Automatic detection and assessment of crack development in ultra-high performance concrete in the spatial and Fourier domains

Abstract

Automatic detection and assessment of surface cracks are beneficial for understanding the mechanical performance of ultra-high performance concrete (UHPC). This study detects crack evolution using a novel dynamic mode decomposition (DMD) method. In this method, the sparse matrix ‘determined’ from images is used to reconstruct the foreground that contains cracks, and the global threshold method is adopted to extract the crack patterns. The application of the DMD method to the three-point bending test demonstrates the efficiency in inspecting cracks with high accuracy. Accordingly, the geometric features, including the area and its projection in two major directions, are evaluated over time. The relationship between the geometric properties of cracks and load-displacement curves of UHPC is discussed. Due to the irregular shape of cracks in the spatial domain, the cracks are then transformed into the Fourier domain to assess their development. Results indicate that crack patterns in the Fourier domain exhibit a distinct concentration around a central position. Moreover, the power spectral density of cracks exhibits an increasing trend over time. The investigation into crack evolution in both the spatial and Fourier domains contributes significantly to elucidating the mechanical behavior of UHPC.

Graphical abstract

Keywords

dynamic mode decomposition / ultra-high performance concrete / crack detection / geometric features / Fourier domain

1 Introduction

Compared with traditional concrete, ultra-high performance concrete (UHPC) has superior strength and durability due to its dense structure and low porosity [1,2]. However, a higher strength may lead to more brittleness. To address this issue, one effective approach is to incorporate various fibers such as steel fibers and polyvinyl alcohol fibers into the concrete to transform the failure mode from brittle to ductile failure that can be attributed to fiber bridging—restraining the initiation and development of cracking at the early stage and sustaining the load after matrix cracking [3,4]. This high ductility is usually accompanied by a high strain capacity. Specifically, UHPC can achieve a tensile strain capacity of 0.8% [5], which is much higher than that of conventional concrete (0.01%). Recently, improvements made by tailoring the fiber and the matrix have been made to attain strain-hardening behavior, multiple cracking, and high energy absorption [2,6]. Therefore, to illustrate the strengthening effect, the crack propagation and generation process of cracks can play a critical role in evaluating the performance of composites [7–9].

For detecting crack development, there are many methods based on conventional image processing techniques [10,11]. For instance, the simplest technique to sort an image into two classes is the intensity thresholding method. Smyl et al. [12] inspected pavement cracks using an improved Ostu thresholding method. Peng et al. [13] developed a twice-threshold segmentation method to detect cracks in airport runway pavement. These efforts focus on the selection of the appropriate threshold value on the basis of the sensitivity to noise and light [14,15]. The edge detection method, which is founded on the brightness discontinuities in the edges, is also widely used in crack detection. Qiang et al. [16] have combined an adaptive Canny algorithm with an iterative threshold segmentation algorithm to detect surface cracks accurately. Morphological component analysis together with the Sobel filter has been demonstrated by Dixit and Wagatsuma [17] for automatic crack detection under some severe conditions. Although the feasibilities of the threshold method and of the edge detection method in detecting cracks have been demonstrated, their performance heavily relies on the configuration of the algorithm parameters that require manual corrections to satisfy various conditions.

Recent development of machine learning techniques in computer vision provides potential methods for automatic crack detection [18,19]. For example, Ahmadi et al. [20] have put forward a hybrid machine-learning model to detect cracks automatically. In addition, deep learning methods are founded on a neural network with three or more layers to extract and learn the characteristics of cracks from a set of training images. Several deep learning methods such as convolutional neural networks [21,22] and the VGG-16 network [23] have been widely applied to crack classification and localization. Although the deep-learning trained model can detect a crack and identify its crack category, the detector boxes that are used to obtain the approximate boundary of the cracks do not provide precise information on the crack geometry, such as the location and shape [24]. Therefore, it is necessary to develop a dense pixel-level crack detection method to extract this precise information regarding cracks [25,26]. Bhowmick et al. [27] developed an improved U-Net neural network to inspect cracks and then quantified their geometry, like length, area, width, and orientation. Barkavi and Chidambarathanu [28] proposed a novel algorithm for measuring the width and length of cracks based on digital images. As presented in those papers, the crack property can be related to mechanical characteristics and can provide a deep understanding of concrete performance. Due to the irregular shape of cracks, it is difficult to find the trend of crack development and prominent patterns in the spatial domain, but transforming the geometric cracks into Fourier domains offers a potential method [29]. In the Fourier domain, each image point represents a specific frequency contained in the spatial domain and the crack information may show a clearer trend than can be identified in the spatial domain. In this study, the properties of cracks are investigated in the Fourier domain based on the detected cracks, and the evolution of cracks with time is further analyzed.

The development of data science in image processing, especially the dynamic mode decomposition (DMD) [30] method, provides a nice framework for motion estimation [31], action recognition [32], and video coding [33]. The basic idea is to decompose the data matrix into low-rank and sparse matrices, in which the low-rank matrix is used for the background reconstruction. Compared to conventional image processing and machine learning method, the DMD method does not require tuning of the algorithm parameters or training the model using quantities of images, and shows good performance and robustness. To the authors’ best knowledge, the application of the DMD algorithm to crack detection and the characteristics of crack development in the Fourier domain has not been reported before. This work aims to explore the feasibility of using DMD for detection of concrete cracks and of the crack development, including geometrical information, based on recorded video.

The contribution of this work is investigation, for the first time, of cracking behavior of UHPC in the spatial and Fourier domains. In comparison with the blurred features of cracks identified in the spatial domain, crack properties analyzed in the Fourier domain present more clear and interesting trends. The DMD method captures the crack progress by extracting information regarding the cracks from a video. The proposed method shows good robustness and precision, compared to results from traditional image process techniques and deep learning methods, because the DMD method uses a sparse matrix to reconstruct the foreground image containing cracks without additional turning parameters or model training from the labeled images. This study also addresses the relation between the cracking behavior and the mechanical characteristic over time, which can help reveal the mechanical performance of the UHPC.

2 Crack detection and assessment over time

2.1 Dynamic mode decomposition for crack detection

The detection of cracks through image processing techniques relies on tracking changes in pixel intensity specifically within crack regions over time, while the remaining regions remain unchanged. According to this principle, the primary objective of crack detection is to distinguish the region exhibiting intensity changes from the static background. The static background typically exhibits a high degree of correlation across frames, represented by a low-rank matrix, while the crack propagation dynamics can be reconstructed using a sparse matrix. Consequently, the data matrix of a video stream (Y) can be decomposed into low-rank (L) and sparse (S) matrices, denoted as follows:

(1)\mathit{Y}=\mathit{L}+\mathit{S}.

In Eq. (1), the low-dimensional subspace L is interpreted as low-frequency dynamical modes because the stationary background hardly changes or changes very slowly. The remaining modes away from the origin are absorbed into a sparse component S.

To distinguish the static background from the dynamic foreground in original images, the DMD serves as a data-driven decomposition method capable of extracting a collection of coherent modes with temporal frequencies from a given video stream. The obtained modes are useful for background reconstruction and crack diagnosis. Supposing the block of video frames consists of p frames and each frame contains q pixels, a data matrix Y can be formed by stacking a sequence of p frames in the column. Each column of Y is vectored by all the q pixels in one frame. Fig.1 illustrates the decomposition procedure of the data matrix, in which the element of y_{pq} represents an intensity of the qth pixel in the pth frame, and {\mathit{y}}_p represents a flattened vector of the pth frame.

The collected data matrix Y is grouped into two sub-matrices. The first matrix {\mathit{Y}}_1 contains (p−1) frames ranging from 1,2,···,(p−1), and the second matrix {\mathit{Y}}_2 sequences 2,3,···,p. The two matrices are formulated as

(2){\mathit{Y}}_1^{p-1}=\left[\begin{array}{c}||||\\ {\mathit{y}}_1{\mathit{y}}_2\cdots {\mathit{y}}_{p-1}\\ ||||\end{array}\right],{\mathit{Y}}_2^p=\left[\begin{array}{c}||||\\ {\mathit{y}}_2{\mathit{y}}_3\cdots {\mathit{y}}_{\mathrm{p}}\\ ||||\end{array}\right].

It is assumed that the matrices of {\mathit{Y}}_1^{p-1} and {\mathit{Y}}_2^p are approximately connected by a linear operator:

(3){\mathit{Y}}_2=\mathit{A}{\mathit{Y}}_1,

where A is a mapping function incorporating the underlying property of video frames. The eigenvectors and corresponding eigenvalues of A describe spatiotemporal coherent patterns, and can be effectively determined by the DMD method. Fig.2 illustrates the DMD procedure to determine the eigenvectors and eigenvalues of a data matrix. The data matrix formed from video images is decomposed into DMD modes and corresponding amplitude, as well as the time evolution. The detailed algorithm of DMD is reported in Algorithm 1.

According to the calculated DMD eigenvalues and eigenvectors, the original data matrix can be reconstructed as

(4)\tilde{\mathit{Y}}\left(t\right)={\sum }_{j=1}^r{b}_j{\varphi }_j\mathrm{e}\mathrm{x}\mathrm{p}\left({\omega }_{j}t\right),

where b_j is an initial amplitude of each mode; the Fourier mode {\omega }_j=\mathrm{l}\mathrm{n}{\lambda }_j/\mathrm{\Delta }t, here \mathrm{\Delta }t is a time step.

According to Eq. (4), these modes can be distinguished by setting a threshold \epsilon for frequencies {\omega }_j. If \left|{\omega }_j\right|\le \epsilon, the corresponding low-rank modes are incorporated into the background; otherwise, the modes belong to crack propagation. This procedure is formulated as

(5)\mathit{Y}=\mathit{L}+\mathit{S}\approx {\sum }_{\left|{\omega }_j\right|\le \tau }{b}_j{\varphi }_j\mathrm{e}\mathrm{x}\mathrm{p}\left({\omega }_{j}t\right)+{\sum }_{\left|{\omega }_j\right|>\tau }{b}_j{\varphi }_j\mathrm{e}\mathrm{x}\mathrm{p}\left({\omega }_{j}t\right).

Equation (5) indicates that the low-rank matrix L picks out the stationary (i.e., zero mode) and potential quasi-stationary (corresponding to near zero modes) to reproduce the background. In practice, the DMD reconstruction is complex, although a real-valued matrix is required. To address this issue, the approximated low-rank matrix \hat{\mathit{L}} is calculated by

(6)\hat{\mathit{L}}=\left|{\sum }_{\left|{\omega }_j\right|\le \tau }{b}_j{\varphi }_j\mathrm{e}\mathrm{x}\mathrm{p}\left({\omega }_{j}t\right)\right|,

where |\cdot | denotes the modulus of each element in the matrix. Taking Eq. (6) into Eq. (5) yields

(7)\hat{\mathit{S}}=\mathit{Y}-\hat{\mathit{L}},

where the approximated sparse matrix \hat{\mathit{S}} may have some negative entries due to the modulus operation in \hat{\mathit{L}}. The negative elements of \hat{\mathit{S}} do not make any sense because they represent pixel intensities. For this reason, the residual negative values are separated from \hat{\mathit{S}} and added back into \hat{\mathit{L}}. The nonnegative sparse matrix \hat{\mathit{S}} is attained through the repetition of Eq. (6).

2.2 Crack feature assessment in the spatial domain

Since the separated foreground mixes several components, such as specimen cracks and the movement of load cell and specimen, it is desirable to inspect cracks without unwanted components, so a specified mask is designed to discard those. The location and shape of unwanted components are identified using Circle Hough transform [35]. In mathematical format, a circle is described by

(8){\left(x-x_{\mathrm{c}}\right)}^2+{\left(y-y_{\mathrm{c}}\right)}^2={r_{\mathrm{c}}}^2,

where x_{\mathrm{c}} and y_{\mathrm{c}} are the coordinates of the central point and r_{\mathrm{c}} is the radius.

Equation (8) has three unknown parameters, which means that a 3D accumulator is required for the Hough transform. In the 3D Hough parameter space, all the circle candidates are produced by voting. The maximum voted circle of the accumulator has the most probability circle of being identified. When the mask is constructed, pure cracks in the binary image are obtained byBoolean operation between the original image and mask.

When the cracks are well identified, it is important to assess the geometric morphology. The gray region in Fig.3(a) represents a random shape of the cracks in a discrete binary image. The total area (A_{\mathrm{c}}) of this crack is defined as

(9)A_{\mathrm{c}}={\sum }_{i=1}^n{\sum }_{j=1}^m{b}_{ij},

where b_{ij} is a value of the pixel in row i and column j.

However, a metric of the total area may not be sufficient to fully describe the evolution of the cracks. For example, the total areas of cracks in Fig.3(a) and Fig.3(b) are the same, but the crack shapes are different. To give a more detailed sense of crack patterns, the crack should be projected in two major directions. The projection of cracks in two major directions is defined by splitting each pixel into bins and finding all the pixels that are perpendicular to each bin. More specifically, the cracks projected on the rows \left(H_i\right) and columns \left(V_j\right) are, respectively, calculated by

\begin{array}{}\text{(10a)}& H_i={\sum }_{j=1}^m{b}_{ij},\end{array}

\begin{array}{}\text{(10b)}& V_j={\sum }_{i=1}^n{b}_{ij},\end{array}

Fig.3 gives an example of different cracks projecting in two major directions. Although the total areas of cracks are the same in Fig.3(a) and Fig.3(b), their projections are different, giving a complete sense of the geometric characteristic of the cracks. In general, projection gives a compact representation perpendicular to the projection direction, which holds much more information.

2.3 Crack property assessment in the Fourier domain

Surface cracks may have a complicated and chaotic structure in the spatial domain; they can be decomposed into different scales, which provides a measure of the magnitude of their characteristics at a specific resolution. To this end, a signal processing technique known as Discrete Fourier Transform (DFT) [36] is adopted, which is defined as

(11)F\left(u,v\right)={\sum }_{x=0}^{\mathrm{M}-1}{\sum }_{y=0}^{\mathrm{N}-1}f\left(x,y\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-\mathrm{i}2\text{π}\left(\frac{u}{M}x+\frac{v}{N}y\right)\right],

where f\left(x,y\right) is the intensity of an image at the point \left(x,y\right) in the spatial domain, F\left(u,v\right) is the corresponding point in the Fourier space, the number of frequencies (u,v) corresponds to the number of pixels (x,y); the exponential term represents the basic function. Equation (11) indicates that the value of F\left(u,v\right) in the Fourier domain is calculated by multiplying f\left(x,y\right) in the spatial domain with the corresponding base function, and making the sum of the result.

Assume that a crack as displayed in Fig.4(a) is detected. The magnitude of the Fourier transform can be obtained by applying the DFT, as shown in Fig.4(b). The whole image is black except for four corners. To observe the crack property in the Fourier domain, the four corners are shifted toward the center (Fig.4(c)). A small bright circle appears in the center and the other parts are black. The transform patterns of cracks in the spatial Fourier domain provide information on how each frequency contributes to the two-dimensional image. Sometimes, it is more straightforward to measure the crack energy in one-dimensional power spectral density (PSD) than in two dimensions. Fig.4(d) presents the PSD of cracks in one dimension, in which P\left(k\right) is the power spectrum and k is the wave number. There are several interesting characteristics in the power spectrum of cracks: most of the power concentrates on the small wave number, which means the image is dominated by the crack; as the wave number increases, the power decreases almost linearly. Cracks in the Fourier domain also have some obvious features, which can be used for the description of crack development.

According to the above analysis and discussion, the proposed framework for automatic detection and assessment of crack development consists of crack detection and crack properties evaluation, as shown in Fig.5. The crack detection process involves the DMD analysis and the global threshold method. It starts to reshape a sequence of video images into the data matrix. The data matrix is decomposed into low-rank and sparse matrices using the DMD technique, in which the sparse matrix is used to construct the foreground containing cracks. Then, the post-processing techniques, such as the Hough transform and global threshold method, are employed to extract cracks from the foreground. When the cracks are detected from a recorded video, it goes into assessing the crack property. The assessment of crack properties is carried out in the spatial and Fourier domains. In the spatial domain, cracks are evaluated in terms of total crack area and projection. In the Fourier domain, cracks are described using equivalent ellipses, and the power spectra of cracks over time are also studied. Analysis of crack features in the spatial and Fourier domains helps find prominent crack patterns and the trend of crack development, providing insight into crack propagation related to mechanical characteristics.

3 Crack detection for ultra-high performance concrete

3.1 Three-point bending tests

To study the flexural properties of UHPC, three-point bending tests were conducted. The UHPC specimens were manufactured by mixing P.O 52.5 Portland cement, Grade-I fly ash, silica fume, and 0.1−1 mm quartz sand. Polycarboxylate superplasticizer was used to reduce water content and improve workability. In addition, micro smooth 13-mm-long steel fibers with 0.12 mm diameter were added with a volume fraction of 1% to enhance ductility. To reduce the randomness of the experiments and to improve the reliability of the results, four samples from the same batch were tested, referred to as Sample A, Sample B, Sample C, and Sample D. These specimens each had a length of 160 mm, with the section of 40 mm × 40 mm (Fig.6(a)). Fig.6(b) describes the setup of the three-point bending test for UHPC specimen. The span of the beam was 100 mm. The load was applied to the upper surface of the mid-span beam, at the middle point of the beam, using an electronic universal testing machine with a maximum load capacity of 100 kN. Thus the distance between the rigid-steel cylinders of the base and the load cell was 50 mm. The test specimen was simply placed on the rigid-steel cylinders of the base so that it could roll along the longitudinal axis of the beam. Displacement control with a constant rate of movement of 0.5 mm/min was employed in the test [37–39]. The load and corresponding displacement were recorded using the load cell itself.

The measured load-displacement curves of four samples are plotted in Fig.7, showing the same trend. In the beginning, force increased rapidly with the increase of displacement. When the displacement was around 1 mm, the force reached the maximum value. Tab.1 reports the mechanical properties of four samples. The maximum force of four samples ranged from 1019.75 to 12459.29 N and the displacements were between 0.80 and 1.03 mm. After the peak force, the force reduced as the displacement increased. The failed displacement varied from 6.28 to 7.11 mm and the force at failure ranged from 1279.15 to 2325.61 N. It indicated that the four samples showed a greater variation in the failed force than the maximum force. The large differences in failure properties for four samples may be attributed to different failure modes.

Fig.8 displays the failure modes of four samples. In general, cracks appeared in the middle of the specimens. The crack width increased from the upper surface of the specimen to the bottom. However, the four samples showed various shapes of cracks and the areas of cracks were also different. The image processing technique was adopted to analyze and further investigate the relation-ship between the crack development and the force–displacement curves, and will be introduced in the next section.

3.2 Crack detection using dynamic mode decomposition technique and threshold method

In the test, the video recording of the crack development and failure modes of each sample was taken using an OPPO Find X3 mobile phone The sampling speed was set to 30 frames/s and the resolution of each frame was 1920 × 1080 pixels. The duration of recorded videos for each sample is reported in the last column of Algorithm. 1. Since color images occupied a large amount of computing memory, they were transformed into grayscale images through a weighted combination of red, yellow, and green channels, which reduced the data volume.

After each frame was converted into grayscales, a set of frames were considered as an assembly to form a data matrix for DMD analysis. To separate the foreground from the background, the imaginary and real parts of eigenvalues for a set are provided in Fig.9. All the eigenvalues were distributed inside the unit circle with horizontal symmetry, representing convergence of the modes. A point with the cyan color was on the circle of unity, which meant that this point was stable for the background reconstruction. Because the eigenvalue {\lambda }_j=1, so the Fourier mode {\omega }_j={\displaystyle \frac{\mathrm{l}\mathrm{n}{\lambda }_j}{\mathrm{\Delta }t}}=0. The amplitude of the first several modes changes along with the frames, as shown in Fig.9(b). The amplitude of a mode (black dot line in Fig.9(b)) was much larger than that of the other modes and had almost no change. This mode corresponds to the eigenvalue {\lambda }_j=1 that is adopted to reconstruct the background. The amplitude of other modes is gradually reduced as the frame grows, which can be used for foreground reconstruction.

Background and foreground reconstruction was performed based on eigenvalue distribution and the change in amplitude. The original grayscale image, reconstructed background, and foreground are shown in Fig.10(a)–Fig.10(c), respectively. Compared with the original image, the background did not have any cracks, while the foreground included any possible changes, such as the movement of the load cell and the cracks. Since the foreground image as seen in Fig.10(c) was a grayscale, a global threshold method was utilized to extract the changes in the foreground. The global threshold is defined as

(12)g\left(x,y\right)=\{\begin{array}{ll}0,& \mathrm{i}\mathrm{f}f\left(x,y\right)>T\\ 1,& \mathrm{i}\mathrm{f}f\left(x,y\right)\le T\end{array},

where g\left(x,y\right) and f\left(x,y\right) are pixel intensity of output and input images, respectively; T is a constant threshold.

When the global threshold with the value T = 85 was applied to the grayscale foreground, a binary image was attained, as depicted in Fig.10(d). It can be seen that the cracks and the movement of the load cell are visible, whereas only the cracks were expected.

To eliminate the motion components of the load cell in the foreground binary image, the processing procedure was carried out as follows. First, the Hough transform was adopted to detect the outline and circle of the load cell, as orange color marked in Fig.11(a). Then, a specific mask of the same size as the original image was constructed (Fig.11(b)), where the orange region was filled with the black color and the other region was filled in white color. It can be noted that the radius of the circles was 10 pixels larger than the detected circle, due to the small gap between the detected circle and the specimen. The specific mask (Fig.11(b)) and the foreground binary image (Fig.10(d)) performed the Boolean operation, and the results are shown in Fig.11(c). The motion components of the load cell were removed and only the detected cracks were retained. For comparison, the detected cracks were superimposed on the original image with most surface cracks covered in red (Fig.11(d)). This means that the cracks could be well identified. Note that the allowable structural crack width in engineering applications is typically much smaller than the widths shown in the tested photos. In this context, the crack width is only used for method verification.

3.3 Crack feature evaluation in the spatial domain

The crack detection process for the four samples followed the procedure outlined in Subsection 3.2. The total crack area of these samples is compared and depicted in Fig.12(a). All the samples started to crack after 35 s and the total area of cracks gradually increased to 75 s. After that, the four samples showed almost a linear increase in total area over time. At the final failure, cracks in sample A and sample D had total areas of 7014 pixel² and 4523 pixel², respectively, showing the largest and smallest values among the four samples. In general, the ability to detect the minimum crack area depended on the image resolution. Higher resolutions enabled the detection of smaller crack widths. Since the loading procedure was a displacement control process, the time evolution could represent displacement, as shown in Fig.12(b). The force and total area are shown on the left and right axes in Fig.12(b). There is no visible crack before a specimen reaches its maximum force. The crack appears after the maximum force is reached. The crack area of four samples shows a linear increase over time, while the force-reduced rate decreases as the displacement increases.

The total surface area of the four samples provides a general sense of crack development, but the detailed shape of the cracks is still unclear. Cracks projected in the vertical and horizontal directions provide supplementary information. Fig.13 presents the development of the projected cracks over time and position. For the horizontal projected cracks of four samples (the left four subfigures in Fig.13), most cracks were concentrated on the horizontal position between 500 and 650 in a single cluster; the number of pixels generally increased with time. For sample C and sample B the maximum numbers of pixels were 165 and 90, respectively, which were the highest and smallest values of the four samples. This indicates that the horizontal cracks of sample B were the most compact among the four samples.

Compared to a single cluster in the horizontal projected cracks, the four samples had several clusters in the vertical projected cracks. The right four subfigures in Fig.13 refer to the vertical projected cracks of four samples. Most vertically projected cracks were located in the vertical position between 200 and 380, their histograms increase over time. This demonstrates that the development of cracks was primarily along the vertical direction, while the horizontal direction changed somewhat. In general, the total area of cracks and their projections in two directions describe the evolution of crack patterns in detail, providing deep insight into the mechanical properties of UHPC.

3.4 Crack property evaluation in the Fourier domain

As the evolution of the cracks in the four samples shows chaotic and irregular patterns in the spatial domain, this work tried to get statistical information about the cracks in the Fourier domain. Fig.14 shows cracks in the Fourier domain with magnitudes of components at failure. The central region of spatial frequency denotes the low-frequency components and peripheral regions denote the high-frequency components. The color map refers to the weight of each Fourier component and provides an intuitive sense of the component magnitude. It can be seen that the central regions of the four samples were brighter than the others, indicating that the low frequencies had the most energy.

To extract the low-frequency components, a global threshold method was employed. Fig.15 shows binary images. It can be seen that the four samples had obvious white regions in the central parts. Compared to the original cracks (Fig.8), the cracks in the Fourier domain show a regular shape. To further describe the evolution of cracks in the Fourier domain, the major length (E_{\mathrm{m}\mathrm{a}\mathrm{j}\mathrm{o}\mathrm{r}}) and minor length (E_{\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{o}\mathrm{r}}) of the equivalent ellipse, as well as the orientation (E_{\theta }), are defined as illustrated in Fig.16. In Fig.16, the white region represents cracks, which can be fitted by an ellipse. The equivalent ellipse is described by the major length (red dot line), minor length (blue dot line), and orientation (E_{\theta }), where orientation is the angle from the horizontal to the major length clockwise.

Fig.17 presents the major and minor lengths of the equivalent ellipse of cracks in the Fourier domain. The major length of the equivalent ellipse of four samples ranged from 32 to 41 pixels, and their minor length was no greater than 12 pixels, indicating that the value of the major length is much larger than that of the minor length. For crack orientations of the four samples, the angle of the major axis varied from 0° to 10° and the angle of the minor axis was between 90° and 100°. As time increased, the angle of the major axis and minor axis of each sample did not change considerably. In general, crack patterns in the Fourier domain show the obvious trends mentioned above, which is quite important for evaluation of the crack characteristics.

The power spectra of cracks over time are shown in Fig.18, plotted on a logarithmic scale to showing the large amplitude of the low frequencies and the subtle change of the high frequencies. Taking sample A as an illustration (Fig.18(a)), the power (P\left(k\right)) ranged from {10}^6 to {10}^9 when the wave number \left(k\right) was less than {10}^1 and it had its highest power at the wave number {10}^{0.48} (approximately equal to 3), which means that the image was dominated by cracks. After the wave number exceeded {10}^1, the power dropped linearly; this is interpreted as the fractal nature of cracks: at a lower resolution, the same type of crack patterns returned but an increasingly lower signal. When the wave number was approximately {10}^{2.74} (approximately equal to 550), the power had a huge spike, which is attributed to the smallest crack features along the edges of the cracks manifested. For different images, when the crack became larger, the power spectrum became greater.

For the different samples, the power spectra in Fig.18(a)–Fig.18(d) show the same trend. Recall that cracks in the spatial domain have intricate and chaotic structures. They are composed of many small cracks and are fractal in appearance. On the other hand, the power spectra measure the strength of various features at different resolutions. When cracks are transformed into the Fourier domain, it is easy to analyze certain frequencies in the image and extract the geometrical structure of the cracks.

4 Discussion

The cracks in UHPC were accurately detected using the DMD method. However, the selection of the threshold frequency for crack detection is subjective and has a substantial impact on the overall accuracy. Therefore, it is crucial to conduct a comprehensive investigation on determining the threshold frequency in future studies. Additionally, to ensure the robustness of the method, it is recommended to employ a larger number of specimens for verification.

The aforementioned investigation revealed the geometric characteristics of cracks in both the spatial domain and the Fourier domain. The results obtained from the Fourier domain analysis exhibited more pronounced trends compared to the crack properties observed in the spatial domain. However, this study primarily focused on qualitative descriptions of the cracks. Further research should aim to provide quantitative analyses to effectively characterize the cracks. Specifically, establishing the relationship between cracks and load−deformation curves would enhance the understanding of crack behavior and its impact on structural performance.

5 Concluding remarks

This work focuses on the automatic detection and assessment of the crack development of UHPC. A novel DMD method combined with a global threshold tool is put forward to inspect the crack development automatically. Based on the detected cracks, the crack features in the spatial and Fourier domains are evaluated. Some conclusions can be drawn as follows.

1) The proposed method of DMD combined with a global threshold technique can detect cracks in three bending tests. The extracted cracks are in good agreement with the original images, demonstrating the robustness and accuracy of the proposed method.

2) The geometric features of the detected cracks are evaluated in terms of crack area and projection, as well as the relationship between area and load−slip curve. Results indicate that the force starts to decrease as the cracks appear and the crack area increases with increasing displacement. The total crack area and projection can comprehensively describe crack development.

3) The crack patterns in the Fourier domain are concentrated in the center position. The PSD of cracks gets larger over time. The crack features in the Fourier domain show a more direct interpretation of the crack development than those in the spatial domain.

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