Frequency selection in absolute phase maps recovery with two frequency projection fringes

In a recent published work we proposed a technique to recover the absolute phase maps of two fringe patterns with different spatial frequencies. It is demonstrated that a number of selected frequency pairs can be used for the proposed approach, but the published work did not provide a guideline for frequency selection. In addition, the performance of the proposed technique in terms of its anti-noise capability is not addressed. In this paper, the rules for selecting the two frequencies are presented based on theoretical analysis of the proposed technique. Also, when the two frequencies are given, the anti-noise capability of technique is formulated and evaluated. These theoretical conclusions are verified by experimental results.


Introduction
Fringe projection profilometry (FPP) is one of the most promising approaches for non-contact 3D shape measurement. A challenging task associated with existing phase measurement techniques in FPP is phase unwrapping operation, which aims to recover the absolute phase maps from the wrapped phase maps falling in ( , ) −π π . Although various phase unwrapping methods have been proposed, such as spatial [1], temporal [2,3] and period coding [4], recovery of absolute phase maps is still a challenging task when the wrapped phase maps contain noise, sharp changes or discontinuities [2].
To achieve reliable and accurate phase unwrapping for FPP, a variety of temporal phase unwrapping approaches have been proposed following work of Huntley and Saldner [2]. The general idea behind the temporal method is the use of multiple fringe patterns that are projected onto the project, yielding a sequence of wrapped phase maps as a function of time t. These phase maps can be considered as a 3D phase map ( , , ) m n t φ , denoting the wrapped phase value at pixel ( , ) m n at the tth phase map (t = 0, 1, 2, …, s). Phase unwrapping can be carried out along any path in the 3D space in order to avoid noise or boundaries and thus achieving correct recovery of the absolute phase map. While the method proposed in [2] is demonstrated to be effective for accurate phase unwrapping, it also suffers from the drawback of requiring many intermediate phase patterns (e.g., 7 sets of fringe patterns were employed in [2]), which is obviously not suitable for fast or real-time measurement. In order to increase the efficiency, Zhao, et al. [3] proposed to use two image patterns, one of which has a very low spatial frequency in contrast to the other. In particular, the low spatial frequency pattern only has a single fringe. Such a pattern has its absolute phase value falling within the range ( , ) π π − , and hence it can be used as a reference to calculate the fringe number of the other fringe pattern, thus yielding its absolute phase map. Li, et al. [5,6] also employ the phase map of single fringe pattern as reference to unwrap high spatial frequency fringe patterns, and it is shown that the spatial frequency of the pattern to be unwrapped is determined by the level of noise. Following the same method in [5], Liu, et al. [7] project a single fringe pattern and a high frequency pattern in one shot to accelerate the speed of 3D measurement. Although these approaches work well in principle, the gap between two spatial frequencies should be restricted within a range based on the noise level or steps in the low frequency phase maps. This is because under the same lighting conditions, fringe patterns with lower frequency are more vulnerable to the influence of noise or interferences [3,6,8], and use of noisy lowfrequency pattern as the reference will lead to mistakes for unwrapping the high-frequency phase maps. Therefore, the techniques proposed in [3,5,6] may not work well when the phase maps are noisy or discontinuous, and multiple intermediate image patterns are still required in order to reduce the frequency gaps among adjacent patterns. This problem is studied again by Saldner and Huntley [9,10], showing that to unwrap a phase map of frequency f, 2 log 1 f + sets of fringe patterns are required. A similar result is also reached by Zhang [8,11], indicating that the spatial frequency can be increased by a factor of 2 between two adjacent patterns. Taking a typical FPP arrangement as an example where the image pattern has 16 fringes, 5 image patterns are still required with these approaches. Towers, et al. [12] propose an optimal frequency selection method to increase the unambiguous range of the measurement, showing that at least three frequencies were needed for a defined reliability in fringe number calculation. Therefore, existing temporal phase unwrapping techniques still require the use of multiple image patterns, and reduction of the number of image pattern while keeping anti-noise capability is still a challenging problem. A set of technologies, which are similar by name to the above and referred to as two (or multiple wavelength) interferometry, are also proposed in the area of traditional interferometry, where use of multiple light sources with different wavelengths have shown to yield significant advantages for distance measurement [13,14]. When a monochromic light with wavelength λ is used, the measurement of a optical path difference (OPD) is suffered from ambiguity of module λ , making λ to be the so-called unambiguous OPD range (UR) for the measurement. The idea behind multiple wavelength interferometry technology is that by using multiple laser beams with different wavelengths in an interferometer, the resulting interferometric pattern is equivalent to the result of using a single laser beam in the same interferometer with a much longer wavelength, implying a significant increase in UR. If the laser beams of different wavelength 1 λ and 2 λ are used in two wavelength interferometry (TWI), an interferometric patterns can be formed with the equivalent wavelength of λ can be made much larger than 1 λ and 2 λ . With the development of digital cameras and computers, the two wavelength interferometry is considerably improved by the introduction of the phase shifting algorithm [15,16]. Houairi, et al. [17] present an analytical algorithm, showing that the actual UR could be much larger than the equivalent wavelength depending on the wavelengths and different sources of error.
In summary of the above, we have seen two classes of research effort. On one hand, temporal phase unwrapping techniques aim to recover the absolute phase map of fringe patters used for FPP. Due to the existence of noise, multiple intermediate fringe patterns must be used. On the other hand, the approaches of multiple wavelength interferometry aim to increase of the UR for distance measurement; they employ an equivalent fringe pattern from the use of multiple laser sources with different wavelengths in the same interferometer setup. The spatial wavelength of the equivalent pattern can be made larger than the individual laser sources, hence leading to increase of the UR for distance measurement. However, it is not guaranteed that the spatial wavelength of the equivalent pattern cover the whole image (that is, the equivalent interferometric pattern contains only a single fringe), and hence they are not yet able to solve the problem of absolute phase map recovery in FPP.
In order to remedy the phase unwrapping problem in FPP described above, the authors of this paper developed a novel approach to recover the absolute phase maps of two image patterns with selected frequencies [18]. Examples were presented to show that both of the two frequencies can be high enough for the applications of FPP. However, a number of issues are still outstanding associated with the proposed technique in [18], namely, the basic rules to select the frequencies and anti-noise capability of the proposed technique, that is, the phase error bound that ensures the correct the recovered absolute phase recovery. This paper aims to address these two important issues with the aim to provide a complete solution for the implementation of the proposed technique in [18]. This paper is organized as follows. In Section 2 we firstly present a brief review of the technique in [18], and then analyze the principle for frequency selection. In Section 3, the phase error bound is given. In Section 4, experiments are presented to validate the principle on frequency selection and the phase error bound. Section 5 concludes the paper.

The technique proposed in [18]
With the approach proposed in [18], two fringe patterns with normalized spatial frequencies which is the pixel index in the horizontal direction and T is the total number of pixels. 1 ( ) are the absolute phase maps to be recovered, 1 ( ) x φ and 2 ( ) x φ are the wrapped phase maps associated with the two fringe patterns with frequencies 1 f and 2 f respectively. As 1 ( ) ( , ) x φ π π ⊂ − and 2 ( ) ( , ) as follows: The relations between 0 ( ) x Φ and 1 ( ) m x , 2 ( ) m x are displayed in Eq. (4)-(5).
π π π π π π π π (4a)  Min m x f m x f The operational principle of the proposed technique can also be explained graphically using an example where 1 5 f = and 2 8 f = . Figure 1 shows the relationship among three absolute phase maps 0 ( ) . Figure

Principle of frequency pair selection
The validity of the approach proposed in [18] relies on the existence of unique mapping from π to a pair of 1 ( ) m x and 2 ( ) m x . That is, 1 f and 2 f must be selected so that such a unique mapping relationship is held.
In order to achieve the above, let us firstly look at the relationship between 0 ( ) x Φ and ( ) On each of the intervals 1 ( ) m x takes a different value. Similarly, Eq.
(5) shows that the range of 0 ( ) x Φ can also be divided into 2 On each of the intervals 2 ( ) m x takes a different value. When 1 f and 2 f are coprime, it is easy to show that f will not be coprime otherwise). Hence we can say that the interval boundaries of the two types of intervals will not coincide. These two types of boundaries together can divide the range of 0 ( ) x Φ into N intervals: Each of the intervals must correspond to an unique pair ( ) The above statement shows when 1 f and 2 f to be coprime, there exists a unique solution for the phase unwrapping problem. In order to show that the proposed approach in [18] is sufficient to work out the solution, we should also have the following: Statement 2: When 1 f and 2 f is coprime, for any two different intervals a p x ⊂ Ω , b q x ⊂ Ω and p q ≠ ,we must have two corresponding pairs of ( ) 1 2 ( ), ( ) m x m x based on Statement 1, which also meet the following: In other words, there exists a unique mapping from ( ) We prove the Statement 2 by reductio ad absurdam. There are three possible scenarios making the two pairs of ( ) . Assume that the following is valid: Equation (7) can be rewritten as: As 1 f and 2 f are coprime, Eq. (8) must be equivalent to the following: where k is an integer and 0 k ≠ . Considering the ranges of 1 ( ) m x and 2 ( ) m x given above, we have 1 1 Comparing Eq. (9) with Eq. (10), it is obvious that 1 k = ± . Hence we have Equation (12) implies that 1 f and 2 f are both even numbers, which is contradict to the fact that 1 f and 2 f are coprime. Hence Eq. (7) will not be true for the case (a)