulations, which in turn is followed by N phase-unwrapping processes, being the final step the sum of N unwrapped-phases into a single 3D digital-rendering [2], [3], [4], [5], [6], [7], [8]. Doing this entire process of boundary estimation, phase-demodulation and phase-unwrapping for N projecting directions is computationally inefficient and multiplies the possible error-sources in 3D-shape profilometry [2], [3], [4], [5], [6], [7], [8]. In contrast, in co-phased profilometry with N-projectors and 1-camera, one “blindly” adds-up the N analytic signals estimated from N projecting directions producing a single wrapped-phase to be processed [9]. In other words, the previous approach where the entire profilometry process had to be applied to each one of the N fringe-patterns [2], [3], [4], [5], [6], [7], [8] is reduced in co-phased profilometry to a single digitally-coherent phase-demodulation, and a single unwrapping process for the whole set of N fringe-patterns [9]. In this paper we work with only 2-projectors and a single camera (for higher complexity configurations, see [9]). In addition to the above-mentioned reduction in computational complexity, we can also reduce quite-drastically the number of phase-maps required to perform a temporal phase-unwrapping [10].

Temporal phase-unwrapping was introduced by Huntley and Saldner to measure optical dynamic wavefronts from a solid sustaining dynamic loading [11]. Temporal unwrapping has also been used to measure dynamic sequences of holograms [12]. As the name implies this unwrapping process is not made in the spatial domain but in the temporal domain. One advantage of using this method is being able to analyze highly discontinuous objects, because each spatial pixel from the interferometric data is unwrapped independently from its neighbors [11]; another feature is that noisy pixels remain isolated and do not spread uncertainties to less noisy regions ruining the entire unwrapping process [11]. In a subsequent paper, Saldner and Huntley [13] applied temporal phase-unwrapping to profilometry of discontinuous 3D objects. They stated that Nyquist׳s sampling-limit allows at most 1-wavelength sensitivity increase per temporal step. Thus, if one wishes to go from $1\lambda$ to $G\lambda$ in phase-sensitivity, a sequence of at least G phase-maps with incremental sensitivity would be needed: $\{1\lambda ,2\lambda ,3\lambda ,\dots ,G\lambda \}$. Nevertheless, we have shown that a similar result can be obtained using only the lower and higher sensitivity phase-maps from this sequence: {$1\lambda ,G\lambda$} [10]. As discussed later in Section 3.3, for the case of static 3D solids, the Nyquist sampling-limit may be surpassed because the profiling surface remains static during the whole 3D profilometry experiment.

Since the publication of references [11], [13], multi-sensitivity profilometry of highly discontinuous solids has been an active research field [10], [14], [15], [16], [17], [18], [19], [20], [21], [22]. This is because discontinuity in industrial objects is the rule rather than the exception, and generally precludes the use of spatial phase-unwrapping algorithms. Several researches have demonstrated algorithms to unwrap temporal phase-maps of static solids having different sensitivities [14], [15], [16], [17], [18], [19], [20], [21], [22]. They have used mainly triangulation and other varied techniques to unwrap higher sensitivity phase-maps from less sensitive phase-maps. As mentioned in the abstract, in this paper we combine two of our previously reported works [9], [10] to generate a computationally efficient and more powerful new 3D profilometer for discontinuous industrial solids. These works are the co-phased profilometry technique [9] and the temporal phase-unwrapping algorithm with 2-sensitivity fringe-patterns [10]. The resulting 3D fringe-projection profilometer is a logical consequence of combining these previous methods that solve the two main difficulties in fringe projection profilometry, namely: self-generated shadows and phase-unwrapping with highly different sensitivities. As far as we know, this new approach to digitize 3D discontinuous solids combining high phase-sensitivity and shadow minimization has not been reported before [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] and its advantages are clearly demonstrated from the experimental results herein presented. This paper also sheds new light on why it is possible to go beyond the Nyquist׳s temporal sampling rate for the 2-step phase-unwrapper. This is related with the Shannon information entropy of the 3D surface under analysis.

The presentation for the rest of this paper is as follows. First, we review the co-phased profilometry technique for two fringe-projectors and one-camera [9]. Next, we analyze the temporal phase-unwrapping method for two phase-maps whose sensitivity differs by many sensitivity wavelengths [10]. Finally, we combine these two techniques into a new high precision, high signal-to-noise ratio 3D profilometer for discontinuous industrial solids. The paper ends up showing an experimental result and conclusions of the compound profilometry technique herein presented.