Preface

The advance in space machineries has created a novel technology for observing and monitoring the Earth from space. Most earth observation remote sensing considerations focus on using conventional image processing algorithms or classic edge detection tools. Nevertheless, these techniques do not implement modern physics, applied mathematics, signal communication, remote sensing data, and innovative space technologies. This book provides readers with methods to comprehend how to monitor coastal environments, disaster areas, and infrastructure from space with advanced talent remote sensing technology to bridge the gaps between modern space technology, image processing algorithms, mathematical models and the critical issue of the coastal and infrastructure investigations. In other words, advanced remote sensing technology, which covers sensor developments, and image processing algorithm modifications, which are based on modern physics, artificial intelligence, and machine learning. In these regards, their applications cover a wide range of coastal observations, for instances, high risk of a tsunami depends on the depth of water, the coastal geomorphology, the direction of the tsunami wave, and the existence of rivers or other water canals. In these circumstances, coastal zones are required for new urban planning and specific infrastructure designing to reduce the impact of such a disaster.

In spite of numerous of synthetic aperture radar (SAR) space technology, the developing country researchers and scientists are still focusing on optical remote sensing technology. In fact, microwave remote sensing require use of mathematics and physics behind the SAR technology. The first chapter introduces a new technology for measuring sea surface current using along-track interferometry of TanDEM-X satellite data. This chapter delivers a novel algorithm to retrieve sea surface current using the multichannel MAP height estimator algorithm, which is considered the first study of ocean current in the coastal waters of Peninsular Malaysia.

The available SAR data increases dramatically with the recent operation of many spaceborne and airborne SAR systems. This makes the joint processing of multiple images for accurate understanding and perception of a scene and target possible. For SAR image pairs acquired from different imaging geometries or by different sensors, there is always a geometrical warp between them, which should be compensated first before any deep application. Image registration is aimed to retrieve the warp function to align the same pixel position in each SAR image to the same target position in the global system. A lot of SAR image registration techniques have been developed hitherto. In the second chapter, the algorithms that conduct registration based on image features, such as contour, region, line, and point are accurately addressed. Contour, region, and line, as well as their combinations, are often used for registration of multi-modality images. For SAR images with geometrical distortion and speckle, point feature is generally much clearer and easier extracted. Tie points, corner, and key points are the commonly-used features in SAR image registration. Tie points usually refer to the features extracted from tie patches in SAR image registration.

**II**

**Chapter 7 125**

**Chapter 8 147**

Utilization of Unmanned Aerial Vehicle for Accurate 3D Imaging

Geo Spatial Analysis for Tsunami Risk Mapping *by Abu Bakar Sambah and Fusanori Miura*

*byYoichi Kunii*

Following the second chapter, the third chapter introduces a novel technology for implementation of interferometry synthetic aperture radar (InSAR) with L-band on monitoring harbor. In fact, L-band SAR and its long-lasting temporal coherence is an advantage to perform precise interferometric coherence analysis.  In this chapter, author presents the basic theory of SAR observation, interferometric coherence analysis for the disaster monitoring and its examples of the harbor facilities.

However, optical remote sensing experts are relying on commercial software and open source codes without fully understanding the mathematical algorithms involved in image processing. In fact, conventional image processing techniques such as image classification are being used. In this view, classifying remote scenes according to a set of semantic categories is a very challenging problem, because of high intraclass variability and low interclass distance.

The most advanced image processing technique is presented in Chapter 4. One of the advanced learning machine algorithms for image processing is Deep convolutional neural networks (CNNs), which have been widely used to obtain high-level representation in various computer vision tasks. Deep CNN models are trained upon a database of more than 1.2 million categorized natural images of 1000+ classes, which serve as the backbone for many segmentation, detection and classification tasks on other data sets.

The fifth chapter presents an image processing technique that is based on the sub-pixel algorithms for modeling a time series of shoreline changes. In fact, the majority of investigations are only used conventional classification or threshold technique to study short periods of coastal erosion. The novelty of the fifth chapter is that the authors implemented eight years of time series of multispectral data with a subpixel technique to reduce the error of shoreline extraction at sub-pixel, pixel and object-based scales.

A different remote sensing technique is introduced in the sixth chapter for monitoring infrastructure. This technique is based on utilization of accelerometer measurements. In fact, infrastructure, including roads, bridges, tunnels, water supply, sewers, electrical grids, and telecommunications, may be exposed to environmentally-induced or traffic-induced vibrations. Some infrastructure, such as bridges and roadside upright structures, may be sensitive to vibration where accelerometers and other types of sensors may be used for their measurement of sensitivity to environmentally-induced loads, such as wind and earthquakes, and traffic-induced loads, such as passing trucks. With data collected by accelerometers, time histories may be obtained, transformed, and then analyzed to determine their modal frequencies and shapes.

A coastal disaster, which is mainly based on a tsunami disaster, requires such advanced technology of geospatial monitoring of the tsunami risk impact. The seventh chapter delivers the integrated approach of raster weighted overlay of all spatial databases of tsunami vulnerability and risk parameters specifying the vulnerability and risk area due to the tsunami and defines the possible area that could be affected by the tsunami and the potential inundated area.

Finally, the book describes a new technology of Unmanned Aerial Vehicle for accurate three-dimensional reconstruction (3-D). In fact, this technology is rapidly growing among the researchers and scientists. Chapter eight presents a novel technology to construct a precise 3-D image using an Unmanned Aerial Vehicle, which is validated

**V**

by ground field measurements during UAV experiments. In this view, there is great potential to use UAV images for 3-D modeling when compared to operational

I wish to convey my appreciation to all authors who contributed novel work to this book.  I am also grateful to the IntechOpen editorial team Ms. Martina Josavac and Ms. Maja Bozicevic who afforded the opportunity to publish

**Prof. Dr. Maged Marghany**

Malaysia

Microwave Remote Sensing expert, Faculty Geospatial and Real Estate, Geomatika University College, Kuala Lumpur, WP Kuala Lumpur,

satellite data.

this book.

by ground field measurements during UAV experiments. In this view, there is great potential to use UAV images for 3-D modeling when compared to operational satellite data.

I wish to convey my appreciation to all authors who contributed novel work to this book.  I am also grateful to the IntechOpen editorial team Ms. Martina Josavac and Ms. Maja Bozicevic who afforded the opportunity to publish this book.

## **Prof. Dr. Maged Marghany**

Microwave Remote Sensing expert, Faculty Geospatial and Real Estate, Geomatika University College, Kuala Lumpur, WP Kuala Lumpur, Malaysia

**IV**

Following the second chapter, the third chapter introduces a novel technology for implementation of interferometry synthetic aperture radar (InSAR) with L-band on monitoring harbor. In fact, L-band SAR and its long-lasting temporal coherence is an advantage to perform precise interferometric coherence analysis.  In this chapter, author presents the basic theory of SAR observation, interferometric coherence analysis for the

However, optical remote sensing experts are relying on commercial software and open source codes without fully understanding the mathematical algorithms involved in image processing. In fact, conventional image processing techniques such as image classification are being used. In this view, classifying remote scenes according to a set of semantic categories is a very challenging problem, because of high intraclass vari-

The most advanced image processing technique is presented in Chapter 4. One of the advanced learning machine algorithms for image processing is Deep convolutional neural networks (CNNs), which have been widely used to obtain high-level representation in various computer vision tasks. Deep CNN models are trained upon a database of more than 1.2 million categorized natural images of 1000+ classes, which serve as the backbone for many segmentation, detection and classification tasks on other

The fifth chapter presents an image processing technique that is based on the sub-pixel algorithms for modeling a time series of shoreline changes. In fact, the majority of investigations are only used conventional classification or threshold technique to study short periods of coastal erosion. The novelty of the fifth chapter is that the authors implemented eight years of time series of multispectral data with a subpixel technique to reduce the error of shoreline extraction at sub-pixel, pixel and object-based scales.

A different remote sensing technique is introduced in the sixth chapter for monitoring infrastructure. This technique is based on utilization of accelerometer measurements. In fact, infrastructure, including roads, bridges, tunnels, water supply, sewers, electrical grids, and telecommunications, may be exposed to environmentally-induced or traffic-induced vibrations. Some infrastructure, such as bridges and roadside upright structures, may be sensitive to vibration where accelerometers and other types of sensors may be used for their measurement of sensitivity to environmentally-induced loads, such as wind and earthquakes, and traffic-induced loads, such as passing trucks. With data collected by accelerometers, time histories may be obtained, transformed,

A coastal disaster, which is mainly based on a tsunami disaster, requires such advanced technology of geospatial monitoring of the tsunami risk impact. The seventh chapter delivers the integrated approach of raster weighted overlay of all spatial databases of tsunami vulnerability and risk parameters specifying the vulnerability and risk area due to the tsunami and defines the possible area that could be affected by the tsunami

Finally, the book describes a new technology of Unmanned Aerial Vehicle for accurate three-dimensional reconstruction (3-D). In fact, this technology is rapidly growing among the researchers and scientists. Chapter eight presents a novel technology to construct a precise 3-D image using an Unmanned Aerial Vehicle, which is validated

and then analyzed to determine their modal frequencies and shapes.

disaster monitoring and its examples of the harbor facilities.

ability and low interclass distance.

and the potential inundated area.

data sets.

Section 1

Synthetic Aperture Radar

Image Processing and

Applications

1

## Section 1


*3.2.1 Adsorbents*

There are many studies that show the properties of adsorbents to separate, concentrate and purify various compounds [66, 67]. Functionality, porosity, irregularities, surface area, tightly bonded impurities, internal porous structure, particle size, ionic strength, pH, and temperature all influence physical adsorption [66]. The temperature influences the adsorption in two ways, increasing the transport

**279**

*Valorization Options of Strawberry Extrudate Agro-Waste. A Review*

nuclear magnetic resonance spectroscopy methods.

*3.2.2 Countercurrent-chromatography*

offered by the different solvent systems [70].

*3.2.3 Two-phase aqueous system*

speed through the outer boundary layer and inside the pores due to the decrease in the viscosity of the solution, and changing the capacity of the adsorbent. However, high temperatures can promote irreversible interactions [67]. Another important parameter for purification with adsorbents is pH. For example, at acid pH, the adsorption of phenolic compounds by different adsorbents increases because the phenols are not dissociated and dispersion interactions predominate [66]. At alkaline pH, the adsorption decreases due to the dissociation of hydroxyl groups and carboxyl groups [66]. There are many types of adsorbents such as activated carbons, mineral adsorbents, synthetic polymeric adsorbents, ion exchange resins, lignin and lignocellulosic materials, adsorbents based on polysaccharides and others [66]. Among the available adsorbents Amberlite XAD adsorbents are widely used in the concentration of polyphenols [68]. Zhang et al. [69] reported the isolation and structural characterization of 10 phenolic compounds from strawberry extracts using a combination of Amberlite XAD-16 and C18 columns, HPLC-UV, and

Countercurrent chromatography is a technique widely used in the purification of natural products [70]. Countercurrent chromatography is a liquid–liquid partition chromatography process in which both the mobile phase and the stationary phase are liquid [70]. The main advantage of countercurrent chromatography, when compared to equivalent techniques such as low pressure liquid chromatography, is that there are no adsorption losses in the stationary phase [70]. The range of selectivity offered by chromatographic resins is equivalent to the range of selectivity

Several studies have shown the importance of countercurrent chromatography

for the purification of bioactive compounds from strawberry. The compound 2,5-dimethyl-4-hydroxy-3[2H]-furanone 6′O-malonyl-β-d-glucopyranoside was isolated from a strawberry glycosidic extract (Fragaria × *ananassa*, cv. Senga Sengana) by countercurrent chromatography [71]. Peonidin-3-glucoside and malvidin-3-glucoside were obtained from grapes in a single step, while in a second step, cyanidin-3-glucoside was isolated [72]. In another research, the separation of anthocyanin monomers of high purity from mulberry fruits was developed [73].

Two-phase aqueous system is a liquid–liquid fractionation technique that is usually formed by mixing two polymers in aqueous media, for example, polyethylene glycol and dextran or maltodextrin, or by a polymer and a salt, such as polyethylene glycol and salts of phosphates, citrates or sulphates [74–76]. This method has advantages over other purification techniques due to a comparatively low consumption of energy and time, as well as the possibility to be designed for a continuous operation. Moreover, two-phase aqueous systems are effective for many types of substances, especially for the concentration and purification of bioactive compounds [74, 75]. Several studies have demonstrated the suitability of this technique for the purification of bioactive compounds such as phenolic compounds from fig fruits (*Ficus carica* L.) [76], or the purification of gallic acid from natural matrices with ionic liquids [77]. Furthermore, two-phase aqueous system has been applied for the purification of polyphenols from a model solution of gallic acid and three real samples of red and white wine, and orange juice in combination with macro and micro extractors [78]. Polyphenols have been also extracted from *Aronia melanocarpa* berries, using ultrasound-assisted extraction in combination with the two-phase aqueous system [79].

*DOI: http://dx.doi.org/10.5772/intechopen.93997*

#### **Table 1.**

*Summary table of characteristics for comparing extraction techniques.*

*Valorization Options of Strawberry Extrudate Agro-Waste. A Review DOI: http://dx.doi.org/10.5772/intechopen.93997*

speed through the outer boundary layer and inside the pores due to the decrease in the viscosity of the solution, and changing the capacity of the adsorbent. However, high temperatures can promote irreversible interactions [67]. Another important parameter for purification with adsorbents is pH. For example, at acid pH, the adsorption of phenolic compounds by different adsorbents increases because the phenols are not dissociated and dispersion interactions predominate [66]. At alkaline pH, the adsorption decreases due to the dissociation of hydroxyl groups and carboxyl groups [66]. There are many types of adsorbents such as activated carbons, mineral adsorbents, synthetic polymeric adsorbents, ion exchange resins, lignin and lignocellulosic materials, adsorbents based on polysaccharides and others [66]. Among the available adsorbents Amberlite XAD adsorbents are widely used in the concentration of polyphenols [68]. Zhang et al. [69] reported the isolation and structural characterization of 10 phenolic compounds from strawberry extracts using a combination of Amberlite XAD-16 and C18 columns, HPLC-UV, and nuclear magnetic resonance spectroscopy methods.

#### *3.2.2 Countercurrent-chromatography*

*Innovation in the Food Sector Through the Valorization of Food and Agro-Food By-Products*

enhanced yields with pulsed electric fields. For instance, a comparison study between a heat treatment at 90°C for 60 or 30 seconds and high intensity pulsed electric field in strawberries juice, showed that strawberry juice treated with high intensity pulsed electric field maintained greater amount of phenolic acids and total anthocyanins than thermally treated juices [63, 64]. Likewise, the recovery of phenols from the shell of the pomegranate by pulsed electric field has been assayed, resulting in a similar antioxidant extraction yields and an energy saving of 50%

compared to an ultrasound extraction technique [65].

target compounds to be recovered or the required extraction yield.

There are many studies that show the properties of adsorbents to separate, concentrate and purify various compounds [66, 67]. Functionality, porosity, irregularities, surface area, tightly bonded impurities, internal porous structure, particle size, ionic strength, pH, and temperature all influence physical adsorption [66]. The temperature influences the adsorption in two ways, increasing the transport

> **Ability to release compounds**

**Degradation of bioactive compounds**

**x x x x**

**x x x x x**

**x x x x x**

**x x x x x x**

**Intracellular attack**

**Breaks of bonds**

**High cost**

**3.2 Purification techniques of bioactive compounds**

**Specificity Possibility of**

*Summary table of characteristics for comparing extraction techniques.*

**x x**

**combination**

*3.1.6 Extraction techniques comparison*

*3.2.1 Adsorbents*

**Extraction technique**

Solvents extraction

Hydrothermal extraction

Microwave assisted extraction

High hydrostatic pressure extraction

Pulsed electric fields extraction

**Table 1.**

Several studies on the extraction of antioxidant compounds in agri-foods show

After reviewing the different extraction techniques that have been applied to the strawberry and strawberry extrudate, a summary describing their most interesting aspects is shown in **Table 1**. The aspects that have been compared are: the specificity of the extraction techniques with the bioactive compounds, the possibility of combining with other extraction techniques, the ability to release bioactive compounds, the potential degradation of bioactive compounds, possibility of intracellular attack, bonds breakage and whether the technique has a high operational and investment cost. The choice of the best technique for the strawberry extrudate is a tailor-made solution for each situation that will depend of the investment capacity,

**278**

Countercurrent chromatography is a technique widely used in the purification of natural products [70]. Countercurrent chromatography is a liquid–liquid partition chromatography process in which both the mobile phase and the stationary phase are liquid [70]. The main advantage of countercurrent chromatography, when compared to equivalent techniques such as low pressure liquid chromatography, is that there are no adsorption losses in the stationary phase [70]. The range of selectivity offered by chromatographic resins is equivalent to the range of selectivity offered by the different solvent systems [70].

Several studies have shown the importance of countercurrent chromatography for the purification of bioactive compounds from strawberry. The compound 2,5-dimethyl-4-hydroxy-3[2H]-furanone 6′O-malonyl-β-d-glucopyranoside was isolated from a strawberry glycosidic extract (Fragaria × *ananassa*, cv. Senga Sengana) by countercurrent chromatography [71]. Peonidin-3-glucoside and malvidin-3-glucoside were obtained from grapes in a single step, while in a second step, cyanidin-3-glucoside was isolated [72]. In another research, the separation of anthocyanin monomers of high purity from mulberry fruits was developed [73].

#### *3.2.3 Two-phase aqueous system*

Two-phase aqueous system is a liquid–liquid fractionation technique that is usually formed by mixing two polymers in aqueous media, for example, polyethylene glycol and dextran or maltodextrin, or by a polymer and a salt, such as polyethylene glycol and salts of phosphates, citrates or sulphates [74–76]. This method has advantages over other purification techniques due to a comparatively low consumption of energy and time, as well as the possibility to be designed for a continuous operation. Moreover, two-phase aqueous systems are effective for many types of substances, especially for the concentration and purification of bioactive compounds [74, 75]. Several studies have demonstrated the suitability of this technique for the purification of bioactive compounds such as phenolic compounds from fig fruits (*Ficus carica* L.) [76], or the purification of gallic acid from natural matrices with ionic liquids [77]. Furthermore, two-phase aqueous system has been applied for the purification of polyphenols from a model solution of gallic acid and three real samples of red and white wine, and orange juice in combination with macro and micro extractors [78]. Polyphenols have been also extracted from *Aronia melanocarpa* berries, using ultrasound-assisted extraction in combination with the two-phase aqueous system [79].


**2. Calderón weights and** *Lp***-weighted inequalities**

A function *ω* defined on *n* is called a *weight* if it is locally integrable and positive almost everywhere. For a measurable set *E* ⊂ *n*, ∣*E*∣ denote its Lebesgue measure, *ω*ð Þ¼ *E* Ð *Eω*, and *Ec* the complement of *E* in *n*. Given a ball *B*, *tB* is the ball with the

same center as *B* and with radius *t* times as long, and *f B* ¼ 1 ∣*B*∣ Ð *Bf*. As usual, *χE* denotes the characteristic function of *E* and *B x*ð Þ ,*r* denotes a ball centered at *x* with radius *r*. Also, *C* denotes a positive constant.

Let *ω* be a weight in *n* and 1≤*p*< ∞. A Lebesgue measurable function *f* belongs to *Lp*ð Þ *ω* if

$$\|f\|\_{L^{p}(\alpha)} = \left(\int\_{\mathbb{R}^{n}} |f|^{p} \, d\nu\right)^{1/p} < \infty.$$

We say that an oprator *T* is a bounded operator on *Lp*ð Þ *ω* if

$$\|Tf\|\_{L^{p}(\alpha)} \leq C \|f\|\_{L^{p}(\alpha)}, \qquad \text{for all} f \in L^{p}(\alpha).$$

Given 1<*p* < ∞, it is said that *ω* is a Calderón weight of class C*p*, that is *ω*∈ C*p*, if the Calderón operator *S* is bounded on *Lp*ð Þ *ω* (see [5]) or, equivalently, if *P* and *Q* are both bounded on *Lp*ð Þ *ω* (see also [6]). It is well known that the class C*p* for *p* >1 is given by the conditions

$$\begin{aligned} M\_p &: \quad \left(\int\_{[0,\infty]} \boldsymbol{\alpha}(t) dt\right)^{1/p} \left(\int\_{[\boldsymbol{x},\infty)} \frac{\boldsymbol{\alpha}^{1-p'}(t)}{t^{p'}} dt\right)^{1/p'} \leq \mathbb{C} \quad \text{for all} \ \boldsymbol{x} > 0;\\ M^p &: \quad \left(\left(\int\_{[\boldsymbol{x},\infty)} \frac{\boldsymbol{\alpha}(t)}{t^p} dt\right)^{1/p} \left(\int\_{[0,\boldsymbol{x}]} \boldsymbol{\alpha}^{1-p'}(t) dt\right)^{1/p'} \leq \mathbb{C} \quad \text{for all} \ \boldsymbol{x} > 0. \end{aligned}$$

The Calderón operator plays an important role in the theory of real interpolation and such theory related to Calderón weights is developed in [5]. On the other hand, in [7], the authors considered a maximal operator *N* on 0, ð Þ ∞ associated to the basis of open sets of the form 0, ð Þ *b* , given by

$$N\!f(x) = \sup\_{b>x} \frac{1}{b} \int\_{[0,b]} |f(t)|dt$$

for measurable functions *f*. Then, for nonnegative functions *f*, we have

$$P(\boldsymbol{\pi}) \le \mathcal{N}\mathcal{f}(\boldsymbol{\pi}) \le \mathcal{S}\mathcal{f}(\boldsymbol{\pi}) \qquad \text{for all } \boldsymbol{\pi} > \mathbf{0}.$$

The classes of weights *ω* associated to the boundedness of *N* on *Lp*ð Þ *ω* are those that satisfy the Muckenhoupt-*Ap* condition, 1≤*p* < ∞, only for the sets of the form ð Þ 0, *b* . These classes are denoted by *Ap*,0 and defined as follows:

$$\begin{aligned} A\_{1,0}: \quad &No(\mathbf{x}) \leq Co(\mathbf{x}) \quad \text{for almost all } \mathbf{x} > \mathbf{0};\\ A\_{p,0}: \quad & \left(\frac{1}{\mathfrak{X}} \int\_{[0,\mathbf{x}]} \boldsymbol{\omega} \right) \left(\frac{1}{\mathfrak{X}} \int\_{[0,\mathbf{x}]} \boldsymbol{\omega}^{1-p'} \right)^{p-1} \leq \mathbf{C} \quad \text{for all } \mathbf{x} > \mathbf{0}, \text{where} \\ &\mathbf{C} \leq C \quad \text{for all } \mathbf{x} > \mathbf{0}, \text{where} \end{aligned}$$

Then, in [7] is proved that *N* and *S* are bounded operators on *Lp*ð Þ *ω* if and only if *ω*∈ *Ap*,0 for 1<*p*< ∞. This result implies, in particular, that the classes of weights *Cp* and *Ap*,0 coincide for 1<*p*< ∞.

*A Brief Look at the Calderón and Hilbert Operators DOI: http://dx.doi.org/10.5772/intechopen.106027*

Taking into account these results it is natural to wonder for the action of the Calderón and Hilbert operators over suitable spaces such as *BMO* or Lipschitz spaces. Also, another interesting question is: which are, in these cases, the Calderón weights in order to obtain weighted inequalities between these spaces?

These problems were treated for instance in the case of the fractional integral operator in [8, 9], which have been the main motivation for the article [10] and for the development of the following sections.


Author details

Joaquín Pérez-Ortega1 \*, Nelva Nely Almanza-Ortega1 , Andrea Vega-Villalobos1 , Rodolfo Pazos-Rangel2 , Crispín Zavala-Díaz3 and Alicia Martínez-Rebollar1

1 National Institute of Technology of Mexico/CENIDET, Cuernavaca, Mexico

\*Address all correspondence to<EMAIL_ADDRESS>
© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The K-Means Algorithm Evolution DOI: http://dx.doi.org/10.5772/intechopen.85447


**4. Results**

This paper proposed using a kernel prediction network and generative adversarial network to construct an end-to-end general denoising network structure, as shown in Figure 1. Our network structure consisted of three parts: the kernel prediction network module, generation adversarial network module, and image reconstruction module. The kernel prediction network module takes the auxiliary feature information image as the input, passes through the source information encoder, the feature information encoder, and the kernel predictor, and finally generates a prediction kernel for each pixel.

The generated adversarial network module is divided into two parts: the generator module and the multiscale discriminator module. The generator module takes the noisy Monte Carlo-rendered image as the input, passes through the symmetric encoder–decoder structure and the residual block structure, and finally outputs the rendered image with preliminary denoising. Then, the prediction kernel and the preliminarily denoised rendered image are sent to the image reconstruction module for reconstruction, and the prediction kernel is applied to the preliminarily denoised rendered image to obtain a preliminarily reconstructed rendered image. To further improve the quality of the result and to be more

robust, the initially reconstructed rendered image undergoes four iterations of filtering for further denoising, and the final denoised image is obtained after four iterations of the image reconstruction module as outputs. Finally, this denoised image is applied to the loss function.

We evaluated the denoising MC renderings based on the KPN-DGAN method to solve the MC noise image problem and the high-frequency detail loss.

The PSNR and SSIM matrices were used as the quantitative indicators of denoising results. Thus, PSNR calculated the reconstruction error between the denoised and real images based on the mean square sum (MSE). Note that the errors of these matrices are sensitive to noise; as long as a certain pixel value changes and regardless of which direction it changes, the PSNR will also change. Thus, the value range of PSNR is not fixed, and the maximum value is related to the image resolution.

Then, we selected the most representative methods in Monte Carlo image denoising in recent years to compare with our experimental results, which are the KPCN work in 2017 [3], AMCD, and DEMC in 2019 [16,17]. Note that the selected scene uses the 4 spp noise image rendered by the tungsten renderer [32]. The results are as follows:

Figure 4 shows the ablation experiment of this paper, and the enlarged area of image details the MC-rendered image with 4 spp, our result against the AMCD result, KPCN result, DEMC result, and the reference rendered image with 4096 spp. The effect of our approach is better in the final denoising result in terms of subjective details and objective indicators, such as the radiator details and the geometrical objects reflecting sharper on a lamp of the automobile scene, maintaining the barrier shape that does not overlap and the lines in the house scene. In the livingroom2 scene, our method performance is also better and enhances sharp edges with greater detail, unlike other methods. The effect of AMCD algorithms is good, but most of the results are a little blurred. The DEMC and KPCN are poor, because the results have many stains. Generally, comparing the results showed that our approach is better at denoising the MC-rendered image, while retaining and restoring the details and structure of the scene.

The PSNR and SSIM index values are reported under each image, and higher values indicate a better result. The network of our approach performed well and reduced the time consumption of denoising. Therefore, we compared our method against prior methods, with similar processing conditions and an equal sample for all methods, and the results of the average SSIM and PSNR scores are as follows:

In order to further observe the results of the method in this paper, more scene models were selected for comparison; we highlighted the difference between diffuse and specular components, and the relationship to high-frequency details. Thus, the following comparison experiments were compared with AMCD, the DEMC work in 2019 [16,17], and the AFGSA work in 2021 [20]; all of these techniques have public released codes and weights. The 3D model is still the tungsten renderer, and the sampling rate is 4 spp. The experimental results are as follows:

Usually, the specular and diffuse components have different noise patterns and are highly dependent upon the smoothness or texture of the surface properties. Figure 5 shows the other methods that led to unsatisfactory results, with disturbing effects on material and glass, such as blur region, glossy reflections, depth of field, area lighting, and global illumination. Therefore, they need to take advantage of auxiliary features in different ways. The material scene showed an erroneous texture, the reflected illumination was poor, the teapot scene had blurred details and was smoother, and there was a glow reflection in the glass with more noise in the coffee scene. All methods accepted our approach.

In all noise reduction tests, our method always performed better than several stateof-the-art solutions. Tables 1 and 2 show the SSIM and PSNR values and time process for all noise reduction results. Accordingly, our method consistently had smaller errors, with higher SSIM values and less time consumption than state-of-the-art methods.

**Figure 4.** The comparison of the results of this paper with AMCD, KPCN, and DEMC.

**Table 1.** The SSIM, PSNR values, and time process results of our approach against the AMCD, KPCN, and DEMC results.

**Table 2.** The SSIM, PSNR values, and time process results of our approach against the AMCD, DEMC, and AFGSA results.

Finally, the KPN-DGAN denoised the Monte Carlo-rendered image with the auxiliary features, which reduced the image noise with a low samples rate, and restored the scene structure details, to improve the quality of rendered images with less time-consuming processing.

**Figure 5.** Comparison of results of our approach against the AMCD, DEMC, and AFGSA results.


**1. Introduction**

In the real world, the linguistic expression is well-suited for the thinking and expressing patterns of human beings. Due to the vagueness of languages and the complexity of decision-making environments, the linguistic fuzzy theory has been well developed in the past decades and shows irreplaceable advantages in the fuzzy decision-making field. Linguistic variables (LVs) were defined for fuzzy reasoning and decision-making [1–4]. Linguistic uncertain variables [5,6] (interval-valued linguistic variables) were then defined to depict uncertain linguistic information in decision-making problems [7,8]. After that, a linguistic intuitionistic fuzzy number (LIFN) [9], which contains two independent LVs to describe the degrees of truth and falsity, respectively, was presented to handle the uncertainty and incompleteness in linguistic decision-making environments [10]. Furthermore, with the wide application of the neutrosophic theory in decision-making [11–13], Fang and Ye [14] proposed a linguistic neutrosophic number (LNN) by adding a new LV to the LIFN for representing the indeterminacy degree to do with the indeterminate and inconsistent linguistic information [15]. Although there exist some research works on LNNs [14,15], existing LNNs cannot depict the hesitancy of decision-makers (DMs) in the linguistic assessment of alternatives.

Concerning the handling of the human hesitant cognition in decision-making environments, many works have been published so far. Torra and Narukawa [16] and Torra [17] originally introduced hesitant fuzzy sets (HFSs) to express the hesitancy by allowing the membership to contain several possible values. Then, for linguistic decision-making problems, the expression of a hesitant fuzzy

linguistic set (HFLS) [18] was obtained based on combining a linguistic term (LT) set with a HFS so as to satisfy the hesitant linguistic evaluation requirements [19,20] of DMs. After that, an interval-valued HFLS [21] was presented as an extension form by combining an interval-valued LT set with a HFS. Recently, Ye [22] proposed the hesitant neutrosophic linguistic number (HNLN) to carry out hesitant decision-making problems with the neutrosophic linguistic number that contains partial determinacy and partial indeterminacy. However, there is no definition or decision-making method for the hesitant sets of LNNs in the existing literature. Additionally, in the hesitant linguistic expressions of DMs, the components between two hesitant sets generally have difference in their length sizes, and thus it is difficult to directly perform measure calculations between hesitant sets. Thus, several researchers have proposed some extension methods to extend the shorter items in the two hesitant sets by adding the minimum values, maximum values, or any values [23,24] to reach their identical length. However, these extension methods depend too much on the subjective preferences and interests of the DMs. To solve this problem, we have already introduced the least common multiple cardinality (LCMC) to extend the hesitant fuzzy elements in our previous research works [22,25], which become more objective for the decision-making calculation of HFSs.

As aforementioned, there is a gap of hesitant LNNs in existing studies. For instance, suppose that we hesitate between two single-valued LNNs, <*h*7, *h*3, *h*4> and <*h*5, *h*3, *h*1>, from the given LT set *H* = {*hs*|*s* ∈ [0, 8]} regarding an evaluated object. However, it is difficult to express the hesitation information and the LNN information of the DMs simultaneously by a unique LNN or a unique HFS. Therefore, for the purposes of satisfying the demand of hesitant decision-making with LNNs and ensuring the objectivity of the measure calculation, this paper aims to (i) define the concept of HLNNs by combining HFSs with LNNs, (ii) present the LCMC-based generalized distance and similarity measures of HLNNs for more objective measure calculation of HLNN information, and (iii) to propose a novel multiple-attribute decision-making (MADM) method based on the proposed LCMC-based similarity measure in the HLNN setting.

In order to do so, Section 2 briefly reviews LNNs. Section 3 defines a HLNN and a HLNN set. Then, in Section 4, the LCMC-based generalized distance and similarity measures of HLNNs are presented. In Section 5, a new MADM method was developed by using the proposed similarity measure of HLNNs. In Section 6, the feasibility of the proposed approach is demonstrated by an investment case. The conclusions and future research of HLNNs are discussed in the last section.

#### **2. Linguistic Neutrosophic Numbers (LNNs)**

Fang and Ye [14] originally presented the following definition of the LNN:

**Definition 1** ([14])**.** *Let H = {h0, h1, ..., hτ} be a LT set, where τ + 1 is an odd cardinality. A LNN can be defined as ϑ =<hT, hU, hF> for hT, hU, hF* ∈ *H and T, U, F* ∈ *[0, τ], where hT, hU, hF represent the degrees of truth, indeterminacy, and falsity, respectively.*

For the comparison of LNNs, the score and accuracy functions of LNNs are defined as follows [14]:

**Definition 2** ([14])**.** *Let ϑ = <hT, hU, hF> be a LNN in H. Then its score function can be given by:*

$$S(\theta) = (2\pi + T - lI - F) / 3\pi \text{ for } S(\theta) \in [0, 1], \tag{1}$$

*and its accuracy function can be expressed as*

$$V(\theta) = (T - F) / \tau \text{ for } V(\theta) \in [-1, 1]. \tag{2}$$

**Definition 3** ([14])**.** *Let ϑα* = < *hTα* , *hUα* , *hFα* > *and ϑβ* = < *hTβ* , *hUβ* , *hFβ* > *be two LNNs in H. There exist the following relations:*

#### **3. Hesitant Linguistic Neutrosophic Numbers (HLNNs) and HLNN Set**

Torra and Narukawa [16] and Torra [17] first defined the HFS as follows:

**Definition 4** ([16,17])**.** *Assume S is a universe set, then a HFS N on S can be given by*

$$N = \{  ~~| s \in \mathcal{S} \}\_s~~$$

*where E(s) is a hesitant component of N containing a set of some values in [0, 1], which represents all possible membership degrees of s.*

By integrating HFS with LNN, we define a HLNN set as follows:

**Definition 5.** *Set a universe of discourse S = {s1, s2,* ... *, sq} and a finite LT set H = {h0, h1,* ... *, hτ}, and then a HLNN set Nl on S can be expressed as*

$$N\_l = \{  |s\_j \in S, j = 1, 2, \dots, q \}$$

*where El(sj) is a set of mj LNNs*, *denoted by a HLNN El*(*sj*) = {< *hTkj* , *hUkj* , *hFkj* > *hTkj* ∈ *H*, *hUkj* ∈ *H*, *hFkj* ∈ *H*, *k* = 1, 2, ··· , *mj*} *for sj* ∈ *S.*

#### **4. LCMC-Based Distance and Similarity Measures of HLNNs**

In most situations, the cardinal numbers (the number of LNNs) of HLNNs evaluated for the same object are usually different. Thus, it is necessary to make the cardinal numbers of the two HLNNs the same to satisfy the distance and similarity measures between them.

We assume that *p* HLNNs on *S =* {*s*1, *s*2, ... , *sq*} are *El*1 (*sj*), *El*2 (*sj*), ··· , *Elp* (*sj*) for *sj* ∈ *S* (*j* = 1, 2, ..., *q*). Then, the HLNNs *Eli*(*sj*) for *i* = 1, 2, . . . , *p* can be given by

$$\begin{split} E\_{l\_1}(s\_j) &= \{ < h\_{\mathbf{T}\_{1j}^2}^2, h\_{\mathbf{L}\_{1j}^2} h\_{\mathbf{F}\_{1j}^2} > \dots \dots < h\_{\mathbf{T}\_{1j}^{m\_1j}} h\_{\mathbf{L}\_{1j}^{m\_1j}} h\_{\mathbf{F}\_{1j}^{m\_1j}} > \}, \\ E\_{l\_2}(s\_j) &= \{ < h\_{\mathbf{T}\_{2j}^2}^2, h\_{\mathbf{L}\_{2j}^2} h\_{\mathbf{F}\_{2j}^2} > \dots < h\_{\mathbf{T}\_{2j}^{m\_2j}} h\_{\mathbf{L}\_{2j}^{m\_2j}} h\_{\mathbf{F}\_{2j}^{m\_2j}} > \}, \\ \vdots \\ E\_{l\_p}(s\_j) &= \{ < h\_{\mathbf{T}\_{pj}^2} h\_{\mathbf{L}\_{pj}^2} h\_{\mathbf{F}\_{pj}^2} > \dots < h\_{\mathbf{T}\_{pj}^{m\_jj}} h\_{\mathbf{L}\_{pj}^{m\_j}} h\_{\mathbf{L}\_{pj}^{m\_j}} > \}, \end{split}$$

where *mij* is the cardinal number of *Eli*(*sj*) (*i =* 1, 2, . . . , *p* and *j =* 1, 2, . . . , *q)*.

Provided that the LCMC of *mij (i* = 1, 2, ..., *p* and *j* = 1, 2, ..., *q)* is *cj* (*j* = 1, 2, ... , *q)*, by increasing the number of LNNs < *hTkij* , *hUkij* , *hFkij* > *(k* = 1, 2, ..., *mij)* in *Eli*(*sj*) depending on *cj* (*j* = 1, 2, ... , *q*), the extended HLNN *Eoli*(*sj*) (*i* = 1, 2, ... , *p* and *j* = 1, 2, ... , *q)* will be obtained by the extension forms:

*Eol*1 (*sj*) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ *cj* < =: ; < *hT*11*j* , *hU*11*j* , *hF*11*j* >, ··· : ;< = *R*1*j* , < *hT*21*j* , *hU*21*j* , *hF*21*j* >, ··· : ;< = *R*1*j* , ··· , < *hTm*1*j* 1*j* , *hUm*1*j* 1*j* , *hFm*1*j* 1*j* >, ··· : ;< = *R*1*j* ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭, *Eol*2 (*sj*) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ *cj* < =: ; < *hT*12*j* , *hU*12*j* , *hF*12*j* >, ··· : ;< = *R*2*j* , < *hT*22*j* , *hU*22*j* , *hF*22*j* >, ··· : ;< = *R*2*j* , ··· , < *hTm*2*j* 2*j* , *hUm*2*j* 2*j* , *hFm*2*j* 2*j* >, ··· : ;< = *R*2*j* ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭, ··· , *Eolp* (*xj*) = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ *cj* < =: ; < *hT*1*pj* , *hU*1*pj* , *hF*1*pj* >, ··· : ;< = *Rpj* , < *hT*2*pj* , *hU*2*pj* , *hF*2*pj* >, ··· : ;< = *Rpj* , ··· , < *hTmpj pj* , *hUmpj pj* , *hFmpj pj* >, ··· : ;< = *Rpj* ⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭,

where *Rij* is the number of LNNs < *hTkij* , *hUkij* , *hFkij* > (*k* = 1, 2, ..., *mij*) in *Eoli*(*xj*) (*i* = 1, 2, ... , *p* and *j* = 1, 2, ... , *q*), calculated by:

$$R\_{ij} = \frac{c\_j}{m\_{ij}}.\tag{3}$$

Additionally, the elements *ϑσ*(*k*) *ij* =< *hTσ*(*k*) *ij* , *hUσ*(*k*) *ij* , *hFσ*(*k*) *ij* > (*k* = 1, 2, ... , *cj*) in *Eoli*(*xj*) are arranged in an ascending order, denoted as *Eoli*(*xj*) = {*ϑσ*(1) *ij* , *ϑσ*(2) *ij* , ··· , *ϑσ*(*cj*) *ij* } (*i* = 1, 2, ... , *p* and *j* = 1, 2, ... , *q*), where *σ* : (1, 2, . . . , *cj*) → (1, 2, . . . , *cj*) is a permutation satisfying *ϑσ*(*k*) *ij* ≤ *ϑσ*(*k*+1) *ij* (*k =* 1, 2, . . . , *cj*).

**Definition 6.** *Let Nl*1 = ,*El*1 (s1), *El*1 (*s*2), ··· , *El*1 (*sq*)/ *and Nl*2 = ,*El*2 (*s*1), *El*2 (*s*2), ··· , *El*2 (*sq*)/ *be two HLNN sets on S = {s1*, *s2*, *...*, *sq}*, *where El*1 (*sj*) *and El*2 (*sj*) *(j = 1*, *2*, ... , *q) are HLNNs in a LT set H = {h0*, *h1*, *...*, *hτ} for hj* ∈ *H. Let f(hj) = j/τ be a linguistic scale function. Then, the normalized generalized distance between Nl*1 *and Nl*2 *can be represented as:*

$$\begin{split} d(N\_{l\_{1}},N\_{l\_{2}}) &= \left\{ \frac{1}{q} \sum\_{j=1}^{q} \left[ \frac{1}{\mathcal{U}\_{l\_{1}^{\prime}}} \left( |f(h\_{T\_{1^{\prime}\_{1}}^{\prime}}) - f(h\_{T\_{2^{\prime}\_{1}}^{\prime}})|^{\rho} + |f(h\_{\mathcal{U}\_{1^{\prime}\_{1}}^{\prime}}) - f(h\_{\mathcal{U}\_{2^{\prime}\_{1}}^{\prime}})|^{\rho} + |f(h\_{\mathcal{U}\_{2^{\prime}\_{1}}^{\prime}}) - f(h\_{\mathcal{U}\_{2^{\prime}\_{1}}^{\prime}})|^{\rho} \right) \right] \right\}^{1/\rho} \\ &= \left\{ \frac{1}{q} \sum\_{j=1}^{q} \left[ \frac{1}{2\epsilon\_{j}^{\prime}\theta} \sum\_{k=1}^{\epsilon\_{j}} \left( |T^{\sigma(k)}\_{1j} - T^{\sigma(k)}\_{2j}|^{\rho} + |L^{\sigma(k)}\_{1j} - L^{\sigma(k)}\_{2j}|^{\rho} + |F^{\sigma(k)}\_{1j} - F^{\sigma(k)}\_{2j}|^{\rho}) \right) \right]^{1/\rho} \text{ for } \rho > 0. \end{split} \tag{4}$$

Obviously, *d*(*Nl*1 , *Nl*2 ) degenerates to the normalized generalized distance of Hamming for *ρ =* 1 and to the normalized generalized distance of Euclidean for *ρ =* 2.

For the generalized distance *d*(*Nl*1, *Nl*2 ), there is a proposition as follows:

**Proposition 1.** *For any two HLNN sets Nl*1 = ,*El*1 (*s*1), *El*1 (*s*2), ··· , *El*1 (*sq*)/ *and Nl*2 = ,*El*2 (*s*1), *El*2 (*s*2), ··· , *El*2 (*sq*)/*, the generalized distance d*(*Nl*1 , *Nl*2 ) *between Nl*1 *and Nl*2 *for ρ > 0 contains the following properties:*

$$\begin{array}{lll} \text{(HP4)} & \text{Let } \mathrm{N}\_{l\_3} = \left\{ E\_{l\_3}(\mathbf{s}\_1), E\_{l\_3}(\mathbf{s}\_2), \dots, E\_{l\_3}(\mathbf{s}\_q) \right\} \text{ be a HLINN set, then } d(\mathrm{N}\_{l\_1}, \mathrm{N}\_{l\_2}) \le d(\mathrm{N}\_{l\_1}, \mathrm{N}\_{l\_3}) \text{ and} \\ & d(\mathrm{N}\_{l\_2}, \mathrm{N}\_{l\_3}) \le d(\mathrm{N}\_{l\_1}, \mathrm{N}\_{l\_3}) \text{ if } \mathrm{N}\_{l\_1} \subseteq \mathrm{N}\_{l\_2} \subseteq \mathrm{N}\_{l\_3}. \end{array}$$

**Proof.** It is obvious that the properties (HP1)–(HP3) are satisfied for *d*(*Nl*1 , *Nl*2 ). Thus, we only need to prove the property (HP4).

Since there is *Nl*1 ⊆ *Nl*2 ⊆ *Nl*3 , there exists *E*0*l*1 (*sj*) ≤ *E*0*l*2 (*sj*) ≤ *E*0*l*3 (*sj*) for *sj* ∈ *S (j = 1*, *2*, *...*, *q)*, which implies *Tσ*(*k*) 3*j* ≥ *Tσ*(*k*) 2*j* ≥ *Tσ*(*k*) 1*j* , *Uσ*(*k*) 3*j* ≤ *Uσ*(*k*) 2*j* ≤ *Uσ*(*k*) 1*j* , *Fσ*(*k*) 3*j* ≤ *Fσ*(*k*) 2*j* ≤ *Fσ*(*k*) 1*j* for *k* = 1, 2, ..., *cj*. It follows that

$$\begin{vmatrix} T\_{1j}^{\sigma(k)} - T\_{2j}^{\sigma(k)} \Big|\_{\rho} \leq \left| T\_{1j}^{\sigma(k)} - T\_{3j}^{\sigma(k)} \right|\_{\rho}^{\rho} & \left| T\_{2j}^{\sigma(k)} - T\_{3j}^{\sigma(k)} \right|\_{\rho} \leq \left| T\_{1j}^{\sigma(k)} - T\_{3j}^{\sigma(k)} \right|\_{\rho} \\ \left| \mathcal{U}\_{1j}^{\sigma(k)} - \mathcal{U}\_{2j}^{\sigma(k)} \right|\_{\rho} \leq \left| \mathcal{U}\_{1j}^{\sigma(k)} - \mathcal{U}\_{3j}^{\sigma(k)} \right|\_{\rho}^{\rho} & \left| \mathcal{U}\_{2j}^{\sigma(k)} - \mathcal{U}\_{3j}^{\sigma(k)} \right|\_{\rho} \leq \left| \mathcal{U}\_{1j}^{\sigma(k)} - \mathcal{U}\_{3j}^{\sigma(k)} \right|\_{\rho} \rho \\ \left| \mathcal{F}\_{1j}^{\sigma(k)} - \mathcal{F}\_{2j}^{\sigma(k)} \right|\_{\rho} \leq \left| F\_{1j}^{\sigma(k)} - F\_{3j}^{\sigma(k)} \right|\_{\rho}^{\rho} & \left| F\_{2j}^{\sigma(k)} - F\_{3j}^{\sigma(k)} \right|\_{\rho} \leq \left| F\_{1j}^{\sigma(k)} - F\_{3j}^{\sigma(k)} \right|\_{\rho} \end{vmatrix}$$

Then there are the following inequalities:

HHH*Tσ*(*k*) 1*j* − *Tσ*(*k*) 2*j* HHH*ρ*+HHH*Uσ*(*k*) 1*j* − *Uσ*(*k*) 2*j* HHH*ρ*+HHH*Fσ*(*k*) 1*j* − *Fσ*(*k*) 2*j* HHH*ρ* ≤HHH*Tσ*(*k*) 1*j* − *Tσ*(*k*) 3*j* HHH*ρ*+HHH*Uσ*(*k*) 1*j* − *Uσ*(*k*) 3*j* HHH*ρ*+HHH*Fσ*(*k*) 1*j* − *Fσ*(*k*) 3*j* HHH*ρ*, HHH*Tσ*(*k*) 2*j* − *Tσ*(*k*) 3*j* HHH*ρ*+HHH*Uσ*(*k*) 2*j* − *Uσ*(*k*) 3*j* HHH*ρ*+HHH*Fσ*(*k*) 2*j* − *Fσ*(*k*) 3*j* HHH*ρ* ≤HHH*Tσ*(*k*) 1*j* − *Tσ*(*k*) 3*j* HHH*ρ*+HHH*Uσ*(*k*) 1*j* − *Uσ*(*k*) 3*j* HHH*ρ*+HHH*Fσ*(*k*) 1*j* − *Fσ*(*k*) 3*j* HHH*ρ*.

Thus, the following relations can be further obtained:

1 3*cjτρ* M *cj* ∑*k*=1 (|(*Tσ*(*k*) 1*j* ) − (*Tσ*(*k*) 2*j* )|*ρ* + |(*Uσ*(*k*) 1*j* ) − (*Uσ*(*k*) 2*j* )|*ρ* + |(*Fσ*(*k*) 1*j* ) − (*Fσ*(*k*) 2*j* )|*ρ*)N ≤ 1 3*cjτρ* M *cj* ∑*k*=1 (|(*Tσ*(*k*) 1*j* ) − (*Tσ*(*k*) 3*j* )|*ρ* + |(*Uσ*(*k*) 1*j* ) − (*Uσ*(*k*) 3*j* )|*ρ* + |(*Fσ*(*k*) 1*j* ) − (*Fσ*(*k*) 3*j* )|*ρ*)N, 1 3*cjτρ* M *cj* ∑*k*=1 (|(*Tσ*(*k*) 2*j* ) − (*Tσ*(*k*) 3*j* )|*ρ* + |(*Uσ*(*k*) 2*j* ) − (*Uσ*(*k*) 3*j* )|*ρ* + |(*Fσ*(*k*) 2*j* ) − (*Fσ*(*k*) 3*j* )|*ρ*)N ≤ 1 3*cjτρ* M *cj* ∑*k*=1 (|(*Tσ*(*k*) 1*j* ) − (*Tσ*(*k*) 3*j* )|*ρ* + |(*Uσ*(*k*) 1*j* ) − (*Uσ*(*k*) 3*j* )|*ρ* + |(*Fσ*(*k*) 1*j* ) − (*Fσ*(*k*) 3*j* )|*ρ*)N.

By Equation (4), there are *d*(*Nl*1 , *Nl*2 ) ≤ *d*(*Nl*1 , *Nl*3 ) and *d*(*Nl*2 , *Nl*3 ) ≤ *d*(*Nl*1 , *Nl*3 ) for *ρ* > 0. Therefore, the property (HP4) can hold. -

If we consider the weight *wj* of an element *sj* ∈ *S* with *wj* ∈ *[0*, *1]* and ∑*qj*=1 *wj* = 1, the generalized weighted distance between *Nl*1and *Nl*2 is

$$\begin{split} d\_{\boldsymbol{w}}(\boldsymbol{N\_{l}},\boldsymbol{N\_{l\_{2}}}) &= \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{\mathcal{X}\_{\boldsymbol{l}\_{1}}} \sum\_{k=1}^{c\_{l}} \left( |f(\boldsymbol{h}\_{\boldsymbol{T\_{1}^{q(k)}\_{1}})} - f(\boldsymbol{h}\_{\boldsymbol{T\_{2}^{q(k)}\_{1}}})|^{p} + |f(\boldsymbol{h}\_{\boldsymbol{L}^{q(k)}\_{1}}) - f(\boldsymbol{h}\_{\boldsymbol{L}^{q(k)}\_{2}})|^{p} + |f(\boldsymbol{h}\_{\boldsymbol{T\_{2}^{q(k)}\_{1}}}) - f(\boldsymbol{h}\_{\boldsymbol{T\_{2}^{q(k)}\_{2}}})|^{p} \right) \right] \right\}^{1/p} \\ &= \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{\mathcal{X}\_{\boldsymbol{l}^{p}}^{q\top}} \sum\_{k=1}^{c\_{l}} \left( |T^{q(k)}\_{1\_{l}^{p}} - T^{q(k)}\_{2\_{l}^{p}}|^{p} + |\mathcal{U}^{r(k)}\_{1\_{l}^{p}} - \mathcal{U}^{r(k)}\_{2\_{l}^{p}}|^{p} + |F^{r(k)}\_{1\_{l}^{p}} - F^{r(k)}\_{2\_{l}^{p}}|^{p} \right) \right] \right\}^{1/p} \text{ for } \rho > 0. \end{split} \tag{5}$$

Since the measures of similarity and distance are complementary with each other, the weighted measure of similarity between *Nl*1 and *Nl*2 can be represented by

$$\begin{split} S\_{\mathbb{W}}(N\_{l\_{1}}, N\_{l\_{2}}) &= 1 - d\_{\mathbb{W}}(N\_{l\_{1}}, N\_{l\_{2}}) \\ &= 1 - \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{\mathcal{K}\_{l}^{\mathbb{W}}} \sum\_{k=1}^{\ell\_{l}} \left( \left| T\_{1\_{l}^{\mathbb{W}}}^{\sigma(k)} - T\_{2\_{l}^{\mathbb{W}}}^{\sigma(k)} \right|\_{\rho} + \left| L\_{1\_{l}^{\mathbb{W}}}^{\sigma(k)} - L\_{2\_{l}^{\mathbb{W}}}^{\sigma(k)} \right|\_{\rho} + \left| F\_{1\_{l}^{\mathbb{W}}}^{\sigma(k)} - F\_{2\_{l}^{\mathbb{W}}}^{\sigma(k)} \right|^{\rho} \right) \right\}^{1/\rho} for \,\rho > 0. \end{split} \tag{6}$$

Similar to the properties (HP1)–(HP4) satisfied by the generalized distance measure in Proposition 1, the similarity measure *Sw*(*Nl*1, *Nl*2) also has the proposition as follows:

**Proposition 2.** *The similarity measure Sw*(*Nl*1 , *Nl*2) *for ρ > 0 contains the following properties:*

*(HP1)* 0 ≤ *Sw*(*Nl*1, *Nl*2) ≤ 1*;*

 *(HP2) Sw*(*Nl*1, *Nl*2) = 1 *if and only if Nl*1= *Nl*2*;*

 *(HP3) Sw*(*Nl*1, *Nl*2 ) = *Sw*(*Nl*2, *Nl*1)*;*

*(HP4) Let Nl*3 *be a HLNN set*, *then there are Sw*(*Nl*1 , *Nl*2 ) ≥ *Sw*(*Nl*1 , *Nl*3 ) *and Sw*(*Nl*2 , *Nl*3 ) ≥ *Sw*(*Nl*1 , *Nl*3 ) *if Nl*1 ⊆ *Nl*2 ⊆ *Nl*3 .

**Proof.** It is clear that *Sw*(*Nl*1 , *Nl*2 ) satisfies the properties (SP1)–(SP3). Thus, we only prove the property (SP4) here.

According to the proved property (HP4) in Proposition 1, if *Nl*1 ⊆ *Nl*2 ⊆ *Nl*3 , there exists the relations of *dw*(*Nl*1 , *Nl*2 ) ≤ *dw*(*Nl*1 , *Nl*3 ) and *dw*(*Nl*2 , *Nl*3 ) ≤ *dw*(*Nl*1 , *Nl*3 ) for *ρ* > 0. Since the similarity measure is the complement of the distance measure, both *Sw*(*Nl*1 , *Nl*2 ) ≥ *Sw*(*Nl*1 , *Nl*3 ) and *Sw*(*Nl*2 , *Nl*3 ) ≥ *Sw*(*Nl*1 , *Nl*3 ) can be easily obtained. Therefore, the property (SP4) can hold. -

#### **5. MADM Method Using the Similarity Measure of HLNNs**

For a MADM problem in the HLNN setting, some DMs need to evaluate *p* alternatives (denoted by *G =* {*g*1, *g*2, ... , *gp*}) over *q* attributes (denoted by *S =* {*s*1, *s*2, ... , *sq*}) from the LT set *H =* {*h*0, *h*1, ... , *hτ}*. Then, a weight vector *W =* (*ω*1, *ω*2, ... , *ωq*), which is on the conditions of 0 ≤ *ωj* ≤ 1 (*j* = 1, 2, ..., *q*) and ∑*qj*=1 *ωj* = 1, represents the importance of the attributes in *S*. Thus, the HLNN decision matrix *M* can be expressed as:

$$\mathcal{M} = \left(E\_{l\_1}(s\_{\bar{\jmath}})\right)\_{p \times q} = \begin{array}{c} \mathcal{G}\_1 \\ \mathcal{G}\_2 \\ \vdots \\ \mathcal{G}\_p \end{array} \left[ \begin{array}{c} E\_{l\_1}(s\_1) \\ E\_{l\_2}(s\_1) \\ \vdots \\ E\_{l\_p}(s\_2) \end{array} \left( \begin{array}{c} \cdots & E\_{l\_1}(x\_q) \\ E\_{l\_2}(s\_2) & \cdots & E\_{l\_2}(s\_q) \\ \vdots & \vdots & \ddots & \vdots \\ E\_{l\_p}(s\_1) & E\_{l\_p}(s\_2) & \cdots & E\_{l\_p}(s\_q) \end{array} \right) \right]$$

where *Eli*(*sj*) = {< *hT*1*ij* , *hU*1*ij* , *hF*1*ij* >, < *hT*2*ij* , *hU*2*ij* , *hF*2*ij* >, ··· , < *hTmij ij* , *hUmij ij* , *hFmij ij* >} is a HLNN for *sj* ∈ *S*, and *mij*is the number of LNNs in *Eli*(*sj*) (*i* = 1, 2, . . . , *p* and *j* = 1, 2, . . . , *q*).

On the basis of the proposed similarity measure, a novel MADM method of HLNN is presented by the following steps:

**Step 1**: For any HLNN *Eli*(*sj*) (*j* = 1, 2, ... , *q*) in *M*, rank all elements *ϑσ*(*k*) *ij (k =* 1, 2, ... , *mij*) in each HLNN *Eli*(*sj*) (*j* = 1, 2, ... , *q*) in an ascending order according to their score and accuracy functions, then yield the corresponding extended HLNN *Eoli*(*sj*) based on the LCMC *cj* and the occurrence number *Rij* of every LNN in *Eli*(*sj*) obtained by Equation (3). Hence, the extended decision matrix *M*◦ is

$$\mathcal{M}^o = \left( E\_{l\_i}^o(s\_j) \right)\_{p \times q} = \begin{array}{c} \mathcal{G}1 \\ \mathcal{G}2 \\ \vdots \\ \mathcal{G}p \end{array} \left[ \begin{array}{c} E\_{l\_1}^o(s\_1) & E\_{l\_1}^o(s\_2) & \cdots & E\_{l\_1}^o(s\_q) \\ E\_{l\_2}^o(s\_1) & E\_{l\_2}^o(s\_2) & \cdots & E\_{l\_2}^o(s\_q) \\ \vdots & \vdots & \ddots & \vdots \\ E\_{l\_p}^o(s\_1) & E\_{l\_p}^o(s\_2) & \cdots & E\_{l\_p}^o(s\_q) \end{array} \right] \end{array}$$

where *Eoli*(*sj*) = {*ϑσ*(1) *ij* , *ϑσ*(2) *ij* , ··· , *ϑσ*(*cj*) *ij* } (*i* = 1, 2, ... , *p* and *j* = 1, 2, ... , *q*) satisfies *ϑσ*(*k*) *ij* ≤ *ϑσ*(*k*+1) *ij* (*k* = 1, 2, . . . , *cj*).

**Step 2**: Specify an ideal HLNN set as *g*∗ = {*Eol* (*s*1), *Eol* (*s*2), ... , *Eol* (*sq*)} = {{*ϑσ*(1) 1 , *ϑσ*(2) 1 , ... , *ϑσ*(*c*1) 1 }, {*ϑσ*(1) 2 , *ϑσ*(2) 2 , ... , *ϑσ*(*c*2) 2 }, ... , {*ϑσ*(1) *q* , *ϑσ*(2) *q* , ... , *ϑσ*(*cq*) *q* }} for all *ϑσ*(*k*) *j* =< *hτ*, *h*0, *h*0 > (*k* = 1, 2, ..., *cj* and *j* = 1, 2, ..., *q*).

Hence, the similarity measure between *gi* (*i* = 1, 2, . . . , *p*) and *g\** can be calculated by

$$\begin{split} S\_{\mathbb{W}}(g\_{i},\boldsymbol{\varrho}^{\*}) &= 1 - d\_{\mathbb{W}}(\boldsymbol{\varrho}\_{i},\boldsymbol{\varrho}^{\*}) \\ &= 1 - \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{2\mathcal{K}\_{j}} \sum\_{k=1}^{c\_{j}} \left( |f(h\_{T\_{\boldsymbol{a}\_{\parallel}^{\*}}}) - f(h\_{\Gamma})|^{\rho} + |f(h\_{U\_{\boldsymbol{a}\_{\parallel}^{\*}}}) - f(h\_{0})|^{\rho} + |f(h\_{T\_{\boldsymbol{a}\_{\parallel}^{\*}}}) - f(h\_{0})|^{\rho} \right) \right] \right\}^{1/\rho} \\ &= 1 - \left\{ \sum\_{j=1}^{q} w\_{j} \left[ \frac{1}{2\zeta\_{j}v^{\nu}} \sum\_{k=1}^{c\_{j}} \left( |T\_{\boldsymbol{a}\_{\parallel}^{\*}}^{\boldsymbol{\sigma}^{\boldsymbol{1}}}| - \mathbf{r}|^{\rho} + |L\_{\boldsymbol{a}\_{\parallel}^{\*}}^{\boldsymbol{\sigma}^{\boldsymbol{1}}}|^{\rho} + |F\_{\boldsymbol{a}\_{\parallel}^{\*}}^{\boldsymbol{\sigma}^{\boldsymbol{1}}}|^{\rho} \right) \right] \right\}^{1/\rho} for \; \rho > 0. \end{split} \tag{7}$$

**Step 3:** According to the similarity measure results, rank the alternatives in *G =* {*g*1, *g*2, ... , *gm*} in a descending order and choose the best one.


*2.3. Transcriptome De Novo Assembly, Gene Functional Annotation, and Differentially Expressed Genes (DEGs) Analyses*

The raw reads of transcriptome sequencing were filtered using SOAPnuke v.15.2 (BIG, Shenzhen, China), and a set of clean reads was obtained. After de novo assembling, the clean reads were mapped to the pepper reference genome [36]. Genes were functionally annotated based on the NCBI non-redundant (Nr) [37], Gene Ontology (GO) [38], and the Kyoto Encyclopedia of Genes and Genomes (KEGG) databases [39]. For gene expression analysis, the numbers of matched reads were calculated and then normalized to RPKM by RSEM v.1.2.12. Significant differential expression genes (DEGs) were identified as those with a fold-change ≥ 2.0 and an FDR ≤ 0.001. DEGs were clustered using GO-Term Finder software, and pathway enrichment analysis was performed based on terms from the KEGG database.


**3. The measurement technology and characterization method for stress among the APS-prepared TBCs**

The stress plays an important role in the failure of the APS-prepared TBCs. It is important to test and characterize the stress. Up to now, various testing techniques and characterization methods for stress among the APS-prepared TBCs have been developed. According to the working principles, the methods can be divided into physical and mechanical testing methods.

#### **3.1 Physical testing methods**

In the physical testing methods, the information obtained from the interaction (including diffraction, scattering, etc.) between electromagnetic or particle radiation and the material is used to evaluate the stress distribution in the material. The common physical testing methods mainly include X-ray diffraction, neutron diffraction, Raman spectroscopy, photoluminescence piezospectroscopy and so on.

#### *3.1.1 X-ray diffraction (XRD)*

The stress in the coating may cause the crystal plane spacing to change. When X-ray are incident on the coating at different angles, the diffraction angle changes with the incidence angle, which results from the change in the crystal plane spacing. And the change value is related to the stress [32]. In XRD method, X-rays are incident on the surface of the coating at different angles several times, and the changes in the diffraction angle value 2*θ* are measured, as shown in **Figure 5** [18]. The stress σ can be obtained by calculating the slope of 2θ versus sin2ψ, as follows:

$$
\sigma = K \frac{d(2\theta)}{d(\sin^2 \psi)}\tag{3}
$$

*Stress among the APS-Prepared TBCs: Testing and Analysis DOI: http://dx.doi.org/10.5772/intechopen.85789*

**Figure 5.** *The schematic diagram for measuring stress by XRD.*

where, *ψ* is the angle between the normal direction of the diffraction crystal plane and coating surface. *K* are elastic constants.

XRD, as a fast and reliable non-destructive testing technology, has no specific requirements on sample size and shape and is suitable for micro-area stress measurement. It has been widely used in the testing and characterization of stress among TBCs. XRD has been used to study the stress evolution in the ceramic top coat and bond coat after operating at different conditions [33–35]. Meanwhile, Xiao et al. studied the stress evolution of the bond coat during oxidation at 1150°C for different time based on XRD. It was found that the stress in the bond coat mainly generated during the process of cooling from 1150 to 600°C. This may be due to that the *β* phase deposition during cooling changes the volume of the bond coat [34, 35].

Although XRD has been widely used to test stress among TBCs, the penetration depth of X-ray generated by laboratory sources is limited to only a few tens of microns. And the thickness of the coating ranges from 300 to 500 μm. Thus, XRD can only measure the stress state of the near surface zone and not characterize the stress state deep inside TBCs [36].

In order to overcome the problem for small penetration depth of X-ray generated by laboratory sources, synchrotron radiation XRD has been developed. The X-ray generated by synchrotron radiation source is used to test the stress deep inside the material. Comparing with the X-ray generated by the laboratory source, the penetration ability is greatly enhanced [19]. The composition of the local phase and the stress deep inside the thermal spray coating were determined by synchrotron radiation XRD [36]. However, the synchrotron radiation device with the large size, high cost, limited machine time is difficult to promote in the laboratory. The application of synchrotron radiation XRD is limited.

#### *3.1.2 Neutron diffraction*

Neutrons with a strong penetrating ability (up to a few tens of millimeters), can penetrate through most materials. Neutron diffraction can be used to measure stress deep inside TBCs, whose principle is basically similar to XRD. The stress distribution in the depth direction of the coating was studied by neutron diffraction. The influence of various operation conditions on the stress distribution was discussed [37, 38]. Neutron diffraction is also suitable for measuring the stress in a large specimen and obtaining the average stress inside the specimen. However, during neutron diffraction testing, the position of the surface with respect to the neutron beam must be accurately known. The 'center of gravity' of a near-surface gauge measurement volume must be calculated. A correction may be required to allow for any pseudostrain effects. Besides, sufficient data are difficult to obtain in a limited time using a weak neutron source, and a special strong neutron source is generally required [38].

#### *3.1.3 Raman spectroscopy*

When the laser with a certain frequency is incident on the material, the molecules in the material absorb part of the energy, vibrate in different ways and degrees, and scatter light with a lower frequency. This phenomenon is called Raman scattering. After the incident photons collide with the molecules, the vibrational energy or the rotational energy of the molecules and the photon energy superimpose each other to form a Raman spectrum. When there is stress in the material, some stress-sensitive bands may move and deform relative to the stress-free state, and the position of the spectral peak also may move, as shown in **Figure 6** [39]. It is found that there is a linear relationship between Raman frequency shift and the stress. The frequency shift of the spectral peak can be expressed as:

$$
\Delta \alpha \rho = \Pi\_{\vec{\eta}} \sigma\_{\vec{\eta}} \tag{4}
$$

where, ∏*ij* is the piezospectroscopic tensor, which represents the stress sensitivity of Raman band. *σij* refers to the stress tensor. Δ*ω* = *ωs*-*ωo*, with *ωo* being the peak position of the stress-free state and *ωs* being the peak position of the stressed state. Then, the stress value can be obtained by measuring the positions of the Raman spectrum peak when the material is stressed or not.

Raman spectroscopy has been widely used to test and characterize stress among APS-prepared TBCs [40]. It is well known that stress in TBCs is inevitably generated during the preparation process. Therefore, it is difficult to obtain a TBCs specimen coated on the substrate in a stress-free state. Generally, a TBCs specimen without the substrate is chosen as a test piece for stress-free conditions. First, the laser with a certain frequency is incident on the TBCs specimen without the substrate to obtain a Raman spectrum (labeled as spectrum 1) corresponding to a stress-free state. Then, a known external stress is applied on the TBCs specimen without the substrate, and the measured Raman spectrum is labeled as spectrum 2. By comparing the spectrum 1 and 2, the piezospectroscopic tensor of TBCs can be obtained according to Eq. (4). Finally, the laser with the same frequency is incident on the TBCs specimen with the substrate, and Raman spectrum, which is labeled as spectrum 3, is obtained. By comparing the spectrum 1 and 3, the stress state in TBCs with the substrate can be evaluated combining the piezospectroscopic tensor obtained above [39, 40].

**Figure 6.** *The schematic diagram for measuring stress by Raman spectroscopy.*

*Stress among the APS-Prepared TBCs: Testing and Analysis DOI: http://dx.doi.org/10.5772/intechopen.85789*

TBCs is generally constructed of porous structures for thermal insulation. When the laser is incident on the porous region, the signal intensity will decrease rapidly due to the optical focus limitation of the instrument. In order to reduce the occurrence of the above phenomenon, the incident position of the laser is set in the dense region to obtain the maximum signal intensity, thereby reducing the influence of voids and cracks [40]. Raman spectroscopy, as a non-destructive, non-contact stress testing method, has many advantages such as high spatial resolution, large spectral range, and so on. Besides, it can measure the stress in the depth direction by adjusting the focusing parameters. However, Raman spectral peak shift is susceptible to external factors such as focus depth, laser heating effect, temperature stability, and so on. And high precision is difficult to achieve without effective calibrations. Besides, as the spot of the laser is generally small to obtain high intensity, it is impossible to measure the stress in a large area. Only local stress information can be obtained.

#### *3.1.4 Photoluminescence piezospectroscopy*

Since the TGO (mainly composed of α-Al2O3) usually contains a trace amount of Cr3+, the ion may generate fluorescence under laser excitation, and its characteristic fluorescence spectrum is bimodal *R*1 and *R*2. Similar to the Raman spectroscopy, stress can also lead to changes in the peak frequencies of *R*1 and *R*2 spectrum. The stress value in the TGO can be obtained by measuring the change in the peak frequency of the fluorescence spectrum under stress. The above method is called as photoluminescence piezospectroscopy [41, 42].

Up to now, photoluminescence piezospectroscopy is mainly applied to the EB-PVD prepared TBCs. The laser easily penetrates the columnar crystal structure, and the reflected signals are strong. However, the scattering of pores and grain boundaries in the APS-prepared TBCs may weaken reflection signals and make it difficult to perform spectral analysis.

### **3.2 Mechanical testing methods**

In the mechanical testing methods, the mechanical information such as displacement and strain of the coating system under certain operating or external excitation conditions is measured. Based on the theoretical model, the stress distribution in the coating can be evaluated. The common mechanical testing methods include curvature measurement, material removal, indentation and so on.

### *3.2.1 Curvature measurement method*

During the preparation process or service, the deformation of the coating is limited by the substrate, and an interaction force is generated between the substrate and the coating. Meanwhile, a bending moment appears to bend the overall structure, and curvature occurs to balance the stress in TBCs. The curvature of the substrate is measured by optical or mechanical methods. By analyzing the change in the curvature of the substrate, the stress evolution of TBCs can be derived. The curvature measurement method was first proposed by Stoney [43] and later applied to APS-prepared coatings. Kuroda et al. [44] developed an in-situ curvature monitoring method. In this method, the stress evolution of TBCs during preheating, spraying and cooling process were monitored. The quenching stress during the spraying process and thermal mismatch stress during the cooling process were obtained, as shown in **Figure 7**. In the spraying process, the thickness of the coating gradually increases, and the curvature of the substrate also changes continuously. According to the equilibrium conditions, the quenching stress can be calculated.

**Figure 7.** *The schematic diagram for measuring stress by curvature measurement method.*

In the cooling process, the thermal mismatch strain between the coating and substrate changes the curvature of the substrate. Then, the thermal mismatch stress can be estimated. Besides, the complicated service loads also lead to the change in the curvature of the substrate. The stress evolution in service can also be obtained by the curvature measurement method [20, 39, 45].

In the curvature measurement method, the substrate deformation of the single point or multiple points is measured by the displacement measurement method. According to the geometric relationship, the curvature is calculated. The stress value can also be obtained based on the equilibrium equations. There are two key problems in the curvature measurement method: one is the accurate measurement of deformation, and the other is the accurate calculation of stress.

Many methods have been developed to measure the deformation, mainly including contact measurement and non-contact measurement method. In the contact measurement method, the deformation at specific point is measured by the highprecision contact displacement sensors, such as extensometers, linear variable differential transformer and so on [46, 47]. In the non-contact measurement method, the deformation of the substrate is measured by a non-contact displacement measuring device, such as laser displacement sensor, optical microscope, CCD camera, and so on [48, 49]. For example, during the measurement of deformation by a laser displacement sensor. The laser was incident on the surface of the curved substrate. And deformations at regular intervals were measured. Since the deformation (a few microns) was much smaller than the length of the specimen (a few centimeters), the contour of the substrate was approximated as a parabola or arc. The deformation values were fitted to a parabolic or circular equation according to the least squares method. The curvature of the specimen was calculated based on the above equations. This method has higher accuracy compared to simply measuring the deformation of the middle or end position of the specimen [48].

Once the curvature of the substrate is determined, the stress distribution in the coating can be determined according to the equilibrium equation. The relationship between the curvature and stress of film established by Stoney has been widely used in the stress measurement of film system [43]. However, for thicker coating and multilayer film system, the application of the above relationship is controversial. Clyne

*Stress among the APS-Prepared TBCs: Testing and Analysis DOI: http://dx.doi.org/10.5772/intechopen.85789*

et al. proposed a two-layer beam (or plate) bending model [14]. The relationship between the curvature change Δ*κ* and the coating average stress *σr* can be given by:

\*\*20-13\*\*\newline\*\*Beam (or plate)\newline\newline\newline\newline\newline\newline\new
\text{Future change }\Delta\kappa \text{ and the coating average stress }\sigma\_r \text{ can be given by:}\newline\newline\newline\newline\newline\newline\new
\text{For }\sigma\_r = \Delta\kappa\frac{E\_\epsilon^2 h\_\epsilon^4 + 4E\_\epsilon E\_i h\_\epsilon^3 h\_i + 6E\_\epsilon E\_i h\_\epsilon^2 h\_i^2 + 4E\_\epsilon E\_i h\_\epsilon h\_i^3 + E\_i^2 h\_i^4}{6\,h\_\epsilon(1-\nu)\,E\_i h\_i \{h\_\epsilon + h\_i\}}\tag{5}

where, *Ec* and *Es* are the modulus of the coating and substrate, respectively. *hc* and *hs* are the thickness of the coating and substrate, respectively. *v* is the Poisson's ratio of the coating. During the preparation process and service, the material properties change with temperature. Clyne et al. considered the influence of temperature on the material properties and obtained more accurate stress values. In order to measure the curvature accurately, test specimens with regular shape such as beam shape and plate shape are chosen in the curvature measurement method. The stress measurement of the test specimens with irregular shape was limited. Besides, the average stress of the coating is obtained from curvature measurement results. More stress information inside the coating is difficult to get.

#### *3.2.2 Material removal method*

In the material removal method, the stress state is obtained by monitoring the strain change at a specific position before and after removing the material. The common measurement methods for APS-prepared TBCs include hole-drilling and layer-removal method.

#### *<IP_ADDRESS> Hole-drilling method*

In the hole-drilling method, some of the stress-bearing material is removed by drilling holes in the surface of the material. The stress in the remaining material will be redistributed, resulting in a change in the shape of the circular hole. The deformation of the remaining material near the circular hole is measured by strain gauge rosette. The strain distribution can be calculated according to analytical models, and stress distribution can also be estimated [50]. In order to accurately measure the strain change caused by drilling, a specially designed strain gauge rosette is arranged around the drilling point, and the center of the strain gauge rosette should coincide with the drilling point. By analyzing the stress state at the circular hole, the strain measured by different strain gauges can be expressed as:

$$\varepsilon\_{i} = A \left( \sigma\_{\max} + \sigma\_{\min} \right) + B \left( \sigma\_{\max} - \sigma\_{\min} \right) \cos(\mathcal{Z}a\_{i}) \tag{6}$$

where, *A* and *B* are calibration coefficients, which are related to parameters such as elastic modulus, Poisson's ratio, and radius of the hole. *σmax* and *σmin* are maximum principal and minimum principal stress, respectively. *αi* is the angle between the axis of strain gauge and the principal stress axis. By measuring the strains in different directions around the circular hole, the maximum principal and the minimum principal stress at the circular hole can be calculated by Eq. (6). Then the stress state at the circular hole can be estimated. It is worth noting that only when the hole is a through hole and the stress distribution is uniform, *A* and *B* have analytical solutions. There are only numerical solutions in the case where the hole is a blind hole or the stress distribution is non-uniform.

As the stress distribution among the APS-prepared TBCs is not uniform, it is recommended to gradually measure the stress distribution in the thickness direction with the incremental hole drilling depth. Then, the drilling method can be developed to the incremental hole drilling method. In this method, the drilling process

is performed in the form of a small step at a time, and the strain changes caused by each step can be recorded by the strain gauges. It is found that after the first drilling step the measured strain changes were affected by both the current and previous small drilling steps [17, 51, 52]. Meanwhile, the calibration coefficients A and B are also related to the drilling depth increment.

Many theoretical studies have been conducted on the incremental hole drilling method. Among them, the integral method can accurately obtain the stress distribution at each drilling step, accounting for the influence of the current and previous small drilling steps. Thus, the integration method is particularly effective for non-uniform stress distribution, especially when the stress changes abruptly with thickness. The stress gradient in the thickness direction can be calculated [53, 54]. However, it is not easy to analyze strain data by the integral method. Numerical methods are usually required. A general method for calculating calibration coefficients based on the finite element method was developed, and the calibration coefficients were obtained accurately [55].

It is worth noting that the above theories are mostly based on the linear elastic model. But the coating may undergo plastic deformation or cracking damage during the drilling process. The stress value calculated by the above theories may deviate from the true value.

#### *<IP_ADDRESS> Layer-removal method*

Similar to the hole-drilling method, the layer-removal method also measures the strain change caused by the material reduction to analyze the stress state of the material. But the layer-removal method mainly removes the coating material of a specific thickness by mechanical methods such as grinding and polishing or chemical methods such as corrosion. The physical or mechanical testing methods are used to measure the strain change at a specific position of the remaining material. Then the stress state in the coating can be evaluated [56, 57].

Similarly, if plastic deformation or damage occurs during the removal of the coating material, the linear elastic assumption in the theoretical analysis will no longer be suitable. Besides, it is difficult to remove the same thickness of material each time, especially when the specimen has deformed before the removal of material.

#### *3.2.3 Indentation method*

The indentation method, as a common micro-scale mechanical testing method, has been widely used to study the stress in the material. The indentation method mainly includes the nanoindentation and indentation fracture method.

#### *<IP_ADDRESS> Nanoindentation method*

It is found that the stress in the materials may lead to the change in the loaddisplacement curve obtained from the nanoindentation tests. In the indentation method, the tensile stress in the material increases the indentation depth (when the indentation load is chosen) or decreases the indentation load (when the indentation depth is chosen). The compressive stress in the material decreases the indentation depth (when the indentation load is chosen) or increases the indentation load (when the indentation depth is chosen), as shown in **Figure 8**. Then, an indentation method was proposed by Suresh et al. to measure the stress in the materials [58].

Assuming that the stress in the material is an equal-biaxial state, the relationship between the contact area *A0* in a stress-free state and the contact area *A* under the stress *σR* can be obtained based on the superposition principle, as follows:

*Stress among the APS-Prepared TBCs: Testing and Analysis DOI: http://dx.doi.org/10.5772/intechopen.85789*

$$\frac{A}{A\_0} = \left(\mathbf{1} - \frac{\sigma^{\mathcal{R}}\sin\alpha}{H}\right)^{-1} \tag{7}$$

where, *H* is the indentation hardness. *α* is the angle between the cone surface of the equivalent conical indenter and the surface of the material (for the Vickers indenter, *α* = 22°; for the Berkovich indenter, *α* = 24.7°).

In the nanoindentation method, indentation tests are conducted on the coating material in the stress-free and stress state, respectively. After indentation load is chosen, the contact areas in the loading stage under the stress-free and stress conditions can be obtained from the load-displacement curves. Then the stress in the material can be evaluated according to Eq. (7) [9, 59, 60].

Two kinds of coating materials corresponding to a stress-free and stress state are required in this method. As mentioned earlier, it is difficult to obtain a TBCs specimen coated on the substrate in a stress-free state. Besides, the nanoindentation method usually has high requirements on the surface quality of the material. But the APS-prepared TBCs is mostly porous, which limits the application of the nanoindentation method.

#### *<IP_ADDRESS> Indentation fracture method*

In addition to the elastic-plastic behavior of materials, the indentation method can also be used to study the fracture behavior. When the indentation load is large enough, the stress intensity factor will be larger than the fracture toughness of the material, and the fracture will occur, as shown in **Figure 9**. The fracture toughness of the material can be calculated based on the critical conditions. The stress intensity factor *Ks* in the stress-free state can be expressed as:

$$K\_s = \delta \left(\frac{E}{H}\right)^{1/2} \frac{P}{c^{3/2}} \tag{8}$$

where, *δ* is a geometric parameter, which is related with the shape of the indenter. *E* and *H* are the modulus and hardness of the material, respectively. *P* is the indentation load. *c* is the average length of the cracks induced by indentation.

**Figure 8.** *The schematic diagram for measuring stress by nanoindentation method.*

**Figure 9.** *The schematic of indentation fracture.*

The stress in the material may influence the critical load corresponding to the occurrence of facture. Tensile stress promotes the initiation and propagation of cracks. When the stress in the material is tensile, the critical load decreases. Compressive stress inhibits the initiation and propagation of cracks. When the stress in the material is compressive, the critical load increases. Assuming that the stress *σr* in the coating is uniform, the contribution of stress to the stress intensity factor *Kr* can be expressed as:

$$K\_r = \varphi \sigma\_r \sqrt{t} \left\{ 2 - \sqrt{t/c} \right\} \tag{9}$$

where, *t* is the thickness of the coating. ψ = 2/√π is crack geometric parameter.

Based on the superposition principle, combing Eq. (8) and (9), the total stress intensity factor *K* can be expressed as:

$$K = \delta \left(\frac{E}{H}\right)^{1/2} \frac{P}{c^{3/2}} + \mathcal{D}\varphi\sigma\_r t^{1/2} - \varphi\sigma\_r t/c^{1/2} \tag{10}$$

When the stress intensity factor is larger than the fracture toughness *KIC* of the material, the fracture will occur. According to the critical conditions, Eq. (10) can be rewritten as:

$$\frac{P}{c^{3/2}} = \frac{K\_{\rm IC} - 2\,\mu\sigma\_r t^{1/2}}{\mathcal{X}} + \left(\varphi\sigma\_r \mathbf{t}\right) c^{-1/2} \tag{11}$$

where <sup>χ</sup> <sup>=</sup> <sup>δ</sup>(E/H) 1/2 .

According to Eq. (11), there is linear relationship between *P*/*c* 3/2 and *c* <sup>−</sup>1/2. *ψσrt* and *KIC*-2*ψσrt* 1/2 represent the slope and intercept, respectively. Therefore, the indentation fracture tests under different indentation loads need to be conducted. The crack lengths under different indentation loads are measured. By linear fitting, the slope and intercept values can be obtained. Then the residual stress and fracture toughness of the material can be evaluated [61]. The indentation fracture method has been successfully used to measure and analyze the stress in ceramic top coat and at the interface between the ceramic top and the bond coat. [33, 62]. Besides, the indentation fracture method has also been extended to the high temperature environments. The stress evolution behavior at different service temperatures was studied [63].

As TBCs is a porous non-uniform structure, the process of inducing cracks by indentation is random. The cracks are susceptible to microstructures such as voids, and the data are discrete, which provides difficulties for subsequent data analysis. Generally, a large of experiential results is required.


**3. Contribution of AMF to plant productivity and agroecosystems**

Sustainability of agricultural ecosystems can be restored by stimulating soil life and internally regulated ecosystem processes [31]. The real significance of mycorrhizal fungi is that they connect the primary producers of ecosystems and enable the flow of energy-rich compounds required for nutrient mobilization [32]. Mycorrhizal fungi are one of the commonly occurring living organisms in soil providing many ecosystem services, including N-fixation, soil carbon cycling, plant nutrition, soil erosion control by soil binding capacity [13], soil pollutants remediation, biodiversity, plant water economy [42], and enhanced C-sequestration [22]. AMF and rhizobia can act synergistically and stimulate plant productivity by supplying different limiting nutrients to the plant (e.g., N by rhizobia and P by AM fungi) [43].

Enhanced nutrient uptake and stress resistance are some of the mechanisms by which AM fungi can enhance plant productivity. AMF symbiosis is probably more favorable in conservative and sustainable agriculture to having the potentiality of major beneficial functions such as: (1) increased productivity in the range of 16–78% by gaining more N, P, and other less mobile nutrients increased [24, 43]; (2) increased water uptake and water holding capacity that initiate drought tolerance; (3) increased tolerance to other abiotic stresses such as soil salinity, heavy metal toxicity, etc.; (4) overcoming biotic stresses and offering bio-protection against pathogen; (5) improved soil quality; (6) enhanced plant vigor and yield, thus leading to the production of safe and high-quality foods, able to promote human health (**Figure 2**) [16, 44]. This is determined by fungal the strain, climate, soil type, and cultural practices (fertilization level) in which the soil type is the

*Arbuscular Mycorrhizal Fungi (AMF) in Optimizing Nutrient Bioavailability and Reducing… DOI: http://dx.doi.org/10.5772/intechopen.106995*

**Figure 2.** *Schematic representation of brief function of arbuscular mycorrhizal fungi (AMF) (source: [44]).*

fundamental criterion to determine effective AMF symbiosis strains or species [24]. It is also to improve soil fertility, as they produce glomalin upon accumulation in soil that aids soil in soil stabilization [42, 45]. Furthermore, AMF prevent leaching losses, phosphorus (60%), and ammonium (7.5%) from grassland microcosms during periods of heavy rainfall that cause top environmental threats to ecosystems worldwide [46].

AMF fungi receive 100% of their carbon from the plant, and this increase in carbon flow to the roots, estimated at up to 20% of the plant's photosynthate, translates to a huge amount of carbon worldwide that plays a significant role in carbon cycling between the atmosphere and biosphere [34]. Bender et al. [47] have demonstrated that AMF contribute to reducing emissions of N2O by increasing N immobilization into microbial or plant biomass, which results in the reduction of soluble N in the soil and, consequently, in a limitation of denitrification. Thus, AMF could have an indirect influence on potent greenhouse gas (GHG) emissions through change of the physical conditions of soil that influence the production and transport of GHG in soil [26].

The AMF symbiosis can reduce nutrient loss from ecosystems in three main ways: (1) by improving crop nutrient extraction capacity [43] allowing the production of good yield at lower levels of soil fertility; (2) by increasing soil aggregation via physical particle enmeshment and cementing with "sticky" exudates, which results in better soil nutrient storage and retention [48]; and (3) by promoting growth of host crops, thus increasing the size of this desirable nutrient sink [46]. On average, plants inoculated with *G. mosseae* yielded 13% more biomass relative to non-mycorrhizal plants [49]. Therefore, AMF are vital endosymbionts playing an effective role in plant productivity and the functioning of the ecosystem service.