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An improved method which considers the use of Fourier and wavelet transform based analysis to infer and extract 3D information from an object by fringe projection on it is presented. This method requires a single image which contains a sinusoidal white light fringe pattern projected on it, and this pattern has a known spatial frequency and its information is used to avoid any discontinuities in the fringes with high frequency. Several computer simulations and experiments have been carried out to verify the analysis. The comparison between numerical simulations and experiments has proved the validity of this proposed method.

In the last three decades, the idea of extracting the 3D information of a scene from its 2D images has been widely investigated. Several contact and non-contact measurement techniques have been employed in many science and engineering applications to compute the 3-D surface of an object. Basically, the aim is to extract the useful depth information from an image in an efficient and automatic way. Then, the obtained information can be used to guide various processes such as robotic manipulation, automatic inspection, reverse engineering, 3D depth map for navigation and virtual reality applications [

Scarcely used fringe processing methods are the well-known Fourier Transform Profilometry (FTP) method [

Due to the fact that the wavelet transform offers multiresolution in time and space frequency, it is a tool that offers advantages over the Fourier transform [

Different wavelet algorithms are used in the demodulation process to extract the phase of the deformed fringe patterns. They can be classified into two categories: phase estimation and frequency estimation techniques.

The phase estimation algorithm employs complex mother wavelets to estimate the phase of a fringe pattern. The extracted phase suffers from 2π discontinuities and a phase unwrapping algorithm is required to remove these 2π jumps. Zhong

The frequency estimation technique estimates the instantaneous frequencies in a fringe pattern, which are then integrated to estimate the phase. The phase extracted using this technique is continuous; consequently, phase unwrapping algorithms are not required. Complex or real mother wavelets can be used to estimate the instantaneous frequencies in the fringe pattern. Dursun

Most of the previous research is focused on using the Fourier and wavelet transforms separately to obtain the 3D information from an object; pre-filtering the images, extracting the phase information of fringe patterns, using phase unwrapping algorithms, and so on.

In the present research, a simple profilometrical approach to obtain the 3D information from an object is presented. Here, the spatial frequency of the projected fringe pattern is obtained. The mathematical description to obtain the spatial frequency is a contribution in this research. Then, a modified Fourier transform method of an extended 1D wavelet based profilometry is applied. Later, a robust phase unwrapping algorithm is developed and used to obtain the desired 3D information. The main contribution of this work is the methodology. In addition, this novel approach is compared with other similar research and the results are presented. In order to validate the methodology, some virtual objects were created for use in computer simulations and experiments.

As described in the previous section, there are several fringe projection techniques which are used to extract the three-dimensional information from the objects. In this section, a Modified Fourier Transform is explained and the Wavelet Profilometry is introduced.

The image of a projected fringe pattern and an object with projected fringes can be represented by:
_{0}(_{0}_{0}

The phase _{0}_{z}

Considering

This leads to the next equation:

The fringe projection _{n}_{n} exp

Here _{0}_{0}

It is observed that the phase map can be obtained by applying the same process for each horizontal line. The values of the phase map are wrapped at some specific values. Those phase values range between π and −π.

To recover the true phase it is necessary to restore the measured wrapped phase by an unknown multiple of 2π_{0}

As mentioned earlier, the unwrapping step consists of finding discontinuities of a magnitude close to 2π, and then, depending on the phase change, 2π can be added or subtracted to the shape according to the sign of the phase change. There are various methods for doing the phase unwrapping, and the important factor to consider in this step is the abrupt phase changes in the neighbouring pixels. There are a number of 2π phase jumps between two successive wrapped phase values, and this number must be determined. This number depends on the spatial frequency of the fringe pattern projected at the beginning of the process.

This step is the modified part in the Fourier Transform Profilometry originally proposed by Takeda [

The wavelet transform (WT) is considered an appropriate tool to analyze non-stationary signals. This technique has been developed as an alternative approach to the most common transforms, such as Fourier transform, to analyze fringe patterns. Furthermore, WT has a multi-resolution property in both time and frequency domains which solves a commonly know problem in other transforms like the resolution.

A wavelet is a small wave of limited duration (this can be real or complex). For this, two conditions must be satisfied: firstly, it must have a finite energy. Secondly, the wavelet must have an average value of zero (admissibility condition). It is worth noting that many different types of mother wavelets are available for phase evaluation applications. The most suitable mother wavelet is probably the complex Morlet one [

The one-dimensional continuous wavelet transform (1D-CWT) of a row

In this contribution, phase estimation and frequency estimation methods are used to extract the phase distribution from two dimensional fringe patterns. In the phase estimation method, a complex Morlet wavelet will be applied to a row of the fringe pattern. The resultant wavelet transform is a two dimensional complex array, where the phase arrays can be calculated as follows:

To compute the phase of the row, the maximum value of each column of the modulus array is determined and then its corresponding phase value is found from the phase array. By repeating this process on all rows of the fringe pattern, a wrapped phase map results and an unwrapping algorithm is then needed to unwrap it.

In the frequency estimation method, a complex Morlet wavelet is applied to a row of the fringe pattern. The resultant wavelet transform is a two dimensional complex array. The modulus array can be found using

Due to the fact that:

We can re-write the

Then, we consider the analytic function _{0} ∈ A_{0};r) ⊂ A

Therefore:

If :

Moreover, the Morlet Wavelet is defined as

If

By solving

Then the instantaneous frequencies are computed using the next Equation [_{o}

The experimental setup shown in

When using the Fourier method, a robust unwrapping algorithm is needed, followed by an unwrapping algorithm with local and global analysis. The main algorithm for the local discontinuity analysis [_{1},w_{2}, .., w_{n}) to each region, (b) the modulation unit is defined that helps to detect the fringe quality and divides the fringes into regions, (c) regions are grouped from the biggest to the smallest modulation value, (d) next, the unwrapping process is started from the biggest to the smallest region, (e) later, an evaluation of the phase changes is carried out to avoid variations smaller than

If the wavelet transform method is used then a simple unwrapping algorithm is enough to obtain the three-dimensional shape from the object. The final step is to obtain the object reconstruction and in some cases to determine the error (in case of virtual created objects).

In the experimental setup, a high-resolution digital CCD camera can be used. The reference plane can be any flat surface like a plain wall, or a whiteboard. In the reference plane it is important to consider a non reflective-surface to minimize the unwanted reflection effects that may cause some problems for the image acquisition process. The object of interest can be any three-dimensional object and for this work, three objects are considered (

It is also important to develop software able to produce several different fringe patterns. To create several patterns, it is necessary to modify the spatial frequency (number of fringes per unit area), and resolution (number of levels to create the sinusoidal pattern) of the fringe pattern. It may also be necessary to include into the software development a routine capable of performing phase shifting as well as to include the horizontal or vertical orientation projection of the fringe pattern.

An object with a Buddha shape generated by computer is used to test the algorithms. The generated Buddha in shown in

The wavelet transform algorithm is considered o obtain the shape of the Buddha. The resulting wrapped phase and its mesh are shown in

By applying the Wavelet Profilometry, the mesh shown on

As a preliminary conclusion, it is clear that the wavelet transform gives a better performance than the Fourier transform for the selected object shape. Therefore, the wavelet transform is selected and used. The error of the wavelet method is about 1 to 2% and using the Fourier one it is about 3 to 5%.

To validate the methodology, the following experiments were conducted. Several objects with different shapes were created by computer, where the height is known in every point in the object. Then, the Fourier and wavelet based analysis are applied as well as the methodology proposed by Gdeisat

In this paper, a three-dimensional reconstruction methodology was presented. The method is based on the modified Fourier Transform Profilometry or Wavelet Transform Profilometry. In the first part of the proposed method the high frequencies that mostly affect the performance on the phase unwrapping in the Fourier method are obtained analytically. An object generated by the computer was virtually created and a known spatial sinusoidal fringe pattern was projected on it. Both Fourier and wavelet analysis were conducted, showing a good performance. In the comparison, the wavelet method was the one that showed a minimal error. Later, a real object was selected and the wavelet analysis was carried out and an accurate reconstruction of the object was achieved. This methodology could be widely used to digitize diverse objects for reverse engineering, virtual reality, 3D navigation, and so on.

Notice that the method can reconstruct only the part of the object that can be seen by the camera, if a full 3D reconstruction (360 degrees) is needed, a rotating table is can be used and the methodology will be applied

One big challenge is to obtain the 3D reconstruction in real time. As a part of the solution, an optical filter could be implemented to obtain the FFT directly, or else, the algorithm can be implemented into a FPGA to carry out a parallel processing and minimize the processing time.

We would like to thank the Informatics Faculty, Autonomous University of Queretaro for the support. This work was financially supported by the PROMEP/103.5/08/3320 and the FIF-2008-01 project number.

Experimental setup.

Complex Morlet wavelet.

Proposed methodology.

Computer created Buddha and fringes projected on it.

Wrapped phase (image and mesh).

Reconstructed object using our Modified Fourier Transform Profilometry.

Reconstructed object using our Extended Wavelet Transform Profilometry.

Reconstructed object using the methodology proposed by Gdeisat

Wrapped phase (image and mesh).

Real objects (a), (c), and (e), and their respective reconstruction (b), (d) and (f) views by using wavelet transform.

Error table.

| |||
---|---|---|---|

Buddha | 3.499 | 1.455 | 1.901 |

Butterfly | 3.776 | 1.733 | 1.824 |

Pyramid | 4.871 | 1.923 | 2.054 |