The form deviation of tooth flank is one critical characteristic of gears. Laser interferometry is an effective way of measuring it, because of its high efficiency, precision and the fact that it is a noncontact measurement. According to the characteristic of gear tooth flank, the grazing incidence interferometry should be applied [1]. When measuring the form deviation of tooth flank with the interferometry, the captured interference fringe pattern (IFP) contains not only the valid measurement region (VMR), but also the background, as is shown in Fig. 1(a). The tooth flank region contains the information of the measured tooth flank, it is essential to extract it from the IFP. Also, it is the first step in the image-processing of the IFP, its result affects the accuracy of the following steps and the final measurement result of the form deviation [2, 3].

By far the most commonly used method of image extraction is gray thresholding, one of which is the modulation [4]. And the fringe contrast method introduced by Ino and Yatagai can also be used to extract the VMR [5]. Nevertheless, these methods are not well performed in extracting the VMR from the IFP of the gear tooth flank, the segmentation results contain not only the false background region, but also the false valid measurement region, as shown in Fig. 1. It is because the histogram of the interference fringe pattern has only one peak and the gray level of the VMR changes periodically, as well as the effect of noise, hence, the VMR cannot be distinguished precisely with these methods. One more effective method, object-based segmentation, is introduced by our research group [6]. However, with this method, an object image pattern must be captured under the identical experiment conditions, which makes the process complicated, and it is hard to guarantee the identical experiment conditions. Besides, some interference measurement systems could not capture the object image at all. Other popular image segmentation techniques are level-set methods [7], but they cannot work well on IFP.

Several texture feature based segmentation methods have been studied over the past few decades [8-12]. So far, there is no relevant research proposed which works on processing the interference fringe pattern as texture. This paper extracts the VMR with the texture segmentation method, which consists of two steps: identifying the texture feature of the interference fringe pattern by structure tensor and smoothing them with nonlinear diffusion combined with the fringe density, as well as incorporating the feature into the adaptive partitioning model to extract the profile of the VMR.

Recently, a linear structure tensor is proposed to extract the texture feature [13-16]. However, the edges of the tooth flank region will be blurred and dislocated in the results, which leads to ambiguous segmentation results. The nonlinear structure tensor seems to be applicable to improve this problem. Rousson, Brox and Catte et al. models [17-19] control the diffusion speeds in edge regions and non-edge regions. In ref [20], the author proposed a model that makes the diffusion in the direction of the edge protect the boundary of the image. Tang et al. [21] model does diffusion along the fringe orientation. However, these methods are not actually suitable for fringe patterns, especially in the high fringe density edge regions, they blur the edges. Based on nonlinear structure tensor, we propose an improved nonlinear structure tensor constructing method for interference fringe patterns that makes the diffusion be driven by the density of the fringe patterns. According to the calculated nonlinear structure tensor, combined with the segmentation model the VMR could be extracted.

In Section 2, the proposed texture feature extraction method based on structure tensor and nonlinear diffusion is described in detail. In Section 3, the incorporation of the feature into the vector valued Chan-Vese model is introduced. In Section 4, the proposed method is applied to a sequence of interference fringe patterns, the experiment results of the proposed method are also compared with the traditional methods. Finally in Section 5, the conclusion is summarized briefly.

The feasibility of the structure tensor matrix has been manifested in many applications, fields such as corner detection, diffusion filtering and texture analysis. The VMR in IFP could be considered as one kind of texture image. The classical structure tensor which is described in Eq. (1) is used to extract our interference fringe pattern features.

The G_σ here is a Gaussian kernel function with standard deviation σ , u is the original image and the u_x , u_y denote partial derivatives of u. There are several advantages in the structure tensor. First, by introducing an integration scale, the structure tensor is not sensitive to noise when we smooth the resulting matrix. Second, we can get the additional orientation information from the gray value derivatives of the image, as it is possible for the structure tensor to identify texture. Gaussian convolution is applied in the classical structure tensor for smoothing, which will blur and dislocate the edge and lead to the ambiguous segmentation results.

The Gaussian convolution is actually equivalent to linear diffusion [22]. Several addressed methods replace the Gaussian smoothing of the linear structure tensor with a nonlinear diffusion method to improve this problem [23]. Based on this idea, we propose a method with better effect in segmenting VMR from interference fringe patterns.

Based on the edge detection theory Perona and Malik introduced nonlinear diffusion [24]. They replace the heat equation by a nonlinear equation:

In this equation, u(x, y, t) denotes the image under evolution, u₀(x, y) is the initial image. g is a decreasing function with g(0) = 1, g(x) ≥ 0, and g(x) tending to zero at infinity. However, there are some serious practical problems when this model is applied. First, the model is sensitive to noise and will introduce great oscillations to the gradient ∇u. Second, the diffusion effects are equal along different directions in the local region. In order to improve the problem of blurred edges, Alvarez [25] improved the above model, and introduced the equation:

where

To better understand Eq. (3), the local coordinate system has to be introduced as shown in Fig. 2. In the local coordinate system, η denotes the unit vector parallel to the image gradient vector ∇u. And ξ denotes the unit vector in tangent direction orthogonal to η.

Eq. (3) can also be written as:

where

According to the form of Eq. (5) we could see the Eq. (3) is composed of two parts. The term u_ξξ controls the diffusion direction along the orthogonal to its gradient, and the term g(|∇u_σ) is used to enhance the edges.

But the above model has its own limitation. When it is applied in VMR segmentation, density of the fringe pattern is a critical feature of IFP. However, the model does not diffuse well at the edge of IFP, which has a non-uniform density distribution. From the model we could figure that the major work is the selection of the diffusion function g. To weaken the diffusion speed further at the edge and based on our exact application, we propose an improved nonlinear differential equation:

The function g is the diffusion function, in which the density of the fringe is considered as follows:

The term u_ρ = 1−qI_ρ, I_ρ is the gray value of an image, in which the density of IFP is normalized. q is a constant here we call the coefficient of density. The explanation of the terms of the equation are given as follows:

(1) Gσ is a Gaussian smoothing kernel function for noise elimination in the fringe pattern. Generally, there is strong speckle noise in the fringe pattern,, with Gaussian convolution the negative effect introduced by random noise can be removed. (2) The diffusion function g(|Gσ ∗∇u|⋅uρ) controls the speed of the diffusion and enhances the edges of the fringe pattern. q ∈[0,1], as Iρ is the gray value of image, in which the density of IFP is normalized, so uρ = 1−qIρ ∈ [1−q, 1]. The diffusion can be slowed down at the edge of VMR and lower density region, on the contrary, inside the inner VMR and higher density region there is high speed of diffusion. In the equation, q plays a role in balancing the speed of the diffusion between the lower density region and the higher density region.

This equation could only be used with a scalar-valued image so far. Gerig et al. [26] introduced a vector-valued version of the P-M equation, according to which we can derive the vector-valued equation as Eq. (9)

where u_i is a vector channel and N is the number of channels.

Compared with the model Eq. (3), the proposed model has better controlled diffusion speed at the edge region and is more suitable for IFP.

The fringe density is one of the most significant features of the IFP, which could be estimated by the method of accumulated differences along different orientations [27]. Calculating with a fixed rectangle window is a common way to process a fringe pattern. Thus, in a fixed window, the gray value of the fringe differs slightly along the fringe normal direction in the wide fringe region. On the other hand, the gray level of the fringe differs obviously along the fringe normal direction in the high density fringe region.

Suppose that is parallel to the fringe normal direction, and is orthogonal to . The accumulated difference reaches its maximum along direction , while noise introduces the difference along direction which should be zero for ideal fringe. D_N and D_T represent the accumulated gray difference in direction and respectively. So we can get

where f (N, T) is the ideal fringe intensity without noise. As is shown in Fig. 3, for one exact period of fringe, B is the amplitude of fringe. The value of ∑|f(N + 1, T)−f(N, T) under the calculating window of L × L should be 2BL. And the data under the calculating window:

In this equation, n is the number of fringes under the calculating window. The fringe density can be obtained as follows:

Define eight lines in a kernel matrix of W×W pixels whose center is the current point, as shown in Fig. 4.

Calculate the mean value in every direction

where the subscript i and j represent the current position, M denotes the quantity of pixels in a line, N is the number of calculated directions, the superscript denotes the kth direction, is the gray level of the lth point in the kth direction.

Calculate the variance in every calculated direction

As it is above, reaches its minimum value in the local tangent direction of fringe, and reaches its maximum value in the local normal direction of fringe.

Based on the normal and tangent directions of fringe, the fringe density can be estimated as follows:

where D_n is the number of fringes under the window. In practice, the speed of diffusion is affected by the relative density of fringes, hence the absolute amplitude of fringe has no effect on diffusion. Therefore an estimated value could be given based on the relative gray level of the image, such as 145, the fringe density at every point can be mapped in [0,1]. So far, the normalization density of interference fringe pattern image I_ρ can be obtained. Fig. 5 is estimation of the fringe density calculated under the window of 5 × 5 shown as gray level image.

Combined with the fringe density, the nonlinear structure tensor could be calculated by Eq. (9) with u₁ = u, , , , the results are shown in Fig. 6.

Basically, the difference between object region and background could be represented by mean gray value. Based on this kind of image we can define the evolving curve C. In what follows, inside(C) denotes the region Ω₁ and outside(C) denotes the region Ω₂. The mean gray values of these regions show the difference between object region and background, then the curve C could be considered as the boundary of the object region. The additional information, such as the length of the curve C, and the area of the region inside C can be defined in the energy function F(c₁, c₂, C),which is introduced by T. Chan and L. Vese [28], as follows:

where μ ≥ 0, υ ≥ 0, λ₁, λ₂ > 0 are fixed parameters. Therefore, the segmentation process could be considered as the minimization problem:

The final result of this minimization problem is to seek the best approximation u of u₀, as a two-valued function, namely

The minimal problem of this energy function could be solved by level set method. Define the level set function ϕ : Ω → R of curve C, namely C = {x | ϕ (x) ≡ 0}. With the regularization Heaviside function , and , the energy F(c₁, c₂, C) can be written as

However, the Chan-Vese equation can only be used to scalar-valued image. Tony F. Chan [29] introduced an improved Chan-Vese equation for vector-valued images. Based on which the vector version of Eq. (20) can be written as

The Fⁿ(x, y) here is a vector combining the original image and the nonlinear structure tensor, and superscript n is the number of the channels.

The Smooth here denotes the nonlinear diffusion that can be obtained by Eq. (9). Keeping c₁ and c₂ fixed and minimizing the energy F (c₁, c₂, C), after the time t is introduced, the Euler-Lagrange equation on ϕ can be written as:

The finite difference method could be applied to obtain the discretized type of Eq. (23). We could define h as the space step, t as the time step and the grid (x_i, y_j) = (ih, jh), 1 ≤ i, j ≤ M . The discretized equation of Euler-Lagrange equation is given by:

where A₁, A₂, A₃, A₄ respectively, namely

Therefore, all the details of the calculation process are described as above, and the process of the proposed method is as shown in Fig. 7.

To examine the proposed method, a simulated fringe pattern (256 × 256) with an irregular boundary is generated for segmenting its VMR, as shown in Fig. 8(a). The amplitude of the simulated fringe pattern takes a value in the interval [−4 4]. Gaussian white noise, salt and pepper noise and speckle noise are injected to the image, they are shown in Fig. 8(b), Fig. 8(c) and Fig. 8(d), respectively. The segmentation results of Fig. 8 are as shown in Fig. 9 in which the red lines are the results of the calculated boundaries and the white lines are the ground truth. The results are seen to be satisfactory. The red lines are always close to the white boundaries which means the proposed method shows better robustness to noise.

In order to verify the performance of the proposed method further, then an experimental study has been conducted on the IFPs, which are captured from four-step phase-shifting interferometry.

The measured objects are two gears. The first one is an involute helical gear, which the module is 3, face width is 15mm, the helix angle is 20 degree and the number of teeth is 60. Another one is an involute spur gear with module of 2.5, 50 teeth, and a face width of 20 mm. One four-step phase-shifting interference fringe pattern is as shown in Fig. 10(a), and Fig. 11(a) is the interference fringe pattern of another gear tooth flank. First, the estimation density of Fig. 10(a) is calculated under the window of 5 × 5, as shown in Fig. 5(b). Then the calculated nonlinear structure tensors, as shown in Fig. 6, are incorporated into the segmentation model to extract the final VMR, the result are as shown in Fig. 10(b). The main parameters are set as follows: σ=3, µ=0.5, υ=0, h=1, Δt=0.1, ε=1, , , q=0.3, k=1 and the number of iterations is 500. In the same way we got the VMR of involute spur gear, as shown in Fig. 11(b). In Figs. 10(a) and 11(a), the red line boundary is obtained by the proposed method.

During the segmentation process, the initial contour must be set first. In Fig. 12(a), we set a rectangle as the initial contour for segmentation. Different final results can be obtained with different numbers of iterations, as shown in Fig. 12(b) and Fig. 12(c), the number of iterations are 200 and 500, respectively. Fig. 13 shows the segmentation results on some interference fringe patterns of gear tooth flank. From left to right in Fig. 13 are the original interference fringe patterns, estimated density, nonlinear diffusion results and the segmentation results, respectively.

In order to compare the traditional nonlinear diffusion (TND) and traditional object-image-based method (TOIB) [6], these methods are applied to extract VMR of the same involute helical gear and spur gear. Their segmentation results are as shown in Fig. 14 and Fig. 15 respectively. The red boundary is obtained by the proposed method, the yellow is obtained by the traditional object-image-based method, and the green one is the result of the traditional nonlinear diffusion.

In Fig. 15, a large part of non-valid measured region is segmented as VMR by TND method. And in Fig. 14(b), the TND method shows more obvious error. The results of TOIB method are better than the TND shows. However, in Fig. 14(c) a small portion of VMR missing in the root region of the gear will result in the false phase unwrapping value and error of image registration, and finally affect the measuring accuracy of shape deviation. On the contrary, in Fig. 14(a) and Fig. 15 the red boundary encloses the exact measurement region of the gear even the root part of the gear which is more difficult in VMR segmentation. In these experiments, the proposed method always performs better than TOIB especially in the tough region. It is well known that in many interference measurement systems, it is difficult or even impossible to obtain the object image. The proposed method breaks through the limit of TOIB and even shows better segmentation results without this extra object image. This makes the measurement process simpler and meanwhile guarantees the precision of the measurement results.

By comparing Fig. 14 and Fig. 15, it can be seen that the proposed method performs more precisely and robustly than TND method, furthermore, it is simpler than the TOIB method in the experiments for extracting the VMR of the spur gear and the helical gear.

In order to make a quantitative evaluation of these experimental results further, we introduce the Probabilistic Rand Index, Global Consistency Error, and Variation of Information, three measures to evaluate the segmentation algorithms. Assume an image X is set for segmentation which has N pixels and its corresponding segmentation results with manual method and algorithm are denoted as S, S_test. {S₁, S₂,..., S_k} is the segmentation regions of S, and is the segmentation regions of S_test.

The Probabilistic Rand Index (PRI) makes statistics for the consistent pixel pairs between the manual segmentation result and the segmentation result through the algorithm[30]. The more number of the consistent pixel pairs it has, the more the result of the algorithm is similar to the manual segmentation result. We mark x_i in S with l_i, and in S_test with . In a good segmentation result, to any pixel pair x_i, x_j, if l_i, and l_j are the same in S, the and should also be the same in S_test . Then the PRI can be written as:

where N is the number of pixels, and p_ij is the ground truth probability that ∏(l_i = l_j). The PRI takes a value in the interval [0,1], the more PRI one result has, the better segmentation it shows.

Global consistency error (GCE) evaluates overlap between the manual segmentation result and the segmentation result through the algorithm [31]. The definition could be described as follows:

For any given pixel X_i, we denote the sets which contain X_i in S and S_test with C(S, X_i) and C(S_test, X_i) respectively. If C(S, X_i) is a subset of C(S_test, X_i), then the pixel lies in an area of refinement, and the local error should be zero. The local refinement error (LRE) is defined as:

where \ denotes set difference, and |x| the cardinality of set x. GCE forces all local refinements to be in the same direction. So it can be defined as:

The output value is in the range [0,1] where zero signifies no error. So the smaller this value attains, the better result it shows.

For a given manual segmentation result, the conditional entropies of the algorithm based segmentation result is calculated to measure the distance of these two segmentation results which is evaluated with this index. The definition of it is as follows:

A clustering S_k is a partition of a data set S into sets {S₁, S₂,..., S_k} called clusters such that

Let the number of data points in S and in cluster S_k be n and n_k respectively. The entropy of S is

The probability of any pixel not only belongs to cluster S_k but also belongs to is denoted as n_kk' / N, where n_kk' is written as

We define I(S, S_test) the mutual information between the clustering S and S_test .

Based on that, the variation of information (VI) can be written as [32]:

VI takes a value in the interval [0,∞], the smaller VI one result has, the better segmentation it shows.

According to these segmentation evaluation indices, Tables 1 and 2 show the segmentation evaluation of spur tooth flank and helical tooth flank with different methods respectively. Based on the definition of these evaluation indices, the proposed method performs best in both experiments.

It is well known that the unwrapped phase value of the continuous surface should be also continuous and have almost the same trend. Based on this theory, we unwrap the data of the experiment with the weighted least-squares phase unwrapping algorithm based on a non-interfering image. In ref [33], it is demonstrated that the phase unwrapping algorithm performs better and has higher unwrapping precision in the field of gear tooth flank interferometric measurement. By applying this phase unwrapping algorithm, two lines of unwrapped data are extracted as shown in Fig. 16. These unwrapped phase values obtained by the object-image-based method, traditional nonlinear diffusion method and the proposed method are shown with blue, green and red lines respectively. The data comes from the proposed method shows smooth phase values, on the contrary, the other two methods have great fluctuation at both ends of points compared with its adjacent points.

Traditional nonlinear structure tensor is applied in a number of segmentation methods. However, these structure tensor methods have several drawbacks we have discussed. To address these problems and according to our application, an improved segmentation method is proposed and applied to segment the valid measurement region of interference fringe pattern. First, the idea of using the optimized nonlinear diffusion based structure tensor for interference fringe pattern features extraction has been analyzed. Second, the features have been incorporated into the vector-valued Chan-Vese model to complete the segmentation.

In order to verify the performance of the proposed method, a series of experiments are carried out, the proposed methods are applied to segment the measured tooth flank region of different gears. And the experimental results show that the valid measurement region of interference fringe pattern can be better segmented compared with the traditional methods, which also prove the feasibility of this method. This method can also be used to segment the measured region of other parts by the grazing incidence interferometry.