1. Introduction

This paper presents a study on the application of 3D digital image correlation (DIC) for examination of the dimensional response of model canvas paintings to changes in environmental conditions, in particular relative humidity (RH). This work was initiated within a larger project on new methods and materials for infilling voids in canvas supports. It came out of the need to complete comparative laboratory tests of properties of materials with examination that would directly show the behavior of traditional and new types of repairs introduced into a structure of a painting and their interaction with the original (the results will be presented elsewhere). It was assumed that it is essential to find a method that would allow the testing of the materials in conditions closest to the natural environment—in paintings on wooden stretchers subjected to changes of climatic conditions that can occur in their environment, optimally also in original paintings.

Canvas paintings are very complex multilayer composites, composed usually of a textile support stretched on a rigid or keyed wooden frame, a glue size, a ground, design layers, and a varnish (Fig. 1). These various and hygroscopic materials independently respond to variations of the RH in their environment [1–5]. The mechanical properties and dimensional stability of a canvas painting depend thus on the composition of the structure and properties of its constituents. It is difficult to specify a general model that could be applied to all canvas paintings, even in frames of a certain period of art history. For instance, advanced research on properties of 19th century paintings based on laboratory tests [6–9] and on the analysis of the relationship between the details of structure and state of preservation of original paintings [10] still shows a great variety of cases within this group. There is an infinite variety of combinations of materials introduced into painting technology that has been used over centuries. Not only the selection of materials but even the way they were applied—cold or hot [1,4,11], thin or thick—can profoundly affect the properties and behavior of a particular painting. Also the different age and degradation processes that paintings undergo in various storage conditions influence their mechanical properties and dimensional response to changes in RH. Usually the structure of the weave and particular threads of canvas [4,12–14], as well as the type and characteristic of the size layer, influence sensibility to climatic changes of the surrounding and extent of global deformation of all the structure, although mostly the composition of ground and paint layers (the type of pigments, fillers, binders, and their ratio) also determines the final behavior of the structure [5,15]. Since canvas paintings are often mounted on a considerably stiff stretcher, expansion of glue size and/or canvas support causes loss of tension of the painting, slacking, sagging, and out-of-plane bulging and waving of all the structure. On the contrary, contraction of these layers causes an increase in tension and inner stress in the structure, leading in extreme situations to such types of damage as plastic deformation of certain layers, formation of tears in canvas, cracks, delamination, and flaking in ground and paint layers. The strain/stress distribution is not uniform on all of the surface of the canvas painting tensioned on the stretcher [16,17]. Moreover, the structure of the composite is not symmetrical over its cross-section. The asymmetry of the composite is responsible for local out-of-plane deformations that occur when continuity/homogenity of particular layers or all the structure is disturbed or lost. The plane of a painting can also be affected by inappropriate treatments in areas of damage. For instance, some repairs: glue joints, patches, or infillings made of materials of parameters different from the ones constituting the original or changing local water vapor permeability and moisture absorption of a painting are able to disrupt the homogeneity of the structure significantly and ultimately disturb the integrity of the paintings (Fig. 2).

 

Fig. 1. Multilayer structure of a canvas painting (exemplary cross-section from 20th century painting). The structure includes: a canvas (a), a glue size layer (b), a ground layer (c), oil paint layers of different thicknesses (here with high impastos) (d), and a varnish (e) [photo W. Grzesik].

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Fig. 2. Fragment of 17th century painting with deformation of the surface: (a) unfitting pattern of deep craquelures, cupping, loss, and delamination of ground and paint layer. Conservation treatment revealed in this spot improper and incompatible repair done in the second-half of 19th century and (b) Thick layer of a rigid gesso (visible after removal of overpainting) and (c) unfitted canvas inserts (visible after removal of secondary gesso) [photo. A. Hadała, M. Zacharska [18]].

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Inhomogenity, anisotropy, and variations in the structure and state of preservation make examination of canvas paintings as well as the choice and evaluation of methods for their conservation very difficult. Such studies demand a universal yet individual and flexible approach and 3D full-field measurements. For this reason, standard point-wise measurement devices, such as strain gauges and optical fiber sensors [19–21] are not sufficient.

Canvas paintings mounted on traditional wooden stretchers in certain conditions can be relatively flexible. The range of displacements resulting from changes of RH can even be of a range of a few millimeters; additionally their out-of-plane displacements are much higher than in-plane displacements [22] and increase with the size of an object. Thus, the main challenge in tracing canvas paintings’ distortions is to capture simultaneously very small local deformations as well as considerably large global ones over the entire surface and in a wide range.

The full-field 3D measurements of an arbitrary displacement vector of a surface of works of art, which are caused by mechanical or small temperature load, are usually performed with holographic interferometry (HI) and electronic speckle pattern interferometry (ESPI). The comprehensive reviews of these methods when applied to works of art were published by Dulieu-Barton et al. [23] and Tornari [24]. In the case of canvas paintings, interferometric techniques are effective for localizing and identifying local defects and discontinuities of the structure, such as cracks or delamination [17,25–27] rather than for full-field measuring of large-scale, real-time deformations. Therefore, due to a high sensitivity, ESPI was very successfully used in laboratory tests to generate maps of in-plane strain/stress distribution in small tensioned canvas painting samples for assessment of conservation methods for local tear mending [27], lining [28], and system of mounting on a stretcher [17]. The method was improved and 3D ESPI was applied permitting simultaneous in- and out-of-plane measurement [29].

However both techniques, HI and ESPI as interferometric ones, require a highly stable optical setup and a stable environment between the object and the measuring instrument, and thus it is difficult to use them in practice for tests of the environmental response of canvases. In such tests, even small thermal or RH gradients that occur during a change of conditions lead to fluctuations of the refractive index of the air and thus a decrease of signal-to-noise ratio. Placing the object in a transparent environmental chamber does not solve this problem entirely due to unwanted reflections from the glass that severely reduce the field of view that can be measured. Specifically, in the full-field measurements performed with ESPI on canvasses on a stretcher, the size of the object is limited and it works best for in-plane displacements. Additionally, global out-of-plane displacements often exceed the measurement range [29], when calculated versus the first reference image. However, this can be extended with a more complicated processing scheme and sufficient sampling frequency over the time of displacement [30].

The above-mentioned problems inspired the search for alternative image-based methods that would satisfy the requirement of the simultaneous measurement of all displacement components in a relatively broad range (up to a few millimeters) and with usually sufficient accuracy (a few micrometers). Since high-resolution CCD and CMOS detectors became available, DIC appeared to be the method to satisfy these requirements. It has been used successfully for examination of strains in tapestries [31–33]. The DIC method [34] enables noncontact, full-field measurements of displacements, simultaneously in two (2D DIC) [35–37] or three dimensions (3D DIC) [38–40], with an accuracy scalable with the field of measurement. The additional advantages of 3D DIC, when compared to the interferometric techniques, is an inexpensive and portable hardware setup consisting of two cameras mounted on a rigid frame, an incoherent light source, and a computer. The significant limitation of the DIC method is the necessity to provide a surface of an object with a stochastic texture. The 3D DIC method was applied for investigation of distortions of canvas paintings for the first time in 2011 [41–43]. Tests described in [42,43] showed promising results of using a natural texture of paint layer as a stochastic pattern permitting DIC measurements. In parallel, an experiment focused on measuring displacements of model canvas paintings induced by changes of RH was performed in a climatic chamber. The homogeneity of these specimens covered with an artificial stochastic texture was disturbed by infillings of regular gaps, mended tears, and of patches on the reverse. The aim of the test was to prove the feasibility of 3D DIC for detection of minor defects in the structure of canvas paintings and evaluation of the correctness of different repair methods. In the case described in [41] the modern, polyester-cotton ($65\%:35\%$) plain weave canvas commercially primed with acrylic ground was used to limit uncertainties connected with manual preparation of the sample and the out-of-plane distortions limited enough so the direct monitoring of displacements and strains in the area of repairs was possible. However, in the case of tests of specimens on a modern linen canvas, the standard analysis of displacements performed with the DIC method shows that the global out-of-plane movement of the support is so significant that it obscures possible deformations caused by repairs in the canvas structure. Such a situation can also take place in the case of old paintings on linen canvases and wooden stretchers when they lack tautness due to a progressive decrease in tension caused by stress relaxation, repeated environmental stress, and creep [44,45]. To bring out the information on how the repairs influence the out-of-plane displacements, a postprocessing method has to be developed.

In this paper, the specimen and the experimental setup are described in Sections 3 and 4. The results and their interpretation are presented in Section 5. Finally the applicability of the proposed experimental data analysis methodology for planning and supporting of conservation works is discussed in Section 6.

2. Description of 3D DIC

3D DIC is a technique that combines 2D DIC (for in-plane displacement analysis) with stereovision (for 3D shape measurement) [34,46]. 2D DIC measurement consists of the acquisition of a sequence of grayscale images of an object, over a certain area of interest (AOI). One of the images serves as a reference for the others (deformed). The information is extracted from changes of intensity in the images. The reference image is divided into small subsets (subimages). The software searches for the most similar subset in all other subsequent images, using the maximum zero-mean normalized sum of squared difference function criterion. The algorithm [34] is insensitive to an offset and a scale in lighting. The location of the center point of the most similar subset found in a deformed image defines the displacement vector. In order to facilitate matching, each subset needs to be sufficiently distinct in the aspect of intensity variations. Therefore, a random speckle pattern is applied (e.g., spray paint, sticker paper, water decals) onto the object within the AOI. By using two cameras for observation of the same AOI, it is possible to obtain the 3D shape of the surface of an object and measure out-of plane displacements.

In general, the accuracy of DIC strongly depends on the phase accuracy of the interpolation filter, used to reconstruct gray values at noninteger locations of the images. In order to minimize the phase error of the interpolation filter, cubic B-spline and optimized eight-tap interpolation filters have been chosen for subset matching (using VIC 3D software). According to [34], this approach minimizes errors introduced by the interpolation filters to less than 0.001 pixels. However, it must be pointed out that in real applications, the accuracy of measurements depends also on other sources of errors, such as image noise and stability of experimental condition. The analysis presented in [34] shows that utilization of optimized interpolation algorithms allows one to achieve 0.01 pixel accuracy of image displacements, for images contaminated by Gaussian noise. The papers [47,48] confirm that a variation in DIC measurement results can be limited by isolating DIC setup from vibrations and isolating cameras from temperature variations. In the case of presented measurements, the experimental conditions were carefully controlled to limit the variability of results: the DIC setup was fixed to the optical table (mechanical unstability less than 2 μm) and LED lights were used in order to avoid temperature changes larger than 1°. It is worthwhile to note that the out-of-plane measurement error is larger than the in-plane one and strongly depends on a stereo angle [34,47,48].

As mentioned above, displacements are the quantities obtained directly from DIC. Displacements in the $x$ direction are given as $u$ in [pixels] and after scaling are expressed as $U$ in [mm]; similarly displacements in the $y$ and $z$ direction are given as $v$ [pixels], $V$ [mm], and $w$ [pixels], $W$ [mm], respectively, while testing mechanical properties of artworks along with displacements and strains at the surface of an object should also be considered [12]. Strains (${\epsilon }_{xx}$, ${\epsilon }_{yy}$—strains along $x$ and $y$ coordinates) can be calculated from displacements, as described in [34,36]:

For strain analysis, Vic-3D 2009 software [49] was used. From available tensor types [Lagrange, Hencky (Logarithmic), Euler-Almansi, Logarithmic Euler-Almansi [50]], the Lagrangian finite strain tensor was chosen, since the velocity and displacement fields of the object have been determined and numerically differentiated in space, at each point of view [34].

3. Model Canvas Painting

As described in the Section 1, the main goal of the described work was to evaluate the possibility of extracting detailed data on small local displacements from strongly dominating global deformation of the canvas. Thus, the specimen chosen to demonstrate the developed postprocessing method (Fig. 3) was made of a new, unbleached, fine plain-weave linen canvas, which was showing considerably little dimensional stability when dampened, despite prior washing and shrinking in boiling water.

 

Fig. 3. Specimen used in experiment. (a) Template of the model painting (orange—fillings in canvas, grey—fillings in ground layer, blue—a patch adhered to the reverse of canvas, red—position of cross-sections analyzed in Fig. 7). (b) Face of the specimen during preparation with repaired defects in canvas support. (c) Backside of the specimen with repairs of the canvas visible. (d) Face of specimen with unified ground layer and a random speckle pattern sprayed on the surface.

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To imitate the common construction of a canvas painting, the textile support was stretched over a rectangular keyed pinewood stretcher with a hand tension. To minimize the creep effect, the weft of the canvas was in a vertical direction. The size of the model $-40\mathrm{cm}$ wide and 30 cm high—was limited by the dimensions of climatic chamber used for the experiment. A cold 10% water solution of hide glue was used as a size and then an emulsive gesso [comprising chalk, methylcellulose (MC), polyvinyl alcohol (PVA) with plasticizers: Venetian turpentine and dammar in oil of turpentine] was applied as a ground layer. This formula has been used and tested for the past 35 years [51] at the Department of Conservation of Painting and Polychrome Sculpture of Nicholaus Copernicus University as a material for infilling emulsive and oil grounds in canvas paintings. It was chosen as a primer since in conservation practice it could cover tested infillings and surrounding original canvas. To create controlled and easily detectable discontinuities in the structure, four sets of regular circular gaps of various sizes ranging from 4 to 12 mm of diameter were cut out symmetrically through the stencil in all the four corners of the specimen out of a zone protected by the wooden stretcher, according to the prepared template [Fig. 3(a)]. Additionally, two L-shaped cuts were made in the central part of the painting. The cuts were fixed and the sets of gaps were filled in with various traditional and experimental materials [Figs. 3(a) and 3(b)] different in respect of structure and assumed various mechanical properties and dimensional stability. Introducing four different kinds of infilling materials was supposed to permit direct comparison of their possibly different interactions with the structure of specimen. The full description of the materials and methods of conservation treatments is given in Table 1. The surface of the painting was unified with fillings made with the same gesso as used for priming. It was decided to limit the layers of specimen to the canvas and a uniform ground layer to permit visual observation of possible deformation in area of repairs and still to minimize uncertainties in the interpretation of results. To allow image correlation, a random speckle pattern was sprayed onto the face of the specimen with a black ink [Fig. 3(d)] and then secured with two layers of varnish: Neutral Firnis (Schmincke) applied in even layer all over the surface.

Table 1. Materials and Methods Used for Repairs in the Canvas of a Specimen

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4. 3D DIC Setup and Climate Control System

Measurements were performed with the 3D DIC system that comprises two CANON EOS 5D Mark II ($5616\times 3774$ pixels) cameras equipped with 28 mm CANON lenses, set on an angle of 30° and pointing to the same area (AOI) at the specimen (Fig. 4). Before performing the experiment, the system was calibrated with utilization of the commercial software VIC 3D. Standard deviation of residuals for 22 views of the calibration target and from both cameras was 0.058. Distortions of the 28 mm lenses were included in the calibration process. A combined distortion model in the form of the second-order polynomial (with two distortion coefficients) was used [34]. The setup and the investigated specimen were mounted on an optical table to provide mechanical stability. In order to ensure sufficient lighting, two 100 W LED lamps (8500 lm) equipped with a light diffuser (“soft box”) were used. The utilization of LED lamps ensured that the investigated object was not being overheated, which is important from the point of view of preventive conservation practice. The capture of images by both cameras was synchronized and pairs of frames were collected every 20 seconds during 1 h and 38 minutes of the experiment. The field of view (FOV) was $0.6\mathrm{m}\times 0.4\mathrm{m}$. The analysis of collected data was carried out with the commercial software VIC-3D [49].

 

Fig. 4. 3D DIC setup with the model painting and orientation of the coordinate system (a) scheme and (b) view.

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The sample was subjected to an environmental stress caused by rapid changes of RH from 21% to 70% (within 57 minutes) and back to 20% (within 41 minutes) in a custom-designed climate control system. It consists of an airtight transparent environmental chamber connected by tubes with a simple manually operated conditioning unit. The unit is equipped with a small fan providing air circulation in the system and a space for placing humidifying and dehumidifying materials. The rise of humidity was achieved by inserting a container of water and a set of air-permeable and water-conducting nonwoven fabrics in the unit. For dehumidification, a set of bags filled with silica gel absorbing moisture was used. The rate of change of humidity was controlled by changing the level of the effective area of input materials and the speed ($0.23–0.89\mathrm{m}/\mathrm{s}$) of an air stream.

The humidity profile was designed according to a measurement (performed by the authors) of climate conditions in historic buildings where transient jumps of RH up to about 10% within 10 minutes were recorded occasionally. The range of humidity was extended over naturally occurring to c. 50% to provide a wide range of conditions in which a large global displacement was expected and in which incompatibility of materials could appear. The temperature was stabilized to a value of 24°C by an air-conditioning system in the laboratory. The temperature and RH inside the chamber were monitored and recorded with a HygroClip S sensor from Rotronic (Switzerland). To enable optical measurements, glass with an antireflective coating [Clear Color Plus (2 mm), Nielsen] was used for the camera facing the chamber wall. The painting (weft of canvas in the vertical direction) was mounted vertically and the coordinate system was assigned to describe the displacements [Figs. 4(a) and 4(b)].

5. Results

From the reference pair of images (captured at the start of the test, at RH: 22.2%) and the sequential pair of images (captured for subsequent RH values), the sets of displacement and strain maps were calculated and linked with humidity data. The sequence of maps was combined into video animations (Media 1). These visualizations are very useful for qualitative inspection and observation of overall deformation.

The shape of reference surface ($z$ axis expanded and false-color-coded) is shown in Fig 5(a). It is worth noting that it shows that already before the experiment the canvas painting was not highly tensioned over the stretcher. Also, a small disturbance of planarity of the surface was recorded, as a result of manual introduction of repairs in canvas and ground layer. For detailed analysis, the displacement and strain maps obtained for the maximum value of RH, i.e., 70,0% (Figs. 5 and 6), will be discussed. The displacement maps (false-color-coded and presented separately for $U$, $V$, $W$ displacement components—a key to symbols of displacements and strains is given in Section 2) are shown in a diagonal perspective as seen from Camera 2, superimposed on the b/w photography from this camera (the rare data)—Figs. 5(b)–5(d). In-plane displacements ($U$, $V$) registered at 70% RH are caused mostly by expansion of the painting due to a rise in RH, although it cannot be interpreted directly as elongation in weft and warp direction for the sample on the relatively rigid wooden stretcher since they are a component of general three-dimensional deformation [Figs. 5(b)–5(d)]. The inversion of canvas reaction that can take place at higher values of RH in linen canvases sized with animal glue size (usually above 70% RH but sometimes at its lower values [8,14]) was not observed. Expansion of the hygroscopic constituents of the structure resulted in a loss of tension and finally waving and bulging of the canvas support in an out-of-plane direction toward the back of the specimen [Fig. 5(d)]. Low-value negative in-plane displacements ($U$, $V$) at the edges of AOI, which were close to the edges of the specimen could also have been caused by expansion (relaxation and/or swelling) of the wooden stretcher but this was not measured independently. Stretching of the specimen at hand tension and uneven keying-out of the stretcher could be the reason for the slightly diagonal orientation of global deformation and some irregularity in displacements along the individual boarders of the stretcher.

 

Fig. 5. (a) Shape of reference surface. Displacement maps at maximum (70,0%) RH, (b) $U$ (in $x$ direction), (c) $V$ (in $y$ direction), and (d) $W$ (in $z$ direction); white lines L1 and L2 indicate position of a cross-section analyzed in Fig. 7, also the location of repairs in canvas: 1E, L, 2E, 3E, 4E is marked in white. The sequence of displacements maps and information of RH changes were combined into video animation Media 1.

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Fig. 6. Strain maps: (a) ${\epsilon }_{xx}$ and (b) ${\epsilon }_{yy}$ at maximum (70,0%) RH. White lines L1 and L2 indicate position of a cross-section analyzed in Fig. 7, also the location of repairs in canvas: 1E, L, 2E, 3E, 4E is marked in white.

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As shown in Fig. 5, the out-of-plane displacement is dominant, and it is an order of magnitude bigger than both in-plane components. It must be noted that on any displacement map there is no clear information how repairs influence the behavior of support. Since the textile was not very tightly tensioned over a keyed wooden stretcher, the global deformation dominates and obscures the local deformation connected with the inserts, mends, and a patch. On the other hand, the strain maps calculated out of $u$ and $v$ in-plane displacement maps (Fig. 6) provide information on the localization of some of the repairs that have an influence on the dimensional response of the canvas painting. Repairs of cuts (L) and inserts in left bottom corner (type 3E) restrain the movement of canvas since strains in these areas (color-coded in violet) are smaller than the surrounding general deformation.

In order to more closely investigate the local behavior of canvas in the areas of repairs, the spatiotemporal maps ($Q$, $t$) were generated (Fig. 7). In this case, the analysis at selected profiles [L1 and L2 in Figs. 5(b)–5(d)] of displacements/strains ($Q=U$, $V$, $W$, ${\epsilon }_{xx}$, ${\epsilon }_{yy}$) as a function in time is performed. The profiles L1 and L2 chosen for this analysis cross all the biggest repairs (1E, 2E, 3E, 4E) and one of the cuts (L), where the biggest deformations were expected. As can be seen from Figs. 7(a)–7(c), the noticeable displacements are related to the global movement of canvas starting and increasing rapidly in the upper levels of RH ($39\%\to 70\%\to 38\%$). The in-plane strains (${\epsilon }_{xx}$, ${\epsilon }_{yy}$) can be calculated using a standard approach [Figs. 7(d) and 7(e)] [34]. Such diagrams visualize development of variations in in-plane strain distribution and are more suitable for indicating inhomogenity in the structure.

 

Fig. 7. Displacements and strains at points along lines L1 and L2 shown as a function of time, (a) displacements $U$ (in $x$ direction), (b) displacements $V$ (in $y$ direction), (c) displacements $W$ (in $z$ direction), (d) strains ${\epsilon }_{xx}$ (in $x$ direction), and (e) strains ${\epsilon }_{yy}$ (in $y$ direction). Corresponding changes in RH are given for reference, the location of repairs in canvas: 1E, L, 2E, 3E, 4E is marked as black lines.

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It must be noted that the in-plane displacements ($U$, $V$) were of an order of magnitude smaller than the observed out-of-plane displacement ($W$). The range of displacement was between $-0.07$ and 0.23 mm for direction $x$ ($U$), $-0.12$ and 0.14 mm for direction $y$ ($V$) and between $-4.04$ and 0.42 mm for direction $z$ ($W$). For conservation-restoration purposes and full evaluation of materials used for repairs in the canvas, it is crucial to get information about the influence of inserts on the out-of-plane deformation of the surface of the painting. However the strain ${\epsilon }_{zz}$ cannot be calculated, as there is no physical meaning of differentiation of $W$ in $z$ direction, since the object surface is a shell projected on $x–y$ plane, without neighborhood in $z$ direction [36].

Therefore, it can be assumed that local $W$ displacements are spatially much smaller than the global out-of-plane deformation, so they have different (higher) spatial frequency characteristics. Such an assumption is fully justified in the case when the repairs are much smaller than the entire AIO. In such a case, it is possible to filter local disturbances using a low pass filter applied in an image or frequency plane [52,53]. This operation will deliver an approximate global displacement map, which can later be subtracted from the original displacement map $W$, yielding the local displacement map ($W^{\prime }$) as shown in Fig. 8.

 

Fig. 8. Scheme of the method for obtaining a local displacement map.

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An average filter that produces the largest edge blurring [52,53] is used as the low pass filter in the procedure shown in Fig 8. The size of the filter depends on the sizes of repairs. In this experiment, the repair sizes were similar, so the filter size was constant over the full field of view. In the case of big differences in sizes of repairs, the size of the filter should be locally adapted according to prior knowledge about an object. The best results were obtained with the filtration mask of $23\times 23$ [points of a displacement map]. It must be noted that the points of a displacement map are not equivalent to pixels in the analyzed images, but they are centers of the subsets (separated here by 9 pixels in the $x$ and $y$ directions). The procedure was performed for all $W$ displacement maps using the Matlab commercial software package. The final local displacement map calculated from the map given in Fig. 5(d) is shown in Fig. 9.

 

Fig. 9. Local displacement map $W^{\prime }$ (in $z$ direction) at maximum (70,0%) RH, with black lines L1 and L2 indicate position of a cross-section analyzed in Fig. 10, also the location of repairs in canvas: 1E, L, 2E, 3E, 4E is marked in black.

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This visualization reveals deformations associated with repairs and also with large-scale bulges of the canvas formed as a response of the canvas to changes in climate conditions. The influence of inserts in canvas on out-of-plane deformation of the surface of specimen can also be evaluated on the basis of the local displacement along lines L1 and L2 shown as a function of time in Fig. 10.

 

Fig. 10. Local displacements $W^{\prime }$ (in $z$ direction) along lines L1 and L2 show as a function of time. Corresponding changes in RH are provided for reference, the location of repairs in canvas: 1E, L, 2E, 3E, 4E is marked as black lines.

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6. Interpretation of the Results

The analysis of deformations (which coincide with the location of the repair in the canvas) may be used to validate the data processing described in this contribution. Inspection of graphs in Fig. 7 compared to respective ones in Figs. 9 and 10 leads to the conclusion that some defects, not detectable by usual analysis of displacements ($U$, $V$, $W$), are seen in strain (${\epsilon }_{xx}$) analysis, and furthermore the defect (2E) not detectable by strain analysis is clearly visible when the local displacement ($W^{\prime }$) is considered. The details are given in Table 2. A plus sign has been placed in the table to mark evident correlation between respective graphs and distortions introduced to the canvas structure.

Table 2. Qualification of the Influence of Particular Repairs in Canvas Support on Deformation of the Surface of the Painting According to Displacements ($\mathsf{U}$, $\mathsf{V}$, $\mathsf{W}$), Strains (${\mathsf{\epsilon }}_{\mathsf{x}\mathsf{x}}$, ${\mathsf{\epsilon }}_{\mathsf{y}\mathsf{y}}$), and Local Out-of-Plane Displacements (${\mathsf{W}}^{\prime }$) (+, Distinct Deformation; −, Neutral Deformation)

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To demonstrate how the presented approach may be used for the analysis, important for the conservation-restoration practice, the exemplary interpretation of the results obtained with the 3D DIC method is given below. The results presented here belong to a broader research on the feasibility of various techniques of tear mending and gap filling. From the results summarized in Table 2, it can be concluded that the inserts 1E (traditional canvas insert—Table 1) and 4E (fibrous pulp insert made of Arbocel 200 and 3%MC, impregnated with 20% Akrylkleber 498 HV) reacted to environmental changes similarly to the original structure of this particular canvas painting—in Figs. 7(a)–7(e) and 10 there is no significant reaction visible. The infilling pulp of repair 2E, varying from 4E just by the type of fibers, in high levels of RH reacted differently than the painting. Temporal deformation of 2E is visible just on the map $W^{\prime }$ (Fig. 10) where it is more concave at high RH than any other place along the cross-section L1 or L2 (hence just $\approx 0,15\mathrm{mm}$ different from surrounding). This could mean that it swells more from the backside that is not covered by gesso. It is still comparably flexible since differences in strains ${\epsilon }_{xx}$ and ${\epsilon }_{yy}$ [Figs. 7(d) and 7(e)] are not distinct. On the contrary, the joint of cut L and the insert 3E are detectable particularly at maps of strains ${\epsilon }_{xx}$ [Fig. 7(d)]. Almost constant and permanent low level of strain at the edge of 3E infilling (dark blue line on the map) can indicate a development of the crack on the right side of insert 3E during the experiment. The sharp border in this place on the map of local displacements $W^{\prime }$ (Fig. 10), and the difference between the area of the insert and the painting in the end of test at the map ${\epsilon }_{yy}$ [Fig. 7(e)] seem to confirm this hypothesis. The appearance of the crack can be caused by an incompatibility of the insert or damage of the joint that happened during execution of repair. The joint of cut L made of nonhygroscopic Polyamid Welding Powder is reacting less than the surrounding area of canvas painting [strain maps ${\epsilon }_{xx}$ at Figs. 6(a) and 7(d)] and its vicinity has a tendency to temporal out-of-plane deformation at high RH levels (Fig. 10).

It is worth mentioning that the proposed analysis of strains and local displacements enabled visualization of deformations that are not visible by the naked eye but indicate the possible danger of development of serious damage or deformation of the surface of the painting in the future. For the purpose of evaluation of different materials for infilling voids in canvas, a further more detailed quantitative analysis of behavior of all kinds and sizes of the repairs localized precisely could be carried out, thanks to the template of specimen (described in Section 3).

7. Conclusions and Future Works

In this paper, the test of applicability of the 3D DIC method for evaluation of restoration treatments for canvas paintings was presented. It delivers quantitative data on displacements and strains appearing in paintings in response to certain changes of the surrounding climate. The presented methodology can support a choice of materials and methods in art conservation.

The proposed modification of data processing in 3D DIC allows the detection of local out-of-plane displacements resulting from conservation treatments even in the presence of significant global displacements of an object. The filtering procedure is simple, time effective, and provides results with accuracy sufficient in most of the cases to recognize the quality of repairs. However in order to serve as an arbitrary tool for evaluation of repairs in composite materials, further work on the filtering algorithm should be conducted in order to provide full adaptivity to the sizes and shapes of the local repairs. The presented methodology, due to proper implementation of local out-of-plane displacement information, supports effectively the analysis based on ${\epsilon }_{xx}$ and ${\epsilon }_{yy}$ strain maps. Also, the proposed spatiotemporal visualization of load displacement/strain changes seems to be an excellent tool for future monitoring of canvas paintings and other composite materials behavior, due to environmental changes.

It is worthwhile to note that in the presented laboratory study, the approach common for the experimental mechanics to apply the stochastic texture by putting additional materials on a surface of the investigated object was used. In the case of actual artworks, this kind of treatment, even with the use of inert and theoretically reversible materials, would be in contradiction to the principle of minimum necessary interference with the original structure. Therefore, in future experiments on original canvas paintings, only the natural texture can be considered as a source of correlation data, provided that the features on a painting have sufficient local contrast so that a subset size can be identified, which can be accurately matched across the image. Some painting surfaces, such as the impressionistic technique with a vibrant texture of optically mixing relatively small, thin, yet visible brush strokes of different colors can prove to fulfill this requirement. Also distinct brush strokes, craquelure pattern, uneven texture of canvas, together with discoloration, dirt deposits, etc., can improve the stochastic pattern of the natural painting surface. The preliminary results [42,43] obtained with selected pictures from the Nicolaus Copernicus University Museum, hold the promise of at least limited applicability of the DIC method to real canvas paintings. The proposed methodology, when linked with analysis of materials and structure of particular objects, could be used to broaden the general knowledge on properties, processes, and phenomena relevant to canvas paintings.