## Abstract

A novel multi-frequency fringe projection profilometry is presented in this paper. Fringe patterns with multiple frequencies are projected onto an object by a digital micro-mirror device projector. The approach involves an improved Fourier transform profilometry method with an additional π phase shifting stage and hence the acquisition of two source images. A peak searching algorithm is then employed to obtain the real height profile of the object together with a mathematical proof of this algorithm. In our method, the height of each point on the object is measured independently and the phase unwrapping procedure is avoided, enabling the measurement of objects with large depth discontinuities, where the phase unwrapping is difficult. The measurement result is given to validate the method in the paper. Our technique has great potential in industrial applications where the measurement of objects with complex shape and large discontinuities is needed.

© 2014 Optical Society of America

## 1. Introduction

Three-dimensional (3D) shape measurement techniques have been developed to measure the profile of objects for many years [1]. In both scientific research and industrial applications, such as solid modeling and automated manufacturing, non-contact and full-field optical surface profilometry is becoming increasingly important. Different methods have been investigated and developed, among which those based on fringe projection and triangulation have attracted a lot of attention and have been applied widely in practical applications because of their superior performance and simple implementation [2]. In a simple fringe projection system, a digital micro-mirror device (DMD) projector is usually used to project structured fringe patterns onto an object and a charge coupled device (CCD) camera is used to record the distorted fringes. The relative phase map of the object can be calculated with different methods such as phase-measuring profilometry (PMP) [3–5], Fourier transform profilometry (FTP) [6]. Compared with PMP and other method used to acquire the relative phase map of an object, FTP needs only one or two images of the deformed fringe pattern, which makes it possible for real-time data processing [7]. During the calculation, the involved arctangent operation makes the value of the relative phase map range from – π to π. Thus, the relative phase map is wrapped and spatial phase unwrapping algorithms must be used to make the relative phase map continuous. However, if the object itself has large-depth discontinuities (corresponding phase of multiple of 2π), it will be difficult to unwrap the phase correctly [8]. Many factors can cause errors in phase unwrapping, such as local shadows or low fringe modulation, irregular surface brightness, fringe discontinuities and under sampling [9]. Moreover, many of the phase unwrapping algorithms are path-dependent, and in this case, the phase errors will propagate through the image. The nonlinear intensity response of the DMD projector and the CCD camera will also cause phase errors and therefore measurement errors [10]. Intensive studies have been devoted to developing reliable phase unwrapping algorithms, and a review on this topic can be found in [9].

In 1993, J. M. Huntley and H. Saldner proposed a temporal phase unwrapping algorithm for measuring objects with surface discontinuities [11–13]. In this method, multiple images are recorded at fringe patterns with different frequencies (from low to high) in time line, and the phase of each pixel is unwrapped along the time axis. The phase map calculated by the fringe patterns with the lowest frequency ranges from –π to π, while the phase map with lower fringe frequencies is used to guide the phase unwrapping process at a higher fringe frequency. In this way, the phase of each pixel is calculated by the same pixel in different phase maps with different fringe frequencies, and phase errors do not propagate through the whole image. Although the temporal phase unwrapping method can be used in measuring objects with large depth discontinuities, it needs strict control of fringe parameters. To carry out temporal phase unwrapping, the first group of fringe patterns must contain only one fringe in the full-field to make the first phase map unwrapped, and the real phase difference of the same point between two adjacent groups of fringe patterns must be less than 2π. Moreover, fringe projection profilometry techniques which use multi-frequency fringe patterns have been researched actively [14] and [15]. In these multi-frequency profilometry techniques, two or more fringe patterns with different frequencies are used in order to get the wrapped phase map of an object. In [14] and [15], the need of phase unwrapping procedure still remains [16]. reports a profilometry using four-step phase-shift method, but these four images are used to calculate the local illumination image function, not wrapped phase maps. It is not a phase-measuring profilometry, but an improved line scanning method. Heterodyne principle is also used in multi-frequency fringe projection profilometry such as [14] and [17]. In order to obtain an absolute phase map, multi-frequency fringe projection profilometry with heterodyne principle needs to control the frequencies used in the measurement. The frequency have to be chosen carefully in order to avoid unwrapping failures caused by phase errors [17]. Overall, it is still a challenge to measure complex shaped objects with large depth discontinuities by fringe projection profilometry.

In this paper, we propose a novel multi-frequency fringe projection profilometry with Fourier transform. Fourier transform is used to obtain the wrapped phase map of the object and a peak searching algorithm is utilized to calculate the height of object. In this novel method, the height of each point on the object is measured independently and the phase unwrapping procedure is avoided, enabling the measurement of objects with large depth discontinuities.

## 2. Principle and method of multi-frequency fringe projection profilometry

In a typical fringe projection profilometry system (Fig. 1), we use a computer to generate fringe patterns with multiple frequencies and a DMD projector to project the patterns onto the object. A CCD camera is used to collect the deformed fringe patterns and send the images to the computer.

#### 2.1 Obtain the wrapped phase map by Fourier transform

The cross section of our system geometry is shown in Fig. 2. *X* axis and *Y* axis form the reference plane. $h\left(x,y\right)$ is the height of the object. After we place the object on the reference plane, the deformed fringe pattern observed through a CCD camera can be written as:

By computing the 1-D Fourier transform in the x-direction of Eq. (1), the Fourier spectra of every line in the object diagram could be obtained, as is shown in Fig. 3. With a suitable filter function used, the fundamental component of the spectra could be obtained. If we use the improved Fourier transform method and acquire another image with π phase shift [19]. We could form that:

Therefore, the differences between Eq. (3) and Eq. (4) is:Here, the Fourier Transform is applied to both Eq. (5) and (6) to obtain the fundamental component with a window function. After that, we apply the inverse Fourier transform to the fundamental component and get a complex signal [19]:

Then, the phase modulated by the height of the object can be expressed as:#### 2.2 Peak searching algorithm for calculating the height of object

Considering the periodicity of a sinusoid, we recognize that the modulus term can be taken into account. Therefore, simply adding the sinusoids together will give a peak signal at the height of the object. We then form the following function:

As was discussed above, we have established a technique of multi-frequency fringe projection to get the height, in which the height of each point of the object is calculated independently and phase unwrapping procedure is avoided, enabling us to measure objects with large depth discontinuities.

#### 2.3 Mathematical proof of the above methods

Considering the expression inside Eq. (15), we can write that:

We can give the following definition according to Fig. 4:

Equation (23) is the length between the two nearest peaks and it is named ambiguity free height. Within this range, the height of the object can be found easily and accurately without encountering extra peaks. The finer the spatial frequency spacing, the greater the ambiguous free height range. Equation (24) is the full width half maximum (FWHM) of the peak and the FWHM is the nominal resolution of this method, which is related to the bandwidth of the selected frequencies as shown in Fig. 4. Therefore, the greater the bandwidth, the finer the resolution.## 3. Experimental results

In the experiment, we used a DMD projector to project the fringe patterns generated by LabVIEW, and a JAI GigE CCD camera (CM-030GE) was used to collect the deformed fringe patterns. Here, we only needed to collect two images with a phase-shift of π for each frequency. The frequencies of the fringe patterns were equally spaced from 7/6 fringes/cm to 1.5 fringes/cm with a common difference of 1/30 fringes/cm.

The object used in the experiment is shown in Fig. 5. The height of the object was measured by Vernier calipers. The higher part of the object is 14.72mm high, while the lower part is 5.14mm high. The image without background intensity and higher spectrum is shown in Fig. 6(a). Figure 6(b) is the wrapped phase map of the object calculated by Fourier Transform.

After carrying out the proposed method mentioned in the last section, we show the curve of Eq. (14) for one point on the object in Fig. 7. Since we used 11 frequencies, there are 10 small peak value between the main maximums in Fig. 7.

The reconstructed 3D shape of the object is shown in Fig. 8. The measured height of the higher part is 14.58mm high while the real height is 14.72mm and the lower part is 4.99mm high while the real height is 5.14mm. Therefore, for the higher part, the measurement error is 0.14mm and 0.15mm for the lower part. The accuracy of measurement is related to several factors such as the gamma effect of the projector and the CCD camera, the selected frequencies, and the resolution of the CCD camera. The accuracy could be improved with more frequencies or by improving the gray level resolution and the spatial resolution of the CCD camera.

The reconstruction time of the 3D shape is short, because only two images with π phase-shift are needed for each frequency. It takes about 60s to complete the measurement process.

## 4. Conclusion

A multi-frequency fringe projection profilometry with Fourier transform is proposed. A peak searching algorithm and its mathematical proof are presented. The measurement results of an object with large discontinuities confirm the validity of the proposed method. Moreover, the measurement range is expanded because of using Fourier Transform profilometry with π phase-shift. The proposed method requires no phase unwrapping process and each point of the object is calculated independently. Therefore, our technique is capable of measuring objects with large depth discontinuities, making it promising in industrial applications.

## Acknowledgments

This work is supported by the National Natural Science Fund Committee and the Civil Aviation Administration of China Jointly Funded Project (U1233116) and Research Fund for the Doctoral Program of Higher Education of China (20120201110032)

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