A Non-Destructive System Based on Electrical

sensors Article

A Non-Destructive System Based on Electrical Tomography and Machine Learning to Analyze the Moisture of Buildings Tomasz Rymarczyk 1 1 2

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ID

, Grzegorz Kłosowski 2, *

ID

and Edward Kozłowski 2

University of Economics and Innovation in Lublin, Research & Development Centre Netrix S.A., 20-209 Lublin, Poland; [email protected] Faculty of Management, Lublin University of Technology, 20-618 Lublin, Poland; [email protected] Correspondence: [email protected]

Received: 15 May 2018; Accepted: 12 July 2018; Published: 14 July 2018







Abstract: This article presents the results of research on a new method of spatial analysis of walls and buildings moisture. Due to the fact that destructive methods are not suitable for historical buildings of great architectural significance, a non-destructive method based on electrical tomography has been adopted. A hybrid tomograph with special sensors was developed for the measurements. This device enables the acquisition of data, which are then reconstructed by appropriately developed methods enabling spatial analysis of wet buildings. Special electrodes that ensure good contact with the surface of porous building materials such as bricks and cement were introduced. During the research, a group of algorithms enabling supervised machine learning was analyzed. They have been used in the process of converting input electrical values into conductance depicted by the output image pixels. The conductance values of individual pixels of the output vector made it possible to obtain images of the interior of building walls as both flat intersections (2D) and spatial (3D) images. The presented group of algorithms has a high application value. The main advantages of the new methods are: high accuracy of imaging, low costs, high processing speed, ease of application to walls of various thickness and irregular surface. By comparing the results of tomographic reconstructions, the most efficient algorithms were identified. Keywords: inverse problem; electrical tomography; moisture inspection; dampness analysis; machine learning; nondestructive evaluation

1. Introduction This article presents the results of research on the development of an effective and non-invasive method for the detection of moisture walls and historical buildings. Humidity is one of the basic, and at the same time undesirable, physical characteristics of building materials [1]. Detection of water inside the walls of buildings and structures made of bricks or lightweight concrete and brick blocks is one of the most frequently performed tests. All buildings are exposed to various factors that erode the material of which they are made. Such factors include water [2]. Many buildings have damp walls and foundations [3]. This is evidenced by often appearing molds and fungi, dark spots, and detachment of plasters or paint coatings. This is especially factual for older buildings. One of many reasons for this is the technology and materials that were formerly used in construction—for example, lack of insulation. Moisture in the walls significantly reduces their durability. The bricks and mortar that contain water are significantly weakened and are less resistant to compression, which is particularly true for lime mortar. This results in both deteriorations of the building’s operational conditions and safety. In addition, the water accumulated

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in the walls significantly worsens the thermal insulation properties of the walls and contributes to their gradual erosion. Another important problem related to moisture in walls is their harmful impact on inhabitant’s health. Microclimate, which arises in rooms and walls with high humidity, often causes the formation of mold fungi that can cause respiratory diseases and intoxication. It can be argued that the main factor causing the destruction of the walls is just moisture. In combination with daily temperature changes, moisture has the greatest impact on the overall strength and durability of building structures. The water in the outer wall has a negative effect on the walls regardless of the state of aggregation, i.e., in solid form (ice, snow), as a liquid (rain) and gas (water vapor). Most structural defects, e.g., brick movements, cracking, molds, fungi, and chemical reaction, are initiated and compounded by the presence of moisture. In order to reduce the risks associated with excessive humidity, moisture evaluations are necessary for buildings [4]. Such tests can be helpful in determining the impact of rainwater and groundwater, leakage, and moisture from water supply and sewage systems as well as condensation of water vapor. Permeation of moisture in the walls of old buildings that are in direct contact with the soil due to the lack of a horizontal or vertical barrier separating the walls from water in the soil leads to migration of moisture (Figure 1). This leads to the migration of dissolved in water salt, which is responsible for many construction problems. Building materials, both natural and man-made (e.g., the brick or concrete), are porous. Moisture from bricks and masonries can be drawn by gravity using the capillary effect [5]. The condition of effective prevention of moisture in the building walls is its proper identification. There are many methods that can generally be divided into two groups—destructive and nondestructive methods. For obvious reasons, non-destructive methods are more desirable and have bigger applied value [6,7]. This feature is gaining importance when walls’ humidity needs to be measured in buildings of historical importance. Among the destructive methods can be distinguished, among others, the “drying-weight” or “carbide” method. Unfortunately, this is an invasive examination. The destructive methods consist in taking samples of the material being examined, which is not always possible, especially in the case of historic buildings. In such cases, these types of methods are not recommended, because they involve a violation of the structure of the examined object. Therefore, non-invasive tests may be a better solution in such situations. Non-destructive methods include, for example, thermovision or ultrasound methods. The disadvantage of thermovision is its exterior nature and the impossibility of penetrating under the surface of the investigated structure. The ultrasound approach is of little use due to the high porosity of materials containing cells (pores) filled with air or water. Non-destructive methods also include electrical impedance tomography (EIT), in which electrical measurements are made [8]. This method, thanks to the measuring device and the implemented algorithms, allows for non-invasive spatial determination of the degree of moisture. In the case of impedance tomography, it is a technique for imaging the spatial distribution of conductivity [9]. Previous studies show that electrical resistance can be used to measure the humidity of concrete and masonry walls [10–12]. There is a known relationship between moisture inside a porous building material and its electrical resistivity (Figure 2). Similar relationships can be observed for a brick wall and a lightweight concrete blocks wall. The electrical resistance increases as the moisture content decreases. It can be seen that the smallest change in resistance occurs in the highest range of moisture content. Although current techniques for measuring moisture in concrete and brick walls are accurate, the use of electrical resistance has the advantage of being a relatively simple procedure that can be used with inexpensive equipment [13].

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Figure Figure1. 1.The Therising risingmoisture moistureresulting resultingfrom fromthe thedirect directconnection connectionof ofsoil soilwith withmasonry masonry [14]. [14]. Figure 1. The rising moisture resulting from the direct connection of soil with masonry [14].

Figure2. 2.Relationship Relationshipbetween betweensaturation saturationand andresistivity resistivityof ofconcrete concrete[15]. [15]. Figure Figure 2. Relationship between saturation and resistivity of concrete [15].

Each Each measuring technique has its own own conditions, Thanks to Eachmeasuring measuringtechnique techniquehas hasits conditions, advantages, advantages, and and disadvantages. disadvantages. Thanks Thanksto to this it can be used only in special circumstances. Currently, the main problem in research on the this circumstances. Currently, this it can be used only in special circumstances. Currently, the the main main problem problem in in research research on on the the concentration concentration of moisture in the walls is the lack of method that provides spatial imaging of its concentrationof ofmoisture moisturein inthe thewalls wallsis isthe thelack lackof ofaaamethod methodthat thatprovides providesspatial spatialimaging imagingof ofits its distribution inside the wall without the need to take samples. Most of the available research methods distribution inside the wall without the need to take samples. Most of the available research methods distribution need to take samples. Most of the available research methods allow allow only spot evaluation of moisture, which makes possible to obtain only discrete model [4]. allowonly onlyaaaspot spotevaluation evaluationof ofmoisture, moisture,which whichmakes makesitititpossible possibleto toobtain obtainonly onlyaaadiscrete discretemodel model[4]. [4]. In In most cases, methods based based on on real real data dataare are invasive. invasive. Inmost mostcases, cases,methods This This fact is basic problem with regard to the analysis of thick walls because the moisture inside Thisfact factis isaaabasic basicproblem problemwith withregard regardto tothe theanalysis analysisof ofthick thickwalls wallsbecause becausethe themoisture moistureinside inside each wall is usually a few percent higher than at its surface [16]. The destructive nature each of currently each wall wall is is usually usually aa few few percent percent higher higher than than at at its its surface surface [16]. The The destructive destructive nature natureof ofcurrently currently used used techniques requiring samplecollection collectionis isunacceptable, unacceptable, especiallyin inhistorical historicalbuildings. buildings.In Insuch such usedtechniques techniquesrequiring requiringsample unacceptable,especially such cases, only non-invasive methods may be used. cases, cases, only only non-invasive methods may be used. The The humidity of the walls walls can measured with electric meters. meters. Electrical Electrical moisture Thehumidity humidityof of the can be be directly directly measured measured with electric electric meters. Electricalmoisture moisture meters, and in particular conductivity meters, are sensitive to very low amounts of moisture meters, and in particular conductivity meters, are sensitive to very low amounts of moisture and/or meters, and amounts of moisture and/or and/or some types of contaminants with soluble salt. For example, a free moisture content lower than some 0.1% some types types of of contaminants contaminants with with soluble soluble salt. salt. For For example, example, aa free free moisture moisture content content lower lower than than0.1% 0.1% can can cause high meter readings. Due to the influence of salinity causing change in electrical resistance cancause causehigh highmeter meterreadings. readings.Due Dueto tothe theinfluence influenceof ofsalinity salinitycausing causingaaachange changein inelectrical electricalresistance resistance and and small depth of measurement, evaluations made with the use of electric humidity meters should andaaasmall smalldepth depthof ofmeasurement, measurement,evaluations evaluationsmade madewith withthe theuse useof ofelectric electrichumidity humiditymeters metersshould should be considered as not very accurate [5,16]. They can give an approximate image of the dampness be inside be considered considered as not not very very accurate accurate [5,16]. [5,16]. They They can can give givean anapproximate approximate image image of of the thedampness dampnessinside inside the the walls (Figure 3). 3). thewalls walls(Figure

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Figure 3. An image of building walls’ dampness developed on the basis of the results of an electrical Figure 3. An image of building walls’ dampness developed on the basis of the results of an electrical test. Higher moisture concentrations are depicted by a color of higher intensity [17]. test. Higher moisture concentrations are depicted by a color of higher intensity [17].

The main purpose of this study is to present and compare non-destructive algorithms based on

The maintomography purpose ofthat thisallow studyestimation is to present and compare algorithms based on electrical of humidity not onlynon-destructive on the wall surface but also inside the masonry wall. that allow estimation of humidity not only on the wall surface but also inside electrical tomography With reference to the tomography of building walls, the most commonly used methods are the masonry wall. impedance tomography (EIT), capacitance tomography (ECT), resistive tomography All are With reference to the tomography of building walls, theand most commonly used(ERT). methods the above methods belong to the group of electrical tomography methods, including many impedance tomography (EIT), capacitance tomography (ECT), and resistive tomography (ERT). All the tomographic techniques showing the distribution of electrical parameters in the tested object [18–20], above methods belong to the group of electrical tomography methods, including many tomographic while the EIT shows the spatial distribution of conductivity γ [21,22]. Authors such as Holder or techniques showing the distribution of electrical parameters in the tested object [18–20], while the EIT Karhunen et al. [23,24] in the monograph give the general principles of the EIT, its instruments, showsprocedures, the spatial and distribution of conductivity γ [21,22]. Authors such as Holder or Karhunen et al. [23,24] challenges. in the monograph give the general of thethe EIT,building its instruments, procedures, and challenges. The proposed tomographic principles method in which materials humidity evaluation is an The proposed tomographic whichcharacteristic, the buildingsuch materials humidity evaluation indirect assessment based on amethod differentin physical as resistivity, allows for manyis an measurements approach in [25,26]) characteristic, without the needsuch to damage the tested object.for many indirect assessment(tomographic based on a different physical as resistivity, allows The tomographic approach allows the moisture inside the wall in aobject. digital measurements (tomographic approach inarchiving [25,26]) without the distribution need to damage tested form and comparing it with the next results in the future (moisture monitoring) when it is necessary. The tomographic approach allows archiving the moisture distribution inside the wall in a digital is extremely in buildings requiring the use of a highmonitoring) imaging efficiency in form This and action comparing it withuseful the next results in the future (moisture whenmethod, it is necessary. particular: constant monitoring of wall humidity, control of the effectiveness of the methods used for This action is extremely useful in buildings requiring the use of a high imaging efficiency method, in drying walls and assessment of the moisture condition of load-bearing walls, in particular, thick ones. particular:The constant monitoring of wall humidity, control of the effectiveness of the methods used for main advantages of the proposed measurement system are the non-invasive and drying walls and assessment of theofmoisture condition of load-bearing walls, in particular, thick non-destructive measurement the tested object thanks to specially designed electrodes, theones. The main advantages of the proposed measurement the non-invasive possibility of imaging the moisture distribution not onlysystem on the are surface but also insideand the nondestructive measurement of described the testedresearch object thanks to specially designed possibility investigated object. The uses simulation tools based onelectrodes, the Matlabthe scientific software and scripts indistribution the Python and programming languages. A special role was played by the of imaging the moisture notRonly on the surface but also inside the investigated object. toolbox called EIDORS dedicated to the Matlab software [18]. It has been used for modeling domain The described research uses simulation tools based on the Matlab scientific software and scripts in and topological algorithms using the finite A element method (FEM) to solve problem the Python and R programming languages. special role was played by the theinverse toolbox called (IP). EIDORS The structure of the article was divided into five parts including the introduction. Section 2 dedicated to the Matlab software [18]. It has been used for modeling domain and topological algorithms presents the hardware of the measurement system and algorithms for solving forward and inverse using the finite element method (FEM) to solve the inverse problem (IP). problems. Section 3 presents the results of tests both in relation to simulation experiments and to the The structure of the article was divided into five parts including the introduction. Section 2 reconstruction of the real object. The analysis covered three types of algorithms applicable in EIT: presents the ElasticNET hardware of theANN. measurement systemwas andalso algorithms for solving forward and the inverse LARS, and The real object reconstructed. Section 4 includes problems. Section presents thealgorithms results ofand tests both in relation to simulation experiments comparison of 3three selected discussions in the perspective of previous studies. and It alsoto the reconstruction of the real object. analysis covered threeThe types of algorithms applicable in EIT: refers to other, known methodsThe within the studied issues. possible improvements were also Finally, SectionThe 5 summarizes LARS,suggested. ElasticNET and ANN. real objectthis wasarticle. also reconstructed. Section 4 includes the comparison of three selected algorithms and discussions in the perspective of previous studies. It also refers to Materials and Methods other,2.known methods within the studied issues. The possible improvements were also suggested. This chapter presents athis measurement Finally, Section 5 summarizes article. system that enabled the collection of electrical data used subsequently to solve forward and inverse problems. Then there are short descriptions of four

2. Materials and Methods This chapter presents a measurement system that enabled the collection of electrical data used subsequently to solve forward and inverse problems. Then there are short descriptions of four selected

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algorithms: Least Angle Regression (LARS), ElasticNet, Artificial Neural Networks (ANN), and GaussNewton. The first three methods were used in the experiments on tomographic imaging and compared selected algorithms: Least Angle Regression (LARS), ElasticNet, Artificial Neural Networks (ANN),  with each other. All tested algorithms are classified as machine learning and artificial intelligence and Gauss‐Newton. The first three methods were used in the experiments on tomographic imaging  methods. Thanks to the above algorithms, it was possible to use the supervised machine learning and compared with each other. All tested algorithms are classified as machine learning and artificial  method, which in combination with the parallel computing (multi-core CPU, GPU) allowed us to intelligence methods. Thanks to the above algorithms, it was possible to use the supervised machine  quickly reach effective solutions in the field of building models of reconstructed objects. learning method, which in combination with the parallel computing (multi‐core CPU, GPU) allowed  us to quickly reach effective solutions in the field of building models of reconstructed objects.  2.1. Selected Hardware Issues, Sample Models, and Reconstructions 2.1. Selected Hardware Issues, Sample Models, and Reconstructions  Electrical tomography is a technique for imaging the distribution of conductivity or permeability Electrical tomography is a technique for imaging the distribution of conductivity or permeability  inside an object under investigation based on measurements of the potential distribution on the inside surface. an  object Numerous under  investigation  on  measurements  the  potential  on  the of object’s different based  techniques can be usedof in the processdistribution  of optimization object’s  surface.  Numerous  different  techniques  can  be  used  in  the  process  of  optimization  of  tomographic methods. tomographic methods.  The data collection system collects the measured voltage from the electrodes and then processes The data collection system collects the measured voltage from the electrodes and then processes  the data. Conventional data acquisition systems require voltage, filtering, demodulation and converting the  data.  Conventional  systems to require  voltage,  filtering,  demodulation  and for equipment to be digitizeddata  and acquisition  a signal processor transfer data to a computer. Our device converting  equipment  to  be  digitized  and  a  signal  processor  to  transfer  data  to  a  computer.  Our  measurement in electrical tomography uses two methods: electrical capacitance tomography and device  for  measurement  in  electrical  tomography  uses  two  methods:  electrical  capacitance  electrical impedance tomography. It allows you to take measurements up to 32 channels. Figure 4a tomography  and  electrical  impedance  tomography.  It  allows  you  to  take  measurements  up  to  32  shows the inside of the tomograph, while Figure 4b shows the main panel of the device with sockets channels. Figure 4a shows the inside of the tomograph, while Figure 4b shows the main panel of the  designed to connect the electrodes. This device provides a non-invasive way to test the spatial device with sockets designed to connect the electrodes. This device provides a non‐invasive way to  distribution of moisture. The system includes an additional software solution. The advantage of test  the spatial  distribution  of  moisture.  The  system includes an  additional  software  solution.  The  theadvantage of the system is the simultaneous measurement of voltage and capacitance.  system is the simultaneous measurement of voltage and capacitance.

  Figure 4. The measurement device. (a) inside view, (b) main panel view.  Figure 4. The measurement device: (a) inside view, (b) main panel view.

The measurements were made using the Polar GND EIT method, at 1 kHz frequency and 100uA  excitation  current.  An  object  whose using internal  properties  are  unknown  is  surrounded  The measurements were made theelectrical  Polar GND EIT method, at 1 kHz frequencyby  and electrodes  placed  on  its  edge  and  electrically  excited  in  various  combinations  (see  5).  100uA excitation current. An object whose internal electrical properties are unknown is Figure  surrounded byMeasurements are made for all possible ways of connecting the power source to the next pairs of  electrodes placed on its edge and electrically excited in various combinations (see Figure 5). electrodes. Other electrodes measure voltage drops. Both parts of Figure 5a,b shows two exemplary  Measurements are made for all possible ways of connecting the power source to the next pairs of measuring cycles. In this way a series of measurements is created. For a system with 16 electrodes  electrodes. Other electrodes measure voltage drops. Both parts of Figure 5a,b shows two exemplary (we mark them as variable n), there are 16 possibilities to connect the power source, but because of  measuring cycles. In this way a series of measurements is created. For a system with 16 electrodes (we the symmetry of the system, we accept only half of them (n/2 = 8).  mark them as variable n), there are 16 possibilities to connect the power source, but because of the Input data for the image construction algorithm are voltage measurements made between adjacent  symmetry of the system, we accept only half of them (n/2 = 8). electrodes.  Measurements  made  on  electrodes  with  an  attached  excitation  source  are  omitted  due    Input data for the image construction algorithm are voltage measurements made between adjacent to unknown voltage drop between these electrodes and the tested area. For a system of n = 16 electrodes  electrodes. Measurements made on electrodes with an attached excitation source are omitted due to and  any  projection  angle,  n  −  4  =  12  independent  measurements  can  be  obtained.  Thus,  the  full    unknown voltage drop between these electrodes and the tested area. For a system of n = 16 electrodes number  of  voltages  measurable  between  neighboring  voltage  electrodes  at  n/2  =  8  angles  is:    and any projection angle, n − 4 = 12 independent measurements can be obtained. Thus, the full number (n − 4) (n/2) = 12 × 8 = 96. The method of measuring the inter‐electrode voltages shown in Figure 5  of corresponds to the first and the second projection angle. For subsequent angles, sequential switching  voltages measurable between neighboring voltage electrodes at n/2 = 8 angles is: (n − 4) (n/2) = 12 × 8of the power supply and measurement circuit to the neighboring electrodes takes place.  = 96. The method of measuring the inter-electrode voltages shown in Figure 5 corresponds to the first and thea second angle. For subsequent sequential switching of28  theindependent  power supply For  system projection of  n  =  32  electrodes  and  any  angles, angle  of  projection,  n  −  4  =  and measurement circuit to the neighboring electrodes takes place. measurements  can  be  obtained.  Hence  the  full  number  of  possible  independent  measurements  of  voltages between neighboring voltage electrodes at n/2 = 16 angles is: (n − 4) (n/2) = 28 × 16 = 448. 

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For a system of n = 32 electrodes and any angle of projection, n − 4 = 28 independent Sensors 2018, 18, x FOR PEER REVIEW of 21 measurements can be obtained. Hence the full number of possible independent 6measurements of voltages between neighboring voltage electrodes at n/2 = 16 angles is: (n − 4) (n/2) = 28 × 16 = 448.

Figure 5. Voltage measurement method. (a) first measuring cycle, (b) second measuring cycle.

Figure 5. Voltage measurement method: (a) first measuring cycle, (b) second measuring cycle. Potential values of electrodes depend on the current distribution within the region, and thus also on the distribution conductivity.depend The algorithm of current computerdistribution image reconstruction looking Potential values ofof electrodes on the withinisthe region, and thus iteratively for such a distribution of conductivity, for which the calculated values of inter-electrode also onvoltages the distribution of conductivity. The algorithm of computer image reconstruction is looking are as close as possible to the corresponding measurement values. iterativelyThe formeasured such a distribution of conductivity, which the calculated values of inter-electrode voltage is generally a voltage drop for on the two impedances of the electrodes and the impedance of the object. The voltage drop on the impedances of the spot electrodes can be omitted voltages are as close as possible to the corresponding measurement values. duemeasured to the high impedance the measuring system. In the case of surface electrodes, the of potential The voltage isofgenerally a voltage drop on the two impedances the electrodes and decrease at the electrode is very low, the tested object has low conductivity, while the electrode has the impedance of the object. The voltage drop on the impedances of the spot electrodes can be omitted a high conductivity. Therefore, the voltage drop is negligible, the surface impedance coefficient of the due toelectrode the hightends impedance of the measuring system. case of surface electrodes, to zero (it is negligibly small), but it In is the programmatically included in the the potential reconstruction process.is The contact is included in the model but has a limited effect decrease at the electrode very low,impedance the tested object has low conductivity, while theonelectrode has a the measurement. high conductivity. Therefore, the voltage drop is negligible, the surface impedance coefficient of the Voltage drops are measured on the surface of the tested object, so it is a non-invasive method. electrode tends to zero (it is negligibly small), but it is programmatically included in the reconstruction After applying the power source in the wall, current starts flowing, which has a higher value closer process. Theshore contact impedance is included the from model has a the limited effect on flow the measurement. to the and the power source. The furtherin away the but electrodes, more the current is getting smaller (tends to zero). This is the factor that causes the reconstruction to be more optimal Voltage drops are measured on the surface of the tested object, so it is a non-invasive method. closer to the measuring fromstarts the electrodes, thewhich detection precision mayvalue closer to After applying the power electrodes, source inthe thefurther wall,away current flowing, has a higher be lower (the reconstruction may be worse due to the current depth distribution). Therefore, the shore and the power source. The further away from the electrodes, the more the current flow is measurements on one edge usually give worse quality compared to measurements on two or more gettingedges. smaller (tends however, to zero).only Thisone-sided is the factor that causes the reconstruction to be more optimal closer Sometimes, measurement is possible. The tested physical models of the wall parts contained 16 or 32 electrodes each for precision measuring may be lower to the measuring electrodes, the further away from the electrodes, the detection the wet wall. The electrodes were placed on both sides of the tested wall sample. Electrical impedance (the reconstruction may be worse due to the current depth distribution). Therefore, measurements on tomography is based on the measurement of the potential difference. The ability to determine the one edge usually give worse quality compared to measurements on two or more edges. Sometimes, wall condition results from the unique conductivity value of each material. The necessary equipment however, one-sided measurement is possible. suchonly as electrodes, meters, alternating current generator, multiplexer and a computer with LabVIEW and EIT modules was used for the measurements. Figure 6a,b the partially each immersed The tested physical models of the wall parts contained 16 shows or 32 electrodes for measuring the lightweight concrete block on which the surface electrodes are placed. It can be seen that in the impedance wet wall. The electrodes were placed on both sides of the tested wall sample. Electrical presented picture the block samples have 2 × 8 (a) and 2 × 16 (b) electrodes. tomography is based on the measurement of the potential difference. The ability to determine the wall condition results from the unique conductivity value of each material. The necessary equipment such as electrodes, meters, alternating current generator, multiplexer and a computer with LabVIEW and EIT modules was used for the measurements. Figure 6a,b shows the partially immersed lightweight concrete block on which the surface electrodes are placed. It can be seen that in the presented picture the block samples have 2 × 8 (a) and 2 × 16 (b) electrodes.

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(b) Sample with 2 × 16 electrodes

Figure 6. A tomographic laboratory to study the moisture inside the cellular lightweight concrete block.

Figure 6. A tomographic laboratory to study the moisture inside the cellular lightweight concrete block. As mentioned before, a big problem related to the moisture content testing is the lack of a method allowing us to determine its spatial distribution without the need to take samples. The As mentioned bigisproblem related totomography. the moisture content is the lackthe of a method method thatbefore, enables athis electrical impedance It consists in testing evenly distributing allowing us to determine its spatial distribution without need to take method that electrodes on the tested object and ensuring a good contactthe of their surface withsamples. the tested The surface. Unfortunately, the wall surfaces have a varied shape. Also, the building materialsthe themselves, enables this is electricaloften impedance tomography. It consists in evenly distributing electrodes on the such as brick or plaster, have a certain porosity, which makes measurement difficult. To ensure an tested object and ensuring a good contact of their surface with the tested surface. Unfortunately, often adequate flow of electric current between the individual electrode pairs, a special multilayer electrode the wall surfaces have a varied shape. Also, the building materials themselves, such as brick or plaster, was developed. have a certainEnsuring porosity, makesofmeasurement difficult. ensure animportant adequate flow of electric the which proper contact the electrodes with the wall To is particularly in testing objects with an uneven as well as a rough surface. An example of the use of this type of electrodes is current between the individual electrode pairs, a special multilayer electrode was developed. the moisture condition investigation inside the masonry. The development of effective and efficient Ensuring the proper contact of the electrodes with the wall is particularly important in testing measuring electrodes for impedance tomography has proved to be a serious challenge. In order to objects with an optimal unevencontact as well as a the rough surface. An example of the of this typesurface of electrodes is ensure between electrode and the wall, an electrode withuse a flexible contact the moisture investigation insideFigure the masonry. The of effective and condition articulated mounting was designed. 7a shows how to development attach a set of rubber electrodesand to efficient building wall. 7b showstomography a view of the electrode set, while Figure 7c shows a schematicIn order to measuringtheelectrodes forFigure impedance has proved to be a serious challenge. view of the structure of a single rubber electrode. ensure optimal contact between the electrode and the wall, an electrode with a flexible contact surface A complete electrode consists of three modules: a specific electrode, a PCB with a contact socket, and articulated mounting wasMechanically, designed. Figure 7a shows how totoattach a setbyofmeans rubber electrodes to and a fastening system. the modules are connected each other of two the building wall. Figure 7b shows a view of the electrode set, while Figure 7c shows a schematic view sleeves placed one inside the other. Before separating, they are secured by a collar placed on the upper sleeve. The PCB is made of double-sided 1.54 mm thick laminate with an SMB1251B1-3GT30Gof the structure of a single rubber electrode. 50 socket. A complete electrode consists of three modules: a specific electrode, a PCB with a contact socket, From the active surface, the galvanic plate is connected by means of four leads with a specific and a fastening system. Mechanically, the silicone modules are on connected to each by means of two sleeves electrode made of electro-conducting coated the one hand in theother galvanization process placed onewith inside the other. Before separating, they arecontact secured by a the collar the upper sleeve. a copper layer. Pins and the copper layer allow between PCBplaced and the on electrically The PCB isconductive made ofsilicone. double-sided 1.54 mm thick laminate with an SMB1251B1-3GT30G-50 socket. The specific electrode made of flexible electrically conductive silicone improves contact with From the active surface, the galvanic plate is connected by means of four leads with a specific the surface of the tested object. This feature is particularly useful in examining objects with electrode increased made ofporosity. electro-conducting silicone coated on the one hand in the galvanization process with a copperThe layer. and thetripod copper layermounting allow contact theASB PCB and the electrically thirdPins module is the electrode system. between It is made of in 3D printing technology. conductive silicone.The holder has two parallel channels that allow quick mounting on tripod profiles. The element responsible for the elasticity of the mount is the rubber ring, which task is to adjust the The specific electrode made of flexible electrically conductive silicone improves contact with the position of the electrode to the tested object surface, which aims to eliminate the unevenness and surface of the tested This feature particularly examining objects with increased pressing theobject. electrode to the wall. is The post-retrofit useful version in is equipped with an additional 10 mm porosity. The third module is the electrode It rubber is made thick shock absorber andtripod a flexible connectionmounting between thesystem. conductive andof theASB PCB.in As3D a printing

technology. The holder has two parallel channels that allow quick mounting on tripod profiles. The element responsible for the elasticity of the mount is the rubber ring, which task is to adjust the position of the electrode to the tested object surface, which aims to eliminate the unevenness and pressing the electrode to the wall. The post-retrofit version is equipped with an additional 10 mm thick shock absorber and a flexible connection between the conductive rubber and the PCB. As a result, the electrodes adhere much better to uneven surfaces of the tested object. Newly designed electrode systems have great potential in practical applications.

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result, the electrodes adhere much better to uneven surfaces of the tested object. Newly designed

Sensorsresult, 2018, 18, the2285 electrodes adhere much better to uneven surfaces of the tested object. Newly designed8 of 21 electrode systems have great potential in practical applications.

electrode systems have great potential in practical applications.

Figure 7. The concept of the electrodes. (a) way of fixing rubber electrodes to the building wall, (b)7.setThe of rubber electrodes, (c) structure a single rubber electrode. Figure concept of the electrodes: (a)ofway of fixing rubber electrodes to the building wall, (b) set Figure 7.electrodes, The concept of the electrodes. way of fixing rubber electrodes to the building wall, of rubber (c) structure of a single(a)rubber electrode. 2.2. Wall Tests with the of the Gauss-Newton (GNM) Method (b) set ofMoisture rubber electrodes, (c) Use structure of a single rubber electrode.

Figure 8 presents thethe results of the tomographic imaging usingMethod the Gauss-Newton (GNM) method. 2.2. Wall Moisture Tests with Use of Gauss-Newton (GNM) 2.2. Wall Moisture Tests with the Use the Gauss-Newton (GNM) Method image (a). The differences, The presented reconstruction (b) of deviates somewhat from the pattern Figure 8 presents the results of tomographic imaging using the Gauss-Newton (GNM) method. however, details the contour imaging of the moistened area. You can therefore use this Figure 8 concern presentsonly the the results of of tomographic using the Gauss-Newton (GNM) method. The presented reconstruction (b) deviates somewhat from the pattern image (a). The differences, to roughly estimate the moisture level and area. The method presented reconstruction (b) deviates somewhat from the pattern image (a). The differences, however, concern only the details of the contour of the moistened area. You can therefore use this however, concern only the details of the contour of the moistened area. You can therefore use this method to roughly estimate the moisture level and area. method to roughly estimate the moisture level and area.

(a)

(b)

Figure 8. The geometrical model of the tested wet wall with 32 electrodes: (a) the pattern image, (b) the image reconstructed by Gauss-Newton method.

(a)

(b)

Figure 8. The geometrical model of the tested wet wall with 32 electrodes: (a) the pattern image, Figure 8. The geometrical model of the tested wet wall with 32 electrodes: (a) the pattern image, (b) the (b) the image reconstructed by Gauss-Newton method. image reconstructed by Gauss-Newton method.

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Figure 9 presents the spatial reconstruction of a wall fragment using the GNM method by means Sensors 2018, 18, 9x presents FOR PEERthe REVIEW 9 9b of 21 of 32 measurement electrodes locatedreconstruction around the of object. 9ausing is a reference image.by Figure is the Figure spatial a wallFigure fragment the GNM method means of 32 measurement electrodes located around theboth object. Figure you 9a is can a reference image. Figure 9b isintensity result generated by the use of GNM. Comparing images, see differences in the 9generated presents by thethe spatial reconstruction of a wall fragment using the see GNM method by means theFigure result usereconstructed of GNM. Comparing images, you can differences in the of the color. Brighter colors of the imageboth indicate less intense moisture inside the wall ofintensity 32 measurement electrodes the object.image Figure 9a is aless reference Figure of the color. Brighter located colors ofaround the reconstructed indicate intenseimage. moisture inside9b is compared to the reference image. the result generated by the use of GNM. Comparing both images, you can see differences in the the wall compared to the reference image.

intensity of the color. Brighter colors of the reconstructed image indicate less intense moisture inside the wall compared to the reference image.

(a)

(b)

Figure 9. The geometrical model 3D with 4 × 8 electrodes—the image reconstruction: (a) pattern

Figure 9. The geometrical model 3D with 4 × 8 electrodes—the image reconstruction: (a) pattern model, (b) Gauss-Newton (a) method with Laplace regularization. (b) model, (b) Gauss-Newton method with Laplace regularization. Figure 9. The 3D withof4 a× reconstruction 8 electrodes—the reconstruction: (a) using pattern In Figure 10,geometrical we can seemodel an example of image a damp concrete block 32 model, (b) Gauss-Newton with Laplace electrodes located on bothmethod sides of the testedregularization. object. In order to solve the problem of the In Figure 10, we can see an example of a reconstruction of a damp concrete block using 32 three-dimensional finite element, mesh was prepared. It can be noticed that surfaces of finite elements electrodes on both sides of are thesmall. tested In order solve the problem of the32 threeFigure 10, we see an example of aobject. reconstruction ofto a damp concrete block using thatInlocated are localized nearcan electrodes Hence, the solution of the forward problem is precise. dimensional finite element, mesh was can beInnoticed that surfaces of finiteofelements electrodes located onareboth sides ofprepared. the testedIt object. to solve thetheproblem the The results obtained similar to those obtained by placing 32order electrodes around lightweight three-dimensional finite element,are mesh It can be noticed that surfaces of finite elements that are localized near electrodes Hence, the solution of the forward problem is precise. concrete block with dimensions 10 ×small. 40was × 90prepared. cm. The reconstructed image deviates from the pattern with the too low intensity of colors. that are localized near electrodes are small. Hence, the solution of the forward problem is precise. The results obtained are similar to those obtained by placing 32 electrodes around the lightweight

Theblock resultswith obtained are similar to 40 those obtained byreconstructed placing 32 electrodes thefrom lightweight concrete dimensions 10 × × 90 cm. The image around deviates the pattern concrete block with dimensions 10 × 40 × 90 cm. The reconstructed image deviates from the pattern with the too low intensity of colors. with the too low intensity of colors.

(a)

(b)

Figure 10. The geometrical model 3D with 2 × 16 electrodes—the image reconstruction: (a) pattern model, (b) reconstruction created with the use of Gauss-Newton method with Laplace regularization.

In Figure 11 two special models (a)of the brick cube “wet” and (b)“moist” with 2 × 8 electrodes are presented. The image was reconstructed by Gauss-Newton method with Laplace regularization or Figure 10. The geometrical model 3D with 2 × 16 electrodes—the image reconstruction: (a) pattern Tikhonov regularization. model 3D with 2 × 16 electrodes—the image reconstruction: (a) pattern Figure 10. The model, (b) geometrical reconstruction created with the use of Gauss-Newton method with Laplace regularization.

model, (b) reconstruction created with the use of Gauss-Newton method with Laplace regularization. In Figure 11 two special models of the brick cube “wet” and “moist” with 2 × 8 electrodes are presented. The image was models reconstructed Gauss-Newton method with Laplace or are In Figure 11, two special of thebybrick cube “wet” and “moist” with 2regularization × 8 electrodes Tikhonov regularization.

presented. The image was reconstructed by Gauss-Newton method with Laplace regularization or Tikhonov regularization.

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(b)

Figure 11. The geometrical model 3D with 2 × 8 electrodes—the image reconstruction: (a) pattern

Figure 11. The(b) geometrical model 2 of ×Gauss-Newton 8 electrodes—the reconstruction: model, reconstruction created3D withwith the use method image with Laplace regularization. (a) pattern model, (b) reconstruction created with the use of Gauss-Newton method with Laplace regularization. 2.3. Masonry Humidity Testing by the Least Angle Regression (LARS) Method

order to obtain more andAngle stable Regression reconstruction results in solving the inverse problem 2.3. Masonry In Humidity Testing byaccurate the Least (LARS) Method

in electrical tomography [19–21], a new solution based on the method of the least angle regression

was tested [27]. more There are many methods to solve the optimization problem [26]. Thethe statistical In order to obtain accurate and stable reconstruction results in solving inverse problem methods can be used to reconstruct an image in electrical impedance tomography [28,29]. in electrical tomography [19–21], a new solution based on the method of the least angle regression was The main objective of the tomography is to perform image reconstruction. During the tested [27]. There are many solve the optimization problemare [26]. The correlated statistical methods measurements, we can methods see that theto measured values from some electrodes strongly can be used an image in In electrical tomography [28,29]. (duetotoreconstruct the way of measurement). this case,impedance we have a multicollinearity problem. When the variablesof (predictors) are correlatedis (collinear), the matrix tends reconstruction. to a single matrix. ByDuring the The independent main objective the tomography to perform image means of the least squares method, we obtain large absolute values of some estimators with unknown measurements, we can see that the measured values from some electrodes are strongly correlated (due parameters. Forecasts based on this model are unstable. The most common approach is to reduce the to the way In this the case, wepredictors have a multicollinearity problem. Then, Whenwethe independent setof ofmeasurement). input variables (removing same that apply to multicollinearity). have variablesa(predictors) are correlated (collinear), the matrix tends to a single matrix. By means of the problem with the selection of predictor variables that will be included in the regression model. For example, when comparing the AIC (Akaike Information Criterion) value for linear models with least squares method, we obtain large absolute values of some estimators with unknown parameters. sets of predictors, we can choose the best model. Forecastsdifferent based on this model are unstable. The most common approach is to reduce the set of input Another possible way to reduce the problem of multicollinearity between predictors depends variableson (removing the of same predictors that apply to multicollinearity). we have the application the least angle regression algorithm. This algorithm takes Then, into account only a problem with the selection of predictor variables that included the regression model. For example, causal variables in the linear model (from thewill set ofbe predictors, youin should select the input variables that have athe direct impact on theInformation response variable). In this case, thefor linear model is built by means of when comparing AIC (Akaike Criterion) value linear models with different sets of step forward regression, where the best variable is added to the model in every step. predictors, we can choose the best model. Let the linear system be described by the state equation Another possible way to reduce the problem of multicollinearity between predictors depends on Y  X   (1) the application of the least angle regression algorithm. This algorithm takes into account only causal 𝑛 𝑛×(𝑘+1) ∈ 𝑅 , 𝑋 model ∈𝑅 denote matrices responseselect and input variables, variableswhere in the𝑌linear (from thethe setobservation of predictors, youofshould the input variables that respectively, and 𝛽 ∈ 𝑅𝑘+1 denotes the vector of unknown parameters. When the linear model (1) have a direct impact on the response variable). In this case, the linear model is built by means of step contains the intercept, then the first column of matrix X is a column of ones. The object 𝜀 ∈ 𝑅𝑛 in the forward regression, the best variable added to the model in every step. linear systemwhere (1) presents a sequence of is disturbances, which is usually defined as a vector of independent identically with normal distribution 𝑁(𝑂̃, 𝜎 2 𝐼), in which Let the linear system be distributed describedrandom by thevariables state equation 𝑂̃ ∈ 𝑅𝑛 is a zeros vector, but 𝐼 ∈ 𝑅𝑛 × 𝑛 is an identity matrix. The classical Least Square Method Y = Xβ depends on identification of unknown parameters 𝛽 =+(𝛽ε0 , 𝛽1 , … , 𝛽𝑘 ) in (1) by solution the task

(1)

where Y ∈ Rn , X ∈ Rn×(k+1) denote the observation matrices of response and input variables, respectively, and β ∈ Rk+1 denotes the vector of unknown parameters. When the linear model (1) contains the intercept, then the first column of matrix X is a column of ones. The object ε ∈ Rn in the linear system (1) presents a sequence of disturbances, which is usually defined a vector of  as  2 e independent identically distributed random variables with normal distribution N O, σ I , in which e ∈ Rn is a zeros vector, but I ∈ Rn × n is an identity matrix. The classical Least Square Method O depends on identification of unknown parameters β = ( β 0 , β 1 , . . . , β k ) in (1) by solution the task

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min kY − Xβk2

β ∈ R k +1


If det X T X 6= 0, then the best unbiased linear estimator of unknown parameters β is   −1 βˆ = X T X XY

(2)

(3)

The problem is often when X T X is singular. The following is a short version of the least angle regression algorithm as the workflow. An extended version of LAR has been presented in [30]. 1. 2.

3. 4.

The predictors should be standardized. The intercept β 0 in expression (1) is equal a mean of the response variable and we put β 1 = β 2 = . . . = β k = 0. Active set A (set of predictors) is empty. Calculate the residuals r = Y − β 0 − X( A) β ( A) for the linear model with all predictors from active set A. Determine the predictor Xj (which is not in active set) most correlated with residuals r and attach to the active set A.

 Move coefficient β j from 0 towards its least-squares coefficient X j , r until some other competitor Xk has a much correlation with the current residuals as does Xk . Move β j and β s in the direction defined by their joint least square coefficient of the current residual

 on X j , Xs until some other competitor Xl has a much correlation with the current residual. Go to step 2 and continue in this way until all k predictors have been entered.

2.4. Masonry Humidity Testing by the ElasticNet Method Another way to determine the linear regression when the input variables are collinear depends on the solution of the task
2
1 n min yi − β 0 − xi β0 + λPα β0 , (4) ∑ ( β 0 ,β0 )∈ Rk+1 2n i =1 where xi = ( xi1 , . . . , xik ), β0 = ( β 1 , . . . , β k ) for 1 ≤ i ≤ n and Pα is an elastic net penalty given by  k 
1 0 1−α 2 0 0 Pα β = (1 − α) k β k L + αk β k L = ∑ βj + α βj 2 1 2 2 j =1

(5)

We see that the penalty is a linear combination of norms L1 and L2 of unknown parameters β0. The introduction the penalty function dependent from parameters to the objective function allows to shrink the estimators of unknown parameters. The parameter λ in the task (4) denotes the coefficient of penalty, but the parameter 0 ≤ α ≤ 1 creates the compromise between LASSO (Least Absolute Shrinkage and Selection Operator) and ridge regression. The ridge regression (∝= 0) is called Tikhonov regularization [31] and is one of the most commonly used for regularization of linear models. LASSO (∝= 1) was introduced by Roberta Tibshirani [32,33]. This method performs the variable selection and regularization in linear statistical models [34,35]. For the ridge regression, the penalty is calculated in the norm L1 but for LASSO in L2 . Difference between ridge regression and LASSO is symbolic, only the norms are changed. The ridge regression shrinks coefficients for correlated predictors towards each other. When the correlated predictors depend on any latent factor, then ridge regression allows to uniformly distribute the strength of latent factor on these predictors. Whereas LASSO is indifferent to correlated predictors. This method allows to determine the preferred predictor and to ignore the rest. By applying the LASSO method, we obtain a model, where the many coefficients to be close to zero, and as a result, we receive a sparse model. The elastic net is a connection of ridge regression and LASSO [36,37]. Choosing the appropriate α, we may create the compromise between ridge regression and LASSO. By solution the task (4) for fixed λ and α we estimate the unknown parameters of the linear system (1), where predictors are correlated. Then the prediction based on model (1) is given by the formula

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ˆ where the vector of estimators of unknown parameters βˆ = βˆ 0 , βˆ 1 , . . . , βˆ k is estimated by Yˆ = X β, solution the task (4). 2.5. Masonry Humidity Testing by the Gauss-Newton Method In the electrical impedance tomography in the reconstruction of the image, the so-called Generalized Tikhonov regularization is very often used. In the literature on the subject, this method is also known as the Gauss-Newton algorithm in a generalized form. The Gauss-Newton method is based on the application of the least squares method in which the matrix Z (l ) fulfills the role of matrix X (first partial derivatives relative to fixed approximations β(l ) and observed values of independent variables), and the role of the vector y (observation of the dependent variable) vector e(l ) . It is a vector ofdifferences  between the empirical values of the dependent variable

and the lth of its approximations f xt , β(l ) . The Gauss-Newton algorithm is used to estimate the structural parameters of non-linear models. The general form of the non-linear function is presented below: yt = f ( xt , β) + ε t

(6)

where: yt —observations of the explanatory variable, xt = [ xt ]—P vector of observations for explanatory variables,   β t = β j —K vector of structural parameters, ε t —implementations of random elements (we assume that random components are uncorrelated, have an average of zero and equal, positive and finite variance). In the Gauss-Newton method, the reconstruction of the internal image of the investigated object is related to the determination of the global minimum of the fitness function. In order to carry out quantitative considerations, we assume that the tested object is polarized with an alternating low-frequency current. Then, the electrical material properties can be described by a function with real values. In this case, in the generalized Laplace equation, we neglect the word proportional to the frequency, and this function can be equated with the electrical conductivity (real isotropic admittivity case). 2.6. Masonry Humidity Testing by the Neural Imaging In order to solve the problem of non-invasive imaging of the interior of moist walls, the method of electrical tomography in connection with artificial neural networks was also used. So far, tomographic and neural networks methods have not been widely disseminated in the assessment of the wall. The reason is the low resolution of the reconstructed image and the low accuracy of mappings [7]. To increase the resolution of tomographic reconstructions depicting the degree of internal humidity of walls, a new method was developed based on a set of many separately trained neural networks. The number of neural networks corresponds to the 3D resolution of the lattice dividing the inside of the wall into individual pixels. In the presented experiment a lightweight concrete block with dimensions 10 × 40 × 90 cm was used, which was divided into 8099 points. Using a device called a multiplexer, in short intervals, the tomographic system generates 192 values of voltage drops readings between different electrode pairs. These are the input data for the neural network system. The neural networks are designed in such a way that on the basis of an input vector containing 192 elements, each of the 8099 neural networks generates the value of a single pixel of the output image. Figure 12 shows the mathematical form of the neural model used during simulation experiments. At the model input, there are 192 electric signals generated by 16 electrodes.

values of voltage drops readings between different electrode pairs. These are the input data for the neural network system. The neural networks are designed in such a way that on the basis of an input vector containing 192 elements, each of the 8099 neural networks generates the value of a single pixel of the output image. SensorsFigure 2018, 18,12 2285shows the mathematical form of the neural model used during simulation 13 of 21 experiments. At the model input, there are 192 electric signals generated by 16 electrodes.

Figure pixels. Figure 12. 12. The The scheme scheme of of converting converting electrical electrical signals signals into into the the image image pixels.

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1313ofof2121

The same input vector the basis for training 8099 separate artificial neural networks (ANN). The Thesame sameinput inputvector vectorisis isthe thebasis basisfor fortraining training8099 8099separate separateartificial artificialneural neuralnetworks networks(ANN). (ANN). In this way, from vector of 192 variables representing electrical values, set of neural networks In Inthis thisway, way,from fromaaavector vectorof of192 192variables variablesrepresenting representingelectrical electricalvalues, values,aaaset setof ofneural neuralnetworks networks creates complete lattice of the lightweight concrete block image. The output image created by creates createsaaacomplete completelattice latticeof ofthe thelightweight lightweightconcrete concreteblock blockimage. image.The Theoutput outputimage imageisis iscreated createdby by assigning colors the output values each pixel. The transformation method shown Figure13. 13. assigning assigningcolors colorstoto tothe theoutput outputvalues valuesofof ofeach eachpixel. pixel.The Thetransformation transformationmethod methodisis isshown showninin inFigure Figure 13.

Figure 13. way ofofconverting a areal vector into concrete block’s Figure 13.The The way converting realnumber number vector intothe thelightweight lightweight concrete block’s Figure 13. The way of converting a real number vector into the lightweight concrete block’s spatial image. spatial spatialimage. image.

Eachof of the 8099 8099 neural networks had aamulti-layered perceptron structure with 1010 neurons in the Each Each ofthe the 8099neural neuralnetworks networkshad had amulti-layered multi-layeredperceptron perceptronstructure structurewith with 10neurons neuronsinin hidden layer. The scheme of a single perceptron is shown in Figure 14. the thehidden hiddenlayer. layer.The Thescheme schemeofofa asingle singleperceptron perceptronisisshown shownininFigure Figure14. 14.

Figure Figure14. 14.Structure Structureofof ofthe theselected selectedmulti-layer multi-layerperceptron. perceptron. Figure 14. Structure the selected multi-layer perceptron.

InInorder ordertotocollect collectdata datanecessary necessarytototrain trainthe theneural neuralnetwork, network,physical physicaland andmathematical mathematicalmodels models In order to collect data necessary to train the neural network, physical and mathematical models were weredeveloped. developed.The Thefinite finiteelement elementmethod methodwas wasused usedfor forthis. this.Based Basedon ona amathematical mathematicalmodel, model,a adata data were developed. The finite element method was used for this. Based on a mathematical model, a data set was generated. After that, it was used to train the neural network system. set was generated. After that, it was used to train the neural network system. set was generated. After that, it was used to train the neural network system. To Totrain trainthe thementioned mentionedabove aboveneural neuralnetwork, network,a acollection collectionofof6140 6140historical historicalcases caseswas wasused used(see (see To train the mentioned above neural network, a collection of 6140 historical cases was used (see Table 1). The main set of data has been divided into three separate subsets: a training set, validation Table 1). The main set of data has been divided into three separate subsets: a training set, validation Table 1). The main set of data has been divided into three separate subsets: a training set, validation set, set,and andtesting testingset, set,ininthe theproportions proportionsofof70%, 70%,15%, 15%,and and15%. 15%.This Thismethod methodofofdata datapreparation preparationhas has set, and testing set, in the proportions of 70%, 15%, and 15%. This method of data preparation has been beenused usedfor forallall8099 8099neural neuralnetworks. networks. been used for all 8099 neural networks. Table Table1.1. 1.Training Trainingresults resultsfor forone oneof of8099 8099neural neuralnetworks. networks. Table Training results for one of 8099 neural networks.

Samples Samples

Samples

Training Training set Training set set Validation set Validation set Validation set Testing set

Testing Testingset set

MSE MSE MSE

RR

R

-6 -6−69.99983·10-1 -1 4298 5.31979· 5.31979· 9.99983· 10× 10−1 42984298 5.31979 10 ×10 10 9.99983

921 921

1.68249 × 10−5

9.99947 × 10−1

-1 -1 921 10-510-5−59.99947· 1010 921 1.68249· 1.68249· 9.99947· 2.03645 ×10 9.99934 × 10−1 -5 -5 9.99934·10-1 -1 921 1010 921 2.03645· 2.03645· 9.99934·10

The highest Mean Squared Error (MSE) concerned the testing set and was 0.000020364. In the The highest Error set 0.000020364. InInthe The highestMean Mean Squaredsmaller Error(MSE) (MSE) concerned thetesting testing setand andwas was 0.000020364. the case of validation set, aSquared slightly error concerned was noted.the Mean Squared Error is the average squared case caseofofvalidation validationset, set,a aslightly slightlysmaller smallererror errorwas wasnoted. noted.Mean MeanSquared SquaredError Errorisisthe theaverage averagesquared squared difference differencebetween betweenoutputs outputsand andtargets. targets.Lower Lowervalues valuesmean meanbetter betterperformance. performance.Zero Zeromeans meansno noerror error (excellent performance). The training set was trained with the lowest training error, which is the most (excellent performance). The training set was trained with the lowest training error, which is the most common commonand andcorrect correctsituation. situation.AAlow lowMSE MSEerror errorininthe thetraining trainingset setresults resultsfrom frombetter betternetwork network adaptation to training cases. Another indicator of the quality of network learning was R (Regression). adaptation to training cases. Another indicator of the quality of network learning was R (Regression).

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difference between outputs and targets. Lower values mean better performance. Zero means no error (excellent performance). The training set was trained with the lowest training error, which is the most common and correct situation. A low MSE error in the training set results from better network adaptation to training cases. Another indicator of the quality of network learning was R (Regression). An R value of 1 means a close relationship between pattern and output, and 0 means a random relationship. In all three cases of data sets (learning, validation, and testing), R was close to 1. This also applies to the test and validation set, which is particularly valuable. Values close to 1 indicate a good match of the results obtained by the network (output vectors) to the patterns included in the individual sets (training, validation, and testing). Good indicators (MSE and R) for the training set show the lack of overtraining effect and the ability of the network to knowledge generalization (i.e., correct conversion of input data to output information not only for learning cases). 3. Results This chapter presents the results of wall humidity tests by EIT tomography in combination with the following machine learning algorithms: Least Angle Regression (LARS), ElasticNet, and Artificial Neural Networks. The root mean square error of prediction (RMSE) indicator was used to quantify the quality of the reconstructions obtained using simulation models. Let vector x = ( x1 , . . . , xn ) presents the pattern, which should be reconstructed. After reconstruction we obtain the vector xˆ = ( xˆ 1 , . . . , xˆ n ), which contains the expected values of reconstruction of explored object. The root mean square error of prediction was determined by the formula (7). s RMSE =

1 n ( xi − xˆi )2 n i∑ =1

(7)

In the further part of this paper two variants of images were compared: 2D and 3D. The 2D image lattice consisted of 2908 pixels, while the 3D grid consisted of 8099 pixels. Thus, in the first case (2D), n = 2908, while for the 3D variant, n = 8099. All results enabling the comparison of LARS, ElasticNET, and ANN methods were obtained thanks to the use of computer simulation methods. The simulated data has been used with added noise to SNR = 14 dB. This noise parameter was calculated for a given measurement system configuration and includes measurement patterns and electrode positions. SNR is defined in terms of power as a signal to noise ratio. 3.1. Results of Wall Moisture Tests Obtained Using the Least Angle Regression (LARS) Method Figure 15 presents one of the results of tomographic imaging using the LARS method. The input data was obtained thanks to the use of an EIT tomograph equipped with 16 electrodes (2 × 8). Intense colors indicate areas with higher humidity. It can be seen, the obtained reconstructive image (in the middle) is very close to the reference image (left). The difference image (right) indicates small deviations of the grid points in the reconstructed image from the reference image. The colors in the images reflect the conductance of the individual pixels that each image consists of. The lack of color in the original and reconstruction images testify to the lack of moisture. RMSE for a 3D sample with the use of LARS is 0.0298. Figure 16 shows a case analogous to the previous one, which was presented in Figure 15, but this time a cellular concrete sample with slightly different dimensions (10 × 20 × 60 cm) was used. The reconstruction was carried out in 2D. RMSE for a 2D sample with the use of LARS is 0.122599.

Figure 15 presents one of the results of tomographic imaging using the LARS method. The input data was obtained thanks to the use of an EIT tomograph equipped with 16 electrodes (2 × 8). Intense colors indicate areas with higher humidity. It can be seen, the obtained reconstructive image (in the middle) is very close to the reference image (left). The difference image (right) indicates small deviations of the grid points in the reconstructed image from the reference image. The colors in the images reflect the conductance of the individual pixels that each image consists of. The lack of color Sensors 2018, 18, 2285 in the original and reconstruction images testify to the lack of moisture. RMSE for a 3D sample with the use of LARS is 0.0298.

Original

Reconstruction

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Difference

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Figure 15. The result of Least Angle Regression (LARS) moisture testing of the lightweight concrete

for the of 2 Angle × 8 electrodes. Figure 15. The block result of case Least Regression (LARS) moisture testing of the lightweight concrete Figure 16 shows a case analogous to the previous one, which was presented in Figure 15, but block for thethis case 2 × 8 concrete electrodes. timeof a cellular sample with slightly different dimensions (10 × 20 × 60 cm) was used. The

reconstruction was carried out in 2D. RMSE for a 2D sample with the use of LARS is 0.122599. Sensors 2018, 18, x FOR PEER REVIEW

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Figure 16 shows a case analogous to the previous one, which was presented in Figure 15, but this time a cellular concrete sample with slightly different dimensions (10 × 20 × 60 cm) was used. The reconstruction was carried out in 2D. RMSE for a 2D sample with the use of LARS is 0.122599.

Figure 16. The result of Least Angle Regression (LARS) moisture testing of the lightweight concrete block for of the Least case of 2Angle × 16 electrodes. Figure 16. The result Regression (LARS) moisture testing of the lightweight concrete block for the3.2.case of of 2 Wall × 16Moisture electrodes. Results Tests Obtained Using the ElasticNet Method

Figure 17 presents an example of a tomographic 3D imaging result using the ElasticNet method. The input data was obtained thanks Using to the use of ElasticNet an EIT tomograph equipped with 16 electrodes 3.2. Results of Wall Moisture Tests Obtained the Method (2 × 8). Intense colors indicate higher spots. It can be seen, the obtained reconstructive Figure 16. The result of Least Anglehumidity Regression (LARS) moisture testing of the lightweight concreteimage (middle is comparable reference image (left). The image of residuals (right)the indicates blockimage) foran the case of 2 × 16 electrodes. Figure 17 presents example oftoathetomographic 3D imaging result using ElasticNet method. the occurrence of deviations of the grid points of the image reconstructed from the reference image. The input data3.2. was obtained thanks to the the useoriginal ofElasticNet anand EIT tomograph equipped with 16 electrodes Wall Moisture TestsofObtained the Method TheResults lack ofof color and shades blue in Using reconstruction images mean the lack of (2 × 8). Intensemoisture. colors indicate higher humidity spots. It can be seen, the obtained RMSE for a 3D sample with the use of ElasticNet is 0.0365. Figure 17 presents an example of a tomographic 3D imaging result using the ElasticNet method. reconstructive input data obtained thanks the reference use of an EIT image tomograph equipped with 16 electrodes image (middleThe image) is was comparable to tothe (left). The image of residuals (right) (2 × 8). Intense colors indicate higher humidity spots. It can be seen, the obtained reconstructive image indicates the occurrence of deviations of the grid points of the image reconstructed from the reference (middle image) is comparable to the reference image (left). The image of residuals (right) indicates image. The lacktheofoccurrence color and shadesofoftheblue in theof original and reconstruction images mean the lack of of deviations grid points the image reconstructed from the reference image. The lack of color and shades of blue in the original and reconstruction images mean the lack of moisture. RMSE for a 3D sample with the use of ElasticNet is 0.0365. moisture. RMSE for a 3D sample with the use of ElasticNet is 0.0365.

Original

Reconstruction

Difference

Figure 17. The result of ElasticNet moisture testing of the lightweight concrete block for the case of 2 × 8 electrodes.

Figure 18 presents an example of a tomographic imaging result using the ElasticNet method. The input data was obtained thanks to the use of an EIT tomograph equipped with 32 electrodes (2 × 16). Original Reconstruction Difference Figure 17. The result of ElasticNet moisture testing of the lightweight concrete block for the case of × 8ElasticNet electrodes. moisture testing of the lightweight concrete block for the case of 2 × 8 electrodes. Figure 17. The result2of

Figure 18 presents an example of a tomographic imaging result using the ElasticNet method. The input data was obtained thanks to the use of an EIT tomograph equipped with 32 electrodes (2 × 16).

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Figure 18 presents an example of a tomographic imaging result using the ElasticNet method. The input data was obtained thanks to the use of an EIT tomograph equipped with 32 electrodes Sensors 2018, 18, x FOR PEER REVIEW of 21 (2 × 16). Intense colors indicate higher humidity spots. It can be seen that the obtained16reconstructive image (middle reproduces the reference (left) poorly. The image of differences (right) Intenseimage) colors indicate higher humidity spots. Itimage can be seen that the obtained reconstructive image indicates the occurrence of significant deviations of the grid points of the image reconstructed from (middle image) reproduces the reference image (left) poorly. The image of differences (right) indicates the occurrence of significant deviations of the grid points of the image reconstructed from the reference image. RMSE for a 2D sample with the use of ElasticNet is 0.282520. Compared with Sensors 2018, 18, x FOR PEER REVIEW 16 of 21 the reference image. RMSE for a 2D sample with the use of ElasticNet is 0.282520. Compared with LARS, ElasticNet showed the lower quality (higher RMSE) of reconstruction in this case. LARS, ElasticNet showed the lower quality (higher ofthat reconstruction in reconstructive this case. Intense colors indicate higher humidity spots. It canRMSE) be seen the obtained image (middle image) reproduces the reference image (left) poorly. The image of differences (right) indicates the occurrence of significant deviations of the grid points of the image reconstructed from the reference image. RMSE for a 2D sample with the use of ElasticNet is 0.282520. Compared with LARS, ElasticNet showed the lower quality (higher RMSE) of reconstruction in this case.

Figure 18. The result of ElasticNet moisture testing of the lightweight concrete block for the case of

Figure 18.2The result of ElasticNet moisture testing of the lightweight concrete block for the case of × 16 electrodes. 2 × 16 electrodes. 3.3. Results of Wall Moisture Tests Obtained Using the Neural Imaging

FigureMoisture 19The presents results moisture of Using neural imaging conjunction withblock the for EIT. Figure 18. result ofthe ElasticNet testing of the in lightweight concrete theConductive case of 3.3. Results of Wall Tests Obtained the Neural Imaging

(positive) are shown in shades of red. Non-conductive (negative) areas are shown in blue or 2 × 16 areas electrodes.

they transparent. Comparing the pattern of the damp block (original) with theEIT. output image, we (positive) Figure 19are presents the results of neural imaging in conjunction with the Conductive 3.3. Resultsthat of Wall Usingisthe Neural Imaging conclude theMoisture accuracyTests of Obtained the imaging very high. The right image shows the absolute areas are shown in shades of red. Non-conductive (negative) areas are shown in blue or they are (numerical) inthe theresults values of of individual pixelsin areconjunction minimal. They do notEIT. exceed ±0.05. It Figure differences 19 presents neural imaging with Conductive transparent. Comparing theresults pattern of the damp block (original) with the the outputtoimage, we conclude can be seen that the obtained by the neural imaging method are comparable the results (positive) areas are shown in shades of red. Non-conductive (negative) areas are shown in blue or that the accuracy theprevious imaging isthe very high. The right image the absolute (numerical) obtained byofboth methods. RMSE for a 3D sample with the useshows of ANN is 0.010819 so the they are transparent. Comparing pattern of the damp block (original) with the output image, we quality is better than LARS and differencesconclude in the values of individual do image not exceed 0.05. It can be seen that the accuracy of ElasticNET. the pixels imagingare is minimal. very high. They The right shows ± the absolute (numerical) differences in neural the values of individual pixels arecomparable minimal. They notresults exceed obtained ±0.05. It by both that the results obtained by the imaging method are todo the can be seen that the results obtained by the neural imaging method are comparable to the results previous methods. RMSE for a 3D sample with the use of ANN is 0.010819 so the quality is better than obtained by both previous methods. RMSE for a 3D sample with the use of ANN is 0.010819 so the LARS andquality ElasticNET. is better than LARS and ElasticNET.

Original

Reconstruction

Difference

Figure 19. The result of Artificial Neural Network system (ANN) moisture testing of the lightweight concrete block for the case of 2 × 8 electrodes. Original

Reconstruction

Difference

Figure 19. The result of Artificial Neural Network system (ANN) moisture testing of the lightweight 19.concrete The result Network system (ANN) moisture testing of the lightweight blockof forArtificial the case of Neural 2 × 8 electrodes.

Figure concrete block for the case of 2 × 8 electrodes.

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Figure 20 shows an analog measurement of a 2D sample with two rows of electrodes on both Figure 20 shows an analog measurement of a 2D sample with two rows of electrodes17on Sensors 2018, 18, x FOR REVIEW of both 21 sides ofsides the block ×PEER 16). is the small pixelsininthe thedifferential differential image. of the (2 block (2 × Noteworthy 16). Noteworthy is the smallamount amountof of colored colored pixels image. This indicates high quality imaging, which is confirmed by low RMSE index, which in this case This indicates high an quality imaging, which isofconfirmed bythe the low RMSE which in this case Figure 20 shows analog measurement a 2D sample with two rows index, of electrodes on both equals 0.106301. equals 0.106301. sides of the block (2 × 16). Noteworthy is the small amount of colored pixels in the differential image. This indicates high quality imaging, which is confirmed by the low RMSE index, which in this case equals 0.106301.

Figure 20. The result of Artificial Neural Network system (ANN) moisture testing of the lightweight

Figure 20. The result of Artificial Neural Network system (ANN) moisture testing of the lightweight concrete block for the case of 2 × 16 electrodes. concrete block for the case of 2 × 16 electrodes. Figure 20. The result of Artificial Neural Network system (ANN) moisture testing of the lightweight

3.4. Moisture Test Real Object concrete block forofthe case of 2 × 16 electrodes.

3.4. Moisture Test of Real Object

Figure 21 shows the results of the reconstruction of a block of cellular concrete immersed in 3.4.water, Moisture Test of Real Object using a system of artificial neural networks. Reconstructions on data obtained from real in Figure 21 shows the results of the reconstruction of a block ofbased cellular concrete immersed objects, contrast simulation based on are the most difficult typereal Figure 21 shows the results ofexperiments thenetworks. reconstruction of numerical a block ofmodels cellularon concrete immersed in water, using a in system of to artificial neural Reconstructions based data obtained from of test for tomographic systems. The first image on the left is the result of the direct processing of the of water, using a system of artificial neural networks. Reconstructions based on data obtained from real objects, in contrast to simulation experiments based on numerical models are the most difficult type tomographic data with the useexperiments of ANN. The ambiguous visual models effect is are caused by noise in thetype value objects, in contrast to simulation based on numerical the most difficult test for tomographic systems. The first image on the left is the result of the direct processing of the of the vector, which in the real objects basically unavoidable. of test forinput tomographic systems. Thecase firstofimage on the is left is the result of the direct processing of the

tomographic data with the use of ANN. The ambiguous visual effect is caused by noise in the value of tomographic data with the use of ANN. The ambiguous visual effect is caused by noise in the value the input vector, which in the case of real objects is basically unavoidable. of the input vector, which in the case of real objects is basically unavoidable.

-dump pixels -dry pixels Real object reconstruction

First denoising

Second denoising

-dump pixels -dry pixels Figure 21. The result of the real object Artificial Neural Networks (ANN) moisture testing of the Real objectconcrete reconstruction lightweight block for the case First of 2 × denoising 8 electrodes.

Second denoising

Figure 21. The result of the real object Artificial Neural Networks (ANN) moisture testing of the

Figure 21. The concrete result ofblock the for realtheobject Neural Networks (ANN) moisture testing of the lightweight case ofArtificial 2 × 8 electrodes. lightweight concrete block for the case of 2 × 8 electrodes.

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To show the results in a way that visually identifies the moisture area, the input vector was subjected to a denoising procedure using denoising stacked autoencoders [38]. The results of denoising were presented in the middle image entitled “First denoising.” Finally, the output image was subjected to one more processing using a filtering script whose objective was to cut off the output values, which obviously exceeded the acceptable range. The filtering effects are shown in the image entitled “Second denoising.” Finally, it was possible to reduce the number of incorrectly reconstructed pixels in the central part of the sample, but it was not possible to obtain a “clean image”. Denoising of tomographic data is an important issue because it affects the results of reconstruction. Due to the complexity of this subject, it may be the subject of separate studies aimed at improving the quality of tomographic images. 4. Discussion The imaging results presented in the previous chapter show great application possibilities of the machine learning algorithms combined with EIT. The analysis involved three methods and algorithms converting input vectors (values of voltage drops) into reconstructed images reflecting the conductance: Least Angle Regression (LARS), ElasticNet, and Artificial Neural Networks (ANN). Of the above methods, the best results were obtained using ANN. However, the LARS method in terms of fidelity representation is very similar to ANN. Table 2 presents a summary of the RMSE values for cases of 2D and 3D imaging in relation to 3 methods tested: ANN, LARS and ElasticNET. The lowest values of the indicators, demonstrating the highest imaging quality, were obtained for ANN. Both LARS and ANN can be successfully used in the EIT tomography dedicated to the reconstruction of moisture in masonries and building walls. In comparison to other, previously used algorithms, these methods allow obtaining precise images with sufficient resolution to perform an effective and error-free analysis of the moisture content of the walls. It is worth noting that taking into account the possibilities of spatial image creating, the LARS and ANN methods are more reliable than invasive methods requiring the sampling of masonry. Table 2. Comparison of the quality of different imaging methods.

Method ANN LARS ElasticNET

RMSE 2D Samples

3D Samples

0.106301 0.122599 0.282520

0.010819 0.029800 0.036500

It is also important that the presented algorithms, used in connection with the EIT system and specially designed electrodes, have large application possibilities. Their basic advantages are functionality, reliability, measurement accuracy and reasonable price. The method is universal due to the possibility of applying to masonry and walls of the various structure, thickness and moisture level. An important role is also played by the speed of the computed tomography scanner. Output images are obtained in real time. The electrical impedance tomography proposed in this article enables the creation of a new non-invasive technique for measuring the humidity of building walls. The EIT has been used to determine the conductivity distribution in specially constructed wall models made of light concrete blocks or bricks. The finite element method implemented in the EIDORS environment has been used to solve the problem. The numerous different lattices were used in the presented numerical models. The analyzed measurement systems contained various electrode distributions. Thanks to this, it is sure that the obtained results are not accidental but repeatable while maintaining similar conditions of the measurement environment.

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The research has provided new and promising results. Future work will be continued thanks to the use of regularization techniques in the optimization process and the hybrid measurement system. These types of hybrid measuring system should be even more reliable in practical applications. It would also be interesting to extend the experimental measurements over time by monitoring the walls at regular intervals. Thanks to this, it would be possible to estimate the speed of spreading the moisture inside walls, as well as its sources and propagation directions. 5. Conclusions The main goal of the work was to analyze the solution based on electrical tomography to study the moisture of walls. Non-destructive methods and algorithms have been analyzed and compared, which allow estimation of humidity also inside the wall. A new concept of a non-destructive system based on electrical tomography has been presented. For research purposes, specially designed electrodes were used, which were placed on the tested lightweight concrete and brick blocks. Three machine learning algorithms were tested: Least Angle Regression (LARS), ElasticNet, and Artificial Neural Networks. It was found that all four methods are suitable for practical applications in EIT tomography dedicated to the detection of moisture in building walls, however, the best results were obtained using the LARS method and the specially designed multi-ANNs system. A characteristic feature of the analyzed solution is the division of the modeled object using a specially developed mesh for a set of elements. The color of each individual mesh element corresponds to the conductance value (in the EIT tomograph). Thanks to this approach the number of information determining the reconstructive picture was large enough to guarantee a sufficient resolution of tomography imaging. The presented research results contain relevant information that may contribute to the acceleration of the development of computational intelligence and machine learning methods in EIT. The research contributes to the improvement of the tomographic imaging efficiency of known methods in the aspect of algorithms for processing input information (electrical quantities) into images. In addition, enriching an input vector with values other than electrical is an easy way to develop new, intelligent tomographic hybrid systems. Author Contributions: T.R. has developed a research project and methods for testing the moisture of walls using electrical tomography. As an expert in the work of tomography, he was the originator of most machine learning concepts presented in this study. G.K. carried out research especially in the field of artificial neural networks and was responsible for the substantive and editorial aspects of the presented paper. E.K. has implemented statistical methods and developed mathematical models’ descriptions. Funding: This research received no external funding. Acknowledgments: The authors would like to thank the authorities and employees of the Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland, for sharing supercomputing resources. Conflicts of Interest: The authors declare no conflict of interest.

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