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Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns
Optics and Lasers in Engineering 100 (2018) 111–117
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns Qiuling Zhou a,d, Chen Tang a,∗, Biyuan Li a, Linlin Wang b, Zhenkun Lei c, Shuwei Tang d a
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China Engineering Training Center, Shenyang Aerospace University, Shenyang 110136, China c State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China d Science and Technology on Electro-Optical Information Security Control Laboratory, Tianjin 300308, China b
a r t i c l e
i n f o
Keywords: Image filtering Oriented partial differential equations Orientation coherence Discontinuous optical fringe patterns
a b s t r a c t The filtering of discontinuous optical fringe patterns is a challenging problem faced in this area. This paper is concerned with oriented partial differential equations (OPDEs)-based image filtering methods for discontinuous optical fringe patterns. We redefine a new controlling speed function to depend on the orientation coherence. The orientation coherence can be used to distinguish the continuous regions and the discontinuous regions, and can be calculated by utilizing fringe orientation. We introduce the new controlling speed function to the previous OPDEs and propose adaptive OPDEs filtering models. According to our proposed adaptive OPDEs filtering models, the filtering in the continuous and discontinuous regions can be selectively carried out. We demonstrate the performance of the proposed adaptive OPDEs via application to the simulated and experimental fringe patterns, and compare our methods with the previous OPDEs. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Optical measurement techniques have been widely used in both research and engineering fields [1]. Optical measurement techniques mainly include: Moire Interferometry (MI) [2], Holographic Interferometry (HI) [3], Electronic Speckle Pattern Interferometry (ESPI) [4] and Fringe Projection Profilometry (FPP) [5]. Optical fringe patterns are common results of these optical interferometric techniques, and the measured physical quantities are related with phase and are encoded in fringe pattern, so phase retrieval from fringe patterns is of fundamental importance for the successful application of optical interferometry methods. Computer-aided fringe analysis can be used to quantitatively evaluate the phase term. However, optical fringe patterns are often subject to the contamination of noise, especially for ESPI fringe patterns. Therefore the filtering of optical fringe patterns is a key step at the preprocessing, and can increases the accuracy and robustness of phase retrieval. In the past few decades, there have been a variety of filtering methods proposed to remove noise in optical fringe patterns, such as the wavelet method [6], the windowed Fourier transform method (WFF) [7], the oriented regularized quadratic cost function (ORQCF) method [8], the oriented spatial filter masks (OSFM) method [9], the localized Fourier transform filter (LFF) [10], image decomposition method [11], and the partial differential equations (PDEs) based image filtering meth-
∗
Corresponding author. E-mail address: [email protected] (C. Tang). http://dx.doi.org/10.1016/j.optlaseng.2017.07.018 Received 31 March 2017; Received in revised form 22 June 2017; Accepted 31 July 2017 0143-8166/© 2017 Elsevier Ltd. All rights reserved.
ods. It is worth to mention that PDEs based image filtering methods have been widely used in ESPI fringe pattern denoising. Generally speaking, the PDEs filtering models are divided into two categories: one is non-orientation PDEs filtering models; another is orientation PDEs filtering models. Non-orientation PDEs filtering models mainly include the second-order non-oriented PDE models [12], the coupled non-oriented PDE filtering models [13] and the fourth-order non-oriented PDE models [14]; Orientation PDEs filtering models have the single oriented PDE models and the double oriented PDE models. The single oriented PDE models have second-order oriented PDE models [15], the coupled oriented PDE filtering models [16], the fourth-order oriented PDE model [17] and the combined oriented PDE models [18]. In Ref. [19], numerous possible OPDEs filtering models based on the variational methods were proposed. The previous filtering methods mainly focused on removing noise and preserving all fringes perfectly. For example, oriented filtering methods (filtering only along fringe orientation) were proposed for fringe patterns with high density. However, we sometimes need to deal with discontinuous optical fringe patterns. For example, for fringe projection profilometry, the resultant fringes are discontinuous at the regions with abrupt changes in height. In addition, when moire interferometry, holographic interferometry and electronic speckle pattern interferometry are applied to the specimen with crack or defect, the resultant fringe images are also discontinuous. As shown in Fig. 1, Fig. 1(a) with image sizes of 320 × 191 pixels is a experimentally obtained FPP fringe image, which depicts a plastic model of human face, Fig. 1(b) and (c)
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Optics and Lasers in Engineering 100 (2018) 111–117
Fig. 1. Three experimentally obtained discontinuous optical fringe images.
with image sizes of 498 × 192 pixels are two experimentally obtained ESPI fringe images, which depict the derivatives of the in-of-plane displacement for three-point bending of nuclear graphite under various loading conditions. For the filtering of these discontinuous fringe patterns, it is highly expected to preserve the integrity of fringes for the continuous regions, and at the same time to preserve the discontinuity for the discontinuous regions while filtering. Therefore, the filtering of discontinuous optical fringe patterns is a challenging problem faced in this area. The majority of previous efforts were focused on continuous fringe image processing. For FPP, some effective background and noise removal methods such as variational image decomposition [20], shearlet transform [21] and wavelet transform [22] have been developed. But less attention is directly paid to the filtering of discontinuous fringe patterns. Recently, a local orientation coherence based fringe segmentation method and its cooperation with boundary-aware coherence enhancing diffusion (LOCS_BCED method) for discontinuous fringe pattern denoising were proposed in Ref. [23]. First, orientation coherence indicated by structure tensors was used for discontinuity recognition. Due to the complexity of the discontinuity problem, the detected boundary had missing parts and was not very accurate. Boundary completion by cubic splines and boundary refinement based on partial structure tensors were further performed as the second and third steps, respectively. After the segmentation, the fringe pattern was divided into segments and denoising can be performed in each segment. The details for this method can be found in the published literatures [23]. In this paper, we are concerned with the filtering for discontinuous optical fringe patterns based on OPDEs filtering methods. The main advantage of OPDEs filtering methods is that it is very flexible. For example, the strength of filtering can be controlled by controlling speed function, and the direction of filtering can also be controlled. The previous single OPDE models made the filtering only along the fringe orientation, so they are suitable for high density fringes. The previous double OPDE models made the filtering along directions both parallel and perpendicular to fringe orientation, so they are applicable to variable density fringes. Unfortunately, the previous existing OPDEs cannot directly apply to discontinuous optical fringe patterns because they cannot distinguish the continuous regions and the discontinuous regions. The orientation coherence can be used to distinguish the continuous regions and the discontinuous regions [24], so we introduce the orientation coherence to our controlling speed function. We will redefine a new controlling speed function to depend on the orientation coherence. Then we introduce the new controlling speed function to the previous OPDEs and propose adaptive OPDEs filtering models. According to
our proposed adaptive OPDEs, the filtering in the continuous regions is allowed, and the filtering in the discontinuous regions is forbidden. We test the proposed adaptive OPDEs on the computer-simulated and experimental discontinuous fringe patterns, and compare our models with the previous OPDEs. Our methods differ from the LOCS_BCED method in Ref. [23] on two points: (1) The calculation of orientation coherence is different. (2) In the LOCS_BCED method, the fringe pattern is divided into the segments according to the orientation coherence, and then the filtering based BCED can be performed in each segment. In our adaptive OPDEs, we propose the adaptive OPDEs through a new controlling speed function. According to our proposed adaptive OPDEs, the filtering in the continuous regions is allowed, and the filtering in the discontinuous regions is forbidden. Obviously, our methods can be directly applied to the entire image. And our methods perform in a more straight way as compared with the LOCS_BCED method. The article is organized as follows. In Section 2, we will first briefly describe related previous PDE models. Then, we describe the definition of orientation coherence and redefine a new controlling speed function. Finally, we present two adaptive OPDEs based on the new controlling speed function and discretization of these equations. Section 3 shows experiments and discussion. Section 4 provides the summary for this paper.
2. The description of our method 2.1. Brief reviews of the related PDE models Like above-mentioned, PDEs-based filtering methods are very flexible. In PDEs-based filtering methods, the strength of filtering can be adjusted by controlling speed function. The most commonly used controlling speed function g( · ) is defined by Ref. [25] g(|∇𝑢|) =
1 1 + (|∇𝑢|∕𝑘)2
(1)
where k is a predefined positive constant. The function g(|∇u|) controls the speed of the filtering. It is a nonincreasing function of the image gradient |∇u|, so the filtering at edges (where |∇u| is large) is low and the edge of image is kept, while the filtering is encouraged within regions (where |∇u| is small). A famous anisotropic PDE filtering model (PM) based on this controlling speed function was proposed [25], the PM nonlinear diffusion 112
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Optics and Lasers in Engineering 100 (2018) 111–117 Table 2 Evaluation results and comparison for Figs. 2, 3 and 4. Methods
Images
SNR
ENL
𝜒
SOOPDE ASOOPDE DOPDE ADOPDE DOPDE ADOPDE
Fig. 2(c) Fig. 2(d) Fig. 3(c) Fig. 3(d) Fig. 4(c) Fig. 4(d)
24.95 24.96 23.90 23.90 22.71 22.73
6.32 6.32 6.02 6.02 6.36 6.40
0.4098 0.6576 0.3195 0.5216 0.2005 0.3021
respect to coordinates x and y, uyy is the second order mixed partial derivative. In Ref. [27], Wang et al. further improved the performance of SOOPDE and proposed the double oriented PDE model (named the coherence-enhancing diffusion (CEDPDE)) ) ( 𝜕𝑢 𝜕2 𝑢 𝜕2 𝑢 = 𝜆1 + 𝜆2 = 𝜆1 𝑢𝑥𝑥 sin2 𝜃 + 𝑢𝑦𝑦 cos2 𝜃 − 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡 𝜕 𝜌2 𝜕𝜌2⊥ ) ( +𝜆2 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃
where 𝜌⊥ denotes the direction perpendicular to fringe orientation. The first term 𝜆1 (𝜕 2 𝑢∕𝜕𝜌2⊥ ) on right-hand side of Eq. (5) made diffusion only in fringe orientation 𝜌⊥ with various speeds which are controlled by 𝜆1 . The second term 𝜆2 (𝜕 2 u/𝜕𝜌2 ) made diffusion only in fringe orientation 𝜌 with various speeds which are controlled by 𝜆2 . The 𝜆2 is set to 1 in order to smooth the direction along the fringe orientation always. The 𝜆1 is set according to the iteration time and the distance map adaptively as { 𝛼 𝑛 ≤ 𝑁0 𝜆1 = (6) max (𝛼, ((𝑑 − 𝑡ℎ𝑟)∕ max(𝑑))) 𝑁0 < 𝑛 ≤ 𝑁
Fig. 2. The first computer-simulated discontinuous fringe pattern, its orientation coherence map and filtered images. (a) Noise image; (b) the orientation coherence map for Fig. 2(a); (c) filtered result by SOOPDE; (d) filtered result by ASOOPDE. Table 1 Parameters for Figs. 2, 3, 4, 5 and 6. Methods
Filtered images
Parameters
SOOPDE ASOOPDE DOPDE ADOPDE DOPDE ADOPDE SOOPDE ASOOPDE DOPDE ADOPDE DOPDE ADOPDE
Fig. 2(c) Fig. 2(d) Fig. 3(c) Fig. 3(d) Fig. 4(c) Fig. 4(d) Fig. 5(b) Fig. 5(c) Fig. 6(b-1) Fig. 6(c-1) Fig. 6(b-2) Fig. 6(c-2)
Δ𝑡 = 0.3, 𝑁 = 190 𝑇𝐶 = 0.9, 𝜅 = 0.98, Δ𝑡 = 0.3, 𝑁 = 190 Δ𝑡 = 0.1, 𝑁 = 550 𝑇𝐶 = 0.99, 𝜅 = 0.97, Δ𝑡 = 0.1, 𝑁 = 550 Δ𝑡 = 0.15, 𝑁 = 1500 𝑇𝐶 = 0.99, 𝜅 = 0.99, Δ𝑡 = 0.15, 𝑁 = 1500 Δ𝑡 = 0.3, 𝑁 = 90 𝑇𝐶 = 0.997, 𝜅 = 0.98, Δ𝑡 = 0.3, 𝑁 = 90 Δ𝑡 = 0.1, 𝑁 = 2000 𝑇𝐶 = 0.99, 𝜅 = 0.99, Δ𝑡 = 0.1, 𝑁 = 2000 Δ𝑡 = 0.1, 𝑁 = 2500 𝑇𝐶 = 0.99, 𝜅 = 0.99, Δ𝑡 = 0.1, 𝑁 = 2500
where 𝛼 is a small value (typically 0.01), thr is a thresholding value (empirically defined as 14 pixels). The distance map d represents the inverse of the local fringe density [28]. The model smoothed a fringe pattern along directions both parallel and perpendicular to fringe orientation with suitable diffusion speeds. In Ref. [18], another double oriented PDE (DOPDE) model was proposed ) ( 𝜕𝑢 𝜕2 𝑢 𝜕2 𝑢 = 𝜆1 + 𝜆2 = 𝜆1 𝑢𝑥𝑥 sin2 𝜃 + 𝑢𝑦𝑦 cos2 𝜃 − 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡 𝜕 𝜌2 𝜕𝜌2⊥ ) ( + 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 + 𝑠1𝑥 cos 𝜃 + 𝑠1𝑦 sin 𝜃
2.2. Our adaptive OPDEs based on the new controlling speed function (2)
Obviously, these previous OPDEs filtering models mainly focused on preserving all fringes perfectly and could not deal with discontinuous optical fringe patterns. In this section, we will redefine a new controlling speed function. The orientation coherence can be used to distinguish the continuous regions and the discontinuous regions, so we redefine the new controlling speed function to depend on the orientation coherence. Then, we introduce the new controlling speed function to Eqs. (4) and (7), and proposed two adaptive OPDEs filtering models. According to the proposed adaptive OPDEs filtering models, our methods can preserve the integrity of fringes for the continuous regions, and at the same time to preserve the discontinuity for the discontinuous regions while filtering.
where this equation can preserve the edges of the image while filtering. Furthermore, the direction of filtering can also be controlled conveniently. Alvarez et al. [26] first proposed the degenerate diffusion PDE model that is of the form ( ) 𝜕𝑢 𝜕2 𝑢 ∇𝑢 = = |∇𝑢|𝑑𝑖𝑣 (3) 𝜕𝑡 |∇𝑢| 𝜕𝑇 2 In this model, T denotes the direction of the edge, the filtering is made only in the direction of the edge. To achieve the filtering of optical fringe images, Tang et al used the fringe orientation 𝜌 instead of T in Ref. [15], and proposed second-order oriented PDE model (SOOPDE) 𝜕𝑢 𝜕2 𝑢 = = 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡 𝜕 𝜌2
(7)
𝜕𝑢 where 𝑠1 = 𝑠𝑖𝑔𝑛( 𝜕𝜌 ) = 𝑠𝑖𝑔𝑛(𝑢𝑥 cos 𝜃 + 𝑢𝑦 sin 𝜃).
equation is of the form 𝜕𝑢 = 𝑑𝑖𝑣(𝑔 (|∇𝑢|)∇𝑢) 𝜕𝑡
(5)
2.2.1. Definition of the orientation coherence The orientation coherence was originally proposed for fingerprint enhancement by Chikkerur et al. in Ref. [24]. A window of size W × W with its center point (i, j) is considered. Let (k, l) denote pixel position
(4)
where 𝜃 is the angle between the fringe orientation with x coordinate. uxx and uyy are the second order partial derivatives of u(x, y, t) with 113
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in the window. The orientation coherence C(i, j) of this center point can be estimated by ∑ ′ | ′ | (𝑘,𝑙)∈𝑊 cos |𝜃 (𝑖, 𝑗 ) − 𝜃 (𝑘, 𝑙)| 𝐶 (𝑖, 𝑗 ) = (8) 𝑊 ×𝑊 where 𝜃′(i, j) and 𝜃′(k, l) are the smoothed orientation angles of center point (i, j) and its neighbors point (k, l), respectively. For optical fringe patterns, the key observation is that the value of orientation coherence is significantly different for the continuous regions and the discontinuous regions. Because the orientation of the central point 𝜃′(i, j) is similar to each of its neighbors 𝜃′(k, l), the coherence is high in the continuous regions. Otherwise, the coherence is low in the discontinuous regions. Therefore, the orientation coherence can be used to distinguish the continuous regions and the discontinuous regions for optical fringe pattern. Given a threshold TC , we classify the optical fringe pattern into two clusters according to { 1(𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑝𝑖𝑥𝑒𝑙) 𝑖𝑓 𝐶 (𝑖, 𝑗 ) ≤ 𝑇𝐶 𝜙(𝑖, 𝑗 ) = (9) 0(𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑝𝑖𝑥𝑒𝑙) 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Obviously, the orientation coherence is relative to the orientation angles. Based on Eqs. (8) and (9), it is required that the method for estimating orientation can give the correct results for continuous part, and at the same time can reflect the change in orientation for discontinuous part. Here we use Fourier transform method to obtain the angle 𝜃 of fringe orientation of each pixel. The details for the Fourier transform method can be found in the published literatures in Ref. [29]. What follow is the estimated equation of the method. The orientation angle of this point (i, j) can be estimated by ( ∑∑ ( ) ( )) 𝐸 𝜔𝑘 , 𝜔𝑙 sin 2𝜃𝑘,𝑙 1 𝜃(𝑖, 𝑗 ) = tan−1 ∑ ∑ ( (10) ) ( ) 2 𝐸 𝜔𝑘 , 𝜔𝑙 cos 2𝜃𝑘,𝑙
Fig. 3. The second computer-simulated discontinuous fringe pattern, its orientation coherence map and filtered images. (a) Noise image; (b) the orientation coherence map for Fig. 3(a); (c) filtered result by DOPDE; (d) filtered result by ADOPDE.
where E(𝜔k , 𝜔l ) is power spectrum in the window. 𝜃 k, l is predefined as by 𝜃𝑘,𝑙 = tan−1 (𝑘, 𝑙). Then the estimated orientation is smoothed by a Gaussian lowpass filter. 2.2.2. Our new controlling speed function Here, we redefine a new controlling speed function gC ( · ) that is 𝑔𝐶 (𝑖, 𝑗 ) = 1 − 𝜅𝜙(𝑖, 𝑗 )
(11)
where 𝜅 is a positive constant that is less than 1. According to the new controlling speed function, the filtering speed at continuous regions is large, and the filtering speed at discontinuous regions is low. 2.2.3. Two adaptive OPDEs based on our new controlling speed function In this section, we introduce our new controlling speed functions to the SOOPDE model (4) and the DOPDE model (7), and propose two adaptive OPDEs filtering models. The adaptive SOOPDE model (ASOOPDE) and the adaptive DOPDE model (ADOPDE) are expressed, respectively, as ) ( 𝜕𝑢 = 𝑔𝐶 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡
(12)
) ( 𝜕𝑢 = 𝜆1 𝑢𝑥𝑥 sin2 𝜃 + 𝑢𝑦𝑦 cos2 𝜃 − 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡 ) ( +𝑔𝐶 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 + 𝑠1𝑥 cos 𝜃 + 𝑠1𝑦 sin 𝜃 (13) Eqs. (12) and (13) preserve the integrity of fringes for the continuous regions, and at the same time to preserve the discontinuity for the discontinuous regions while filtering. Therefore, we would expect to see something that shows the uniqueness of our models. In fact, the experimental results, as shown later, demonstrate the performance of our models.
Fig. 4. The third computer-simulated discontinuous fringe pattern, its orientation coherence map and filtered images. (a) Noise image; (b) the orientation coherence map for Fig. 4(a); (c) filtered result by DOPDE; (d) filtered result by ADOPDE.
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Fig. 5. The orientation coherence map, filtered images and the phases for Fig. 1(a). (a) The orientation coherence map for Fig. 1(a); (b) the filtering results of SOOPDE; (c) the filtering results of ASOOPDE; (d) the unwrapped phase image of Fig. 5(b); (e) the unwrapped phase image of (c); (f) the three-dimensional phase of (b); (g) the three-dimensional phase of (c).
2.2.4. Discretization of our adaptive OPDEs The numerical solutions of two adaptive OPDEs based on the new controlling speed function give the filtered image. For computing numerically Eqs. (12) and (13), it is needed to discretize them. It is easy to derive the discrete schemes for Eq. (12) and (13), respectively, (( )𝑛 ( ) ( )𝑛 ( )) 𝑢𝑥𝑥 𝑖,𝑗 cos2 𝜃𝑖,𝑗 + 𝑢𝑦𝑦 𝑖,𝑗 sin2 𝜃𝑖,𝑗 ( )𝑛 ( )𝑛 ( ) ( ) 𝑢𝑛𝑖,𝑗+1 = 𝑢𝑛𝑖,𝑗 + Δ𝑡 𝑔𝐶 𝑖,𝑗 (14) +2 𝑢𝑥𝑦 𝑖,𝑗 cos 𝜃𝑖,𝑗 sin 𝜃𝑖,𝑗
𝑢𝑛𝑖,𝑗+1 = 𝑢𝑛𝑖,𝑗
(( )𝑛 ( ) ( )𝑛 ( )) 𝑢𝑥𝑥 𝑖,𝑗 sin2 𝜃𝑖,𝑗 + 𝑢𝑦𝑦 𝑖,𝑗 cos2 𝜃𝑖,𝑗 ⎛ ⎞ ( ) ( ) ( ) 𝜆 𝑛 ⎜ 1 −2 𝑢 ⎟ 𝑥𝑦 𝑖,𝑗 cos 𝜃𝑖,𝑗 sin 𝜃𝑖,𝑗 ⎜ ( )𝑛 ( ) ( )𝑛 ( ) ⎟ 2 2 ⎜ ⎞ ⎛ 𝑢 𝑖,𝑗 cos 𝜃𝑖,𝑗 + 𝑢𝑦𝑦 𝑖,𝑗 sin 𝜃𝑖,𝑗 ⎟ + Δ𝑡 ⎜ ( )𝑛 ⎜ 𝑥𝑥 ( )𝑛 ( ) ( ) ⎟⎟ ⎜+ 𝑔𝐶 𝑖,𝑗 ⎜+2 𝑢𝑥𝑦 𝑖,𝑗 cos 𝜃𝑖,𝑗 sin 𝜃𝑖,𝑗 ⎟⎟ ( ) ( ) ( ) ( ) 𝑛 𝑛 ⎜ ⎜+ 𝑠 ⎟⎟ ⎝ 1𝑥 𝑖,𝑗 cos 𝜃𝑖,𝑗 + 𝑠1𝑦 𝑖,𝑗 sin 𝜃𝑖,𝑗 ⎠⎠ ⎝
(
𝑠1𝑦
)𝑛 𝑖,𝑗
=
( )𝑛 ( )𝑛 𝑠1 𝑖,𝑗+1 − 𝑠1 𝑖,𝑗−1 2
(17)
3. Experiments and discussion In this section, to show the performance of our proposed adaptive OPDEs, we test our methods on three computer-simulated and three experimentally obtained discontinuous fringe patterns. We compare the filtered results of our adaptive OPDEs (ASOOPDE and ADOPDE) with SOOPDE and DOPDE. The filtering results of all OPDE models are relative to discrete time step Δt and iteration time n. Both DOPDE and ADOPDE also are relative to the parameters 𝛼, thr and N0 in Eq. (6). Our ASOOPDE and ADOPDE also are relative to the parameters TC and 𝜅. As far as we know in literatures, there isn’t the explicit formula or a method to determine these parameters. These parameters are chosen based on the better performance by trial. In our all implementation, the chosen parameters in the DOPDE and ADOPDE models are 𝑡ℎ𝑟 = 15, 𝛼 = 0.01, and 𝑁0 = 200. According to our extensive experiments, the appropriate value of TC is somewhere between 0.9 and 1, and the appropriate value of 𝜅 is somewhere between 0.97 and 0.99. In Table 1, we list the parameters used in the four filtering models for each testing image.
(15)
where 𝑢𝑛𝑖,𝑗 is the numerical solution, the subscripts i, j denote the pixel position in a discrete two-dimensional grid, the superscript n denotes iteration time, then the discrete time 𝑡𝑛 = 𝑛Δ𝑡, Δt is time step. In Eqs. (14) and (15), all the spatial derivatives are approximated using central differences. For example, ( )𝑛 ( )𝑛 𝑠1 𝑖+1,𝑗 − 𝑠1 𝑖−1,𝑗 ( )𝑛 (16) 𝑠1𝑥 𝑖,𝑗 = 2 115
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Fig. 6. The orientation coherence maps and filtered images for Fig. 1(b) and (c). (a-1) The orientation coherence map for Fig. 1(b); (b-1) the filtering results of DOPDE for Fig. 1(b); (c-1) the filtering results of ADOPDE for Fig. 1(b); (a-2) the orientation coherence map for Fig. 1(c); (b-2) the filtering results of DOPDE for Fig. 1(c); (c-2) the filtering results of ADOPDE for Fig. 1(c).
edge preservation measure 𝜒, ΔI and Δ𝐼̂represent the Laplace operator performed on the original image and the filtered image, respectively. Also, Δ𝐼𝑖 and Δ𝐼̂𝑖 represent the mean value in the discontinuous region of interest of ΔI and Δ𝐼̂, respectively.
Firstly, we take three computer-simulated discontinuous fringe patterns with image sizes of 390 × 390 pixels as the tested images. Fig. 2(a) and (b) shows the first computer-simulated discontinuous fringe pattern and its orientation coherence map, respectively. Fig. 2(c) and (d) shows the filtered results of SOOPDE and ASOOPDE, respectively. Fig. 3(a) and (b) shows the second computer-simulated discontinuous fringe pattern and its orientation coherence map, respectively. Fig. 3(c) and (d) shows the filtered results of DOPDE and ADOPDE, respectively. Fig. 4(a) and (b) shows the third computer-simulated discontinuous fringe pattern and its orientation coherence map, respectively. Fig. 4(c) and (d) shows the filtered results of DOPDE and ADOPDE, respectively. One can find that our proposed adaptive OPDEs filtering models can preserve not only all fringes perfectly for the continuous regions but also the discontinuity for the discontinuous regions while filtering. Secondly, in order to quantitatively evaluate the performance of our proposed adaptive OPDE models, we calculate the signal-to-noise ratio (SNR), equivalent number of looks (ENL) and edge preservation 𝜒. We select four regions of interest (ROIs) marked in white boxes in Figs. 2(a), 3(a) and 4(a), including 2 of which are the continuous regions, 2 of which are the discontinuous regions. The smoothness of the continuous regions of interest is measured by SNR and ENL, and the edge preservation of the discontinuous regions of interest is measured by 𝜒. The image quality indices are presented below: ( ( ) ) (18) SNR = 10 log max 𝐼 2 ∕𝜎𝑡2 ENL =
𝜇𝑡2
The averaged SNR and ENL values are obtained for ROIs 1 and 2 in Figs. 2(a), 3(a) and 4(a). The averaged edge preservation 𝜒 value is obtained for ROIs 3 and 4 in Figs. 2(a), 3(a) and 4(a). As listed in Table 2, the best result for each column is marked in bold. From the results shown in Table 2, it can be seen that our proposed adaptive OPDEs filtering methods have the largest 𝜒 values for Figs. 2(a), 3(a) and 4(a). This means that our methods can preserve the discontinuity for the discontinuous regions better than the previous OPDEs. In addition, our methods and the previous OPDEs provide similar results in terms of SNR and ENL values, which means that our methods can also significantly suppress noise. Subsequently, SOOPED and ASOOPDE are applied to Fig. 1(a). Fig. 5(a) shows the orientation coherence map for Fig. 1(a). Fig. 5(b) and (c) shows the filtered results of SOOPDE and ASOOPDE, respectively. Fig. 5(d) and (e) shows the unwrapped phase images of Fig. 5(b) and (c) by using wavelet transform algorithm, respectively. Fig. 5(f) and (g) shows the three-dimensional phases of Fig. 5(b) and (c), respectively. Finally, DOPED and ADOPDE are applied to Fig. 1(b) and (c). Fig. 6(a-1) and (a-2) shows the orientation coherence maps of Fig. 1(b) and (c), respectively. Fig. 6(b-1) and (c-1) shows the filtered results of DOPDE and ADOPDE for Fig. 1(b), respectively. Fig. 6(b-2) and (c-2) shows the filtered results of DOPDE and ADOPDE for Fig. 1(c), respectively. From the filtered results of experimentally obtained ESPI fringe patterns, the following conclusions can be made:
(19)
𝜎𝑡2 )( ) 𝑁 ( ∑ Δ𝐼𝑖 − Δ𝐼𝑖 Δ𝐼̂𝑖 − Δ𝐼̂𝑖
𝜒= √
𝑖=1 𝑁 ( ∑ 𝑖=1
)( Δ𝐼𝑖 − Δ𝐼𝑖
)∑ )( ) 𝑁 ( Δ𝐼𝑖 − Δ𝐼𝑖 Δ𝐼̂𝑖 − Δ𝐼̂𝑖 Δ𝐼̂𝑖 − Δ𝐼̂𝑖
•
(20)
𝑖=1
𝜎𝑡2
where I, 𝜇 t and in SNR and ENL denote amplitude data, the mean and variance of the continuous region of interests, respectively. In the
•
116
By using SOOPDE and DOPDE, the noise is effectively eliminated and the fringes are well preserved, but the discontinuous regions are blurred. SOOPDE is suitable for high density fringes, and DOPDE is applicable to variable density fringes. The ASOOPDE and ADOPDE preserve the integrity of fringes for the continuous regions, and at the same time to preserve the
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discontinuity for the discontinuous regions while filtering. Therefore, our proposed adaptive OPDEs filtering models are suitable for discontinuous optical fringe patterns.
[7] Qian K. Windowed Fourier transform for fringe pattern analysis. Appl Opt 2004;43:2695–702. [8] Villa J, Quiroga JA, Rosa IDL. Regularized quadratic cost function for oriented fringe-pattern filtering. Opt Lett 2009;34:1741–3. [9] Tang C, Gao T, Yan S, Wang L, Wu J. The oriented spatial filter masks for electronic speckle pattern interferometry phase patterns. Opt Express 2010;18:8942–7. [10] Li C, Tang C, Yan H, Wang L, Zhang H. Localized Fourier transform filter for noise removal in electronic speckle pattern interferometry wrapped phase patterns. Appl Opt 2011;50:4903–11. [11] Fu S, Zhang C. Fringe pattern denoising via image decomposition. Opt Lett 2012;37(3):422–4. [12] Tang C, Zhang F, Yan H, Chen Z. Denoising in electronic speckle pattern interferometry fringes by the filtering method based on partial differential equations. Opt Commun 2006;260:91–6. [13] Tang C, Zhang F, Li B, Yan H. Performance evaluation of partial differential equation models in electronic speckle pattern interferometry and the 𝛿-mollification phase map method. Appl Opt 2006;45:7392–400. [14] Cheng L, Tang C, Yan S, Chen X, Wang L, Wang B. New fourth-order partial differential equations for filtering in electronic speckle pattern interferometry fringes. Opt Commun 2011;284:5549–55. [15] Tang C, Han L, Ren H, Zhou D, Chang Y, Wang X, et al. Second-order oriented partial-differential equations for denoising in electronic-speckle-pattern interferometry fringes. Opt Lett 2008;33:2179–81. [16] Tang C, Han L, Ren H, Gao T, Wang Z, Tang K. The oriented-couple partial differential equations for filtering in wrapped phase patterns. Opt Express 2009;17:5606–17. [17] Wang Y, Ji X, Dai Q. Fourth-order oriented partial differential equations for noise removal of two-photon fluorescence images. Opt Lett 2010;35:2943–5. [18] Tang C, Yang N, Yan H, Yan X. The new second-order single oriented partial differential equations for optical interferometry fringes with high density. Opt Lasers Eng 2013;51:707–15. [19] Tang C, Mi Q, Si Y. Numerous possible oriented partial differential equations and investigation of their performance for optical interferometry fringes denoising. Appl Opt 2013;52(35):8439–50. [20] Zhu X, Tang C, Li B, Sun C, Wang L. Phase retrieval from single frame projection fringe pattern with variational image decomposition. Opt Lasers Eng 2014;59:25–33. [21] Li B, Tang C, Zhu X, Su Y, Xu W. Shearlet transform for phase extraction in fringe projection profilometry with edges discontinuity. Opt Lasers Eng 2016;78:91–8. [22] Li S, Su X, Chen W. Hilbert assisted wavelet transform method of optical fringe pattern phase reconstruction for optical profilometry and interferometry. Optik 2012;123(1):6–10. [23] Wang H, Qian K. Local orientation coherence based segmentation and boundary-aware diffusion for discontinuous fringe patterns. Opt Express 2016;24(14):15609–19. [24] Chikkerur S, Cartwright AN, Govindaraju V. Fingerprint enhancement using STFT analysis. Pattern Recogn 2007;40:198–211. [25] Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell 1990;12:629–39. [26] Alvarez L, Lions P, Morel JM. Image selective smoothing and edge-detection by nonlinear diffusion. II.. SIAM J Numer Anal 1992;29:845–66. [27] Wang K, Qian K, Gao W, Seah H, Lin F. Fringe pattern denoising using coherence enhancing diffusion. Opt Lett 2009;34(8):1141–3. [28] Wang H, Qian K. Comparative analysis on some spatial-domain filters for fringe pattern denoising. Appl Opt 2011;50(12):1687–96. [29] Tang C, Wang Z, Wang L, Wu J, Gao T, Yan S. Estimation of fringe orientation for optical fringe patterns with poor quality based on Fourier transform. Appl Opt 2010;49(4):554–61.
4. Conclusion In the paper, we propose two adaptive OPDEs filtering models for discontinuous optical fringe patterns. This paper first redefines a new controlling speed function to depend on the orientation coherence. Then we introduce the new controlling speed function to the previous SOOPDE and DOPDE, and propose ASOOPDE and ADOPDE, respectively. According to our adaptive OPDEs filtering models, the filtering in the discontinuous regions is allowed, and the filtering in the discontinuous regions is selectively carried out. Furthermore, we test the proposed adaptive OPDEs on three computer-simulated and three experimentally obtained discontinuous fringe patterns, and compare our methods with the previous OPDEs filtering models. From the experimental results, it can be clearly seen that the proposed adaptive OPDEs can preserve the integrity of fringes for the continuous regions, and at the same time to preserve the discontinuity for the discontinuous regions while filtering. In addition, the new controlling speed functions can be extended automatically to any other OPDEs filtering models. Acknowledgment The authors would like to thank the editor and reviewers for their valuable comments on the manuscript. This work was supported by the National Natural Science Foundation of China (NNSFC) (grant nos. 61177007, 61405146). References [1] Robinson DW, Reid GT, Groot PD. Interferogram analysis: digital fringe pattern measurement techniques. Phys Today 1994;47(8):66. [2] Liu H, Basaran C, Cartwright AN, Casey W. Application of Moire interferometry to determine strain fields and debonding of solder joints in BGA packages. IEEE Trans Compon Packag Technol 2004;27(1):217–23. [3] Brown GM, Grant RM, Stroke GW. Theory of holographic interferometry. J Acoust Soc Am 2005;45(5):1166–79. [4] Bavigadda V, Jallapuram R, Mihaylova E, Toal V. Electronic speckle-pattern interferometer using holographic optical elements for vibration measurements. Opt Lett 2010;35(19):3273–5. [5] Zheng D, Da F. Gamma correction for two step phase shifting fringe projection profilometry. Optik 2013;124(13):1392–7. [6] Federico A, Kaufmann GH. Comparative study of wavelet thresholding techniques for denoising electronic speckle pattern interferometry fringes. Opt Eng 2001;40:2598–604.
117
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Adaptive oriented PDEs filtering methods based on new controlling speed function for discontinuous optical fringe patterns Qiuling Zhou a,d, Chen Tang a,∗, Biyuan Li a, Linlin Wang b, Zhenkun Lei c, Shuwei Tang d a
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China Engineering Training Center, Shenyang Aerospace University, Shenyang 110136, China c State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China d Science and Technology on Electro-Optical Information Security Control Laboratory, Tianjin 300308, China b
a r t i c l e
i n f o
Keywords: Image filtering Oriented partial differential equations Orientation coherence Discontinuous optical fringe patterns
a b s t r a c t The filtering of discontinuous optical fringe patterns is a challenging problem faced in this area. This paper is concerned with oriented partial differential equations (OPDEs)-based image filtering methods for discontinuous optical fringe patterns. We redefine a new controlling speed function to depend on the orientation coherence. The orientation coherence can be used to distinguish the continuous regions and the discontinuous regions, and can be calculated by utilizing fringe orientation. We introduce the new controlling speed function to the previous OPDEs and propose adaptive OPDEs filtering models. According to our proposed adaptive OPDEs filtering models, the filtering in the continuous and discontinuous regions can be selectively carried out. We demonstrate the performance of the proposed adaptive OPDEs via application to the simulated and experimental fringe patterns, and compare our methods with the previous OPDEs. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Optical measurement techniques have been widely used in both research and engineering fields [1]. Optical measurement techniques mainly include: Moire Interferometry (MI) [2], Holographic Interferometry (HI) [3], Electronic Speckle Pattern Interferometry (ESPI) [4] and Fringe Projection Profilometry (FPP) [5]. Optical fringe patterns are common results of these optical interferometric techniques, and the measured physical quantities are related with phase and are encoded in fringe pattern, so phase retrieval from fringe patterns is of fundamental importance for the successful application of optical interferometry methods. Computer-aided fringe analysis can be used to quantitatively evaluate the phase term. However, optical fringe patterns are often subject to the contamination of noise, especially for ESPI fringe patterns. Therefore the filtering of optical fringe patterns is a key step at the preprocessing, and can increases the accuracy and robustness of phase retrieval. In the past few decades, there have been a variety of filtering methods proposed to remove noise in optical fringe patterns, such as the wavelet method [6], the windowed Fourier transform method (WFF) [7], the oriented regularized quadratic cost function (ORQCF) method [8], the oriented spatial filter masks (OSFM) method [9], the localized Fourier transform filter (LFF) [10], image decomposition method [11], and the partial differential equations (PDEs) based image filtering meth-
∗
Corresponding author. E-mail address: [email protected] (C. Tang). http://dx.doi.org/10.1016/j.optlaseng.2017.07.018 Received 31 March 2017; Received in revised form 22 June 2017; Accepted 31 July 2017 0143-8166/© 2017 Elsevier Ltd. All rights reserved.
ods. It is worth to mention that PDEs based image filtering methods have been widely used in ESPI fringe pattern denoising. Generally speaking, the PDEs filtering models are divided into two categories: one is non-orientation PDEs filtering models; another is orientation PDEs filtering models. Non-orientation PDEs filtering models mainly include the second-order non-oriented PDE models [12], the coupled non-oriented PDE filtering models [13] and the fourth-order non-oriented PDE models [14]; Orientation PDEs filtering models have the single oriented PDE models and the double oriented PDE models. The single oriented PDE models have second-order oriented PDE models [15], the coupled oriented PDE filtering models [16], the fourth-order oriented PDE model [17] and the combined oriented PDE models [18]. In Ref. [19], numerous possible OPDEs filtering models based on the variational methods were proposed. The previous filtering methods mainly focused on removing noise and preserving all fringes perfectly. For example, oriented filtering methods (filtering only along fringe orientation) were proposed for fringe patterns with high density. However, we sometimes need to deal with discontinuous optical fringe patterns. For example, for fringe projection profilometry, the resultant fringes are discontinuous at the regions with abrupt changes in height. In addition, when moire interferometry, holographic interferometry and electronic speckle pattern interferometry are applied to the specimen with crack or defect, the resultant fringe images are also discontinuous. As shown in Fig. 1, Fig. 1(a) with image sizes of 320 × 191 pixels is a experimentally obtained FPP fringe image, which depicts a plastic model of human face, Fig. 1(b) and (c)
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Optics and Lasers in Engineering 100 (2018) 111–117
Fig. 1. Three experimentally obtained discontinuous optical fringe images.
with image sizes of 498 × 192 pixels are two experimentally obtained ESPI fringe images, which depict the derivatives of the in-of-plane displacement for three-point bending of nuclear graphite under various loading conditions. For the filtering of these discontinuous fringe patterns, it is highly expected to preserve the integrity of fringes for the continuous regions, and at the same time to preserve the discontinuity for the discontinuous regions while filtering. Therefore, the filtering of discontinuous optical fringe patterns is a challenging problem faced in this area. The majority of previous efforts were focused on continuous fringe image processing. For FPP, some effective background and noise removal methods such as variational image decomposition [20], shearlet transform [21] and wavelet transform [22] have been developed. But less attention is directly paid to the filtering of discontinuous fringe patterns. Recently, a local orientation coherence based fringe segmentation method and its cooperation with boundary-aware coherence enhancing diffusion (LOCS_BCED method) for discontinuous fringe pattern denoising were proposed in Ref. [23]. First, orientation coherence indicated by structure tensors was used for discontinuity recognition. Due to the complexity of the discontinuity problem, the detected boundary had missing parts and was not very accurate. Boundary completion by cubic splines and boundary refinement based on partial structure tensors were further performed as the second and third steps, respectively. After the segmentation, the fringe pattern was divided into segments and denoising can be performed in each segment. The details for this method can be found in the published literatures [23]. In this paper, we are concerned with the filtering for discontinuous optical fringe patterns based on OPDEs filtering methods. The main advantage of OPDEs filtering methods is that it is very flexible. For example, the strength of filtering can be controlled by controlling speed function, and the direction of filtering can also be controlled. The previous single OPDE models made the filtering only along the fringe orientation, so they are suitable for high density fringes. The previous double OPDE models made the filtering along directions both parallel and perpendicular to fringe orientation, so they are applicable to variable density fringes. Unfortunately, the previous existing OPDEs cannot directly apply to discontinuous optical fringe patterns because they cannot distinguish the continuous regions and the discontinuous regions. The orientation coherence can be used to distinguish the continuous regions and the discontinuous regions [24], so we introduce the orientation coherence to our controlling speed function. We will redefine a new controlling speed function to depend on the orientation coherence. Then we introduce the new controlling speed function to the previous OPDEs and propose adaptive OPDEs filtering models. According to
our proposed adaptive OPDEs, the filtering in the continuous regions is allowed, and the filtering in the discontinuous regions is forbidden. We test the proposed adaptive OPDEs on the computer-simulated and experimental discontinuous fringe patterns, and compare our models with the previous OPDEs. Our methods differ from the LOCS_BCED method in Ref. [23] on two points: (1) The calculation of orientation coherence is different. (2) In the LOCS_BCED method, the fringe pattern is divided into the segments according to the orientation coherence, and then the filtering based BCED can be performed in each segment. In our adaptive OPDEs, we propose the adaptive OPDEs through a new controlling speed function. According to our proposed adaptive OPDEs, the filtering in the continuous regions is allowed, and the filtering in the discontinuous regions is forbidden. Obviously, our methods can be directly applied to the entire image. And our methods perform in a more straight way as compared with the LOCS_BCED method. The article is organized as follows. In Section 2, we will first briefly describe related previous PDE models. Then, we describe the definition of orientation coherence and redefine a new controlling speed function. Finally, we present two adaptive OPDEs based on the new controlling speed function and discretization of these equations. Section 3 shows experiments and discussion. Section 4 provides the summary for this paper.
2. The description of our method 2.1. Brief reviews of the related PDE models Like above-mentioned, PDEs-based filtering methods are very flexible. In PDEs-based filtering methods, the strength of filtering can be adjusted by controlling speed function. The most commonly used controlling speed function g( · ) is defined by Ref. [25] g(|∇𝑢|) =
1 1 + (|∇𝑢|∕𝑘)2
(1)
where k is a predefined positive constant. The function g(|∇u|) controls the speed of the filtering. It is a nonincreasing function of the image gradient |∇u|, so the filtering at edges (where |∇u| is large) is low and the edge of image is kept, while the filtering is encouraged within regions (where |∇u| is small). A famous anisotropic PDE filtering model (PM) based on this controlling speed function was proposed [25], the PM nonlinear diffusion 112
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Optics and Lasers in Engineering 100 (2018) 111–117 Table 2 Evaluation results and comparison for Figs. 2, 3 and 4. Methods
Images
SNR
ENL
𝜒
SOOPDE ASOOPDE DOPDE ADOPDE DOPDE ADOPDE
Fig. 2(c) Fig. 2(d) Fig. 3(c) Fig. 3(d) Fig. 4(c) Fig. 4(d)
24.95 24.96 23.90 23.90 22.71 22.73
6.32 6.32 6.02 6.02 6.36 6.40
0.4098 0.6576 0.3195 0.5216 0.2005 0.3021
respect to coordinates x and y, uyy is the second order mixed partial derivative. In Ref. [27], Wang et al. further improved the performance of SOOPDE and proposed the double oriented PDE model (named the coherence-enhancing diffusion (CEDPDE)) ) ( 𝜕𝑢 𝜕2 𝑢 𝜕2 𝑢 = 𝜆1 + 𝜆2 = 𝜆1 𝑢𝑥𝑥 sin2 𝜃 + 𝑢𝑦𝑦 cos2 𝜃 − 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡 𝜕 𝜌2 𝜕𝜌2⊥ ) ( +𝜆2 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃
where 𝜌⊥ denotes the direction perpendicular to fringe orientation. The first term 𝜆1 (𝜕 2 𝑢∕𝜕𝜌2⊥ ) on right-hand side of Eq. (5) made diffusion only in fringe orientation 𝜌⊥ with various speeds which are controlled by 𝜆1 . The second term 𝜆2 (𝜕 2 u/𝜕𝜌2 ) made diffusion only in fringe orientation 𝜌 with various speeds which are controlled by 𝜆2 . The 𝜆2 is set to 1 in order to smooth the direction along the fringe orientation always. The 𝜆1 is set according to the iteration time and the distance map adaptively as { 𝛼 𝑛 ≤ 𝑁0 𝜆1 = (6) max (𝛼, ((𝑑 − 𝑡ℎ𝑟)∕ max(𝑑))) 𝑁0 < 𝑛 ≤ 𝑁
Fig. 2. The first computer-simulated discontinuous fringe pattern, its orientation coherence map and filtered images. (a) Noise image; (b) the orientation coherence map for Fig. 2(a); (c) filtered result by SOOPDE; (d) filtered result by ASOOPDE. Table 1 Parameters for Figs. 2, 3, 4, 5 and 6. Methods
Filtered images
Parameters
SOOPDE ASOOPDE DOPDE ADOPDE DOPDE ADOPDE SOOPDE ASOOPDE DOPDE ADOPDE DOPDE ADOPDE
Fig. 2(c) Fig. 2(d) Fig. 3(c) Fig. 3(d) Fig. 4(c) Fig. 4(d) Fig. 5(b) Fig. 5(c) Fig. 6(b-1) Fig. 6(c-1) Fig. 6(b-2) Fig. 6(c-2)
Δ𝑡 = 0.3, 𝑁 = 190 𝑇𝐶 = 0.9, 𝜅 = 0.98, Δ𝑡 = 0.3, 𝑁 = 190 Δ𝑡 = 0.1, 𝑁 = 550 𝑇𝐶 = 0.99, 𝜅 = 0.97, Δ𝑡 = 0.1, 𝑁 = 550 Δ𝑡 = 0.15, 𝑁 = 1500 𝑇𝐶 = 0.99, 𝜅 = 0.99, Δ𝑡 = 0.15, 𝑁 = 1500 Δ𝑡 = 0.3, 𝑁 = 90 𝑇𝐶 = 0.997, 𝜅 = 0.98, Δ𝑡 = 0.3, 𝑁 = 90 Δ𝑡 = 0.1, 𝑁 = 2000 𝑇𝐶 = 0.99, 𝜅 = 0.99, Δ𝑡 = 0.1, 𝑁 = 2000 Δ𝑡 = 0.1, 𝑁 = 2500 𝑇𝐶 = 0.99, 𝜅 = 0.99, Δ𝑡 = 0.1, 𝑁 = 2500
where 𝛼 is a small value (typically 0.01), thr is a thresholding value (empirically defined as 14 pixels). The distance map d represents the inverse of the local fringe density [28]. The model smoothed a fringe pattern along directions both parallel and perpendicular to fringe orientation with suitable diffusion speeds. In Ref. [18], another double oriented PDE (DOPDE) model was proposed ) ( 𝜕𝑢 𝜕2 𝑢 𝜕2 𝑢 = 𝜆1 + 𝜆2 = 𝜆1 𝑢𝑥𝑥 sin2 𝜃 + 𝑢𝑦𝑦 cos2 𝜃 − 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡 𝜕 𝜌2 𝜕𝜌2⊥ ) ( + 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 + 𝑠1𝑥 cos 𝜃 + 𝑠1𝑦 sin 𝜃
2.2. Our adaptive OPDEs based on the new controlling speed function (2)
Obviously, these previous OPDEs filtering models mainly focused on preserving all fringes perfectly and could not deal with discontinuous optical fringe patterns. In this section, we will redefine a new controlling speed function. The orientation coherence can be used to distinguish the continuous regions and the discontinuous regions, so we redefine the new controlling speed function to depend on the orientation coherence. Then, we introduce the new controlling speed function to Eqs. (4) and (7), and proposed two adaptive OPDEs filtering models. According to the proposed adaptive OPDEs filtering models, our methods can preserve the integrity of fringes for the continuous regions, and at the same time to preserve the discontinuity for the discontinuous regions while filtering.
where this equation can preserve the edges of the image while filtering. Furthermore, the direction of filtering can also be controlled conveniently. Alvarez et al. [26] first proposed the degenerate diffusion PDE model that is of the form ( ) 𝜕𝑢 𝜕2 𝑢 ∇𝑢 = = |∇𝑢|𝑑𝑖𝑣 (3) 𝜕𝑡 |∇𝑢| 𝜕𝑇 2 In this model, T denotes the direction of the edge, the filtering is made only in the direction of the edge. To achieve the filtering of optical fringe images, Tang et al used the fringe orientation 𝜌 instead of T in Ref. [15], and proposed second-order oriented PDE model (SOOPDE) 𝜕𝑢 𝜕2 𝑢 = = 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡 𝜕 𝜌2
(7)
𝜕𝑢 where 𝑠1 = 𝑠𝑖𝑔𝑛( 𝜕𝜌 ) = 𝑠𝑖𝑔𝑛(𝑢𝑥 cos 𝜃 + 𝑢𝑦 sin 𝜃).
equation is of the form 𝜕𝑢 = 𝑑𝑖𝑣(𝑔 (|∇𝑢|)∇𝑢) 𝜕𝑡
(5)
2.2.1. Definition of the orientation coherence The orientation coherence was originally proposed for fingerprint enhancement by Chikkerur et al. in Ref. [24]. A window of size W × W with its center point (i, j) is considered. Let (k, l) denote pixel position
(4)
where 𝜃 is the angle between the fringe orientation with x coordinate. uxx and uyy are the second order partial derivatives of u(x, y, t) with 113
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in the window. The orientation coherence C(i, j) of this center point can be estimated by ∑ ′ | ′ | (𝑘,𝑙)∈𝑊 cos |𝜃 (𝑖, 𝑗 ) − 𝜃 (𝑘, 𝑙)| 𝐶 (𝑖, 𝑗 ) = (8) 𝑊 ×𝑊 where 𝜃′(i, j) and 𝜃′(k, l) are the smoothed orientation angles of center point (i, j) and its neighbors point (k, l), respectively. For optical fringe patterns, the key observation is that the value of orientation coherence is significantly different for the continuous regions and the discontinuous regions. Because the orientation of the central point 𝜃′(i, j) is similar to each of its neighbors 𝜃′(k, l), the coherence is high in the continuous regions. Otherwise, the coherence is low in the discontinuous regions. Therefore, the orientation coherence can be used to distinguish the continuous regions and the discontinuous regions for optical fringe pattern. Given a threshold TC , we classify the optical fringe pattern into two clusters according to { 1(𝑑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑝𝑖𝑥𝑒𝑙) 𝑖𝑓 𝐶 (𝑖, 𝑗 ) ≤ 𝑇𝐶 𝜙(𝑖, 𝑗 ) = (9) 0(𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑝𝑖𝑥𝑒𝑙) 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Obviously, the orientation coherence is relative to the orientation angles. Based on Eqs. (8) and (9), it is required that the method for estimating orientation can give the correct results for continuous part, and at the same time can reflect the change in orientation for discontinuous part. Here we use Fourier transform method to obtain the angle 𝜃 of fringe orientation of each pixel. The details for the Fourier transform method can be found in the published literatures in Ref. [29]. What follow is the estimated equation of the method. The orientation angle of this point (i, j) can be estimated by ( ∑∑ ( ) ( )) 𝐸 𝜔𝑘 , 𝜔𝑙 sin 2𝜃𝑘,𝑙 1 𝜃(𝑖, 𝑗 ) = tan−1 ∑ ∑ ( (10) ) ( ) 2 𝐸 𝜔𝑘 , 𝜔𝑙 cos 2𝜃𝑘,𝑙
Fig. 3. The second computer-simulated discontinuous fringe pattern, its orientation coherence map and filtered images. (a) Noise image; (b) the orientation coherence map for Fig. 3(a); (c) filtered result by DOPDE; (d) filtered result by ADOPDE.
where E(𝜔k , 𝜔l ) is power spectrum in the window. 𝜃 k, l is predefined as by 𝜃𝑘,𝑙 = tan−1 (𝑘, 𝑙). Then the estimated orientation is smoothed by a Gaussian lowpass filter. 2.2.2. Our new controlling speed function Here, we redefine a new controlling speed function gC ( · ) that is 𝑔𝐶 (𝑖, 𝑗 ) = 1 − 𝜅𝜙(𝑖, 𝑗 )
(11)
where 𝜅 is a positive constant that is less than 1. According to the new controlling speed function, the filtering speed at continuous regions is large, and the filtering speed at discontinuous regions is low. 2.2.3. Two adaptive OPDEs based on our new controlling speed function In this section, we introduce our new controlling speed functions to the SOOPDE model (4) and the DOPDE model (7), and propose two adaptive OPDEs filtering models. The adaptive SOOPDE model (ASOOPDE) and the adaptive DOPDE model (ADOPDE) are expressed, respectively, as ) ( 𝜕𝑢 = 𝑔𝐶 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡
(12)
) ( 𝜕𝑢 = 𝜆1 𝑢𝑥𝑥 sin2 𝜃 + 𝑢𝑦𝑦 cos2 𝜃 − 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 𝜕𝑡 ) ( +𝑔𝐶 𝑢𝑥𝑥 cos2 𝜃 + 𝑢𝑦𝑦 sin2 𝜃 + 2𝑢𝑥𝑦 sin 𝜃 cos 𝜃 + 𝑠1𝑥 cos 𝜃 + 𝑠1𝑦 sin 𝜃 (13) Eqs. (12) and (13) preserve the integrity of fringes for the continuous regions, and at the same time to preserve the discontinuity for the discontinuous regions while filtering. Therefore, we would expect to see something that shows the uniqueness of our models. In fact, the experimental results, as shown later, demonstrate the performance of our models.
Fig. 4. The third computer-simulated discontinuous fringe pattern, its orientation coherence map and filtered images. (a) Noise image; (b) the orientation coherence map for Fig. 4(a); (c) filtered result by DOPDE; (d) filtered result by ADOPDE.
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Fig. 5. The orientation coherence map, filtered images and the phases for Fig. 1(a). (a) The orientation coherence map for Fig. 1(a); (b) the filtering results of SOOPDE; (c) the filtering results of ASOOPDE; (d) the unwrapped phase image of Fig. 5(b); (e) the unwrapped phase image of (c); (f) the three-dimensional phase of (b); (g) the three-dimensional phase of (c).
2.2.4. Discretization of our adaptive OPDEs The numerical solutions of two adaptive OPDEs based on the new controlling speed function give the filtered image. For computing numerically Eqs. (12) and (13), it is needed to discretize them. It is easy to derive the discrete schemes for Eq. (12) and (13), respectively, (( )𝑛 ( ) ( )𝑛 ( )) 𝑢𝑥𝑥 𝑖,𝑗 cos2 𝜃𝑖,𝑗 + 𝑢𝑦𝑦 𝑖,𝑗 sin2 𝜃𝑖,𝑗 ( )𝑛 ( )𝑛 ( ) ( ) 𝑢𝑛𝑖,𝑗+1 = 𝑢𝑛𝑖,𝑗 + Δ𝑡 𝑔𝐶 𝑖,𝑗 (14) +2 𝑢𝑥𝑦 𝑖,𝑗 cos 𝜃𝑖,𝑗 sin 𝜃𝑖,𝑗
𝑢𝑛𝑖,𝑗+1 = 𝑢𝑛𝑖,𝑗
(( )𝑛 ( ) ( )𝑛 ( )) 𝑢𝑥𝑥 𝑖,𝑗 sin2 𝜃𝑖,𝑗 + 𝑢𝑦𝑦 𝑖,𝑗 cos2 𝜃𝑖,𝑗 ⎛ ⎞ ( ) ( ) ( ) 𝜆 𝑛 ⎜ 1 −2 𝑢 ⎟ 𝑥𝑦 𝑖,𝑗 cos 𝜃𝑖,𝑗 sin 𝜃𝑖,𝑗 ⎜ ( )𝑛 ( ) ( )𝑛 ( ) ⎟ 2 2 ⎜ ⎞ ⎛ 𝑢 𝑖,𝑗 cos 𝜃𝑖,𝑗 + 𝑢𝑦𝑦 𝑖,𝑗 sin 𝜃𝑖,𝑗 ⎟ + Δ𝑡 ⎜ ( )𝑛 ⎜ 𝑥𝑥 ( )𝑛 ( ) ( ) ⎟⎟ ⎜+ 𝑔𝐶 𝑖,𝑗 ⎜+2 𝑢𝑥𝑦 𝑖,𝑗 cos 𝜃𝑖,𝑗 sin 𝜃𝑖,𝑗 ⎟⎟ ( ) ( ) ( ) ( ) 𝑛 𝑛 ⎜ ⎜+ 𝑠 ⎟⎟ ⎝ 1𝑥 𝑖,𝑗 cos 𝜃𝑖,𝑗 + 𝑠1𝑦 𝑖,𝑗 sin 𝜃𝑖,𝑗 ⎠⎠ ⎝
(
𝑠1𝑦
)𝑛 𝑖,𝑗
=
( )𝑛 ( )𝑛 𝑠1 𝑖,𝑗+1 − 𝑠1 𝑖,𝑗−1 2
(17)
3. Experiments and discussion In this section, to show the performance of our proposed adaptive OPDEs, we test our methods on three computer-simulated and three experimentally obtained discontinuous fringe patterns. We compare the filtered results of our adaptive OPDEs (ASOOPDE and ADOPDE) with SOOPDE and DOPDE. The filtering results of all OPDE models are relative to discrete time step Δt and iteration time n. Both DOPDE and ADOPDE also are relative to the parameters 𝛼, thr and N0 in Eq. (6). Our ASOOPDE and ADOPDE also are relative to the parameters TC and 𝜅. As far as we know in literatures, there isn’t the explicit formula or a method to determine these parameters. These parameters are chosen based on the better performance by trial. In our all implementation, the chosen parameters in the DOPDE and ADOPDE models are 𝑡ℎ𝑟 = 15, 𝛼 = 0.01, and 𝑁0 = 200. According to our extensive experiments, the appropriate value of TC is somewhere between 0.9 and 1, and the appropriate value of 𝜅 is somewhere between 0.97 and 0.99. In Table 1, we list the parameters used in the four filtering models for each testing image.
(15)
where 𝑢𝑛𝑖,𝑗 is the numerical solution, the subscripts i, j denote the pixel position in a discrete two-dimensional grid, the superscript n denotes iteration time, then the discrete time 𝑡𝑛 = 𝑛Δ𝑡, Δt is time step. In Eqs. (14) and (15), all the spatial derivatives are approximated using central differences. For example, ( )𝑛 ( )𝑛 𝑠1 𝑖+1,𝑗 − 𝑠1 𝑖−1,𝑗 ( )𝑛 (16) 𝑠1𝑥 𝑖,𝑗 = 2 115
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Fig. 6. The orientation coherence maps and filtered images for Fig. 1(b) and (c). (a-1) The orientation coherence map for Fig. 1(b); (b-1) the filtering results of DOPDE for Fig. 1(b); (c-1) the filtering results of ADOPDE for Fig. 1(b); (a-2) the orientation coherence map for Fig. 1(c); (b-2) the filtering results of DOPDE for Fig. 1(c); (c-2) the filtering results of ADOPDE for Fig. 1(c).
edge preservation measure 𝜒, ΔI and Δ𝐼̂represent the Laplace operator performed on the original image and the filtered image, respectively. Also, Δ𝐼𝑖 and Δ𝐼̂𝑖 represent the mean value in the discontinuous region of interest of ΔI and Δ𝐼̂, respectively.
Firstly, we take three computer-simulated discontinuous fringe patterns with image sizes of 390 × 390 pixels as the tested images. Fig. 2(a) and (b) shows the first computer-simulated discontinuous fringe pattern and its orientation coherence map, respectively. Fig. 2(c) and (d) shows the filtered results of SOOPDE and ASOOPDE, respectively. Fig. 3(a) and (b) shows the second computer-simulated discontinuous fringe pattern and its orientation coherence map, respectively. Fig. 3(c) and (d) shows the filtered results of DOPDE and ADOPDE, respectively. Fig. 4(a) and (b) shows the third computer-simulated discontinuous fringe pattern and its orientation coherence map, respectively. Fig. 4(c) and (d) shows the filtered results of DOPDE and ADOPDE, respectively. One can find that our proposed adaptive OPDEs filtering models can preserve not only all fringes perfectly for the continuous regions but also the discontinuity for the discontinuous regions while filtering. Secondly, in order to quantitatively evaluate the performance of our proposed adaptive OPDE models, we calculate the signal-to-noise ratio (SNR), equivalent number of looks (ENL) and edge preservation 𝜒. We select four regions of interest (ROIs) marked in white boxes in Figs. 2(a), 3(a) and 4(a), including 2 of which are the continuous regions, 2 of which are the discontinuous regions. The smoothness of the continuous regions of interest is measured by SNR and ENL, and the edge preservation of the discontinuous regions of interest is measured by 𝜒. The image quality indices are presented below: ( ( ) ) (18) SNR = 10 log max 𝐼 2 ∕𝜎𝑡2 ENL =
𝜇𝑡2
The averaged SNR and ENL values are obtained for ROIs 1 and 2 in Figs. 2(a), 3(a) and 4(a). The averaged edge preservation 𝜒 value is obtained for ROIs 3 and 4 in Figs. 2(a), 3(a) and 4(a). As listed in Table 2, the best result for each column is marked in bold. From the results shown in Table 2, it can be seen that our proposed adaptive OPDEs filtering methods have the largest 𝜒 values for Figs. 2(a), 3(a) and 4(a). This means that our methods can preserve the discontinuity for the discontinuous regions better than the previous OPDEs. In addition, our methods and the previous OPDEs provide similar results in terms of SNR and ENL values, which means that our methods can also significantly suppress noise. Subsequently, SOOPED and ASOOPDE are applied to Fig. 1(a). Fig. 5(a) shows the orientation coherence map for Fig. 1(a). Fig. 5(b) and (c) shows the filtered results of SOOPDE and ASOOPDE, respectively. Fig. 5(d) and (e) shows the unwrapped phase images of Fig. 5(b) and (c) by using wavelet transform algorithm, respectively. Fig. 5(f) and (g) shows the three-dimensional phases of Fig. 5(b) and (c), respectively. Finally, DOPED and ADOPDE are applied to Fig. 1(b) and (c). Fig. 6(a-1) and (a-2) shows the orientation coherence maps of Fig. 1(b) and (c), respectively. Fig. 6(b-1) and (c-1) shows the filtered results of DOPDE and ADOPDE for Fig. 1(b), respectively. Fig. 6(b-2) and (c-2) shows the filtered results of DOPDE and ADOPDE for Fig. 1(c), respectively. From the filtered results of experimentally obtained ESPI fringe patterns, the following conclusions can be made:
(19)
𝜎𝑡2 )( ) 𝑁 ( ∑ Δ𝐼𝑖 − Δ𝐼𝑖 Δ𝐼̂𝑖 − Δ𝐼̂𝑖
𝜒= √
𝑖=1 𝑁 ( ∑ 𝑖=1
)( Δ𝐼𝑖 − Δ𝐼𝑖
)∑ )( ) 𝑁 ( Δ𝐼𝑖 − Δ𝐼𝑖 Δ𝐼̂𝑖 − Δ𝐼̂𝑖 Δ𝐼̂𝑖 − Δ𝐼̂𝑖
•
(20)
𝑖=1
𝜎𝑡2
where I, 𝜇 t and in SNR and ENL denote amplitude data, the mean and variance of the continuous region of interests, respectively. In the
•
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By using SOOPDE and DOPDE, the noise is effectively eliminated and the fringes are well preserved, but the discontinuous regions are blurred. SOOPDE is suitable for high density fringes, and DOPDE is applicable to variable density fringes. The ASOOPDE and ADOPDE preserve the integrity of fringes for the continuous regions, and at the same time to preserve the
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discontinuity for the discontinuous regions while filtering. Therefore, our proposed adaptive OPDEs filtering models are suitable for discontinuous optical fringe patterns.
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4. Conclusion In the paper, we propose two adaptive OPDEs filtering models for discontinuous optical fringe patterns. This paper first redefines a new controlling speed function to depend on the orientation coherence. Then we introduce the new controlling speed function to the previous SOOPDE and DOPDE, and propose ASOOPDE and ADOPDE, respectively. According to our adaptive OPDEs filtering models, the filtering in the discontinuous regions is allowed, and the filtering in the discontinuous regions is selectively carried out. Furthermore, we test the proposed adaptive OPDEs on three computer-simulated and three experimentally obtained discontinuous fringe patterns, and compare our methods with the previous OPDEs filtering models. From the experimental results, it can be clearly seen that the proposed adaptive OPDEs can preserve the integrity of fringes for the continuous regions, and at the same time to preserve the discontinuity for the discontinuous regions while filtering. In addition, the new controlling speed functions can be extended automatically to any other OPDEs filtering models. Acknowledgment The authors would like to thank the editor and reviewers for their valuable comments on the manuscript. This work was supported by the National Natural Science Foundation of China (NNSFC) (grant nos. 61177007, 61405146). References [1] Robinson DW, Reid GT, Groot PD. Interferogram analysis: digital fringe pattern measurement techniques. Phys Today 1994;47(8):66. [2] Liu H, Basaran C, Cartwright AN, Casey W. Application of Moire interferometry to determine strain fields and debonding of solder joints in BGA packages. IEEE Trans Compon Packag Technol 2004;27(1):217–23. [3] Brown GM, Grant RM, Stroke GW. Theory of holographic interferometry. J Acoust Soc Am 2005;45(5):1166–79. [4] Bavigadda V, Jallapuram R, Mihaylova E, Toal V. Electronic speckle-pattern interferometer using holographic optical elements for vibration measurements. Opt Lett 2010;35(19):3273–5. [5] Zheng D, Da F. Gamma correction for two step phase shifting fringe projection profilometry. Optik 2013;124(13):1392–7. [6] Federico A, Kaufmann GH. Comparative study of wavelet thresholding techniques for denoising electronic speckle pattern interferometry fringes. Opt Eng 2001;40:2598–604.
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