Applicability of colour transfer techniques in Twelve fringe photoelasticity (TFP)

Optics and Lasers in Engineering 127 (2020) 105963

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Applicability of colour transfer techniques in Twelve fringe photoelasticity (TFP) Sachin Sasikumar, K Ramesh∗ Dept. of Applied Mechanics, Indian Institute of Technology Madras, India

a r t i c l e

i n f o

Keywords: Digital photoelasticity Twelve fringe photoelasticity Colour transfer Colour adaptation Image processing Principal component analysis

a b s t r a c t Colour adaptation techniques are useful in extending the method of Twelve fringe photoelasticity (TFP) to solve problems in industries as one can minimise the generation of specific calibration tables for each experiment. Image artefacts affect the functioning of existing colour adaptation techniques affecting the fringe order demodulation. Colour transfer is a popular technique in the field of image processing to tackle colour mismatch between images. In this paper, the applicability of two colour transfer strategies in the context of TFP is explored, of which one is selected for implementation. A comparative study on the performance of the novel colour transfer method with the existing colour adaptation techniques is carried out.

1. Introduction Twelve Fringe Photoelasticity (TFP) [1] uses a single isochromatic image of the application specimen acquired under white light for its whole field fringe order estimation. TFP has proved to be an extremely useful technique for a large class of problems where one would require only isochromatic information from the field viz., determination of stress intensity factors (SIF) in fracture analysis, stress concentration factor (SCF) at the critical regions of engineering components, parameters in contact mechanics problems etc. It has also been successfully applied for the analysis of transient thermal problems [2] where multiple image acquisitions as demanded by phase shifting techniques (PST) poses a challenge. The preliminary step in TFP is to establish a calibration table which is an evolved form of the colour table made use of in conventional white light photoelasticity. The fringe order at any point in the application specimen is obtained by comparing its R, G, B intensity values with that of the calibration table in a least squares sense. Since colour information is the sole criterion for fringe order estimation in TFP, tint variation between the application and calibration images can lead to erroneous fringe order demodulation. Tint variation in a photoelastic experimental setup can be contributed from illumination source, optical elements, specimen material and camera characteristics [3]. Colour adaptation techniques [4–7] have been originally put forward to tackle the issue of small tint variation between the application and calibration images. First effort towards this was made by Madhu et al. [4] in 2007 when they proposed one-point colour adaptation where the zero-load bright field image of the application and calibration speci-

∗

mens were utilized to modify the calibration table. Neethi Simon and Ramesh [5] came up with two-point colour adaptation that performed the adaptation from a single isochromatic image of the application specimen itself. Two-point adaptation is a linear interpolation scheme which makes use of the maximum and minimum intensity information from the application image and the calibration table. Neethi et al. [6] also studied the effect of ambient illumination on two-point adaptation and demonstrated its robustness with problems having varying fringe densities. Three-point colour adaptation [7], which is a quadratic interpolation scheme that makes use of the mean of intensities of the application image and the calibration image (or table) in addition to their maximum and minimum intensities was proposed by Swain et al. in 2015. They recommended it to be used in conjunction with a theoretically generated calibration table. Swain et al. [8] also proposed a normalization scheme for the application image that uses a calibration table generated theoretically. The pitfalls in using theoretically generated calibration table were brought out by Ramesh and Ashutosh [9] thereby reestablishing the importance of acquiring experimental calibration images. The success of colour adaptation in both two-point and three-point schemes depend on the accuracy of recording intensity values from the application and calibration images. In industrial scenarios where it is common to have external marks on the specimen [9] or in cases where bright or red spots appear on the image as lighting/ photographic artefacts (Section 3.2), the recording of minimum and maximum intensities are highly susceptible to error. Such cases can corrupt the results yielded by existing colour adaptation schemes. This opens up the possibility for exploring new colour adaptation methodologies that are more generic, and which yields better refined fringe order data. In this paper two colour transfer strategies [10], one using principal component

Corresponding author. E-mail address: [email protected] (K. Ramesh).

https://doi.org/10.1016/j.optlaseng.2019.105963 Received 15 March 2019; Received in revised form 7 October 2019; Accepted 21 November 2019 0143-8166/© 2019 Elsevier Ltd. All rights reserved.

S. Sasikumar and K. Ramesh

Optics and Lasers in Engineering 127 (2020) 105963

analysis (PCA) and the other that matches the means and standard deviations of the images are explored. A comparative study with error analysis of two-point and three-point adaptation schemes with that of the newly proposed methodology is performed for the benchmark problem of a circular disc under diametral compression and the problem of a biaxially loaded cruciform specimen having an inclined crack. A stepwise visualization of the new algorithms with respect to the pixel clouds of application and calibration images is presented to get an intuitive understanding of the methodology. 2. Methodology of TFP and existing colour adaptation techniques In TFP, the R, G, B intensity information at each pixel position in the application specimen is compared with that of the experimentally obtained calibration table for whole field fringe order estimation. The most straight forward approach for comparison would be to calculate the least squares error term (ei ) for each row i in the calibration table and find the fringe order corresponding to the minimum ei value. Depending on the number of colour planes used in the calculation of the error term ei , there can be different colour difference formulae viz., RGB, RG, HSV and RGBHSV [1]. The incorporation of HSV information in addition to RGB in the colour difference formula would extend the fringe order estimation beyond three [9] and therefore becomes useful in demodulating fringe orders up to 12 and beyond. The RGBHSV colour difference formula used for fringe order estimation is given by, √ ( )2 ( )2 ( )2 ( )2 ( )2 ( )2 𝑅 − 𝑅𝑖 + 𝐺− 𝐺𝑖 + 𝐵 − 𝐵𝑖 + 𝐻 − 𝐻𝑖 + 𝑆 − 𝑆𝑖 + 𝑉 − 𝑉𝑖 𝑒𝑖 = (1) The initial fringe order estimate after least squares analysis is bound to have fringe order discontinuities due to repetition of colours, that appears as streaks of noise [11]. Fringe order continuity is established through the process of refining that employs a suitable scanning algorithm. Vivek and Ramesh [12] had proposed fringe resolution guided scanning in TFP (FRSTFP) that had successfully worked for a plethora of problems with only a single seed point. In FRSTFP, the scanning scheme is guided by fringe resolution such that low fringe resolution zones are resolved only towards the end thus minimizing error propagation from these zones. In this paper, since the maximum resolvable fringe order considered is 3, RGB colour difference formula is used in the initial fringe order estimation which is then refined using FRSTFP scanning scheme with a single seed point. The accurate fringe demodulation in TFP depends on the similarity in colour characteristics between the application and calibration images or in other words the similarity in pixel distribution of the application and calibration images in RGB colour space. The colour mismatch or tint variation can be emanating from (i) light source – its emission spectra, colour rendering index (CRI), dispersion of birefringence (wavelength dependence), non-uniformity in illumination (ii) optical elements – transmission spectra of polarizing elements, quarter wave plate error (iii) specimen material – material response, transmission spectra, composition, ageing effects, annealing and stress freezing (iv) camera – relative spectral response, calorimetric characterization, dynamic range, spectral exposure level, and white balance [3]. In one-point colour adaptation technique, Madhu et al. [4] made use of bright field image corresponding to zero load of the application and calibration specimens to modify the calibration table. To avoid slight variations that may be present in the R, G, B values, they resorted to use the average of individual R, G, B intensities inside a small tile within the image. Let 𝑅az , 𝐺za , 𝐵za denotes the averaged RGB values for the application specimen and 𝑅cz , 𝐺zc , 𝐵zc denotes the same for the calibration specimen. Then the modified intensity values of the calibration table are given by, 𝑅cmi =

𝑅az

𝐺za

𝑅z

𝐺z

c 𝑅c ; 𝐺mi = c i

c 𝐺c ; 𝐵mi = c i

𝐵𝑧𝑎 𝐵𝑧𝑐

𝐵ic

(2)

c , 𝐵 c indicate the RGB values at the ith row in the modified where 𝑅cmi , 𝐺mi mi calibration table and 𝑅ci , 𝐺ic , 𝐵ic are the RGB values at the ith row in the original calibration table. In the subsequent discussions, the superscripts a and c denote the application and calibration specimens. The subscript m denotes the entry in the modified calibration table. In two-point colour adaptation proposed by Neethi Simon and Ramesh [5], let the maximum and minimum intensity in the applicaa , 𝐵 a , 𝑅a , 𝐺 a , 𝐵 a and that tion image be represented as 𝑅aMax , 𝐺Max Max Min Min Min c c c , 𝑅c , 𝐺 c , 𝐵 c . The ith in the calibration table be 𝑅Max , 𝐺Max , 𝐵Max Min Min Min row entries in the original calibration table 𝑅ci , 𝐺ic , 𝐵ic gets modified as c , 𝐵 c given by, 𝑅cmi , 𝐺mi mi ] [( a ) ( c ) 𝑅 −𝑅aMin Max c 𝑅cmi = × 𝑅 − 𝑅 + 𝑅aMin 𝑐 c i Min 𝑅 −𝑅 Max

[( c = 𝐺mi

[( c = 𝐵mi

Min

a −𝐺 a 𝐺Max Min c −𝐺 𝑐 𝐺Max Min a −𝐵 a 𝐵Max Min c −𝐵 𝑐 𝐵Max Min

) )

] ) ( c a × 𝐺ic − 𝐺Min + 𝐺Min

(3)

] ( ) c a × 𝐵ic − 𝐵Min + 𝐵Min

Swain et al. [7] reported that the mean intensities of colours in the application image and the calibration table has a nonlinear dependency and proposed a three-point colour adaptation to address this nonlinearity between the dependent and independent variable. Let c , 𝐼c 𝐼Min , 𝐼 c be the independent variables which denotes the minMean Max imum, mean and maximum intensities of the calibration table and a , 𝐼a 𝐼Min , 𝐼 a be the dependent variables that corresponds to the minMean Max imum, mean and maximum intensities of the application image, where I indicates any of the R, G, B intensities. For an entry at the ith row of c is given by, the calibration table 𝐼ic , the modified value 𝐼mi c a 𝐼mi = 𝐼Min

c c ) (𝐼ic − 𝐼Mean )(𝐼ic − 𝐼Max c − 𝐼c c − 𝐼c ) (𝐼Min )(𝐼Min Mean Max

a + 𝐼Max

(𝐼ic c (𝐼Max

a +𝐼Mean

c )(𝐼 c − 𝐼 c − 𝐼Min ) Mean i c )(𝐼 c c − 𝐼Min − 𝐼Mean ) Max

c )(𝐼 c − 𝐼 c ) (𝐼ic − 𝐼Min Max i c c )(𝐼 c c ) (𝐼Mean − 𝐼Min − 𝐼Max Mean

(4)

In specific situations, the colour adaptation techniques [5–7] can fail. Consider Fig. 1, which contains a bright red spot which may be ‘hot pixels’ that develop at long exposure or high ISO photography. These kinds of noise can develop due to lighting or photographic artefacts. The positions of maximum and minimum R, G, B intensity values are marked in Fig. 1. On scrutiny, one can see that the maximum red intensity (𝑅aMax ) a ) are wrongly identified at the red and minimum blue intensity (𝐵Min spot. Moreover, commonly developed external marks on the specimen in industrial scenario, and the inclusion of background pixels due to

Fig. 1. The presence of photographic artefact (hot pixels) as red spot in biaxially loaded cruciform specimen with an inclined crack. The positions of maximum and minimum colour channel intensities are marked on the image along with the magnified view of the red spot. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

S. Sasikumar and K. Ramesh

erroneous image domain mask selection can alter the minimum intensity value. In these cases, the existing colour adaptation schemes would lead to error (Section 5.2) since they use maximum and minimum intensity values as the interpolation parameters. This calls for the exploration of new colour matching strategies. Colour transfer [10] is the counterpart to colour adaptation used in the field of image processing to match colour characteristics between natural scenes. In the subsequent sections of the paper, the applicability of two colour transfer strategies in the context of TFP is explored. 3. Colour transfer using principal component analysis (PCA) Colour transfer [10] is a technique that is widely used in the field of image processing, where the colour characteristics of one image (source) is transferred to another image (target). This is achieved by employing one or more of geometrical transformations like translation, scaling and rotation [13]. In this work, by considering application image as the source and calibration image as the target, the utility of colour transfer in TFP is explored. Previous works in colour transfer had suggested the usage of ⟨l, 𝛼, 𝛽⟩ colour space [14] that is in tandem with the human perception system. In this colour space, a colour is expressed in terms of three numerical values, ⟨l⟩ for lightness, ⟨𝛼⟩ and ⟨𝛽⟩ for (green – red) and (blue – yellow) colour components respectively. The ⟨ l, 𝛼, 𝛽 ⟩ colour space was used by Reinhard et al. [10] for colour transfer between natural scenes. Histogram matching was also recommended as an additional step to achieve a complete transfer of distribution of images [15]. Reinhard and Tania [16] had carried out a comprehensive study on the suitable choice of colour spaces to be used for colour transfer. Some authors have recognized that it is impractical to recommend a colour space that would work globally for all colour transfer problems and has suggested to apply PCA [17] or independent component analysis [18] to derive colour spaces that are image specific. In 2018, Zhang et al. [13] proposed a colour transfer method that employed singular value decomposition (SVD). In their method, mean values of pixels was used for translating the pixel distribution of target image. Scaling factor and orientation were extracted by performing SVD on the covariance matrix of colour channel intensities. Performing SVD on the covariance matrix is one of the methods for accomplishing PCA. The result was a colour transferred target image that shared the same mean, spread, and orientation as that of the source image in RGB colour space. Revisiting Eq. (3) of two-point colour adaptation, the term c ) can be interpreted as translation of calibration pixel cloud (Ri c − RMin with respect to its minima, which is then scaled by a scaling factor of Ra

−Ra

Min ( RMax ) and finally translated with respect to the minimum intenc −Rc Max

Min

sity of application image cloud RaMin . Eq. (4) of three-point adaptation also constitutes of only translation and scaling steps. The concept of orientation matching of pixel clouds as explored by Zhang et al. was not considered in the previous colour adaptation schemes. Following the work of Zhang et al., in the current study translation of pixel clouds is dictated by mean values of the pixels. Initially, scale and orientation matching are done by PCA of colour channel intensities. Fig. 2 shows the dark field isochromatic of an epoxy disc (diameter = 60 mm, thickness = 6 mm, load = 492 N, F𝜎 = 12.16 N/mm/fringe) under diametral compression (Fig. 2(a)) captured using a Sony XC003P three-CCD camera. A white fluorescent lamp (Philips- UAE Essential 18W CDL B22 1CT/12) is used as the light source. The calibration specimen of polycarbonate material shown in Fig. 2(b) is captured using the same light source and image grabber. Polycarbonate is fully transparent, and epoxy has a yellow tinge, and these are taken as extreme case examples to illustrate the methodology. In the process of calibration table generation from a calibration image, integer fringe orders are allotted to the minima of green channel intensities [1]. The R, G, B variation along the mean line of the calibration image shows that there are three

Optics and Lasers in Engineering 127 (2020) 105963

minima in green channel which is used for the generation of a 0–3 calibration table. An image domain mask shown in Fig. 2(c) is generated by carefully selecting the boundaries of the specimen and assigning the value of 255 to all pixels inside the boundary, while allocating the value of 0 to all pixels outside the boundary. The mask is adopted to eliminate the higher fringe order zones near the loading point as the fringe resolution is lower than 10 pixels/ fringe order in this zone. The pixel distribution (in RGB colour space) of the application and the calibration images along with their principal components are shown in Fig. 2(d) and (e) respectively, which are superimposed on each other in Fig. 2(f). The superimposed pixel clouds show that the two images have different pixel distributions. 3.1. Algorithm for colour transfer employing PCA in TFP Let the calibration image intensity be Ic and the application image intensity be Ia represented respectively as, c ⎛𝑅(𝑥1 ,𝑦1 ) ⎜𝐺 c 𝐼 c = ⎜ (c𝑥1 ,𝑦1 ) ⎜𝐵 ⎜ (𝑥1 ,𝑦1 ) ⎝ 1

⋮

𝑅c(𝑥 ,𝑦 ) i

i

𝐺(c𝑥 ,𝑦 ) i i 𝐵(c𝑥 ,𝑦 ) i i 1

a ⎞ ⎛𝑅 ( 𝑥 1 , 𝑦 1 ) ⎟ ⎜ ⋮ 𝐺a ⎟; 𝐼 a = ⎜ (𝑥1 ,𝑦1 ) ⎟ ⎜𝐵 a ⎟ ⎜ (𝑥1 ,𝑦1 ) ⎠ ⎝ 1

𝑅a(𝑥 ,𝑦 ) i

⋮

i

𝐺(a𝑥 ,𝑦 ) i

i

i

i

𝐵(a𝑥 ,𝑦 ) 1

⎞ ⋮⎟ ⎟ ⎟ ⎟ ⎠

The column vectors in the matrices indicate the R, G, B intensities of the pixels. A fourth row of ones is added to the matrices for the purpose of homogeneous representation. If Q represents the transformation matrix, the colour transfer methodology can be summarized as 𝐼mc = 𝑄. 𝐼 c

(5)

The transformation matrix Q comprises of translation (T), scaling (Y) and rotation (Z) matrices [13] corresponding to calibration and application images given by, 𝑄 = 𝑇 a. 𝑍a. 𝑌 a. 𝑌 c. 𝑍c. 𝑇 c

(6)

Let the means of R, G, B intensities of the application image be a a c c (𝑅aMean , 𝐺Mean , 𝐵Mean ) and the calibration image be (𝑅cMean , 𝐺Mean , 𝐵Mean ), then the translation matrices can be represented as: ⎛ ⎜ ⎜ 𝑇c = ⎜ ⎜ ⎜ ⎝

1

0

0

0

1

0

0

0

1

0

0

0

−𝑅cMean ⎞ ⎛ 1 ⎟ ⎜ c −𝐺Mean ⎟ ⎜ 0 a and T = ⎟ ⎜ c −𝐵Mean ⎟ ⎜ 0 ⎟ ⎜ ⎠ ⎝ 0 1

0

0

1

0

0

1

0

0

𝑅𝑎Mean ⎞ ⎟ 𝑎 𝐺Mean ⎟ ⎟ 𝑎 𝐵Mean ⎟ ⎟ ⎠ 1

(7)

To compute the scaling and rotation matrices, PCA can be performed on Ic and Ia . The PCA algorithm (Appendix A) can be performed using an inbuilt function pca() available in Matlab®. For a given input dataset X, pca() returns three outputs given by, [coefficient, score, latent] = pca(X); The output latent is a column vector which contains the eigen values of the covariance matrix of the input dataset X arranged in descending order. Their corresponding eigen vectors are available as column vectors in the coefficient matrix. In PCA, the eigen vectors of the covariance matrix are called the principal components. These principal components are mutually independent and orthogonal such that the first principal component is the axis that captures the maximum variation of the input dataset, the second principal component capturing the second largest variation and so on. Eigen values corresponding to each principal component is an index for the variation captured by it. The coefficient matrix which represents the dominant orientations of the input data set is actually its rotation matrix. In the context of colour transfer, if Cc and Ca denote the coefficient matrices obtained by applying PCA on Ic and Ia respectively, the rotation matrices Zc and Za can be written as [13]: 𝑍 𝑐 = (𝐶 𝑐 )−1 , 𝑍 𝑎 = 𝐶 𝑎

(8)

S. Sasikumar and K. Ramesh

Optics and Lasers in Engineering 127 (2020) 105963

Fig. 2. (a) Dark field isochromatic of circular disc under diametral compression, (b) dark field isochromatic of the top half of central zone of four point bend calibration specimen with its R, G, B variation along line AB, (c) application image domain mask, pixel distribution of (d) Image ‘a’ (within domain mask) with its principal components (e) Image ‘b’ with its principal components (f) Images ‘a’ and ‘b’ super imposed.

Similarly, if the latent vector obtained after applying PCA on Ic be {𝜆c1 , 𝜆c2 , 𝜆c3 }T and on Ia be {𝜆a1 , 𝜆a2 , 𝜆a3 }T , then the scaling matrices Yc and Ya can be represented as [13]: 1

⎛√ c ⎜ 𝜆1 ⎜ 0 𝑌c = ⎜ ⎜ ⎜ 0 ⎜ ⎝ 0

0

0

√1 c

0

0

√1 c

0

0

𝜆2

𝜆3

√ 0⎞ ⎛ 𝜆a1 ⎟ ⎜ 0⎟ ⎟ and 𝑌 a = ⎜⎜ 0 ⎟ ⎜ 0 0⎟ ⎜ ⎟ ⎝ 0 1⎠

0 √ 𝜆a2

0

0

0 √ 𝜆a3

0

0

0⎞ ⎟ 0⎟ ⎟ 0⎟ ⎟ 1⎠ (9)

3.2. Step wise visualization of the algorithm From Eq. (6) it can be understood that the algorithm comprises of five geometrical transformation steps. Step 1 translates the calibration image pixel cloud such that it has zero mean. Rotation in step 2 aligns the dominant orientation of the cloud along the RGB coordinate axes. Step 3 scales the pixel cloud. It should be noted that because of step 1, even after step 3 the pixel clouds would be having zero mean. Rotation in step 4 aligns the scaled pixel cloud along the dominant orientation of the application image pixel cloud. In final step 5, the zero mean pixel cloud

of the calibration image is translated to match the mean of application image pixel cloud. Fig. 3(a)–(e) show the modification of the calibration image in Fig. 2(b) after translation (step 1), rotation (step 2), scaling (step 3), rotation (step 4), and translation (step 5) as pixel cloud representation. The principal components of the final colour transferred image in Fig. 3(e) is same as that of the application image (Fig. 2(d)). This implies that orientation matching is achieved through PCA. Fig. 3(f) is the image representation of final colour transferred calibration pixel cloud (Fig. 3(e)). One of the important observations is that three fringe orders are not identified in the final colour transferred calibration image. This is due to the fact that the modulation of each of the colour channel intensity has got shifted through colour transfer using PCA. As the reason for the modulation shift is not apparent from the pixel cloud representation, the intermediate calibration images at each stage of the geometrical transformations is obtained as shown in Fig. 4(a)– (d). It is to be noted that after translation step, the intensities of pixels with lesser than mean intensity value become negative and they would be thresholded as zero intensity in the image representation. Therefore, the intermediate images are considered for visual representation purpose only. On scrutiny of Fig. 4(b) and (d), it is visually evident that the rotation step changes the colour characteristics of the

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Optics and Lasers in Engineering 127 (2020) 105963

Fig. 3. Pixel distribution of polycarbonate calibration image (Fig. 2(b)) after (a) translation, (b) rotation, (c) scaling, (d) rotation, (e) translation, (f) Image representation of Fig. 3(e) with its R, G, B variation plotted along line AB. Image ‘f’ represents the final modified calibration image after colour transfer using PCA.

making it non-feasible in applying to TFP. In view of it, further studies on applying PCA is not carried out in this paper. The understanding gained is used to develop a new colour transfer method which is discussed in the next section. 4. New colour transfer methodology for TFP

Fig. 4. Polycarbonate calibration image in Fig. 2(b) after each stage of geometric transformations: (a) translation, (b) rotation, (c) scaling, (d) rotation.

calibration image which is seen as modulation shift in colour channel intensities. Therefore, even though the orientation matching of pixel clouds was achieved by PCA, the modulation shift of green channel intensity (along with other colour channels) due to rotation makes the colour transferred table unable to identify all the fringe orders as the original table and

From the study on colour transfer using PCA, it can be inferred that a successful colour transfer for TFP should perform only translation and scaling of the calibration pixel clouds, and not rotation. But the translation and scaling parameters must be modified such that it can work even in cases where the previous colour adaptation schemes lead to error (Section 5.2). In the new colour transfer method proposed, mean and standard deviation of intensities of the application specimen image a a a , 𝐵 a ) and calibration table (or image) (𝑅aMean , 𝐺Mean , 𝐵Mean , 𝑅aSD , 𝐺SD SD c c c c c , 𝐵 c ) are used to modify the entries in the (𝑅Mean , 𝐺Mean , 𝐵Mean , 𝑅SD , 𝐺SD SD calibration table (𝑅ci , 𝐺ic , 𝐵ic ) to generate a modified calibration table c c c (𝑅mi , 𝐺mi , 𝐵mi ) given by, 𝑅cmi =

c 𝐺mi =

𝑅aSD 𝑅cSD a 𝐺SD c 𝐺SD

(𝑅ci − 𝑅cMean ) + 𝑅aMean

(10)

c a (𝐺ic − 𝐺Mean ) + 𝐺Mean

(11)

S. Sasikumar and K. Ramesh

Optics and Lasers in Engineering 127 (2020) 105963

Fig. 5. Polycarbonate calibration image (Fig. 2(b)) after each stage of geometric transformations: (a) translation, (b) scaling, (c) translation, pixel distribution of (d) Image ‘a’, (e) Image ‘b’, (f) Image ‘c’. Image ‘c’ represents the final modified calibration image after the new colour transfer method.

c 𝐵mi

=

a 𝐵SD c 𝐵SD

(𝐵ic

−

c 𝐵Mean )

+

a 𝐵Mean

(12)

The choice of mean and standard deviation, which are statistical quantities would better capture the overall pixel distribution of the images. 4.1. Step wise visualization of the proposed method and its performance Translation of pixel cloud in steps 1 and 3 of the proposed method is same as that of the method discussed in Section 3.1, but scaling factor is decided by the ratio of standard deviation of colour channel intensities. Fig. 5 tracks the changes occurring to the polycarbonate calibration image (Fig. 2(b)) as a result of the proposed colour transfer method. Fig. 5(a)–(c) show the calibration images after translation (step 1), scaling (step 2) and translation (step 3) respectively. The modifications in pixel distribution of the calibration image, shown in Fig. 2(e) in response to the geometrical transformations are shown in Fig. 5(d)–(f). It is evident from Fig. 5(c) that the modulation of colour channel intensities is not

shifted as it is able to obtain all fringe orders present in the original calibration table. The modified calibration table is used to demodulate the fringe order in Fig. 2(a) and the result is shown in Fig. 6(a). Fig. 6(b) represents the superimposed pixel distribution of the application image (Fig. 2(a)) and the colour transferred calibration image (Fig. 5(c)) after the proposed colour transfer methodology. The superimposed pixel clouds give a visual understanding of the pixel cloud matching achievable through the new colour transfer method. The whole field fringe order data in Fig. 6(a) is compared with analytical solution for circular disc and the absolute error plot is shown in Fig. 6(c). It is found to have a mean absolute error (MAE) of 0.08 fringe orders and a standard deviation (SD) of 0.11 fringe orders.

5. Comparative study of the proposed colour transfer method with the existing colour adaptation schemes In this section a comparative study on the performance of the proposed colour transfer scheme with respect to two-point and three-point adaptation schemes is performed. For this, two problems are considered

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Optics and Lasers in Engineering 127 (2020) 105963

Fig. 6. (a) Colour map of refined fringe order of application image by proposed colour transfer method, (b) Pixel distribution of application image (Fig. 2(a)) and colour transferred calibration image (Fig. 5(c)) superimposed, (c) Absolute error between image ‘a’ and analytical solution.

Table 1 Summary on percentage of resolved pixels after least squares analysis. Method used for modifying the experimental calibration table

Percentage of resolved pixels (with absolute error in N less than 0.1)

Two-point colour adaptation Three-point colour adaptation Proposed Colour transfer

59.74 64.92 68.02

– 1. circular disc under diametral compression, being the benchmark problem, 2. the problem of a biaxially loaded cruciform specimen having an inclined crack (bi-axial ratio (Fx /Fy ) = −0.5, thickness 6 mm) representing a problem from the field.

8(f). Table 2 summarizes the MAE and SD (in terms of fringe orders) after refining, for the three methodologies.

5.1. Circular disc under diametral compression

Fig. 9(a) shows a region from a biaxially loaded cruciform specimen made of epoxy having an inclined crack. The image is captured using the same light source and image grabber as discussed in Section 3. The image domain mask is shown in Fig. 9(b) which is selected to eliminate the low fringe resolution zone near the crack tip. Fig. 9(c) shows the application image within the domain mask. For obtaining analytical solution for this problem, the crack tip stress field equation for the case of mixed mode loading in terms of positional coordinates (r, 𝜃) with respect to the crack tip reported by Atluri and Kobayashi [19] and later corrected by Ramesh et al. [20] is made use as given by,

The circular disc shown in Fig. 2(a) is considered as the application image. In Section 3, calibration image made of polycarbonate was used to demodulate fringe orders from the application image made of epoxy to bring out the robustness of the proposed methodology and for getting clarity in step wise visualization of the algorithms. Originally colour adaptation schemes are recommended to be used to circumvent only small tint variations between application and calibration images. Keeping this in mind, calibration specimen (Fig. 7(a)) is made of the same material and is captured using the same light source and image grabber as that of the application specimen, but at different points in time. The small tint variation is due to different aging of calibration and application specimens. A calibration table is generated from Fig. 7(a) which is separately modified by two-point adaptation, three-point adaptation, and the proposed colour transfer method. The modified calibration images with their R, G, B intensity variation along the mean line are shown in Fig. 7(b)–(d). Fig. 7(e)–(g) show the whole field fringe order obtained when two-point adapted, three-point adapted, and the colour transferred calibration tables are respectively made use in the least squares analysis of the application image. Table 1 summarizes the percentage of resolved pixels with absolute error in fringe order less than 0.1 for the three methodologies. The error is calculated with respect to the analytical solution of circular disc. The streaks of noise which represent fringe order discontinuity are eliminated by refining using FRSTFP scanning scheme. In Fig. 8, the whole field fringe order obtained after refining Fig. 7(e)–(g) are represented as Fig. 8(a)–(c) respectively. The absolute error plots of the refined results with respect to analytical solution are shown in Fig. 8(d)–

5.2. Biaxially loaded specimen with inclined crack

⎧𝜎 ⎫ ∞ ∞ ∑ 𝑛−2 𝑛−2 ⎪ 𝑥⎪ ∑𝑛 𝑛 𝐴In 𝑟 2 𝑓 (𝑟, 𝜃) − 𝐴IIn 𝑟 2 𝑔(𝑟, 𝜃) ⎨𝜎 𝑦 ⎬ = 2 2 ⎪𝜏𝑥𝑦 ⎪ 𝑛=1 𝑛=1 ⎩ ⎭

(13)

where the coefficients AI1 , AI2 , … and AII1 , AII2 , … are the unknown mode I and mode II parameters respectively. The stress intensity factors √ (SIFs) for the crack are determined from the coefficients as 𝐾I = 𝐴I1 2𝜋 √ and 𝐾II = −𝐴II1 2𝜋. For an application specimen having a thickness h and material stress fringe value F𝜎 , if N represents the experimental fringe order obtained through TFP, then an error function g can be defined for the mth data point as: { } { } 𝜎𝑥 − 𝜎𝑦 2 𝑁m 𝐹𝜎 2 𝑔m = + (𝜏𝑥𝑦 )2m − (14) 2 2ℎ m Initial estimates of the unknown coefficients do not yield zero gm due to inaccuracy in estimation. The estimates are then incremented and

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Optics and Lasers in Engineering 127 (2020) 105963

Fig. 7. (a) Dark field isochromatic of the top half of central zone of four-point bend calibration specimen, Fig. 7(a) after: (b) two-point colour adaptation, (c) three-point colour adaptation, (d) proposed colour transfer method. Images ‘a’ to ‘d’ contain the R, G, B intensity variation along the centre line AB given alongside the calibration images. Colour map of the fringe order after least squares analysis of Fig. 2(a) by (e) two-point colour adaptation, (f) three-point colour adaptation, (g) proposed colour transfer method.

Table 2 Summary on MAE and SD after refining with FRSTFP. Method used for modifying the experimental calibration table

MAE (in fringe orders)

SD (in fringe orders)

Two-point colour adaptation Three-point colour adaptation Proposed Colour transfer

0.10 0.09 0.09

0.12 0.11 0.10

Table 3 Summary on percentage of resolved pixels after least squares analysis. Method used for modifying the experimental calibration table

Percentage of resolved pixels (with absolute error in N less than 0.1)

Two-point colour adaptation Three-point colour adaptation Proposed Colour transfer

26.79 13.46 40.88

Table 4 Summary on MAE and SD after refining with FRSTFP. Method used for modifying the experimental calibration table

MAE (in fringe orders)

SD (in fringe orders)

Two-point colour adaptation Three-point colour adaptation Proposed Colour transfer

0.36 0.12 0.09

0.38 0.17 0.16

corrected iteratively until the fringe order error minimization criterion is met for the desired convergence error as given by, ∑

|𝑁theory − 𝑁exp |

total no. of data points

≤ convergence error

(15)

where Nexp corresponds to the fringe order of selected points from experimental image obtained by TFP and Ntheory is the theoretical fringe order calculated for the same set of points in each iterative step. Since a priori one does not know how many parameters would appropriately model the stress field, it is recommended to start with two parameters

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Optics and Lasers in Engineering 127 (2020) 105963

Fig. 8. Colour map of refined fringe order of Fig. 2(a) by (a) two-point colour adaptation, (b) three-point colour adaptation, (c) proposed colour transfer method, Absolute error between analytical solution and (d) Image ‘a’, (e) Image ‘b’, (f) Image ‘c’.

and a convergence error of 0.5. Then the number of parameters is progressively increased with reduced convergence error until a convergence error of 0.05 or less is achieved. Following the above procedure, data points are collected from the whole field fringe order data evaluated by TFP for Fig. 9(c) which are echoed back on the theoretically reconstructed fringe pattern as shown in Fig. 9(d). The solution required 10 parameters (Appendix B) and the √ stress field parameters evaluated are KI = 0.585 MPa m and KII = 0.043 √ MPa m. The convergence error obtained is 0.025. The theoretically reconstructed fringe pattern is employed for comparing the performance of the three methodologies. A 0–4 fringe order calibration table of epoxy which maintains the same image grabbing condition as that of the application specimen is used for effecting different colour adaptation schemes.

Fig. 10 shows the whole field fringe order obtained when two-point adapted (Fig. 10(a)), three-point adapted (Fig. 10(b)), and the colour transferred (Fig. 10(c)) calibration tables are made use in the least squares analysis of the application image. Table 3 summarizes the percentage of resolved pixels with absolute error in fringe order less than 0.1. In Fig. 11, the whole field fringe order obtained after refining Fig. 10(a)–(c) is represented as Fig. 11(a)–(c) respectively. The absolute error plots of the refined results with respect to analytical solution are shown in Fig. 11(d)–(f). Table 4 summarizes the MAE and SD (in terms of fringe orders) after refining, for the three methodologies. The high MAE and SD for two-point scheme is due to the presence of the red spot (hot pixel) in the application image (Fig. 1) as discussed in Section 2.

S. Sasikumar and K. Ramesh

Optics and Lasers in Engineering 127 (2020) 105963

Fig. 9. (a) Dark field isochromatic of bi-axially loaded cruciform specimen with an inclined crack, (b) Image domain mask, (c) Image ‘a’ within the domain mask, (d) theoretically reconstructed fringe pattern with data points echoed back.

Fig. 10. Colour map of fringe order after least squares analysis of Fig. 9(c) by (a) two- point colour adaptation, (b) three-point colour adaptation, (c) proposed colour transfer method.

S. Sasikumar and K. Ramesh

Optics and Lasers in Engineering 127 (2020) 105963

Fig. 11. Colour map of refined fringe order of Fig. 9(c) by (a) two-point colour adaptation, (b) three-point colour adaptation, (c) proposed colour transfer method, Absolute error between theoretically reconstructed fringe pattern (Fig. 9(d)) and (d) Image ‘a’, (e) Image ‘b’, (f) Image ‘c’.

6. Conclusions

Supplementary materials

In this paper, the potential of two colour transfer methodologies for tackling small tint variation between application and calibration images in TFP is explored. For visualization of changes occurring to the calibration image along with its pixel distribution during each step of the colour transfer algorithms, application and calibration images are treated as 3-D pixel clouds in RGB colour space. This provided an intuitive understanding of the methodologies. The first algorithm that uses PCA is found to be not feasible in the context of TFP as it shifts the modulation of colour channel intensities due to rotation of pixel clouds involved in the algorithm. A new colour transfer methodology that uses the means and standard deviations of colour channel intensities as translating and scaling parameters is proposed. A comparative study of the proposed methodology with the existing two-point and three-point colour adaptation schemes has been done for the benchmark problem of a circular disc under diametral compression and the problem of a biaxially loaded cruciform specimen with an inclined crack. The results show that the proposed method leads to a larger percentage of resolved pixels after the least squares analysis. It also works well even in the presence of image artefacts and yields better whole field fringe order data after refining which is reflected in its lesser mean absolute errors (MAE) and standard deviations (SD).

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.optlaseng.2019.105963.

Declaration of Competing Interest

In the function pca() available in Matlab®, eigen values sorted in descending order is obtained as the column vector ‘latent’ and the correspondingly arranged eigen vector is available as the ‘coefficient’ matrix.

The authors have no conflict of interests to declare.

Appendix A To apply PCA to an (n × m) dataset X 1 Normalize X, i.e., for each column of X subtract the mean of that column from each entry. Let the normalized data matrix be called N. 2 Compute the covariance matrix of N. i.e., NT N 3 Calculate the eigen values and the corresponding eigen vectors of NT N. The eigen decomposition is where NT N is decomposed into PDP− 1 , where P is the eigen vector matrix and D is the diagonal matrix containing eigen values as the diagonal entries and all other values as zero. The eigen values on the diagonal of D can be associated with the corresponding column in P i.e., eigen value in (1,1) position in D traces to the first column in P, Eigen value in (2,2) position corresponds to second column of P and so on. 4 Sort the eigen values 𝜆1 , 𝜆2, 𝜆3 …. 𝜆m in the descending order and correspondingly rearrange the columns in P (for example, if 𝜆3 is the largest eigen value, bring the third column in P to the position of first column). This sorted matrix of eigen vectors can be called as P∗ .

S. Sasikumar and K. Ramesh

Optics and Lasers in Engineering 127 (2020) 105963

Appendix B

Table B1 Stress field parameters for theoretically generated fringe pattern in Fig. 9(d). Mode I parameters

Mode II parameters √

AI1 = 7.375967 MPa mm AI2 = 0.288365 MPa AI3 = - 0.199015 MPa(mm)−1/2 AI4 = 0.012779 MPa(mm)−1 AI5 = −0.001984 MPa(mm)−3/2 AI6 = −0.000003 MPa(mm)−2 AI7 = 0.000039 MPa(mm)−5/2 AI8 = - 0.000002 MPa(mm)−3 AI9 = 0.000001 MPa(mm)−7/2 AI10 = 0.000000 MPa(mm)−4

√ AII1 = −0.546962MPa mm AII2 = 0.000000 MPa AII3 = 0.068140 MPa(mm)−1/2 AII4 = −0.035680 MPa(mm)−1 AII5 = 0.005148 MPa(mm)−3/2 AII6 = −0.001107 MPa(mm)−2 AII7 = 0.000047 MPa(mm)−5/2 AII8 = - 0.000017 MPa(mm)−3 AII9 = −0.000004 MPa(mm)−7/2 AII10 = 0.000001 MPa(mm)−4

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