Parametric Evaluation of Errors Using Isolated Dots for ... - MDPI

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Parametric Evaluation of Errors Using Isolated Dots for Movement Measurement by Image Cross-Correlation Belen Ferrer 1, * 1 2

*

ID

and David Mas 2

Civil Engineering department, University of Alicante, Carretera San Vicente del Raspeig, s/n, 03690 Sant Vicent del Raspeig, Spain Institute of Physics Applied to Sciences and Technologies, University of Alicante, Carretera San Vicente del Raspeig, s/n, 03690 San Vicente del Raspeig, Spain; [email protected] Correspondence: [email protected]; Tel.: +34-965-903-400

Received: 22 December 2017; Accepted: 6 February 2018; Published: 9 February 2018

Abstract: Digital Image Correlation (DIC) is a common tool for assessing the movement of objects in a scene. Among others, one of the most popular techniques consists of tracking a dotted texture imitating speckle patterns. In this work, we analyzed the individual dots that form this pattern in order to propose an optimum size, shape, and dynamic range that allows minimizing the tracking error. Tracking was accomplished by using normalized cross-correlation with peak interpolation in order to obtain subpixel accuracy. For the models here used, we show that dot radii of 30–40 px with 150 gray levels are enough to obtain an accurate subpixel tracking resolution. Also, we show that 0.002 px is the performance limit of this technique, being this limit in accordance with the experimentally achievable subpixel limit. Keywords: normalized cross-correlation; subpixel tracking accuracy; simulated speckle; error; movement measurement

1. Introduction In the latest years, image correlation has been widely used in a broad spectrum of fields, such as geothecnics [1], structural health monitoring [2], particle velocimetry [3], material characterization [4], or medicine [5], among others. In most of these applications there is not a clear target to follow, so the surface texture is used instead. In some cases, the surface is soft and the texture does not have enough contrast to allow an effective application of tracking methods; therefore, an artificial pattern is painted on the surface. This pattern has to be complex enough to allow the detection of local movements and deformations through the whole specimen under study. Thus, dotted, speckle-like patterns are often used (see Figure 1). These patterns (often called “simulated speckle”) can be easily painted on a surface by using some paint with an atomizer. This easy preparation, together with the uniformity of the points and the high contrast between points and background, makes this pattern suitable for dynamic correlations. In fact, some commercial software has been developed using Digital Image Correlation (DIC) and speckle patterns added or painted in the surface, such as GOMCorrelate [6], Ncorr [7] or GeoPIV [8].

Sensors 2018, 18, 525; doi:10.3390/s18020525

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Figure 1. (a) Texture resulting after painting with an atomizer on a metallic surface, after [9];

Figure 1. (a) Texture resulting after painting with an atomizer on a metallic surface, after [9]; (b) detail (b) detail of the texture with isolated dots. (b) of the texture with isolated dots.(a)

When recording a series of images during the movement of an withsurface, a texture, the[9]; details Figure 1. (a) Texture resulting after painting with an atomizer on object a metallic after in the(b)surface together with theduring object and thus they appear in consecutive differentthe images When recording series of images the movement of an object with a texture, details in detail ofmove theatexture with isolated dots. with the only together differencewith of their withthus respect toappear the image edges. DIC calculation the surface move theposition object and they in consecutive different between images with two consecutive frames provide thewith relative displacement of one pattern with respect to thethe previous When recording a series of images during thetomovement of edges. an object with a texture, details two the only difference of their position respect the image DIC calculation between one, the movement description canobject thus be obtained. in theand surface move together with the and thus they appear in consecutive different images consecutive frames provide the relative displacement of one pattern with respect to the previous one, DICdifference is definedof ontheir discrete sets, with the tracking is limited one pixel, which is not with Since the only position respect resolution to the image edges. to DIC calculation between and the movement description can thus be obtained. enough for the majority of applications. Subpixel resolution can pattern be achieved interpolation which two consecutive frames provide the relative displacement of one with by respect to the previous Since DICeither ismovement defined on discrete sets, the tracking resolution is limited to onefunction pixel, which is is applied on the images that are prior to calculate the correlation [10] one, and the description canbeing thus compared be obtained. not enough for the majority of applications. Subpixel resolution can be achieved by interpolation or onSince the DIC correlation function itself,sets, in order to increase the accuracy location is defined on discrete the tracking resolution is limitedoftothe onemaximum pixel, which is not which is applied either images that are being compared prior toby calculate the correlation function [11]. The first on approach is commonly used for assessing in solid materials since enough for the majority ofthe applications. Subpixel resolution candeformations be achieved interpolation which function [10] an or on the correlation function itself, in order the accuracy of theofmaximum it allows efficient implementation deformation mappings. The second one is faster and easy is applied either on the images that areofbeing compared priortotoincrease calculate the correlation function [10] implementation in non-deforming subsets. is tracking isolated orlocation sparse or on the correlation function itself, in order to usually increasepreferred the accuracy of the maximum location function [11]. The first approach isItcommonly used for for assessing deformations in solid objects through image cross-correlation and it is often used in particle tracking and velocimetry. function [11]. The first approach is commonly used for assessing deformations in solid materials since materials since it allows an efficient implementation of deformation mappings. The second one is faster In implementation [12], the authors proposed a of technique for measuring the in faster atracking concrete probe it allows an efficient implementation deformation mappings. Thedeformation second one is and ofisolated easy or and of easy in non-deforming subsets. It is usually preferred for under loading–unloading cycles by subsets. tracking It theisdisplacements of theforsurface defects. These implementation in non-deforming usually preferred tracking isolated or defects sparse sparse objects through image cross-correlation and it is often used in particle tracking and velocimetry. were due to small bubbles that remainedand added to theused formwork surface during concrete hardening objects through image cross-correlation it is often in particle tracking and velocimetry. In [12], the authors proposed a technique for measuring the deformation in a concrete probe (see Figure used method consisted in dividing the image small subsets a mosaic In [12],2). theThe authors proposed a technique for measuring theindeformation in forming a concrete probe under cycles bybytracking ofthe the surface defects. defects soloading–unloading thatloading–unloading each of the subsets contained a small the number of hollows.ofThe test machine andThese the These temporal under cycles tracking the displacements displacements surface defects. defects were were due to small bubbles that remained added the formwork surface during concrete hardening resolution the camera were adjusted orderto allow very subtle changes between consecutive due toofsmall bubbles that remainedin added totothe formwork surface during concrete hardening (see Figure 2). used method consisted in dividing theimage image in small subsets forming a mosaic frames. By The comparing each subset with the one ininthe following frame, theaauthors (see Figure 2). The used method consisted incorresponding dividing the small subsets forming mosaic so that each of the subsets contained a asmall number of hollows. hollows. The test machine the temporal determined the relative shift of the bubbles thus obtained theThe probe deformation asand a function of so that each of the subsets contained smalland number of test machine and the temporal time. Tracking the concrete surface was done by calculating the normalized cross-correlation between resolution of the camera were adjusted in order to allow very subtle changes between consecutive resolution of the camera were adjusted in order to allow very subtle changes between consecutive twoBy consecutive subsets and then with performing a local peak interpolation infollowing order to achieve frames. By comparing each subset withthe the corresponding corresponding one following frame, thesubpixel authors frames. comparing each subset oneininthethe frame, the authors accuracy. The proposal resulted in an accurate and fast method for the analysis of concrete determined the relative shift thebubbles bubblesand and thus thus obtained deformation as a as function of determined the relative shift of of the obtainedthe theprobe probe deformation a function of deformation. time. Tracking the concrete surface was done by calculating the normalized cross-correlation between time. Tracking the concrete surface was done by calculating the normalized cross-correlation between two consecutive subsets and then performing a local peak interpolation in order to achieve subpixel two consecutive subsets and then performing a local peak interpolation in order to achieve subpixel accuracy. The proposal resulted in an accurate and fast method for the analysis of concrete accuracy. The proposal resulted in an accurate and fast method for the analysis of concrete deformation. deformation.

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Figure 2. (a) Concrete surface pattern; (b) enlarged detail of a surface defect. Notice the degraded intensity surrounding the central dot and the low definition of the image at this scale.

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In Figures 1b and 2a, one can notice that the textures of the painted speckle and concrete are similar. At a local scale, both textures show a distribution of isolated dots and therefore both textures can be analyzed following similar methods. We would like to underline that this pattern is not only found in simulated speckle or concrete surfaces but also in other materials like granite, aerogels, particles in liquids, or aggregates like terrains. In [12], we noticed that some characteristics of the single dots, like the object size, the definition, and the brightness, play an important role in the final accuracy of the tracking method. To our knowledge, the specific characteristics of dotted texture images that provide the best results have not yet been analyzed in detail. Recently, a related work from LePage has analyzed the best way of performing the painting of speckle. There, it is shown that using a white paint as a background with black spots on it gives a better accuracy than the opposite, because of the best hiding power of the black paint versus the white paint [13]. From our point of view, this result could also be linked to the number of gray levels included in each case. However, the dots size and the intensity gradient surrounding each point, which is connected to the object definition (see Figure 2b), were not conveniently analyzed. The point size is linked not only to the way in which a dotted pattern is created, but also to the appearance of the recorded image. Different magnifications may provide different point sizes. Additionally, the image resolution will influence the gradient of intensities that appear around the central point. This gradient is indirectly connected to the dynamic range of the registered image and thus is connected to the expected accuracy of the method [14]. Regarding the errors obtained using artificial speckle images and image correlation, a study by Bornet et al. [15] assessed the errors in the measurement of deformations by using three different speckle sizes among other parameters, such as the subset size and the shape function. However, their results did not show conclusive assessments about the speckle sizes or the intensity gradients that appeared around each pattern point. Other studies focused on the errors due to intensity interpolation [16,17] on undermatched subset shape functions [18,19] or on overmatched shape functions [20,21]. The light intensity influence on errors was analyzed by [22] among other parameters, showing that the error potentially decreased as the mean image intensity increased. The influence of the subset size was studied by [23], concluding that larger subsets improve the tracking error. The same conclusion was given in [24]. Notice that all works cited in the above written paragraph refer to speckle and propose a mathematical analysis based on this particular texture, which has a defined range of sizes and spatial statistics. From our point of view, the parameters defining this texture may change with the optical system. High-resolution sensors with high zooming produce patterns in which the characteristics of the individual spots defining the texture become relevant for the analysis, and thus, other points of view may be necessary for an optimum tracking of the surface. In this work, we propose the study of the influence of three main parameters defining the individual objects in a dotted pattern: the point size, the gradient of intensities around the point, and the peak luminance. These parameters have been selected since they can be directly connected with the experimental implementation of a tracking experiment. The brightness is directly related to the illumination of the sample, and the intensity gradient around the central dot is linked to the optical system and the charge-coupled device (CCD) resolution. The dot size is determined by the texture, but it can be modified in the image by changing the magnification through the optical system or by changing the distance from the object to the camera. Other parameters that also affect the errors, such as the type of movement registered, the intensity interpolation, the shape functions, the subset size, or the interpolation function for peak refinement, are maintained fixed. Numerical experiments are used in order to permit changing all parameters and performing arbitrarily small subpixel shifts. The final aim of the paper is to find those configurations that would allow the maximum subpixel resolution under ideal circumstances and thus help to proper configure an experimental test.

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2. Materials and Methods Sensors 2018, 18, x FOR PEER REVIEW

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2.1. Object Design

resolution under ideal circumstances and thus help to proper configure an experimental test.2.

ToMaterials study the influence of all the above mentioned parameters in the detection of movement using and Methods image cross-correlation, different numeric test objects were created. These objects were created so that Object some Designof the characteristics that would influence the results, that is, the size of the point, one can2.1. control the dynamic andinfluence the gradient of intensities [25]. parameters in the detection of movement Torange, study the of all the above mentioned Points were cross-correlation, here approximated bynumeric circles,test since thewere basic pointThese shape is not far from a using image different objects created. objects werevery created so that one can control some of the characteristics that would influence the results, that is, the size circle (see Figures 1 and 2). The design of the object was done by stacking 255 concentricofcircles of point, the dynamictheir range,diameters. and the gradient of intensities value 1the and diminishing In this way, we[25]. built a peak with a top plateau value of Points were here approximated by circles, since the basic point shape is not very far from a circle 255, corresponding with a white color in 8-bits images, which is the most common codification in (see Figures 1 and 2). The design of the object was done by stacking 255 concentric circles of value 1 commercial grayscale their cameras. The size of way, the plateau, by the smallest one on the and diminishing diameters. In this we built defined a peak with a top plateaucircle value (the of 255, top of the figure having highest value) wasimages, used aswhich the point corresponding with the a white color in 8-bits is thesize. most common codification in commercial grayscalefrom cameras. of the defined thetop smallest circle (the onethe on gradient the The radii variation the The firstsize circle inplateau, the bottom toby the one represents of top of the figure having the highest value) was used as the point size. intensities that can be found around the point in the image. This variation was designed as a Gaussian The radii variation from the first circle in the bottom to the top one represents the gradient of curve (Figure 3). The design of the point in this way allowed the parameters of point size and gradient intensities that can be found around the point in the image. This variation was designed as a Gaussian to be changed independently. The gradient is linked to the number of intensity levels; notice that curve (Figure 3). The design of the point in this way allowed the parameters of point size and gradient too stepped gradients will not include all the 256 intensity levels. The width (and slope) to be changed independently. The gradient is linked to the number of intensity levels; notice that of toothis halo aroundstepped the central plateau is connected with definition sharpness of an object. Narrow halos gradients will not include all the 256the intensity levels.ofThe width (and slope) of this halo around the central plateau is connected with the definition of sharpness of an object. Narrow halos represent well focused objects with sharp borders, while extended areas may represent a full variety represent well focused sharpaffected borders, while extended areas may represent a fullundersampled variety of objects, like objects withobjects a softwith profile, by diffraction, unfocused, or even of objects, like objects with a soft profile, affected by diffraction, unfocused, or even undersampled (see Figures 1b and 2b). (see Figures 1b and 2b).

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Figure 3. Point design: (a) upper view; (b) lateral view of its level contour lines.

Figure 3. Point design: (a) upper view; (b) lateral view of its level contour lines. Another interesting parameter to be considered in the analysis is the ratio between the area of the upper part and the area occupied the slope down to the base ofisthe determine Another interesting parameter to bebyconsidered in the analysis theobject. ratio To between thethe area of the radius of the base, the whole figure was binarized using Otsu’s method [26]. If R is the radius of the upper part and the area occupied by the slope down to the base of the object. To determine the radius point and Rb is the radius of the base, the area ratio is: of the base, the whole figure was binarized using Otsu’s method [26]. If R is the radius of the point and Rb is the radius of the base, the area=ratio is:− (1) = −1 2

2

2

πRb −ofπR R was done in positive luminances. The The reader should notice that the definition the target AR = = 2b − 1 2 analysis that is presented below can be easily πRextended Rto negative images just by taking the complement image, i.e., 255-L, being L the luminance of the original positive dot.

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The reader should notice that the definition of the target was done in positive luminances. 2.2. Image Processing The analysis that is presented below can be easily extended to negative images just by taking the complement image, i.e., 255-L, being L the luminance of the original positive dot.

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2.2. Image Processing A series of dot images were used as test benches. Each test was composed of a sequence of frames showing a moving dot defined as explained above. Each frame showed the same dot, shifted 0.005 px from the initial position. The full sequence was composed of 200 frames, thus the total displacement was 1 px. The analytical definition of the object given in the previous section allowed to control Sensors 2018, 18, x FOR PEER REVIEW 5 of 14 small and constant displacements between consecutive images. Also, the frame size was fixed to A series dot imagesthat werethe usedobject as test benches. Each test was a sequence of frames 1500 × 1500 px, thusofensuring representation wascomposed accurateof[25]. a moving as explained Each frame showed the same dot, shifted The showing displacement of dot thedefined dot was checked above. through normalized cross-correlation between the 0.005 px from the initial position. The full sequence was composed of 200 frames, thus the total original frame and the subsequent frames which contained the shifted object. To this end, we used the displacement was 1 px. The analytical definition of the object given in the previous section allowed normxcorr2 algorithm in Matlab.between Noticeconsecutive that this algorithm to control small implemented and constant displacements images. Also,requires the framethat size the was template frame is smaller than the original image. Therefore, the frames with the shifted objects were cropped fixed to 1500 × 1500 px, thus ensuring that the object representation was accurate [25]. The displacement of the dot was checked through normalized cross-correlation between the by removing an external band of 8 pixels in all image sides. No intensity interpolation was done originaland frame the subsequent whichi.e., contained the shifted object. this end,between we used the two on the images noand shape function frames was used, no deformation wasTo allowed the normxcorr2 algorithm implemented in Matlab. Notice that this algorithm requires that the images. In order to achieve a subpixel tracking resolution, the correlation peak was interpolated in a template frame is smaller than the original image. Therefore, the frames with the shifted objects were 3 × 3 pixels neighborhood around theofmaximum, the described in [11]. cropped by removingarea an external band 8 pixels in allfollowing image sides. Noideas intensity interpolation was The peak done on thedone imagesby and no shape was used, i.e., no deformation was allowed between the the peak, interpolation was using the function first terms of a Taylor polynomic expansion around In order to achieve subpixelthe tracking resolution, correlationpoint peak was interpolated resulting two in aimages. quadratic surface, andafinding location of thethe maximum of this analytic surface. in a 3 × 3 pixels neighborhood area around the maximum, following the ideas described in [11]. The Once the complete movement history was obtained, it was compared with the theoretical peak interpolation was done by using the first terms of a Taylor polynomic expansion around the displacement. The error defined as theand difference between andpoint the of theoretical peak, resulting in was a quadratic surface, finding the locationthe of calculated the maximum positionsthis for analytic each one of the 200 steps. Here, we studied the maximum error and the standard deviation surface. Once the sequence. complete movement history was obtained, it was compared with the theoretical of the errors in a full 3.

displacement. The error was defined as the difference between the calculated and the theoretical for each one of the 200 steps. Here, we studied the maximum error and the standard Resultspositions and Discussion deviation of the errors in a full sequence.

For all studied cases, the calculated error showed a rotational symmetry with respect to a 3. Results and Discussion displacement of 0.5: For all studied cases, the calculated showed symmetry with respect to a Error (derror Error (a1 rotational − d) (2) )=− displacement of 0.5:

being d the displacement which is defined in the interval [0, 1] pixels. When the error is large, the (2) ( )=− (1 − ) curve tends to a sinusoidal shape with small oscillations due to error fluctuations, as it is shown in being d the displacement which is defined in the interval [0, 1] pixels. When the error is large, the Figure 4a. In this last case, the maximum absolute error could be found at displacements about 0.25 px curve tends to a sinusoidal shape with small oscillations due to error fluctuations, as it is shown in and 0.75 Figure px. However, error decreased, the be noise inatthe curve increased 4a. In this as lastthe case,simulation the maximum absolute error could found displacements about until it completely smooth shape of the curve, as showntheinnoise Figure 4a, andincreased it became 0.25distorted px and 0.75the px. However, as the simulation error decreased, in the curve untilsimilar to completely distorted4b. theThe smooth shape of as shown in Figure 4a,be and it became similar the graphit shown in Figure location ofthe thecurve, maximum could now anywhere, but symmetry graph shown in Figure 4b. regardless The locationofofthe theerror maximum could now besymmetry anywhere, is butdue to the expressedtointhe Equation (2) remains true, magnitude. That symmetry expressed in Equation (2) remains true, regardless of the error magnitude. That symmetry fact that, iswhen a full pixel displacement is accomplished, the displaced object is exactly equivalent to due to the fact that, when a full pixel displacement is accomplished, the displaced object is exactly the original one but in aoriginal different equivalent to the one position. but in a different position.

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Figure 4. Example of the curve shape in the errors obtained: (a) big error, for a point size of 25 px, 43 intensity levels, and area ratio of 0.28, the maximum error is 0.01174 px; (b) small error, for a point size of 25 px, 232 intensity levels ,and area ratio of 3.84, the maximum error is 0.001315 px.

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Figure 4. Example of the curve shape in the errors obtained: (a) big error, for a point size of 25 px, 43 intensity levels, and area ratio of 0.28, the maximum error is 0.01174 px; (b) small error, for Sensors 2018, 18, 525of 25 px, 232 intensity levels ,and area ratio of 3.84, the maximum error is 0.001315 px. 6 of 14 a point size

The behavior shown in Figure 4a is known as peak-locking or bias locking and it is described in The behavior shown in Figure 4a is known as peak-locking or bias locking and it is described [27]. Briefly, this happens because the peak neighborhood in the correlation function is not in [27]. Briefly, this happens because the peak neighborhood in the correlation function is not symmetric. The use of symmetric interpolant functions makes the result (i.e., the maximum of the symmetric. The use of symmetric interpolant functions makes the result (i.e., the maximum of the interpolant function) tend towards the highest values of the original samples neighborhood. Using a interpolant function) tend towards the highest values of the original samples neighborhood. Using a non-symmetric interpolant function allows to have non-symmetric surroundings without affecting non-symmetric interpolant function allows to have non-symmetric surroundings without affecting the the location of its maximum. location of its maximum. Although different algorithms have been proposed to compensate this effect, this effect is Although different algorithms have been proposed to compensate this effect, this effect is inherent inherent to the method and thus it always appears [28]. In our case, we decided to use the simplest to the method and thus it always appears [28]. In our case, we decided to use the simplest and and most used method (quadratic interpolation), despite the known limitations. Since all the tests most used method (quadratic interpolation), despite the known limitations. Since all the tests were were calculated through the same algorithm, the bias affected all of them in the same way. The use calculated through the same algorithm, the bias affected all of them in the same way. The use of an of an error reduction algorithm shifted the curves and produced lower errors, but the comparison error reduction algorithm shifted the curves and produced lower errors, but the comparison among among different cases remained the same. different cases remained the same. 3.1. 3.1. Influence Influence of of the the Point Point Size Size The The point point size size cannot cannot be be analyzed analyzed as as an an independent independent parameter, parameter, as as itit is is linked linked to to the the gradient gradient and to the number of levels. With large enough point sizes and variances, all 256 intensity and to the number of levels. With large enough point sizes and variances, all 256 intensity levels levels were were present. However, while the point size decreased, for the same variance, the flank width was present. However, while the point size decreased, for the same variance, the flank width was not not enough enough to to include include all all levels. levels. Therefore, Therefore, each each point point size size had had its its own own minimum minimum Gaussian Gaussian variance variance that that allowed allowed the the inclusion inclusion of of all all the the 256 256 intensity intensity levels. levels. In Ingeneral, general,the theminimum minimumvariance variancedecreased decreased with with the point size, but there were local maxima (Figure 5). the point size, but there were local maxima (Figure 5).

Figure 5. Relation Relation between between the thepoint pointsize sizeand andthe the minimum Gaussian variance needed to have Figure 5. minimum Gaussian variance needed to have 256 256 intensity levels. intensity levels.

Furthermore, for different sizes, two points of view may be considered with respect to the Furthermore, for different sizes, two points of view may be considered with respect to the gradient gradient of intensities surrounding the central point. The first one is to consider the area ratio as fixed, of intensities surrounding the central point. The first one is to consider the area ratio as fixed, letting letting the involved number of levels change. The second approach is to fix the dynamic range and the involved number of levels change. The second approach is to fix the dynamic range and let the let the area ratio change. Both points of view were analyzed in the following sections. area ratio change. Both points of view were analyzed in the following sections. 3.1.1. 3.1.1. Influence Influence of of the the Point Point Size Size for for the the Same Same Area Area Ratio Ratio and and Different Different Dynamic Dynamic Ranges Ranges The of the thepoint pointsize sizefor fora aconstant constant area ratio was analyzed in this section. The influence influence of area ratio was analyzed in this section. The The onlyonly way way to maintain constant the area ratio for a changing point size is decreasing the number of levels to maintain constant the area ratio for a changing point size is decreasing the number of levels with with the point size. led That to poorer halos andresolutions. point resolutions. In 6, Figure 6, four different point the point size. That toled poorer halos and point In Figure four different point sizes for sizes for the same area ratio are shown, as an example. Different visualization sizes were also used the same area ratio are shown, as an example. Different visualization sizes were also used in order in to order clearly thepoint different point resolutions. For aradius pointofsize of 100 px,levels a total of clearlytoshow the show different resolutions. For a point size 100 radius px, a total of 256 were 256 levels were so this point was the largest obtained, so thisobtained, was the largest size in ourpoint tests.size in our tests.

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Four different different examples of points with different point sizes and an area ratio of 1. Figure 6. Four 1. Figure 6. Four different examples of points with different point sizes and an area ratio of 1. Figure 6. Four different examples of points with different point sizes and an area ratio of 1.

Theresults results showed that the error quickly decreased asthe the point sizeincreased increased (Figure 7).This This The resultsshowed showedthat that the error quickly decreased as point the point size increased (Figure 7). The the error quickly decreased as size (Figure 7). The results showed that the error quicklyerror decreased as the the standard point sizedeviation. increased In (Figure 7).8,This behavior was found both for the maximum and for Figure it This behavior was found both for the maximum error and for the standard deviation. In Figure 8, it is behavior was found both for the maximum error and for the standard deviation. In Figure 8, it isis behavior wasasfound bothsizes for the maximum error and for the standard deviation. Ingray Figure 8, italso is shown that, the point increased, having a fixed area ratio of 1, the number of levels shownthat, that,as asthe thepoint pointsizes sizesincreased, increased,having havingaafixed fixedarea arearatio ratioof of1,1,the thenumber numberof ofgray graylevels levelsalso also shown shown that,Therefore, as the point sizes increased, having a fixed the areaerror ratiowas of 1,decreasing the number of grayof levels also increased. one may question was whether because thepoint point increased. Therefore,one onemay mayquestion questionwas waswhether whetherthe theerror errorwas wasdecreasing decreasingbecause becauseof ofthe the point increased. Therefore, increased. Therefore, question was whether the error was decreasing because ofthe the point sizeor orbecause because of the theone risemay ofthe the levelsnumber number [14].Therefore, Therefore, inthe the following section, number size or because of rise of levels [14]. inin following section, the of size of the levels number [14]. Therefore, the following section, thenumber number size or because of the rise of the levels number [14]. Therefore, in the following section, the number of levels was fixed to 256 and the influence of the point size was analyzed. Of course, the only way levels was fixed toto 256 and thethe influence ofof thethe point size was analyzed. OfOf course, the only way to of levels was fixed 256 and influence point size was analyzed. course, the only way of levels was fixed tolevels 256 and the influence ofsizes the was pointtosize was analyzed. Of course, the only way tohave have always 256 for different point have different area ratios. have always 256 levels forfor different point sizes was to to have different area ratios. to always 256 levels different point sizes was have different area ratios. to have always 256 levels for different point sizes was to have different area ratios.

Figure 7. Influence of the point size when the area ratio was set to 1. Figure 7.7.Influence ofofthe point size when the area ratio was set to 1.1. Figure7. Influenceof thepoint pointsize sizewhen whenthe thearea arearatio ratiowas wasset setto to1. Figure Influence the

Figure 8. Number of levels involved over point size for an area ratio of 1. Figure 8. Number of levels involved over point size for an area ratio of 1. Figure Figure8.8.Number Numberofoflevels levelsinvolved involvedover overpoint pointsize sizefor foran anarea arearatio ratioof of1.1.

3.1.2.Influence Influenceof ofthe thePoint PointSize Sizefor forthe theSame SameDynamic DynamicRange Rangeand andDifferent DifferentArea AreaRatios Ratios 3.1.2. 3.1.2. Influence of the Point Size for the Same Dynamic Range and Different Area Ratios

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8 of 14 3.1.2. Influence of the Point Size for the Same Dynamic Range and Different Area Ratios To analyze the point size influence with all possible luminance levels, i.e., 256, different objects To analyze the point size influence with all possible luminance luminance levels, levels, i.e., i.e., 256, 256, different objects objects To were created in which the number of levels was set to 256. In all cases, the variation of the halo width In all all cases, cases, the the variation variation of of the the halo halo width width were created in which the number of levels was set to 256. In was adjusted to be minimum for the 256 intensity levels. Figure 9 shows some points having different be minimum minimum for the the 256 256 intensity intensity levels. levels. Figure Figure 99shows showssome somepoints pointshaving having different different was adjusted to be sizes for the same number of gray levels, as examples of all the considered points in this section. In sizes for the of of allall thethe considered points in this section. In this sizes the same samenumber numberofofgray graylevels, levels,asasexamples examples considered points in this section. In this case, the area ratio decreased while the point size increased (Figure 10). case, the area ratio decreased while the point size increased (Figure 10). this case, the area ratio decreased while the point size increased (Figure 10).

Figure 9. Four different examples of points with different point sizes and the same number of levels. levels. Figure 9. 9. Four Fourdifferent different examples examples of of points points with with different different point point sizes sizes and and the the same same number number of of Figure levels.

Figure 10. Area ratio value over point size when 256 levels were involved having the minimum Figure Figure 10. Area Area ratio ratio value value over over point point size size when when 256 256 levels levels were were involved involved having having the the minimum minimum possible Gaussian variance, according to Figure 3. possible possible Gaussian variance, according to Figure 3.

Contrary to the previous analyzed case, the calculated error seemed to be slightly decreasing, Contrary to the previous analyzed case, the calculated error seemed to be slightly decreasing, Contrary the previous case, error seemed be slightly but with a verytonoisy behavior analyzed (Figure 11). Forthe thecalculated standard deviation, this to trend was alsodecreasing, observed. but with a very noisy behavior (Figure 11). For the standard deviation, this trend was also observed. but with a very noisy behavior (Figure 11). For the standard deviation, this trend was also In Figure 12, we compared these values with those obtained for an area ratio of 1 (Figure 7).observed. One can In Figure 12, we compared these values with those obtained for an area ratio of 1 (Figure 7). One can In Figure 12, we compared these values with the those obtained for an areaa ratio of 1 (Figure 7). of One can see that there was a big difference between errors obtained with constant area ratio 1 and see that there was a big difference between the errors obtained with a constant area ratio of 1 and see that there was a big256 difference between the errors a constant area ratiobecause of 1 andof those those obtained using levels up to a point size ofobtained radius ofwith 35 px. This happened the those obtained using 256 levels up to a point size of radius of 35 px. This happened because of the obtained using 256 levels up to a point size of radius of 35 px. This happened because of the reduced reduced number of levels for the points with area ratios of 1. Notice that for points of larger sizes reduced number of levels for the points with area ratios of 1. Notice that for points of larger sizes number of levels for thehad points withvalues area ratios of 1.for Notice foralmost points256 of larger 35 px, than 35 px, both errors similar because thosethat sizes levelssizes werethan involved, than 35 px, both errors had similar values because for those sizes almost 256 levels were involved, both errors had similar values because for those sizes almost 256 levels were involved, even for the even for the area ratio of 1 (Figure 8). even for the area ratio of 1 (Figure 8). area ratio of 1 (Figure 8).

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intensity levels. levels. Figure11. 11.Influence Influenceofofthe thepoint pointsize sizefor for256 256intensity intensity Figure levels.

Figure 12. Comparison between the errors obtained for an area ratio of 1 and for 256 levels Figure Figure12. 12. Comparison Comparisonbetween betweenthetheerrors errorsobtained obtainedforforananarea arearatio ratioofof1 1and andfor for256 256levels levels (superposition of Figures 7 and 11). (superposition of Figures 7 and 11). (superposition of Figures 7 and 11).

Therefore,regarding regardingthe thepoint pointsize sizeinfluence influenceon onerrors, errors,one onecan canconclude concludethat thatititisismainly mainlydue duetoto Therefore, Therefore, regardinglevels the point sizeincluded influencefor oneach errors, onesize, can and conclude that it is mainly due the number of intensity that are point not to the point size itself. the number of intensity levels that are included for each point size, and not to the point size itself. to the number of intensity levels that are included for each point size, and not to the point size itself. Also,notice noticethat thatthe themaximum maximumerror errorfor foran anarea arearatio ratioofof1 1could couldbebeadjusted adjustedtotoa apotential potentiallaw lawand and Also, Also, noticeathat theradius maximum error for an areadid ratio of improve 1 could be adjusted to awith potential law andsizes. that, that, from point of 35 px, the error not significantly larger point that, from a point radius of 35 px, the error did not improve significantly with larger point sizes. from a point of ratio 35 px, the the error did notpoint improve significantly with larger sizes. Therefore, Therefore, forradius anarea area optimal radius wasbetween between and4040point px.Larger Larger sizesmay may Therefore, for an ratio ofof1,1,the optimal point radius was 3030and px. sizes for an area ratio of 1, the optimal point radius was between 30 and 40 px. Larger sizes may require requiremore morecomplicated complicatedand andexpensive expensiveimaging imagingsetups setupsand andwill willonly onlyresult resultininslight slightimprovements improvements require more complicated and expensive imaging setupsmethods. and willRegarding only resultthe in slight improvements in the in the errors, thus resulting in high cost-effective dynamic range, 256levels levels in the errors, thus resulting in high cost-effective methods. Regarding the dynamic range, ifif256 errors, thus resulting in high cost-effective methods. Regarding the dynamic range, point if 256 size levels were were involved, the error had a slight but constant decreasing trend with increasing values. were involved, the error had a slight but constant decreasing trend with increasing point size values. involved, the error had a slight but constant decreasing trend with increasing point size values. 3.2.Influence Influenceofofthe theGradient Gradient 3.2. 3.2. Influence of the Gradient Thevariation variationininthe thegradient gradientwas wasanalyzed analyzedfor fora afixed fixedpoint pointsize. size.AAradius radiusofof2525px pxwas wasselected selected The The to variation in the gradient was related analyzed for apoint fixed size, pointassize. A radius of 25previous px was selected in order have a middle error value to the explained in the section. in order to have a middle error value related to the point size, as explained in the previous section. in order to the have a middle error value related to the point size,of asthe explained inincreased the previous section. Increasing halo radius by means of increasing the variance Gaussian the number Increasing the halo radius by means of increasing the variance of the Gaussian increased the number Increasing the halo radius by means of increasing the variance of the Gaussian increased the number intensitylevels levelsand andalso alsothe theratio ratiobetween betweenareas. areas.Therefore, Therefore,the theinfluence influenceofofthe thegradient gradienterror errorwas was ofofintensity of intensity levelsthe and alsoratio. the ratio between areas. Therefore, the influence of thethe gradient error area was studied through area Figure 13 shows some examples of objects having same inner studied through the area ratio. Figure 13 shows some examples of objects having the same inner area studied through the area ratio. Figure 13 shows some examples of objects having the same inner area butdifferent different surrounding areas. but surrounding areas. but different surrounding areas.

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Figure different examples of points the with same the pointsame size and different surrounding Figure 13. 13.FourFour different examples of with points point size and different Figure 13. Four different examples of points with the same point size and different surrounding gradients. surrounding gradients. gradients.

For a number of levels below 256, the error decreased quickly with the increasing area ratio, with For number of below 256, the error decreased quickly with the area ratio, with Foraapoint number of levels levels below error quickly with the increasing increasing a singular in the area ratio of256, 4.43the that haddecreased an error higher than expected (Figurearea 14).ratio, Fromwith an aasingular point in the area ratio had error higher than expected (Figure 14). From singular ratioof of4.43 4.43that thaterror hadan an error higher than 14). Froman an area ratio ofpoint 1.35 in on,the thearea average maximum was around 0.002 px,expected and the (Figure standard deviation area ratio of 1.35 on, the average maximum error was around 0.002 px, and the standard deviation was area0.0007 ratio px. of 1.35 thestabilization average maximum error px was 0.002 px, and standard deviation was Thison, error around 0.002 is around in accordance with thethe above described case 0.0007 px. This errorerror stabilization around 0.0020.002 px is px in accordance with with the above described case and was 0.0007 px. This stabilization around is in accordance the above described case and also with the hypothesis in [14]. also the hypothesis in [14]. and with also with the hypothesis in [14].

Figure 14. Influence of the area ratio for a point radius of 25 px (from 0 to 256 levels). Figure 14. 14. Influence Influence of of the the area arearatio ratiofor foraapoint pointradius radiusof of25 25px px(from (from00to to256 256levels). levels). Figure

For area ratios larger than 12.42, the number of levels was constant and equal to 256. Therefore, For meaningful area ratios larger than 12.42, the number of levels was constant and in equal to 256. Therefore, the only representation for objects above this area ratio order to avoid the For area ratios larger than 12.42, was the number of levels was constant and equal to 256. Therefore, the only meaningful representation was for objects above this area ratio in order to avoid the influence of the number of levels in the analysis. In this representation, the behavior changed in the only meaningful representation was for objects above this area ratio in order to avoid the influence influence of the number of levels in the analysis. In this representation, the behavior changed comparison with the previous explained cases, and the errorthe increased (Figure 15).inNotice that, ifwith thein of the number of levels in the analysis. In this representation, behavior changed comparison comparison with the previous explained cases, and the error increased (Figure 15). Notice that, if the base of the object enlarged while the(Figure size of15). theNotice upper that, part of thebase pointofas well as the previous explained cases, andkeeping the errorconstant increased if the the object base of the object enlarged while keeping constant the size of the upper part of the point as well the number of levels, theconstant error increased until a value thatasobtained having onlyas enlarged while keeping the size of the reaching upper part of thesimilar point astowell the number of levels, the number levels,that the error increasedthat untilimproved reaching the a value to that obtained having only 43 Thisofmeans the parameter errorsimilar when point size was constant thelevels. error increased until reaching a value similar to that obtained havingthe only 43 levels. This means 43 levels. This means that the parameter that improved the error when the point size was constant was ofthat levels in the image andwhen not the Therefore, the optimum valueofof the that the number parameter improved the error the area pointratio. size was constant was the number levels was the number of levels in the image and not the area ratio. Therefore, the optimum value of the area ratio (equivalent to the gradient) was between 1.35 and value 4. Larger area ratios, spite of having in the image and not the area ratio. Therefore, the optimum of the area ratioin (equivalent to the area ratio (equivalent to the gradient) was between 1.35 and 4. Larger area ratios, in spite of having higher number of levels, did not improve the error obtained. higher number of levels, did not improve the error obtained.

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Figure 15. Influence of the area ratio for a point radius of 25 px (all having 256 levels). Figure 15. 15. Influence Influence of of the the area area ratio ratio for for aa point point radius radius of of 25 25 px px (all (all having having256 256levels). levels). Figure

3.3. Influence of the Illumination (Maximum Intensity and Dynamic Range) 3.3. Influence Influence of of the the Illumination Illumination (Maximum (Maximum Intensity Intensity and and Dynamic Dynamic Range) Range) 3.3. Different illuminations were simulated here for the same point size and same area ratio. The Differentilluminations illuminationswere weresimulated simulated here the same point size same and sameratio. area ratio.effect The here forfor the point size and effectDifferent of having more or less illumination was that thesame maximum value in each area point was The higher or effect of having more or less illumination was the maximum in each point wasorhigher or of having more or lessFor illumination wasillumination, that thethat maximum value invalue each point was higher smaller, smaller, respectively. an optimum the maximum intensity value would be 256, but, smaller, respectively. For an optimum illumination, the maximum intensity value would be 256, but, respectively. For an optimum illumination, intensity value would be 256, but, with with lower intensities, the maximum value the andmaximum the number of levels involved may decrease. This with lower intensities, the maximum value and the number of levels involved may decrease. This lower the maximum valuethe andwhole the number of levels involved may decrease. lower This effect effect intensities, was simulated by downsizing image intensity with a new maximum, than was 256. effect was by simulated by downsizing the whole image intensity a new maximum, than 256. simulated downsizing the whole intensity a newwith maximum, thanlower 256. Figure 16 Figure 16 shows some examples of image variation in the with maximum intensity forlower the same point size and Figuresome 16 shows someof examples ofinvariation in the intensity maximum intensity for the size same point size and shows examples variation the maximum for the same point and area ratio. area ratio. area ratio.

Figure 16. 16.Four Fourdifferent differentexamples examplesofof points with same point same surrounding gradient, Figure points with thethe same point size,size, same surrounding gradient, and Figure 16. Four different examples of points with the same point size, same surrounding gradient, and different maximum intensity levels. different maximum intensity levels. and different maximum intensity levels.

Since the illumination level is linked to the number of intensity levels, the error showed a fast Since Since the the illumination illumination level level is is linked linked to to the the number number of of intensity intensity levels, levels, the the error error showed showed aa fast fast descending trend (Figure 17). As it was observed for the influence of the point size for a constant area descending (Figure 17). 17).As Asititwas wasobserved observed influence of the point a constant descending trend (Figure forfor thethe influence of the point sizesize for afor constant area ratio, once 150 levels were involved, the error descended very slowly with the increment of the area levels were involved, error descendedvery veryslowly slowlywith withthe the increment increment of the ratio,ratio, onceonce 150 150 levels were involved, thethe error descended the number of levels. This result matched with that obtained by Haddadi et al. [22] in which the error number of levels. This result matched with that obtained by Haddadi et al. [22] in which the error due number of levels. This result matched with that obtained by Haddadi et al. [22] in which the error due to luminosity decreased with the rise of the mean intensity values. In our work, as the to luminosity decreased with the risethe of the intensity In our work, thework, background due to luminosity decreased with risemean of the mean values. intensity values. In as our as the background had a constant intensity, the increase of the maximum intensity led to an increase of the background had a constant intensity, the increase of the maximum intensity led to an increase of the mean intensity as well. mean intensity as well.

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Figure17. 17.Influence Influenceofofillumination illuminationfor foraapoint pointradius radiusofof25 25px pxand andan anarea arearatio ratioof of12.42. 12.42. Figure

4.4.Conclusions Conclusions In Inthis thispaper, paper, the the errors errors obtained obtained in in measuring measuring aa rigid rigid displacement displacement(without (withoutdeformation deformationor or rotation) rotation)through throughimage imagecross-correlation cross-correlationwere wereanalyzed. analyzed.The Thefocus focuswas wasset setininthe theanalysis analysisof ofone one single more complex dotted texture pattern. TheThe object waswas designed to singleobject objectthat thatcould couldbelong belongtotoa a more complex dotted texture pattern. object designed have control over its size, both in its andand lower intensity areas, as well as over the the number of to have control over its size, both in higher its higher lower intensity areas, as well as over number levels involved and the maximum current intensity. The patterns used here were similar to those of levels involved and the maximum current intensity. The patterns used here were similar to those commonly commonlyused usedin inDIC, DIC,but butin inour ourcase, case,the thefocus focuswas wasset seton onthe theparticular particularelements elementsthat thatformed formedthe the structure structureand andnot noton onthe thetexture textureitself, itself,with withtheir theirparticular particular links links and and statistical statistical properties. properties. The Theresults results showed showed that, that, when when analyzing analyzingthe the individual individualdots dotsforming formingthe thetexture, texture, the the main main parameter parameter that that influences influences the the error error was was the the number number of of levels levels involved. involved. However, However, itit was was also also demonstrated demonstratedthat, that,above above150 150levels, levels,increasing increasingthe thenumber numberof ofthe thelevels levelsdid didnot notproduce produceaasignificant significant improvement improvementof ofthe theerror. error. Regarding Regardingthe thepoint pointsize, size,for forarea arearatios ratiosclose closeto to1,1,the theoptimum optimumvalue value corresponded radiusbetween between3030and and Below these values, the error increased quickly, corresponded to to aa radius 4040 px.px. Below these values, the error increased quickly, while, while, these values, thedid error not improve significantly (Figure 12). For higher ratios above above these values, the error not did improve significantly (Figure 12). For higher area ratiosarea (including (including all 256 levels), this optimum value was not so clear, because for smaller point sizes the all 256 levels), this optimum value was not so clear, because for smaller point sizes the error increase error increase was so big as in the previous case.including The studyall including all was 256 levels wasto included was not so big as not in the previous case. The study 256 levels included analyze to analyze separately the influence of each parameter, but, in real images from digital cameras, separately the influence of each parameter, but, in real images from digital cameras, images showing images showing the 256each levels around pointtowere difficult to obtain. Images likeinthose shown all the 256 levelsall around point wereeach difficult obtain. Images like those shown Figure 6 are in Figure 6 are much common in standard applications than those shown in Figure 9; therefore, we much common in standard applications than those shown in Figure 9; therefore, we can state, without can state, without lack of generality, that the optimum value for the point radius was between lack of generality, that the optimum value for the point radius was between 30 and 40 px. 30 andThe 40 px. study of the intensity gradient around the point (measured as the area ratio between the The study thethe intensity gradient around pointit) (measured the area ratioofbetween top top planar areaofand descending sloped ringthe around led to an as optimal value the areathe ratio of planar area and the descending sloped ring around it) led to an optimal value of the area ratio of 1.35. 1.35. Higher intensity gradients between the point and the background than that obtained for an area Higher between the point and the background than that an area ratio ratio ofintensity 1.35 not gradients only did not improve the results, but worsened them veryobtained quickly,for while smoother of 1.35 not only did not improve the results, but worsened them very quickly, while smoother gradients did not affect the error significantly. Regarding the influence on errors of the maximum gradients did notto affect the error significantly. Regarding themaximum influence level, on errors of the the maximum intensity (linked the illumination level), the higher was the the higher number intensity to the level), the was the maximum number of levels (linked involved, and,illumination as a consequence, the higher error decreased. However,level, as forthe thehigher rest ofthe parameters, of levels involved, and, asthe a consequence, themaximum error decreased. However, as forof thethe rest of parameters, once reached 150 levels, increase of the value (or, equivalently, number of levels) once reached 150 levels, the increase of the maximum value (or, equivalently, of the number of levels) did not improve the results. did not improve the values results. here obtained for the different parameters may be used to design the The reference The reference values herethe obtained for the different parameters may be used accuracy to designinthe measurement procedure and illumination setup in order to have the maximum the measurement procedure and the illumination setup in order to have the maximum accuracy in the results at a minimum cost. results at a minimum cost. Acknowledgments: This work was partially supported by “Generalitat Valenciana” through the project nr. PROMETEO II/2015/015. Acknowledgments: This work was partially supported by “Generalitat Valenciana” through the project nr. PROMETEO II/2015/015.

Author Contributions: B.F. and D.M. had the initial idea of the concept; D.M. conceived the design of the object to follow; B.F. performed the numerical experiments; B.F. and D.M. analyzed the results, and B.F. wrote the paper.

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Author Contributions: B.F. and D.M. had the initial idea of the concept; D.M. conceived the design of the object to follow; B.F. performed the numerical experiments; B.F. and D.M. analyzed the results, and B.F. wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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