Application of Segmentation and Fuzzy Classification Techniques (TSK) in Analyzing the Composition of Lightweight Concretes Containing Ethylene Vinyl Acetate and Natural Fibers Using Micro-Computed Tomography Images

Abstract

The reuse of ethylene vinyl acetate (EVA) discarded from the sports and footwear industries as a partial substitute for gravel in concrete is a way of reducing anthropogenic environmental impacts by enabling the production of lighter structures with similar and superior resistance to those built with traditional concrete. Several studies have been published replacing gravel with EVA and natural fibers, resulting in lighter, more resistant, cheaper, and more ecological concrete. However, there is no methodology to characterize the composition and internal structure of these materials accurately and efficiently, which is vital for quality control in mass-produced pre-molded shapes. In this study, an automated system was developed to measure the percentage of each component in test cores using micro-computed tomography (Micro-CT). For this, (1) Micro-CT images were obtained for concrete test cores made with coarse aggregate consisting of gravel, EVA, and natural fibers in different proportions; (2) the images were segmented differentiating the gravel from the rest of the aggregate, while the remainder was further segmented with the cementitious matrix as background, and the pores, EVA fragments, and fibers as objects against this background; and (3) a Takagi–Sugeno–Kang-type fuzzy inference system was built to classify the objects in the foreground as pores, EVA, and fiber. The tool developed in this manner estimates the percentages of each concrete component and also provides an estimate of the porosity.

1. Introduction

Concrete is one of the most important materials in the construction industry and is the most used structural element in the world. It can be defined as a composite material that consists of a binding medium within which particles or fragments of the aggregates are incorporated. It is usually composed of Portland cement, inorganic material, aggregates (granular materials divided into coarse and fine according to the fineness modulus of the particles), and water (responsible for the kneading and chemical reactions in the cement, causing it to harden and increase the strength of the material [1,2]).

Ethylene vinyl acetate (EVA) is a polymeric material made from oil and extensively utilized in the sport and footwear industry. In these sectors, the disposal can account for up to 20% of the total EVA consumption. Depending on the vinyl concentration, EVA becomes thermoset, rendering it unsuitable for recycling, as it cannot be melted or remolded. Consequently, the environmental impact of discarding EVA into the environment, like many other plastic materials, is a grave concern. Given the substantial volume of EVA waste, extensive storage spaces are required, often leading to the use of open landfills, which increases the risk of environmental contamination, including the emission of toxic gases resulting from EVA combustion [3].

One viable solution to mitigate this environmental issue is the reuse of EVA in other industries, particularly in civil construction. EVA possesses favorable characteristics, such as good processability, thermal stability, impact resistance, fatigue resistance, resilience, toughness, and flexibility [4,5,6]. As a result, numerous studies have been conducted to explore its potential as an aggregate in the civil construction sector, aiming to produce lightweight concretes with good acoustic and thermal performance [7,8,9].

However, it is worth noting that one drawback of lightweight concrete is a reduction in its compressive and tensile strength [10,11]. To address this limitation, one of the alternatives is to incorporate fibers that optimize the distribution of internal stresses in the concrete, thereby bolstering its resistance to demands on the material [2,3,12,13].

As new materials are introduced and incorporated into production processes, the need arises to characterize these materials and comprehend how their internal characteristics influence the material properties. X-ray computed microtomography (Micro-CT) is one of the non-destructive analysis techniques employed for this propose. Micro-CT allows the acquisition of internal images of objects in two and three dimensions, enabling the analysis and characterization of materials through the use of digital image processing (DIP) techniques. In the literature, numerous studies demonstrate the frequent use of DIP to analyze and characterize features like pores in concrete samples. These studies not only show the properties, performance and efficiency of the concrete but also detect internal elements, such as fibers, in the cement matrix [14,15,16,17,18,19,20,21].

However, the characteristics of the constituent elements of lightweight concrete, namely, pores, EVA grains, and piassava fibers, exhibit a diffuse nature that poses challenges when classifying them using Micro-CT images and conventional DIP techniques [22]. In this context, the utilization of a fuzzy-logic-based system becomes necessary, as it accommodates degrees of uncertainty associated with object detection, unlike crisp-logic-based systems [23]. A fuzzy system considers the fuzzy nature of object detection [16,24,25,26].

This study aims to apply image processing techniques and a fuzzy inference system to analyze and classify lightweight concrete containing EVA and piassava fibers. The proposed methodology encompasses three key steps: (1) image segmentation; (2) feature extraction; and (3) development of a fuzzy system to classify the internal objects within the samples.

One of the primary challenges in this task is the absence of precise reference values for assessing the accuracy of the estimates related to different composition features. This is due to the spatial heterogeneity of specimens, stemming from the variable morphology and granularity of the concrete components. Although there is a concrete homogenization process during production, the volumetric density of each component within a specimen is not constant. Consequently, the composition of an extracted core may vary depending on its extraction location and have significant lower or higher volumetric fractions than the average expected from the concrete mix. Therefore, quantitative evaluation of estimation errors is not feasible; we can only assess weather estimated values are “similar” to the theoretical values expected based on the mix.

The primary contribution of this work is the presentation of a classification model based on the Takagi–Sugeno–Kang (TSK) inference system, wherein the inputs are fuzzy and the output is crisp. Additionally, this approach’s application in material characterization, specifically in lightweight concrete, is unprecedented.

The structure of this article is as follows: Section 2 describes the materials used to carry out the research, along with the segmentation, feature extraction, and fuzzy inference system steps. Section 3 presents and discusses the obtained results, and Section 4 provides the conclusions of this research stage and offers suggestions for future studies.

2. Materials and Methods

2.1. Fabricating and Characterizing the Samples

The specimens were prepared by [27] using CP II E 40 RS Portland cement, water, fine sand as fine aggregate, gravels 0 (4.8 mm to 9.5 mm) and 1 (9.5 mm to 19 mm) as coarse aggregates, superplasticizer, EVA grains, and piassava fibers. The piassava fibers were extracted from Attalea funifera Martius plants, which are palm trees from Amazon forest and, mostly, from the southern region of Bahia, Brazil.

The EVA aggregates were divided into two categories, according to diameter. Elements with diameters between 2.36 mm and 1.18 mm were designated coarse EVA (EG), and EVA aggregates with diameters less than 850 µm were designated fine EVA (EF). The length adopted for the piassava fibers was a uniform 2 cm long. The superplasticizer used was the MaSterGlenium 54, in a content of 0.5% of the cement mass.

The specimens were molded in cylindrical molds 100 mm in diameter and 200 mm in height, according to NBR 5738 [28].

The mixture for the reference concrete (RC) was defined by the ABCP (Associação Brasileira de Cimento Portland) method [29] as 1:1.70:0.98:2.41 with a 0.62 water/cement ratio. This RC has 44.2% gravel (coarse aggregate), while in the lightweight concrete, the non-conventional aggregates partially replace the gravel in volume values. All the other concrete constituents’ values are preserved in all specimens.

Table 1. Acronyms used in this work for each concrete mixture and percentages in volume of gravel substitution for non-conventional aggregates.

		EVA Grain
		Thin EVA			Thick EVA
Fiber	0%	5%	15%	25%	15%
0%	CR	EF5	EF15	EF25	EG15
1%	-	1EF5	1EF15	1EF25	1EG15

lists the different concretes studied, with the acronyms and percentages used in the manufacture of the specimens.

Table 2. Reference volume percentages of each aggregate (gravel, EVA and fiber).

Probe	EVA-Ref %	Fiber-Ref (%)	Gravel-Ref (%)	Pore-Ref (%)
CR	-	-	44.2	5.41
EF5	2.20	-	41.9	5.95
EF15	6.60	-	37.6	8.98
EF25	11.0	-	33.1	12.3
EG15	6.60	-	37.5	11.2
1EF5	2.20	0.98	41.5	0.86
1EF15	6.60	0.98	37.2	15.3
1EF25	11.0	0.98	32.8	10.4
1EG15	6.60	0.98	37.2	7.20

lists the calculated percentages of gravel, EVA, and fiber considering the mix in each specimen type.

Porosity Estimation

The concrete mix defines the volumetric composition of the specimens but the porosity is caused by the air that is stuck inside the cementitious mass. For this reason, specimen porosity is usually estimated using standardized techniques in the construction industry. The porosities of the specimens analyzed in this study were obtained using water absorption method [27] and the calculated reference values in volume are shown in last column of Table 2.

The porosity reference values measured in the samples with lightweight additives were higher than the porosity measured in the reference concrete (5.41%). This indicates a higher air retention associated with the addition of EVA and fiber. This finding was also supported by [30,31], who reported porosities in standard concrete samples ranging between 2.5% and 6.2%. However, after the incorporation of lightweight aggregates, they observed values ranging from 4% to 15.28%. Similarly, other authors, such as [15], who analyzed similar samples, estimated the porosity ranging from 4.28% to 15.46%. Thus, the porosity results used in this study align with those reported in the literature, confirming an increase in concrete porosity resulting from the addition of light aggregates.

2.2. Obtaining Images

The microtomographic images were obtained using a SkyScan^®1173 microtomograph, version 1.6, from the Laboratório de Instrumentação Nuclear (Nuclear Instrumentation Laboratory) of Universidade Federal de Rio de Janeiro, Brazil. To be able to adjust the specimens to the tomography scanner, it was necessary to extract a core of 25 mm in diameter and 40 mm in height from the specimens. The projections were reconstructed using NRecons^®software, version 1.7.7.0. For each core, the assays generated around 2100 images of their cross section, all in grayscale (8 bits) with 2240 × 2240 pixels each.

Figure 1a and Figure 2a show images generated by the microtomography scanner.

2.3. Processing the Images

2.3.1. Image Segmentation

To quantify the volumetric fractions of the different coarse aggregates (gravel, EVA, and fibers) using the Micro-CT images, firstly, grayscale segmentation was performed. This process consists of dividing the image into different regions with colors ranging in a certain number of intervals (classes) in which the different constituents are found, such that it is possible process the regions of interest individually.

The most used techniques to segment grayscale images are based on thresholds. The Otsu and K-means [32,33] methods stand out among thresholding techniques. In any given case, once the number of $\mathcal{K}$ classes is determined, the method assigns to each pixel p of color $x_p\in [0,255]$ (grayscale), one of the chosen colors $X_k,k=1,2,\dots ,\mathcal{K}$ to represent the segmented image. The way in which a class $k\in [1,\mathcal{K}]$ is assigned to each pixel p depends on the thresholding method.

The K-means method iteratively searches for the $\mathcal{K}$ centroids (the colors) $C_k$, $k=1,2,\dots ,\mathcal{K}$ of the classes and assigns to each pixel p of color $x_p$ the class k whose centroid $C_{\kappa }$ is closest, i.e., $|x_p-C_{\kappa }|={min}_{k\in [1,\mathcal{K}]}|x_p-C_k|$. When all pixels have been assigned to a class, their centroids are updated as the mean of the colors of all pixels belonging to each class. Every time a centroid is updated (changes its value), the pixel distribution per class is repeated. The iterative process ends when no centroids are updated [33].

The Otsu method searches exhaustively for the $\mathcal{K}-1$ thresholds between the classes $l_k,k=1,2,\dots ,\mathcal{K}-1$, such that $l_1<l_2<\dots <l_{\mathcal{K}-1}$, and assigns class k to a pixel p of color $x_p$, if $l_{k-1}\le x_p<l_k$, for a $k\in [1,\mathcal{K}]$.

The grayscale boundaries are denoted by $l_0=0$ (black) and $l_{\mathcal{K}}=255$ (white). The choice between the $\mathcal{K}(\mathcal{K}-1)/2$ color combinations of the $\mathcal{K}$ thresholds $l_1,l_2,\dots ,l_{\mathcal{K}}$ is made by choosing the color combination which maximizes the interclass variance of the image [32].

After several unsuccessful attempts to make a single segmentation with 5 classes representing: (1) cement mass, (2) pores, (3) gravel, (4) EVA, and (5) fiber, the decision was made to use two binary segmentations:

• Segmentation gravel + remainder = differentiation of the densest aggregate from the remaining aggregates, as shown in Figure 1b where the gravel is in white and the remainder in black. This segmentation was performed in two steps. First, the K-means method with 3 classes was used since 3 dominant shades of gray were identified. In this context, the centroids were $C_1$ = dark gray, $C_2$ = gray, and $C_3$ = light gray. The color between the light gray and gray centroids was then used as the binary segmentation threshold, that is, we defined the $l_{gravel}=(C_2+C_3)/2$, so that pixels with colors greater than $l_{gravel}$ were considered gravel and lesser values are part of the remainder.
• Segmentation (pores, EVA, fiber, and exterior) + remainder = differentiation of the light aggregates, air, and pores from the remainder, as shown in Figure 2c, where pores, EVA, and fiber + area outside the sample (air) are black and the remainder of the image is white. This segmentation was performed using the range-constrained Otsu method [26] which lead to optimal threshold. A first attempt was made with the original Otsu method [32], but the result was unsatisfactory due to a known feature of the method of shifting the thresholds towards the class with the greatest variability. The result obtained with the classic Otsu method for the example image is shown in Figure 2b. There is a lower density of dark regions within the sample and a smaller average size of these regions; this indicates that the method was not successful in capturing the smallest pores present in the sample at the maximum resolution possible.

2.3.2. Segmentation Post-Processing

The images segmented with the range-constrained Otsu method (Figure 1c) were processed to:

• Calculate the area ($pixe{l}^2$) ) of the core (of the circular region) in each image $i=1,2,\dots ,2100$, which was denoted $A_i$. For this, binarization based on the active contour model (snakes) [34] segmentation method was used.
• Identify and index the dark regions inside the core and label the pixels of each one with the index assigned to the region. The methods for detecting regions (blobs) in images identify regions that differ in brightness or color from the surrounding environment ([35]). A blob is a region where the brightness or color is approximately constant, such that all the points in a blob can be considered similar.
• Calculate the white area $b_i,i=1,2,\dots ,2100$ ($pixe{l}^2$) to calculate the percentage of cement paste in each cross-section as described below.

In parallel, after binarizing the images highlighting the gravel (Figure 1b) as described in the previous section, it was necessary to perform morphological opening and closing operations ([36]) to filter noise and connect the contiguous regions of the image. The result of these operations is shown in Figure 1c. Improved definition of the contours of the gravel regions and uniformity of the color inside those regions can be observed. The number of white pixels in each of the 2100 axial images of the binarized and corrected core enable estimation of the volumetric percentage of gravel. The white area in image i is denoted by $B_i,i=1,2,\dots ,2100$ ($pixe{l}^2$) ), and the volumetric percentage of gravel in the i-th section of the core can be estimated by the ratio $B_i/A_i$. Thus, the volumetric percentage of gravel, G, in the entire core can be approximated by the mean of the percentages of gravel in the 2100 cross-sections of the core, i.e.,

$$G=\frac{100}{N}\sum _{i=1}^N\frac{B_i}{A_i}(\%),$$

(1)

where $N=2100$ in this case. Finally, in this post-processing phase, the percentage of cement paste in the core was calculated. The area of white pixels $b_i,i=1,2,\dots ,2100$ calculated in each cross-section in the image of Figure 2c represents the area of cement paste and gravel, since the area of black pixels corresponds to pores, EVA, and fiber. As we calculated the gravel area $B_i,i=1,2,\dots ,2100$ in the image of Figure 1c, the cement paste area in each cross-section can be calculated as $c_i=b_i-B_i$, such that the cement paste percentage in the i-th section is given by r $c_i/A_i$. The percentage of cement paste in the core, C, can be estimated using the mean of the cement paste areas in the 2100 cross-sections, that is,

$$C=\frac{100}{N}\sum _{i=1}^N\frac{c_i}{A_i}(\%).$$

(2)

At this point, after estimating the percentages of gravel (G) and cement paste (C), it remained necessary to estimate the percentages of the light, coarse aggregates (EVA and fiber) and the pores. The pipeline executed thus far is illustrated in Figure 3.

2.3.3. Extraction of Characteristics of Light Coarse Compounds and Pores

This section describes the processing of segmented images using the Otsu method (Figure 2c). The aim is to classify the internal black regions identified in the post-processing segmentation as pores, EVA, or fibers. To accomplish this, the first step is to calculate the characteristics or descriptors of each region (which we will call the “object” identified by the blob detection method) to subsequently estimate to which class each region belongs using a fuzzy inference algorithm.

Of the several most used region descriptors, area (A), eccentricity (E), and fractal dimension (D) were selected. The fractal dimension is one of the defining characteristics of fractals; this characteristic represents the level of occupation complexity in Euclidean space and, therefore, the higher the occupancy level, the more complex the fractal structure. This feature enables use of the fractal dimension for complexity analysis, especially to describe regions in image processing [37,38]. One of the best known and most used methods to obtain the fractal dimension of an object is box counting, which consists of covering the image with a mesh of squares and counting N squares that are necessary to cover the shape on a scale S [39]. The process is repeated on a smaller scale, and the fractal dimension D is estimated through the slope $\alpha$ dof the diagram [40], given by Equation (3)

$$D=\frac{ln\left(N\right)}{ln(1/s)}.$$

(3)

Once the descriptors are defined, using the contour of the circular region obtained in the previous step with the snake method [34], the objects inside the core in each image are selected, and the 3 descriptors of each object are calculated and stored in a descriptor repository.

2.4. Fuzzy System with Takagi–Sugeno–Kang Inference

Fuzzy inference matches continuous numerical input variables (crisp values) to fuzzy output variables that are the bases for decision-making (pattern recognition) through defuzzification. Fuzzy inference, after assigning a certain degree of truth to the input values using membership functions, applies an operator based on a defined set of if-then rules, to assign values to the output vector [41].

Between the two main types of fuzzy inference systems, Mamdani-type [42] and Sugeno-type [43], which differ in the way the outputs are determined, the latter was chosen, as it does not require the defuzzification step to generate the output.

The general flow of a fuzzy inference system is shown in Figure 4.

The input vector $(x_1,x_2,\dots ,x_n)$ contains n crisp variables $x_i,\forall i\in [1,n]$, each of them is transformed into K fuzzy sets using K membership function values for each input variable $x_i$, given in the model as $F_{k,i}\left(x_i\right)\in [0,1],k=1,2,\dots ,K$.

In other words, the fuzzification step generates a $K\times n$ matrix of $F_{k,i}\left(x_i\right),i=1,2,\dots ,n$, $k=1,2,\dots ,K$ values that vary in the interval $[0,1]$. These functions are used to calculate the weight of $K^n$ decision rules $r=1,2,\dots ,K^n$ such that

$$\begin{array}{cc}\hfill w_r& =F_{k(r,1),1}\left(x_1\right)\Delta F_{k(r,2),2}\left(x_2\right)\Delta \hfill \\ \hfill \dots & \Delta F_{k(r,n),1}\left(x_n\right),\hfill \end{array}$$

(4)

where $k(r,i)\in [1,K]$ is the fuzzy set assigned to the input variable $x_i$ in rule r and $\Delta$ represents a T-norm [23].

The output Y of the inference consists of the weighted mean of the functions activated by each rule, which is denoted by $y_r,r=1,2,\dots ,K^n$, of the form

$$Y=\frac{{\sum }_{r=1}^{K^n}{w}_r(x_1,x_2,\dots ,x_n)y_r}{{\sum }_{r=1}^{K^n}{w}_r(x_1,x_2,\dots ,x_n)}.$$

(5)

If the functions activated by the $y_r$ rules are a fuzzy subset, the inference output Y needs to be defuzzified to translate the fuzzy output Y into a crisp output z.

In the Takagi–Sugeno–Kang inference system, the functions $y_r$ are constants or linear functions, by which Y is crisp and no defuzzification is necessary. On the other hand, the T-norm used to calculate the weights $w_r$ is the norm of the minimum [23], i.e.,

$$\begin{array}{cc}\hfill w_r& =min(F_{k(r,1),1}\left(x_1\right),F_{k(r,2),2}\left(x_2\right),\hfill \\ \hfill & \dots ,F_{k(r,n),1}\left(x_n\right)).\hfill \end{array}$$

(6)

2.5. Application of the TSK Inference System to Classify Objects

Each of the object descriptors (area A, fractal dimension D, and eccentricity E) obtained in the first stage of the image processing pipeline (Figure 3) was considered a crisp input variable of the fuzzy system. That is, the input vector with $n=3$ variables in Figure 4 is given by $(x_1,x_2,x_3)=(A,D,E)$. Each of these variables was fuzzified using the C-means [44] fuzzy method with 3 classes ($K=3$) representing $k=1$ high (H), $k=2$ medium (M), and $k=3$ low (L). Membership functions obtained in this way for the descriptor D (fractal dimension) are shown in Figure 5. The 3 membership functions of the other two descriptors were obtained in a similar fashion.

A rule base was built considering the three descriptors with the 27 rules shown in

Table 3. Rule base considering the descriptors of area A, eccentricity E, and fractal dimension D. Each rule $R_r,r=1,2,\dots ,27$ defines a non-repeated classification of the 3 descriptors as (L) low, (M) medium, and (H) high. Column $y_r$ its outputs 1, 2, and 3 corresponding to pore, EVA, and fiber, respectively.

${\mathit{R}}_{\mathit{r}}$	A	E	D	${\mathit{y}}_{\mathit{r}}$	${\mathit{R}}_{\mathit{r}}$	A	E	D	${\mathit{y}}_{\mathit{r}}$
$R_1$	L	L	L	1	R15	M	M	H	3
$R_2$	L	L	M	2	R16	M	H	L	1
$R_3$	L	L	H	1	R17	M	H	M	3
$R_4$	L	M	L	1	R18	M	H	H	1
$R_5$	L	M	M	2	R19	H	L	L	3
$R_6$	L	M	H	1	R20	H	L	M	2
$R_7$	L	H	L	1	R21	H	L	H	3
$R_8$	L	H	M	1	R22	H	M	L	3
$R_9$	L	H	H	1	R23	H	M	M	2
$R_{1}0$	M	L	L	1	R24	H	M	H	3
$R_{1}1$	M	L	M	3	R25	H	H	L	1
$R_{1}2$	M	L	H	3	R26	H	H	M	2
$R_{1}3$	M	M	L	1	R27	H	H	H	2
$R_{1}4$	M	M	M	3

. The rules were defined according to previous knowledge on the morphology and granulometry of the additive materials (EVA and fiber) and the pores in the analyzed specimens. The outputs of the inference rules $y_r,r=1,2,\dots ,27$ are constant functions with 3 discrete values: $1,2$, and 3, representing the classification of the object as pore, EVA, and fiber, respectively. The processing described in this section up to this point is illustrated in Figure 6.

With the rule base defined, the $w_r$ weights of each rule r for each object using Formula (6) were calculated, where the sub-indexes $k(r,i)$ are the fuzzy sets $L,M$, or H of the attribute $i=A,D,E$ (column) in the rule $r=1,2,\dots ,27$ (row) of Table 3.

Once the weights of each rule are calculated, the output of the system Y is calculated according to Equation (5) using the output functions $y_r$ in the last column of Table 3. As the weights vary in the interval $[0,1]$, the output Y is a real number that varies between 1 and 3 and is necessary to transform the output into a discrete variable with value $1,2$, or 3 representing the type of component, pore, EVA, or fiber, respectively.

In this study, the closest class is assigned, i.e., the output class, will be $z\in [1,3]$ if $|Y-z|={min}_{j\in [1,3]}|Y-j|$. However, it is possible that the output is the exact mean value between two classes. In this case, z is assigned to the smaller class, which is the more frequent of the two in concrete due to their ordering. In other words, the most frequent class in the concretes studied is pore (1), followed by EVA (2) and fiber (3). Therefore, if $Y=1.5$, $z=1$ is assigned as the pore is more frequent than EVA, and if $Y=2.5$, $z=2$ is assigned as EVA is more frequent than fiber.

The final part of the processing, which consists of applying the TSK inference engine to classify the objects, is illustrated in Figure 7.

Since all objects (blobs) in each image $i=1,2,\dots ,2100$ are classified in one of 3 classes, pore, EVA, or fiber, the area (number of pixels) of each $A_{pore,i}$, $A_{eva,i}$, and $A_{fib,i}$, is calculated using the blob inventory. The percentage of each component in the cross section i of the core is calculated by dividing the area of the section $A_i$, and the percentage in the whole core as the arithmetic mean of the percentages per section, i.e.,

$$P=\frac{100}{N}\sum _{i^{=}1}^N\frac{A_{pore,i}}{A_i},$$

(7)

$$EVA=\frac{100}{N}\sum _{i^{=}1}^N\frac{A_{eva,i}}{A_i},$$

(8)

and

$$F=\frac{100}{N}\sum _{i^{=}1}^N\frac{A_{fib,i}}{A_i}.$$

(9)

3. Results

As briefly outlined in the introduction, in this problem the true values of the variables that are estimated with the developed method are not available, so it is not possible to calculate the error of the estimates necessary to evaluate the accuracy of the method. This is due to it being a core extracted from a highly heterogeneous concrete specimen that is scanned in the tomography scanner at core-size scale, so the actual composition of the core is unknown. For the analysis the specimen, volumetric compositions were used as reference value.

The difference between the values estimated in the core with our method and the reference values of the specimen consists of two components that are not measurable in practice; the first component is due to imprecision of the method used, and the second component is due to the difference between the composition of the core and the specimen.

In this context, the objective of the analysis was divided into three parts:

• Quantification of the magnitude of the differences between estimated and reference values, supporting the establishment of the range of variation of the different measured variables. Therefore, when a core is analyzed and the percentage of each component is estimated, the interval to which its percentage belongs in the specimen can be inferred.
• Identification of probable systematic deviations that may signal bias in the classification method. The identification and characterization of systematic deviations serves two purposes: (1) to signal the need to improve the method in specific processes, and (2) to carry out a calibration of the reading that corrects the identified deviation.
• Assessment of the degree of correlation between the estimated and reference values. If there is no reasonable degree of correlation between the estimated and reference values, it is not feasible to infer the composition of the specimen from the estimated value in the core.

Accordingly, the reference values and estimates of the percentages of objects (EVA + pore + fiber) and gravel resulting from the first step of the pipeline (Figure 3) are shown in

Table 4. Estimated percentages of objects and gravel using the Micro-CT image binarization and gravel reference values in the specimens.

Probe	Objects (EVA, Pores and Fibers) (%)	Gravel Estimated (%)	Gravel Reference (%)
CR	5.40	45.2	44.2
EF5	8.20	52.9	41.9
EF15	15.6	44.2	37.6
EF25	23.4	42.5	33.1
EG15	17.9	54.8	37.5
1EF5	4.10	55.5	41.5
1EF15	22.9	49.3	37.2
1EF25	22.4	38.7	32.8
1EG15	14.8	54.7	37.2

, and the reference values and the estimates of percentages of EVA, pore, and fiber, resulting from the last pipeline step (Figure 7) are shown in

Table 5. Comparison of the percentages of pore, EVA, and fiber estimated with the TSK inference system operating on attributes obtained from the digital processing of Micro-CT images of the cores, with the calculated reference values of the specimens from the concrete mix.

	Estimated			Reference
Probe	Pore	EVA	Fiber	Pore	EVA	Fiber
CR	2.63	1.76	0.10	5.41	0.00	0.00
EF5	5.71	3.12	0.50	5.95	2.22	0.00
EF15	7.83	5.54	0.50	8.98	6.59	0.00
EF25	10.57	7.72	2.50	12.31	11.04	0.00
EG15	7.13	2.78	1.10	11.21	6.65	0.00
1EF5	1.34	0.97	0.20	0.86	2.22	0.98
1EF15	16.54	6.84	0.20	15.28	6.59	0.98
1EF25	12.56	7.37	1.70	10.39	11.04	0.98
1EG15	6.00	4.75	1.00	7.20	6.65	0.98

.

The estimated and reference percentages of gravel, EVA, pore, and fiber shown in Table 4 and Table 5 are plotted in Figure 8 to obtain a qualitative assessment of the results. The rhombuses represent the reference values and the squares represent the estimated values of the percentages of gravel, pore, EVA, and fiber. In this type of radar or spider web graph, the plotted variable grows radially, such that the differences between the estimated and reference values are measured along the radius.

The method overestimated the content of gravel in all the samples with additives, while it underestimates the EVA content in most samples. Although a more detailed analysis needs to be performed, this suggests that in the current configuration, a part of the EVA is classified as gravel. If this is confirmed, changes should be made to correct this problem in segmentation with K-means (see Figure 3). On the other hand, from a practical point of view, it represents a systematic deviation that can be corrected with adequate calibration.

Estimates of porosity and fiber content do not show a systematic deviation, indicating that the difference between the predicted and reference porosities are the smallest of the four variables studied, while in the case of fiber the largest differences are obtained. In the latter, the estimated values are not correlated with the reference values, which leads to such large differences. The probable cause of this lack of correlation between the estimated and reference values is that 1% of fiber is below the detection threshold of our method, such that the estimated values lack practical value. For this reason, subsequent analysis focuses on the gravel, EVA, and pore content. Determining the fiber detection threshold is a future study to be carried out.

Figure 9 shows the estimated values corrected by calibration and the reference values for crushed gravel and EVA contents. The calibration was performed by translation, that is, adding the constants $c_{gravel}=-11$.$73\%$ and $c_{eva}=1$.$74\%$ to the estimated percentages of gravel and EVA, respectively. The constants were calculated using the formula

$$c=-\frac{{\displaystyle \sum _{i=1}^n\left(v_{e,i}-v_{r,i}\right)}}{n},$$

(10)

where $v_{e,i}$ and $v_{r,i}$ are estimated and reference values of the variable v in core i, respectively, and n is the number of cores studied. The corrected estimates of the variable v are calculated as $v_{c,i}=v_{e,i}+c$, fulfilling ${\sum }_{i=1}^n(v_{c,i}-v_{r,i})=0$.

In addition to enabling correction of the systematic deviation in the measured variables, the applied calibration procedure enables measurement of the size of the absolute deviation; that is, we know that our method overestimates the gravel content by approximately 12% and underestimates the EVA content by approximately 2%. In the remainder of the article, when referring to the estimated gravel and EVA contents, we use the calibrated values. The porosity estimates were not calibrated. In

Table 6. Comparison of estimated and reference percentages of pore, EVA, and gravel contents after calibration.

	Estimated (%)			Reference (%)
Probe	Pore	EVA 1	Gravel 1	Pore	EVA	Gravel
CR	2.630	3.50	33.45	5.410	0.000	44.16
EF5	5.710	4.860	41.22	5.950	2.220	41.95
EF15	7.830	7.280	32.46	8.980	6.590	37.55
EF25	10.57	9.460	30.76	12.31	11.04	33.12
EG15	7.130	4.520	43.06	11.21	6.650	37.53
1EF5	1.340	2.710	43.78	0.860	2.220	41.54
1EF15	16.54	8.580	37.58	15.28	6.590	37.18
1EF25	12.56	9.110	26.96	10.39	11.04	32.80
1EG15	6.000	6.495	43.01	7.200	6.650	37.16

${}^1$ Calibrated value.

, we list the estimated and reference values that will be the object of subsequent analysis in this study.

The differences between estimates and reference values of the volumetric contents in the lightweight concrete components are not only out of the imprecision of the developed component quantification method. Nevertheless, it is necessary to establish a resolution concept for the method from the difference between the estimates and the reference values. In other words, we will assume that resolution of the method increases when the difference between estimates and reference values decreases. Thus, the higher the resolution, the lower the likely range of variation in the composition of the specimen studied around the values estimated from the analysis of its core with the method developed.

It is important to emphasize that even in the ideal case of null imprecision of the method of analysis of the core, there will be a difference between the estimates and the reference values due to the compositional difference between the core and the specimen, such that the ideal resolution does not imply the absence of uncertainties.

To compare the estimates with the references, it is necessary to establish a relative measurement that considers the difference in absolute value between the estimate and its reference in relation to the reference value. According to this, we define the mean absolute percentage difference (MAPD) using the same notation as Equation (10), of the form (the formulation is similar to the mean absolute percentage error (MAPE)).

$$MAPD=\frac{100}{n}\sum _{i=1}^n\mid \frac{v_{e,i}-v_{r,i}}{v_{r,i}}\mid (\%).$$

(11)

The MAPD calculated for each variable measured in our method is an adequate measure to characterize the quality of our method in the range of variation of the composition of the specimens, that is, with EVA contents ranging between 0% and 25%, fiber content ranging between 0% and 1%, and volumetric gravel content equal to 44.2% for EVA and fiber contents (see Table 2).

However, MAPD has a natural limitation, which is that it can only handle cases in which the reference variable is not null, to avoid division by zero. Therefore, the case studies in which $v_R=0$ were not included in the calculation of the mean differences or in the variation of the differences in what follows. That is, this analysis leaves out four (CR, EF5, EF15, EF25, and EG15) of the nine test cases for fiber and one test case (RC) for EVA.

Table 7. Mean absolute percentage difference (MAPD, Equation (11)), rate of decrease of the relative difference with ACA (a), standard deviation of the relative differences with respect to its decreasing linear trend with ACA (sd), and the correlation coefficient between the relative difference and the content of EVA + fiber (r) for the estimates of pore volume, EVA, and gravel.

Component	MAPD (%)	a (%/%)	sd (%)	r
GRAVEL	11.27	−0.475	6.200	−0.492
EVA	30.97	−2.842	28.00	−0.585
PORE	24.49	−1.053	15.50	−0.490

ACA = Increase in Additive Content (EVA + fiber).

presents the mean absolute percentage difference (MAPD) between the estimated and reference percentages of the volume of pores, gravel, and EVA, considering the nine samples (except EVA where the reference concrete was not included, leaving only eight samples). The MAPD for fiber was not calculated for the reasons explained above.

MAPD varies between 11.3% for the gravel volume and 31.0% for the EVA content, which defines an acceptable range, taking into account that the composition of the cores scanned with Micro-CT can differ significantly from the composition of the specimen from which it was extracted. Establishing a dependence between MAPD and the quality Q of the estimates, such that $Q=100-MAPD$, we would have qualities ranging between 69.0% and 88.7%. It is worth highlighting that in this study, we describe the results of the first version of the method, which may be subject to future improvements.

Furthermore, it was found that the absolute percentage difference between the reference percentages and those estimated for EVA, pore, and gravel, decreases with the increase in the content of additives (EVA + fiber), as shown in Figure 10. That is, the quality is superior when analyzing concretes with more content of light aggregates, which is the objective in civil construction practice: to add the largest quantity of light aggregates possible without compromising the mechanical strength characteristics of the structures.

The percentage decrease rates of the relative differences (coefficient a, in the linear model equation y = a*x + b) vary between $a=-0.475$ for gravel content and $a=-2.842$ for EVA volume. In other words, the relative difference between the reference values and those estimated for the volume of gravel decreases 0.475%, on average, for each 1% increase in the EVA + fiber content, decreasing 2.842%, on average, in the case of the EVA volume.

These rates of decrease are statistically supported by the significant anti-correlation observed, with coefficients varying between $r=-0.49$ for pore content and $r=-0.585$ for EVA volume. The standard deviation of the differences in relation to the linear model was confined to the range of $sd$ $=6.2\%$ for gravel content to $sd$$=28.0\%$ for EVA content. The values of the rates of decrease in the relative differences between the estimated and the reference values, as well as the correlation with the content of additives (EVA + fiber) and the standard deviation in relation to the adjusted linear model, are listed in Table 7. These results confirm that the relative difference between estimates and references decreases as the proportion of light aggregates increases.

From a practical point of view, it is also very important that there is a good correlation between the estimated and reference values. To assess the degree of correlation between the reference volumes and the estimates of the lightweight concrete components in the nine specimens studied, we used three different metrics:

• The linear correlation coefficient $r_{xy}\in [0,1]$, defined as:

  $$r_{xy}=\frac{s_{xy}}{s_x{s}_y},$$

  (12)

  where

  $${\displaystyle s_{x,y}=\frac{1}{n-1}\sum _{i=1}^n\left(v_{r,i}-{\overline{v}}_r\right)\left(v_{e,i}-{\overline{v}}_e\right)}$$

  (13)

  is the covariance of the data, and ${\overline{v}}_r$ and ${\overline{v}}_e$ are the means of the reference and estimated values, respectively;

  $$s_x=\sqrt{{\displaystyle \frac{1}{n-1}\sum _{i=1}^n{\left(v_{r,i}-{\overline{v}}_r\right)}^2}}$$

  (14)

  is the root of the standard deviation of the reference values; and

  $$s_y=\sqrt{{\displaystyle \frac{1}{n-1}\sum _{i=1}^n{\left(v_{e,i}-{\overline{v}}_e\right)}^2}}$$

  (15)

  is the root of the standard deviation of the estimated values.
  The coefficient $r_{xy}$ varies between −1 and 1 where the positive sign indicates that the independent variable y increases when the independent variable x increases and vice versa for the negative sign. However, absolute values close to 1 indicate that the points align, forming a line, increasing the dispersion of the points in relation to the line when the absolute value of $r_{xy}$ decreases. In the present study, positive values close to 1 indicate a good alignment of the reference and estimated values, which is one of the requirements for the application of the developed method.
• The root of normalized root mean square deviation nRMSD >= 0, defined as

  $$nRMSD=\frac{RMSD(v_e,v_r)}{{\overline{v}}_r},$$

  (16)

  where

  $$RMSD(v_e,v_r)=\sqrt{{\displaystyle \frac{1}{n}\sum _{i=1}^n{\left(v_{e,i}-v_{r,i}\right)}^2}}$$

  (17)

  is the root mean square deviation and ${\overline{v}}_r$ is the mean of the reference values.
  The RMSD serves to aggregate the magnitudes of the differences between the estimated and reference values for several specimens in a single measurement. It is a widely used measurement to assess the accuracy of predictions, but it depends on the scale. For this reason, in this study, we used the version normalized by the mean value of the reference, which generally varies between 0 and 1 in different applications, although it can exceed 1 in cases with a very low correlation between the estimated and expected data. In our case, we require low values of $nRMSD$, much lower than 1.
  The concomitant use with $r_{xy}$ is due to the fact that although both measurements measure the distance between the estimated and reference values, they accomplish this differently: with bivariate $r_{xy}$ and univariate $nRMSD$ as they capture different characteristics of the compared data.
• Finally, after correcting for systematic deviations, the estimated values must be proportional to the reference values and vice versa, and the proportionality coefficient must be close to 1. In other words, the coefficient of the proportional model $v_r=a{v}_e$ must be close to 1. Applying minimum squares, the formula for a was obtained, such that

  $$a=\frac{{\displaystyle \sum _{i=1}^n{v}_{e,i}{v}_{r,i}}}{{\displaystyle \sum _{i=1}^n{\left(v_{e,i}\right)}^2}}.$$

  (18)
  Including the proportionality coefficient is strictly necessary, as neither $r_{xy}$ or $nRMSD$ are affected by the dependence slope or the displacement of the line that represents the relationship between the estimated and reference values. In other words, two linear dependence functions $y=a_{1}x+b_1$ and $y=a_{2}x+b_2$, with $a1\ne a2$ and $b1\ne b2$, can have the same $r_{xy}$ and $nRMSD$ correlation coefficients. This indicates that these two metrics do not carry information on the type of linear dependence that relates the two variables. In this case, the dependence between the reference values and the estimated values being proportional is a requirement to validate the developed method, which implies that considering the null intercept, $b=0$, the linear coefficient of the regression is close to 1, $a\to 1$.

The calculated metrics are shown in

Table 8. Metrics for evaluating the linear correlation between estimated and reference values of porosity and EVA and gravel contents.

Component	a (%/%)	${\mathit{r}}_{\mathit{xy}}$	nRMSE	Q
GRAVEL	0.958	0.672	0.147	0.873
EVA	1.018	0.853	0.253	0.913
PORE	1.022	0.910	0.211	0.925
MEAN	0.999	0.812	0.204	0.904

, and in Figure 11 we plot the reference values on the vertical axis and the estimated values for the pore, gravel, and EVA contents on the horizontal axis, together with the proportional model obtained by regression in each case.

Analyzing the component with the best of each of the metrics, it can be noted that the EVA estimates have the closest coefficient to 1 ($a=1.0178$), the porosity estimates have the highest correlation ($r_{xy}=0.9099$), while the gravel volume estimates have the lowest $nMSD=0.1473$. Under these conditions, it is difficult to conclude which of the three compositional variables was the best estimated. Therefore, we introduce a quality metric of estimates, $Q\in [0,1]$, which considers the three basic metrics in an integrated manner. The quality is defined according to the harmonic mean of the deviations from the optimal value of each metric, as follows:

$$Q=1-\left(\right|a-1|\ast (1-r_{xy}){\ast nRMSD)}^{1/3}.$$

(19)

This metric varies between 0 and 1 and gives equal weight to all the integrated metrics, serving as an indicator to compare the performance of concrete characterization methods.

It can be seen that the developed method achieves significant quality values, varying between $Q=0$.8732 for gravel content and $Q=0$.9247 for porosity, with a mean quality of $0.9035$.

Despite the satisfactory obtained results, some drawbacks can be considered in the proposed methodology. First, the descriptor set can be increased to include some other features, such as color and texture. Also, a hierarchical classification between the descriptors must be made since now all features have the same weight.

4. Conclusions

Our findings have important implications for the construction industry and sustainable development. By leveraging Micro-CT imaging technology, we successfully obtained detailed images of concrete test cores containing various proportions of gravel, EVA, and natural fibers. These images were then meticulously segmented to differentiate between the components, with the cementitious matrix serving as the background against which pores, EVA fragments, and fibers were identified as distinct objects. This segmentation process laid the foundation for the creation of a Takagi–Sugeno–Kang type fuzzy inference system, which, in turn, facilitated the classification of objects within the foreground as pores, EVA, or fibers. The application of this kind of inference is unprecedented in fiber lightweight concrete characterization and can be applied in other material characterizations using DIP.

The tool we have developed not only provides precise estimates of the percentages of gravel and EVA within a concrete core but also offers valuable insights into the porosity of the material. This breakthrough enables a deeper understanding of the internal structure and performance potential of these alternative concrete mixtures.

In conclusion, our study represents a notable milestone in the characterization of composite concrete, lending crucial support to the adoption of sustainable construction materials. Through the introduction of an innovative methodology for assessing concrete blends incorporating EVA and natural fibers, we equip the industry with the tools to manufacture structures that are not only lighter, more durable, and cost-efficient but also environmentally conscious. However, further research is imperative to enhance the precision and sensitivity required for accurately quantifying low amounts of fibers in this particular class of concrete formulations.