Massimo Nespolo

The correct symmetry of the sixfold twin in β-Ca 11 B 2 Si 4 O 22

Keywords: Twinning, polychromatic symmetry, twin point groups

The correct symmetry of the six-fold twin in

β-Ca11B2Si4O22

Introduction

In a recent article [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF], an interesting case of multiple twinning in the borosilicate -Ca11B2Si4O22 has been presented, where unfortunately the description of twinning is partially incorrect. -Ca11B2Si4O22 and the closely related mineral spurrite, Ca5(SiO4)2CO3, were interpreted as derivative structures of -Ca2SiO4, which is commonly found in ordinary portland cement.

-Ca11B2Si4O22 and spurrite crystallize in the same type of space group (No. 14), with similar cell parameters: P121/c1, a = 14.059, b = 6.834, c = 10.597 Å,  = 100.735°, V = 1000.333 Å 3 for the former [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF], P121/a1, a = 10.484, b = 6.712, c = 14.156 Å,  = 101.27°, V = 976.930 Å 3 for the latter [START_REF] Grice | The structure of spurrite, tilleyite and scawtite, and relationships to other silicate-carbonate minerals[END_REF]. If we ignore the small numerical differences, we can convert one unit cell to the other by a change of basis vectors c,-b,a, which also converts the two settings of the space group, P121/c1 and P121/a1. Both structures are derivatives of the β-Ca2SiO4, obtained by partially substituting SiO4 with BO3 or CO3 respectively [START_REF] Midgley | The crystal structure of β dicalcium silicate[END_REF].

The possibility of interpreting the structure of both -Ca11B2Si4O22 and spurrite as derivative of that of -Ca2SiO4 and the fact the latter is obtained through a phase transformation from the hexagonal polymorph -Ca2SiO4 has led the authors [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF] to hypothesize a similar hexagonal aristotype for derivatives of -Ca2SiO4. Unfortunately, the proposed transformations are incorrect and incompatible with the corresponding space groups.

The cell parameters of -Ca11B2Si4O22 and spurrite can be obtained from those of -Ca2SiO4 by the change of basis vectors 2a-b,-c,a+2b and a+2b,c,2a-b respectively (Fig. 10 in [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF]); these two transformations are related by a change of basis vectors c,-b,a. The results are a = 14.620, b = 7.307, c = 9.571Å,  = 109.11°, V = 1022.517 Å 3 (with a and c exchanged for the second transformation). The closeness of the computed and experimental parameters simply shows the existence of a metric relation, but does not guarantee at all the existence of a group-subgroup relation. Indeed, the corresponding transformation matrices have determinant 5, which means that the metric relation implies an enlargement of the unit cell by a factor 5. The reported basis transformations show that the hexagonal c axis becomes the monoclinic unique axis, so that one should expect a group-subgroup relation in which 63/m becomes the monoclinic 21/c with a cell enlargement of 5 in the hexagonal (0001) plane, i.e. the monoclinic (010) plane. The maximal subgroup of P63/mmc in which the 63 screw rotation is transformed to a 21 screw rotation is of type Cmcm, whose full symbols is C2/m2/c21/m. *Correspondence author (e-mail:massimo.nespolo@univ-lorraine.fr) To obtain a subgroup in which 21/m becomes 21/c (in the b-unique setting) one needs at least doubling of the area in the (010) plane, but 2 is not a factor of 5.

Moreover, the symmetry of the twin is incorrectly described by a hypothetical dichromatic group 6'/mm'm' which does not exist. If we take the index 2 achromatic subgroup of a dichromatic group by removing the chromatic operations we should get the symmetry of the untwinned crystals. 6'/m can be obtained by adding a chromatic twofold rotation to either 3 or 6; because in the hypothetical dichromatic group the mirror reflection normal to [001] is not primed, it must be a symmetry operation of the crystal, ruling out 3. Further, if the mirror reflection normal to both 100 and 110 are chromatic, because the inversion is also chromatic (being absent from 6), then the twofold rotations along the same directions must be achromatic, leading to 622, which is not a crystallographic point-group type. The twin reported in [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF] must actually be described by a hexachromatic group, due to the presence of six twinned individuals. Because of the relative rarity of this type of multiple twinning, we think it is useful to show how to correctly describe this type of twin. Details about how to describe twinning are available in [START_REF] Nespolo | Twin point groups and the polychromatic symmetry of twins[END_REF] and [START_REF] Nespolo | Tips and traps on crystal twinning: how to fully describe your twin[END_REF].

The sixfold twin of -Ca 11 B 2 Si 4 O 22

The twin operations are described in Table 2 in [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF], although they are incorrectly called "twin elements". These are given as 1, 2z, 3 + y, 2x-z, 3 -y and 2xz. The first operation is the identity and not a twin operation; the other five operations map the orientation of an individual taken as reference to the orientation of the other five individuals. The 3-fold rotations about [010] and the three twofold rotations about directions in the (010) plane show that the hexagonal twin lattice is described in an unconventional b-unique setting adopted to comply with the 12/m1 monoclinic setting of the individual taken as reference. The construction of the twin point group is straightforward if we adopt the conventional c-unique setting of the hexagonal twin lattice, although this means to describe the reference individual in the c-unique monoclinic setting 112/m. The twin lattice can be obtained by the following transformations: The sixfold rotation about the hexagonal [001] direction exchanges three colours (individuals) and is therefore trichromatic, symbol 6 (3) ; the mirror perpendicular to it is a symmetry element for the individuals and is therefore achromatic, as is the inversion centre; all the twofold rotations in the (0001) plane, and the mirror planes normal to it, exchange all the colours in pairs, so that the symbol is 2 (2) /m (2) . The symbol of the hexachromatic twin point group is therefore:

K (6) =             6 3 2 2 2 2 6 2 2 m m m         (1)
Fig. 1 shows the construction of the stereographic projection of the hexachromatic group (1).

If W is the matrix representation of a twin operation and P the matrix expressing the change of the basis vectors 3c,-2a-2c,-b from the monoclinic to the hexagonal reference obtained above, the transformation is given by Wm = PWhP -1 , where the subscripts m and h stand for monoclinic and hexagonal, respectively. Table 1 gives the matrix representation of a twin operation in each twin law (coset representative) with respect to the hexagonal basis vectors, and in the monoclinic setting P121/c1 reported in [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF]. The results are the transposed matrices of those published in Table 2 of [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF],

showing that the hexachromatic point group (1) is consistent with the structure refinement of the twin reported in [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF]. To be noted that the directions of the twin elements given in [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF], namely [010] (y), [001] (z), [101] (xz) and [101] (x-z) transform to [001], [100], [010] and [430] respectively; the latter is not a symmetry direction of the hexagonal lattice and is therefore incorrect. The matrix given as 2x-z in [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF] corresponds to 2[110] in the hexagonal setting but the hexagonal direction [110] corresponds to [201] in the P121/c1 setting, so that the operation should have been listed as 22x-z. As we have already emphasized previously [START_REF] Nespolo | Unravelling the diffraction pattern of a twin[END_REF], the representation of the twin operations in the basis of the individual makes these operations much less easy to read and understand; the representation in the basis of the twin lattice is always to be preferred.

Table 1. Matrix representation of the twin operations in the hexagonal setting of the twin lattice and in the monoclinic setting of the individual. The latter coincides with the transpose of the representation in [START_REF] Volkov | β-Ca11B2Si4O22: six-fold twinning, crystal structure and thermal expansion[END_REF]. Hexagonal representation Along the [001] direction a two-fold symmetry (achromatic) axis (black) is co-axial with a six-fold twin (trichromatic) axis (blue, red, green), which is perpendicular to a symmetry (achromatic) mirror plane (black). Along the six symmetry directions of the hexagonal lattice a twin (dichromatic) two-fold axis perpendicular to a twin (dichromatic) plane occur; twin elements bearing the same colour relate the same pairs of individuals (same colour-exchange code).

0 1 0 1 1 0 0 0 1         1 1 0 1 0 0 0 0 1         1 1 0 0 1 0 0 0 1         0 1 0 1 0 0 0 0 1         1 0 0 1 1 0 0 0 1         Monoclinic representation 2 2 3 3 7 1 6 3 0 0 1 0 0         1 2 3 3 7 2 6 3 0 0 1 0 0         1 2 1 0 0 0 1 0 0 1         1 2 3 3 4 1 3 3 0 0 1 0 0         2 2 3 3 5 2 6 3 0 0 1 0 0        

1 )

 1 c,-a-c,-b from P121/c1 (b-unique) to P1121/a (c-unique) with cell parameters a = 10.597, b = 15.952, c = 6.834 Å,  = 90,  = 90,  = 120.010°; 2) 3a,2b,c from the above setting, leading to a = 31.791, b = 31.903, c = 6.834 Å,  =  = 90;  = 120.01°. This is very close to the hexagonal unit cell obtained from the data collection in [1], i.e. a = 31.819, c = 6.833 Å, The global transformation from the basis vector of P121/c1 to those of the twin lattice is 3c,-2a-2c,-b. In this setting, the twin operations are 3 + [001], 3 -[001], 2[010], 2[110], 2[100], and the construction of the twin point group is obtained in the following way: {1, 2[001], 1, m[001]}3 + [001]{1, 2[001], 1, m[001]}3 -[001]{1, 2[001], 1, m[001]}2[010]{1, 2[001], 1, m[001]}2[110]{1, 2[001], 1, m[001]}2[100]{1, 2[001], 1, m[001]} = {1, 2[001], 1, m[001]}{3 + [001], 6 -[001], 3 + [001], 6 -[001]}{3 -[001], 6 + [001], 3 -[001], 6 + [001]}{2[010], 2[210], m[010], m[210]}{2[110], 2[110], m[110], m[110]}2[100], 2[120], m[100], m[120]}.

Figure 1 .

 1 Figure 1. (a) Stereographic projection of a general form (left) and of the symmetry elements (right) of the point group 6/mmm. (b) Stereographic projection of a general form (left) and of the symmetry elements (right) of the point group 112/m. (c) Stereographic projection of a general form (left) and of the symmetry elements (right) of the hexachromatic group isomorphic to 6/mmm in (a) obtained as extension of the point group 112/m in (b). Left: stereographic poles of faces oriented North and South are differentiated by their size; poles with the same colour belong to the same individual (domain state) and are related by the symmetry (achromatic) operations of the individuals. Poles of different colours belong to different individuals (domains) and are related by the twin (chromatic) operations of the individuals.Along the [001] direction a two-fold symmetry (achromatic) axis (black) is co-axial with a six-fold twin (trichromatic) axis (blue, red, green), which is perpendicular to a symmetry (achromatic) mirror plane (black). Along the six symmetry directions of the hexagonal lattice a twin (dichromatic) two-fold axis perpendicular to a twin (dichromatic) plane occur; twin elements bearing the same colour relate the same pairs of individuals (same colour-exchange code).

Acknowledgements.

Critical remarks by two anonymous reviewers are gratefully acknowledged.


Michael O'byrne
email: obyrnemj@tcd.ie

Bidisha Ghosh
email: bghosh@tcd.ie

Vikram Pakrashi
email: v.pakrashi@ucc.ie

Franck Schoefs
email: franck.schoefs@univ-nantes.fr

Image Processing based Damage Detection of Offshore Infrastructural Elements using Texture Information

Keywords: Structural Health Monitoring, Damage Detection, Image Processing, Texture Analysis, Offshore Marine Structures, Grey Level Co-occurrence Matrix (GLCM), Support Vector Machines (SVM)

The current approach to Structural Health Monitoring (SHM) for offshore infrastructural elements is largely based on feedback provided by divers who may lack the desired level of inspection training, resulting in an undependable perspective of the damage present. There is thus a requirement for a robust, reliable and reproducible means for non-destructive testing. This paper presents a texture analysis based image segmentation using Support Vector Machine (SVM) classification for the purposes of detecting and classifying damaged regions on the surface of infrastructural elements. For the texture analysis stage, a Grey Level Co-occurrence Matrix (GLCM) was employed to calculate seven texture measures, namely; contrast, homogeneity, correlation, angular second moment (ASM), skewness, entropy and kurtosis. SVM was then used to classify regions as being either damaged or undamaged. As an example, the technique was applied to images of a corroding metallic surface in marine environment. Texture analysis is especially useful when regions of interest in an image are more distinguishable from the background based on their texture rather than pixel intensity. The effectiveness of this Non-Destructive Testing (NDT) technique is evaluated through the use of performance coordinates in the Receiver Operating Characteristic (ROC) space.

INTRODUCTION

Regular inspections of structures are vital to ensure their safety and operability. Currently, many structures are assessed using a regime of visual inspections. The levels of inspection cover a range of detail. It can be a cursory check, a principal inspection involving detailed examination of all surfaces, or special inspections where the employment of invasive, semi-invasive and noninvasive tests may become necessary. Most inspection still depends on visual observations and the quality of data collected largely depends on the ability of the inspectors to observe and objectively record details of damage. Output from visual inspections can markedly vary and can have a lower level of accuracy and repeatability of detection compared to other inspection methods [START_REF] Agin | Computer Vision Systems for Industrial Inspection and Assembly[END_REF]. Furthermore, in the case of offshore structures, inspections are typically carried out by divers who may lack the desired level of inspection training and are unlikely to be an engineer. This has the potential to introduce additional sources of error. An image processing based detection approach not only acts as a relatively inexpensive tool to facilitate maintenance management, but also offers a far greater reproducible and measurable performance over visual inspection techniques. This approach allows for captured images to be archived for possible monitoring and assessment of degradation over time. Image processing based techniques have received a lot of attention in many other fields but have not been explored to their full potential for Structural Health Monitoring (SHM).

Image based analysis for SHM is a form of Non-destructive testing (NDT) that can provide useful information on the state of the structure, often keeping it in operation. This information can also be a part of an Infrastructure Management System (IMS). The output from an IMS can assist in determining the course of action the engineer or owner should take, as well as enabling informed decisions to be made when prioritising resources towards the correction of deficiencies. Furthermore, information acquired from NDT techniques increase understanding of defect initiation and propagation and provide a fundamental basis for both improving ageing laws for real structures, and for updating existing models to predict the behaviour of a particular structure and thus achieve a more reliable and suitable management strategy. This aspect has attracted a growing interest in recent years as the importance of life cycle optimisation and the related financial benefits continue to be recognised [START_REF] Sheils | Development of a two-stage inspection process for the assessment of deteriorating bridge structures[END_REF][START_REF] Schoefs | Comparison of additional costs for several replacement strategies of randomly ageing reinforced concrete pipes[END_REF] . The most suitable choice of NDT for a given application will largely depend on the damage to be detected and will require an in-depth knowledge of the advantages and limitations associated with each option. The measure of the on-site performance of a NDT tool is still an outstanding question in most cases [START_REF] Schoefs | Quantitative evaluation of NDT method performance: application example based on contactless impact echo measurements for void detection in tendon duct[END_REF]. Image processing based techniques in conjunction with powerful detection algorithms can be considered as an accessible, inexpensive, and easily deployable technique for visual inspection. The primary limitations are the lack of penetration below the surface of the material and the requirement of good visibility and lighting conditions. Previous image processing based methods have relied on colour information [START_REF] Iyer | Segmentation of Pipe Images for Crack Detection in Buried Sewers[END_REF][START_REF] Tsai | Automatic Detection of Deficient Video Log Images Using a Histogram Equity Index and an Adaptive Gaussian Mixture Model[END_REF], textural information [START_REF] Cord | Automatic Road Defect Detection by Textural Pattern Recognition Based on AdaBoost[END_REF], or a combination of the two [START_REF] Wu | Enhanced Roadway Inventory Using a 2-D Sign Video Image Recognition Algorithm[END_REF] to segment and extract regions of interest in images.

Texture and colour based segmentation approaches are the primary modes of segmentation employed for image analysis. While both approaches have important applications in image processing methods, the colour based methods have been researched to a much greater extent. Texture may be considered as an innate property of surfaces. It may be qualified by terms such as fine, coarse, smooth, rippled, molled, irregular, or lineated [START_REF] Haralick | Textural Features for Image Classification[END_REF]. Texture segmentation finds particular relevance in cases where the regions of interest are more separable from the background based on their texture than colour. This paper presents an enhanced texture based detection technique [START_REF] Iyer | Segmentation of Pipe Images for Crack Detection in Buried Sewers[END_REF] involving Grey Level Co-occurrence Matrix (GLCM). GLCM is a well-established texture analysis method and offers a highly accurate approach for quantifying the perceived texture in an image. Support Vector Machines (SVM) are subsequently used to classify regions based on the statistics derived from the GLCM. While GLCM had been used previously in conjunction with SVM classification to classify woven fabrics [START_REF] Xian | An identification method of malignant and benign liver tumors from ultrasonography based on GLCM texture features and fuzzy SVM[END_REF][START_REF] Ben Salem | Automatic recognition of woven fabrics based on texture and using SVM[END_REF], the proposed algorithm incorporates a number of additional statistics which produces a feature vector that is well-rounded and offers a better described representation of the texture in the image.

Detection of corrosion on the surface of infrastructural elements in marine environment has been chosen as the illustrative application in this paper for several reasons. Firstly, offshore structures typically represent large initial investments and are exposed to harsh environmental conditions. This means that stakeholders are likely to place an increased emphasis on SHM in order to maximise the structure's operational lifetime. Secondly, damage detection in itself is an important field of research and has significant financial and safety implications as demonstrated in a recent study. Thirdly, texture segmentation is especially relevant in cases where the regions of interest are more separable from the background based on their texture than colour. This scenario is encountered to a greater extent when considering underwater or near-surface infrastructural elements, where challenging lighting conditions and reflections introduce inherent chromatic and luminous complexities. These complexities tend to have a greater effect on colour based segmentation techniques where only pixel intensity values are considered rather than texture based segmentation techniques where it is the relationship between adjacent pixel intensity values that is considered [START_REF] Shevell | Color in Complex Scenes[END_REF].

Methodology

An image based damage detection algorithm has been proposed in this paper. The algorithm involves two steps; the first step is to develop a texture characteristics map of a colour image of a damaged surface using a GLCM approach. GLCM is primarily designed to work on greyscale images, however, it may be readily extended to colour images which consist of three channels by operating on each channel separately. The second step is classifying the damaged regions in the image using a SVM classifier. These two steps are discussed in detail in the following subsections.

Texture Characteristics Map

A texture characteristics feature vector {v f } a,b,c has to be generated for each pixel within the original image, I, for each separate colour channel, c, where f indicates the index of the vector element, and a and b indicate the spatial coordinates of the pixel. Each element of {v f } a,b,c is obtained by computing statistics derived from a GLCM. The process in which the GLCM is created is illustrated in Figure 1.

Figure 1-Overview of the GLCM process

The GLCM is a matrix of frequency values of paired combinations of pixel intensities as they appear in certain specific spatial arrangements within an image or sub-image. In this paper, the GLCM for each pixel is generated through a sub-image that is a sliding window, SW, centred on the pixel. The centre is indicated as (a,b) and the size of the window (N-pixel x N-pixel) is optimised for best performance.

Each element, g ij , of the GLCM corresponds to the number of occurrences of a paired grey level combination where i denotes the grey level in the reference pixel, j denotes the grey level in the destination pixel; the GLCM is defined for a certain interpixel distance, d, and a certain angle of offset, θ between neighbouring pixels. The grey levels are defined using integer values between 1 and G. In this paper, the grey levels are defined on a scale of 1 -12 (G =12) instead of a typical scale of 1 -255. Quantizing in this manner allows computational parsimony and avoids sparse matrices.

Paired combinations of intensities of pixels that are spatially close together tend to be more relevant than combinations that involve spatially distant pixels. Accordingly, a value of 1 was chosen for d to ensure certain level of spatial proximity. The angle along which the interpixel distance was counted was defined as the angles of offset and the four angles for the offset that were chosen are, θ = 0°, θ = 45°, θ = 90°, θ = 135°. So, this generated a set of 4 GLCMs (d =1; θ = 0°, 45°, 90°, 135°) for each colour channel at each pixel in this paper.

The GLCM for each pixel is populated as:

( ) , , 11
1 if and where 0 otherwise

d NN uv uv ij d uv w iw j g AA θ θ = =  = = = =   ∑∑ (1)
where, w uv , is the pixel intensity expressed in quantised grey levels for the reference pixel located at row u and column v within the sliding window;

ij d d g pi j NN θ θ = - ( 1)
There are seven texture features calculated from the GLCM, namely: Angular Second Moment (ASM), homogeneity, contrast, correlation, skewness, entropy, and kurtosis. They are defined as follows:

1. Angular Second Moment (ASM) represents the uniformity of distribution of grey level in the image.

{ } 2 1, , 11
( ) (, ) GG fd d ij v pi j θ θ = = = = ∑∑ (3)
ASM ranges from 1/G 2 to 1. A value of 1 indicates a constant image.

2. Homogeneity gives a measure of the similarity in the image. ( ,) where

GG fd d ij v n p i j n i-j θ θ = = = = ⋅ = ∑∑ (4)
Homogeneity ranges from 0 to G-1. A value of 0 indicates a strong similarity in the image.

3. Contrast is a measure of the local variations present in an image. If there is a high amount of variation the contrast will be high.

Contrast ranges from 0 to (G-1) 2 . A value of 0 indicates a constant image.

4.

Correlation is a measure of image linearity. Correlation will be high if an image contains a considerable amount of linear structure. ( ) (, ) ()

GG d fd ij ij p i j v θ θ µµ σσ = = = - = ∑∑ (6)
where 1 µ , 2 µ , 1 σ and 2 σ are the means and standard deviations of the marginal probability matrices, 1 p and 2 p , obtained by summing the rows and columns of p(i,j) d,θ respectively.

Correlation ranges from -1 to 1. A value of 1 indicates a perfectly positively correlated image. An undefined value is returned in the case of a constant image.

5.

Entropy is a statistical measure of randomness.

) l o g (, )

GG fd d d ij v pi j pi j θ θθ = = = = -∑∑ (7)
Entropy can range from 0 to infinity.

6. Skewness is a measure of the asymmetry of the data around the sample mean.

An estimate for the skewness is: where m and σ are the mean and standard deviation over G 2 observations. Skewness ranges from 0 to (G-1) 2 . A value of 0 indicates a constant image.

Kurtosis is a measure of how outlier prone a distribution is. A positive value for kurtosis

indicates that the distribution has a greater peakedness than that predicted by a normal distribution, while a negative value indicates that the distribution is less peaked than predicted by a normal distribution. An estimate for the kurtosis is given by:

4 7, , 2 4 11 1 ( ) ( (, ) ) 3 ( 1) GG fd d ij v pi j m G θ θ σ = = = = -- - ∑∑ (9)
Kurtosis ranges from 0 to (G-1) 2 . A value of 0 indicates a constant image.

In certain cases undefined values may result for some statistics such as correlation, skewness and kurtosis when the intensity values in the window are perfectly uniform, i.e. when the standard deviation is equal zero. These undefined values are ignored by the SVM classifier.

Each texture measure in the feature vector is an average of the four texture measure values as calculated in the four GLCM matrices for each combination of d and θ.

( ) ,

1 ff d d vv R θ θ  =   ∑∑ (10)
where R is the number of all possible combinations of d and θ.

Some of texture features relate to certain texture characteristics in the image such as homogeneity or contrast. Other features describe aspects such as image complexity or the transition of pixel intensity values. However, in spite of each of the aforementioned features containing some degree of information about the texture characteristics of the image, it is difficult to establish which textural trait is represented by each feature.

Linear SVM Classification

Support Vector Machines (SVM) are used to classify pixels as being either damaged or undamaged, based on the feature vector assigned to each pixel. SVM is a supervised learning classifier based on statistical learning theory. The linear SVM lies on both maximum margin hyperplane and kernel function concepts. The later is selected for the computational challenge of products in high dimensional space [START_REF] Boser | A training algorithm for optimal margin classifiers[END_REF][START_REF] Cortes | Support-vector networks[END_REF][START_REF] Cristianini | An Introduction to Support Vector Machines and other Kernel-based Learning Methods[END_REF]. The goal of SVM is to produce a model based on the training data which predicts the target values of the test data given only the test data attributes. In this case, the pixels are the test data while the feature vector corresponding to each pixel provides the test data attributes. The first stage in the classification process involves training in the data. Representative samples from damaged and undamaged regions in each image were selected and the texture features corresponding to these regions provided the basis for separation. Each pixel in the image was then classified based on the texture features corresponding to that pixel. A schematic of the methodology is presented in Figure 2.

EVALUATION OF TECHNIQUE

The proposed technique was performed on three images of pitting corrosion all taken from the same structure (Figure 3).

1 Classification Enhancement

It is necessary to enhance the classification results following the SVM stage in order to remove small insignificant regions that are classified as damaged regions. With knowledge of the nature of the damage type, it is clear that these regions of negligible area are merely spurious artefacts as oppose to actual damaged regions so they may be removed by specifying that regions below a certain size are to be discarded. For illustrative purposes, the detected regions for Image (a) directly after the classification stage are shown in Figure 4

Performance Evaluation

The final detected regions using the proposed technique are presented in Figure 5.

Figure 5 -Detected Regions from the Proposed Technique

The performance of the proposed technique is evaluated by plotting performance points in the Receiver Operating Characteristic (ROC) space. The ROC space provides a common and convenient tool for graphically characterising the performance of NDT techniques and its usage has recently been extended to image detection [START_REF] Pakrashi | ROC dependent event isolation method for image processing based assessment of corroded harbour structures[END_REF]. A box counting approach similar to the procedure described in [START_REF] O'byrne | Receiver Operating Characteristics of a Modified Edge Detection for Corrosion Classification[END_REF] was employed to calculate the Detection Rate (DR) and Misclassification Rate (MCR) for each image. This involved visually segmenting the images (Figure 6) through box counting and comparing them with the SVM classified images. The DR and the MCR associated with each image provide the coordinates for the performance point in the ROC space which allows for a graphical representation of the performance of the technique. A convenient way for comparing and analysing the performance points is achieved through the use of the alpha-delta method [START_REF] Schoefs | The αδ method for modelling expert Judgment and combination of NDT tools in RBI context: application to Marine Structures, Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and performance (NSIE)[END_REF][START_REF] Baroth | Construction Reliability[END_REF]. Essentially, this method relies on calculating the angle, α, and the Euclidean distance, δ, between the best performance point (coordinates (0,100%)) and the considered point to give a measure of the performance of the considered point. For this situation only the delta, δ, parameter is required. A low value for δ is indicative of a strong performing technique. The detection and misclassification rates are summarized in Table 1 and the associated points in the ROC space, along with a fitted curve, are illustrated in Figure 7. The proposed technique performed well both in terms of identifying the location of damaged regions as well as defining their shape and size. It may be observed from Figure 4 that the detected regions are relatively homogenous, with only minor speckles of undetected regions being enclosed by the damaged areas. Although only three sample images are considered, the tight nature of the scatter points in the ROC space (Figure 7) suggests that the technique demonstrates a high degree of repeatability.

CONCLUSION

A texture based image processing technique to detect damage on the surface of infrastructural elements has been proposed. Although popular in other fields, image processing techniques have received little attention in the field of Structural Health Monitoring (SHM). As a case study, the proposed technique was successfully performed on three images of a corroded metallic surface in marine environment. There was a strong correlation between the detection and misclassification rates for each of the images which demonstrates the high level of repeatability of this approach.

The texture is described at each point in the image by a feature vector that contains seven texture related statistics derived from a Grey Level Co-occurrence Matrix (GLCM). This large feature vector ensures that a well-rounded description of the texture is created. This aspect also increases the robustness of the technique as some statistics which may be good at differentiating regions in one image may not necessarily be particularly useful in another image.

The use of Support Vector Machines (SVM) as a classifier in this two-class task is particularly suited to the large data sets. A procedure for removing small spurious regions formed in the SVM classification stage has been outlined.

  pixel intensity expressed in quantised grey levels for the destination pixel located at an interpixel distance d along an angle θ from the reference pixel. The GLCMs are normalised as

Figure 2 -

 2 Figure 2 -Flowchart of the proposed methodology

Figure 3 -

 3 Figure 3 -Sample Images of Pitting Corrosion from the Same Structure

  (a). A cleaner and more accurate image is presented in Figure4(b) which shows the smaller sections removed.

Figure 4 -

 4 Figure 4 -(a) Detected Regions with smaller sections, (b) detected regions with smaller sections removed

Figure 6 -

 6 Figure 6 -Visually Identified Damaged Regions

Figure 7 -

 7 Figure 7 -Performance Coordinates in the ROC Space for the Proposed Technique

Table 1 -

 1 Performance of the Proposed Technique.

	Image	DR	MCR	δ
	Image (a)	94.78%	5.27%	0.07
	Image (b)	91.24%	4.18%	0.10
	Image (c)	93.41%	9.17%	0.11

ACKNOWLEDGMENTS

The authors wish to thank the Irish Research Council for Science, Engineering and Technology (IRCSET) for providing a grant to support this research and CAPACITES/IXEAD society for the practical and financial assistance.