Abstract

The Shadow Moiré Method using the Phase Shifting Technique and Digital Image Processing: Computational Implementation and Application to the 3D-Reconstruction of a Buckled Plate

Guilherme Cremasco Coelho

Departamento Engenharia Mecatrônica. Pontifícia Universidade Católica de Minas Gerais. Av. Dom José Gaspar, 500. Bairro Coração Eucarístico. 30535-610 Belo Horizonte. MG. Brasil

Introduction

Among the techniques for measuring the surface structures of an object, two important groups can be selected, one based on interferometrical techniques and another on the Moiré technique. The methods of the first group can be used in objects that possess small relief variation, because an immense number of fringes can appear, hindering or disabling the analysis (Mauvoisin, Gérard et al., 1993). In this technique, the equipment use monochrome light and have high sensibility. Moreover, the surfaces should be appropriately polished. The equipment are quite sophisticated and are usually employed only in scientific applications in very equipped laboratories, with effective vibrational isolation.

The Shadow Moiré or Projection Moiré techniques belongs to the second group (Cloud, Gary L., 1995; Dally, James W., et al. 1991). It is convenient to mention the Reflection Moiré technique (Cloud, Gary L., 1995; Sciammarella, César A, 1982; James W et al. 1991), that provides the surface inclinations map, differently from other mentioned methods, that map the displacements in the direction perpendicular to the plane. The Shadow Moiré technique uses the interference between a pattern and its shadow projected on a surface. Compared to the interferometric methods used for out-of-plane displacement analysis, the Shadow Moiré method has limited sensibility, which is caused mainly by the diffraction phenomenon. Its sensibility, in an usual approach, is approximately equal to the master grid pitch, which is greater than 0,1 mm. However, it is simpler and more easily applicable to analysis of large displacements, is therefore more adapted for industrial applications.

The Shadow Moiré technique is not only an important tool in experimental mechanics, but also in assembly lines of industries, in the surfaces measurement and control. It has also been applied in medicine, in the obstetrics, orthopedical and biostereometric area. (Cloud, Gary, 1995).

Principles of Shadow Moire

The Shadow Moiré method uses the overlap of a pattern with its own shadow. A beam of collimated light with oblique incidence to a surface passes through a master grid, positioned in front of the surface to be analyzed, as shown in Figure1. In this Figure, p is the master-grating pitch, W is the distance between the surface and the master grid, î is the light incidence angle and is the light reflection angle in the observer's direction.

A bit of geometrical analysis for an observer located at infinite, leads to the classic equation (1):

where N is the fringe order.

Buitrago and Durelli (1978) generalized the illumination and observation conditions for points at finite distances, but the displacement interpretation is not simple. However, the difference between the equations developed by them and Equation (1) is small when the distance between the observer and the master grid is much larger than W. For this reason, under such conditions, Equation (1) can still be used without loss of precision.

The basic experimental arrangement is shown in Figure 2 and consists of a collimated light source, a master grid and, usually, a photographic camera or video camera as observer. For the sake of simplicity, the distance c is taken as zero, so that the observer's axis is perpendicular to master grid. In this way, the light reflection angle in the observer's direction becomes zero and Equation (1) is reduced to:

It is supposed that the pattern assumes a sinusoidal form (Mauvoisin, et al., 1993) given by:

As shown in Figure 1, the light passes through the master grid in a point x₁, reaches the surface and is reflected, going again through the master grid, via point x₂. The luminous pattern that reaches the observer, formed by the interference of the master grid with its shadow is given by (Mauvoisin et al. 1993):

The parameters a and b depends on the surface quality, the intensity of the luminous source and the contrast applied to the photographic camera or video-camera; whereas f is the phase. The phase is related with the fringe order by:

Phase Shifting Method

In the Phase Shifting method, some modified images of the same surface are used. The question returns to the method of generation of that Phase Shifting and its relationship with the luminous pattern phase.

There are different ways to change the phase. The parameters that can be modified are, according to Figure 1 and Figure 2: b, h, p, and W. The parameters will be modified by, respectively, db, dh, dp,dW and consequently, the phase will be shifted by, dfb, dfh, dfp e dfw. The modification of the parameters b, h, p, and W consists in, respectively, changing the distance between the observer and the luminous source; moving both the luminous source and the observer axially to the master grid; changing the master grid pitch; and move the master grid axially to the surface.

Using Equations (2) and (5), and making the parameters change, the obtained phase differences are:

(6.b)

(6.c)

(7)

It can be observed from Equation (7) that the only df that is independent of the surface format (or, in other words, that is independent of W), is the one produced in W (when W is transformed in W+dW). So, the phase shift is the same in all points of the surface. Therefore, it is the only useful.

For instance, to generate a phase shift of 2p/3 radian, for î=45º and using a master grating of 50 lines/inch (p=0,508 mm), the master grid should be at a distance dw from the surface. Using Equation (7), dw would be 0,508 mm /3 @ 0,17 mm.

Displacement Determination

In the phase shifting method, three fringe patterns were recorded by a video camera. The phase shifting method can also use 4, 6 or more fringe patterns (Sullivan, John L., 1991) however in this work preference was given to the use of 3 images, because the processing time is shorter and it needs less computer storage. The images are digitalized in luminous intensity levels that vary from 0 to 1. The obtained patterns will have the form

In the expressions df12 is the phase difference, in radians, applied to the pattern of image 1 to generate the pattern of image 2 whereas df13 it is the phase difference, in radians, applied to the pattern of the image 1 to generate the pattern of the image 3.

It is important to note that one cannot just extract the arctangent of tanf , obtained by Equation (10), and after substitute f in Equation (5). The arctangent function have domain in the interval between -p /2 and p /2, so, the phase field has discontinuities which must be removed adding ± p in the discontinuities places. This operation is known as phase unwrapping and can be difficult in patterns with very concentrated fringes or when there is a bad image resolution. After the phase unwrap procedure, the obtained continuous phase was then substituted in Equation (10), and finally the displacement was obtained.

Application to the 3D Surfaces Reconstruction

A software was developed to analyze Shadow Moiré patterns using the phase shifting method. The program contains image processing procedures and functions, such as filters and region-of-interest routines; graphical interface and numerical functions. The applications were made firstly in simulated images, to evaluate the methods error accurately, and after in a buckled plate.

Application to Simulated Images

A software was also developed to simulate Shadow Moiré patterns using the equation of the surface that one wants to simulate. For an application, a simulated image of a fringe pattern was generated. The corresponding surface is:

with –5<x<5 e –5<y<5. In the simulation, the pitch was considered p=0,2mm and î=45º. Three fringe pattern images were generated for the same surface, but with a phase difference between them. The phase difference between the image pattern 0 and the image 1 was of 2p/3 and the phase difference between the image pattern 0 and image 2 was 4p/3.

The obtained pattern can be seen on the screen of the program, illustrated in Figure 3. The phase pattern before the unwrap operation can be seen in an image format, in Figure 4 with a gray scale representing the phase values in radians. In this image, one can observe the phase discontinuity represented by fast changes in color in the neighborhood of some points.

The obtained displacement can also be visualized as an image with a gray scale representing the displacement value W, as shown in Figure5. Finally the displacement W in a surface format can be seen in Figure 6.

The original surface (the one that generated the pattern analyzed by the software) is shown in Figure 7, and the error presented between the displacements of the surface calculated by the method and the original displacement is shown in Figure 8. The average standard deviation of the error presented between the theoretical values and the calculated values was 0,0013 mm, demonstrating the effectiveness of the method. It is important to point out that the image did not possess any noise or any other source of experimental error.

Application to a Buckled Plate

The second application of the method was to a buckled plate. A PVC plate was placed in a small load system, and buckled. The pitch of the pattern was p = 0,508 mm (50 lines per inch) and the incidence angle î=45º. A CCD video camera with 512 x 512 pixels resolution and 256 gray levels, with a data acquisition system connected to a PC was used to collect the images. The distance applied to the pattern of the image 1 in relation to the pattern of the image 0 was dw21=0,17 mm, producing a difference of phase df21=2p/3 radians .The distance applied to the pattern of the image 2 in relation to the pattern of the image 0 is dw31=0,34 mm, producing a difference of phase df31 =4p/3 radians. The distance produced dw was measured by 3 micrometers with a precision of 0,01 mm.

The obtained images were treated by a process to increase contrast, filtered and noise elimination. The filters used were Gaussian 3x3 neighborhood, Average 3x3 neighborhood, Median 7x7 neighborhood and Median 14x14 neighborhood. In the Figure 9 the software’s main screen can be observed with the 3 patterns. There, one can see the images with a reasonable contrast and filtered. Figure 10 shows the pre-unwrap phase. The displacements are shown under a gray scale image shown in Figure 11 and in surface format in Figure 12.

Conclusion

The Phase Shifting method described in this work and the software developed for the Shadow Moiré technique provided a fast method to obtain information codified in the moiré fringes. In the examples presented in the study, three or more fringe patterns of the same surface were used. They differ from each other by a phase shift. The same method also can be used for photoelastic fringes or any other type of fringe pattern . A constant phase shift for the whole field can just be obtained distancing the surface analyzed from the master grid. The method was applied firstly to images obtained by simulation, to evaluate the error of the method. Then a buckled plate was studied. The results were excellent. The technique seems to be a powerful tool in the resolution of problems in the area of stress analysis, materials and non destructive tests.

Manuscript received: August 1999. Technical Editor: Alisson Rocha Machado.

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