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Crack Detection Method Applied to 3D Computed Tomography Images of Baked Carbon Anodes Donald Picard 1 , Julien Lauzon-Gauthier 2 , Carl Duchesne 2 , Houshang Alamdari 3, *, Mario Fafard 1 and Donald P. Ziegler 4 1 2 3 4

*

Département de Génie Civil, Aluminium Research Centre—REGAL, Université Laval, Québec, QC G1V 0A6, Canada; [email protected] (D.P.); [email protected] (M.F.) Département de Génie Chimique, Aluminium Research Centre—REGAL, Université Laval, Québec, QC G1V 0A6, Canada; [email protected] (J.L.-G.); [email protected] (C.D.) Département de Génie des Mines, de la Métallurgie et des Matériaux, Aluminium Research Centre—REGAL, Université Laval, Québec, QC G1V 0A6, Canada Alcoa Primary Metals, Alcoa Technical Center, Alcoa Center, PA 15069-0001, USA; [email protected] Correspondence: [email protected]; Tel.: +1-418-656-7666

Academic Editor: Hugo F. Lopez Received: 13 September 2016; Accepted: 3 November 2016; Published: 9 November 2016

Abstract: Carbon anodes used in the aluminium industry were imaged through destructive and non-destructive testing (NDT) methods. For the latter case, computed tomography (CT), which has previously been used to map the 3D apparent density distribution, was extended to crack detection. Previous work has shown how to overcome technical hurdles related to crack detection by using percolation-based algorithms operating on low-resolution images of full-scale baked carbon anodes. The previous application to 2D images was extended here to the 3D case. The crack detection algorithm has been performed on anode slices containing several independent macro cracks with different morphologies. Keywords: carbon anodes; computed tomography; crack detection; percolation algorithm

1. Introduction Of all materials used in an electrolysis cell, only carbon anodes (approximately 30 anodes per cell) could be considered consumable items requiring regular replacement. These anodes are consumed during electrolysis and replaced after approximately 26 days of operation, depending on a number of factors including cell technology. Hence a large number of anodes, and consequently a large quantity of raw materials, are required to operate a plant. To stay economically competitive and considering the large amount of anodes consumed each year, aluminium producers need to reduce the cost of their anode production. Part of the solution is to reduce the cost of the raw materials. The drawback is that producers now have to deal with continuous changes in raw material properties due to shortages of traditional quality anode coke [1]. This results in wide variations in the physical properties of baked anodes and could lead, e.g., to electrical resistivity variations [2,3] and anode cracking [4–6]. This latter type of defect is generally related with green anode production processes (mixing, forming and cooling) or baking processes (release of induced stresses, pitch devolatilisation and final baking temperature) [6] and could lead, though not exclusively, to an increase of energy consumption (increase of electrical resistivity) during the electrolysis process or ultimately to anode failure (corner, vertical or horizontal cracks) [6]. One solution to minimize the effect of variations in raw materials properties is to use numerical simulation methods to model the manufacturing process and/or to improve the process control. The objective is to predict the anode characteristics, to control the process parameters more efficiently,

Metals 2016, 6, 272; doi:10.3390/met6110272

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and to take corrective actions before the anode is produced. To achieve this goal, a series of experimental data must be collected first in order to validate the models, particularly for crack quantification. Several image processing techniques are available for crack detection and have been developed and/or used by several authors [7–9]. Torok et al. [9] reviewed the 2D and 3D techniques applied for inspection of building materials and highlighted that most techniques have been applied for the 2D case. 3D image generation techniques, such as X-ray computed tomography, have opened the way for the development of 3D crack detection techniques. These 3D techniques are generally based on region growing or template matching methods [10]. Considering the wide variety of 3D crack patterns in carbon anodes (no single template would be appropriate), the former technique has been chosen for its relative simplicity. Different region growth algorithms are available in the literature [8,10]. However, few of them are applicable for the 3D case as highlighted by Ehrig et al. [10], who investigated two different region growth techniques: percolation and Hessian-driven percolation [11]. Ehrig et al. [10] also argued that the percolation method could be too computer-time consuming and that the Hessian-driven percolation algorithm [11] is the most suited method for crack identification except in the case of thin cracks (crack of few pixels in width). In the present case, an algorithm able to identify cracks a few pixels in width is needed, as explained later in Section 2.1. Therefore, the region growing technique based on the liquid permeation model, namely the percolation method, has been investigated as an alternative to Hessian-driven percolation. The basic idea behind a percolation algorithm is to fill a crack with pixels/voxels as water would fill it from a starting point. A lot of effort has already been put into increasing the efficiency of the percolation algorithm [7,12–14]. This method has both advantages and disadvantages. In addition to its simplicity, the method has the potential to overcome an important hurdle in this investigation, i.e., the low resolution of the images. The main downside is that the chosen method proposed by Yamaguchi and Hashimoto [12,13] has been improved only for 2D images [7,12–14]. Therefore, an extension of the percolation algorithm to the 3D case was needed and has been developed. The proposed algorithm has then been used to identify cracks in various industrial anodes samples. 2. Materials and Methods 2.1. X-ray Computed Tomography X-ray computed tomography has been used for decades now in science to reveal the internal structures of objects from living beings to engineering materials (e.g., [15–18]) including the carbon materials used in the aluminium production industry [19–23]. Its main advantage is that it is a non-destructive technique. Thus, it can preserve object integrity such as cracks inside materials. Hence, carbon anode images have been obtained by scanning a whole baked anode block using computed tomography (3D non-destructive testing (NDT) imaging tool). Unfortunately, the scanning area of the X-ray apparatus used in this investigation (Siemens Somatom Sensation 64, maximum resolution of 0.09 × 0.04 × 0.09 mm3 ) is limited to objects of less than 40 × 40 mm2 cross-section, which is smaller than the anode section (Figure 1). The anode block was thus sliced in 52 pieces of 50 mm thick [4] as illustrated in Figure 2. Care was taken to ensure that the cracks of interest were not affected by the slicing operation. Furthermore, in order to implement the method in the industry, cost analysis (scanning and volume reconstruction times) had to be kept low. Hence, image resolution had to be adapted to the object size [24] resulting in a voxel (volumetric pixel) resolution of 0.7 × 0.6 × 0.7 mm3 . This means that most visible cracks (1–2 mm width) have a width of 2–3 voxels only. Using this resolution, a typical anode slice contains approximately 70 × 468 × 1068 voxels (approximately 35 million voxels). Therefore, contrast between most anode cracks (1–2 mm width) and background carbon is low.

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Figure 1. Siemens Somatom Sensation 64 designed designed for the human body but adapted for material Figure 1. Siemens Somatom Sensation forthe thehuman human body adapted for material Figure 1. Siemens Somatom Sensation64 64 designed for body butbut adapted for material research. Courtesy of INRS-ETE, Quebec, QC, Canada. research. Courtesy of INRS-ETE, Quebec, research. Courtesy of INRS-ETE, Quebec,QC, QC,Canada. Canada.

Figure 2. Slicing scheme of the investigated anode (approximately 0.6 × 0.6 × 1.5 m33).

Figure 2. Slicing scheme investigatedanode anode (approximately × 0.6 × 1.5×m1.5 ). m3 ). Figure 2. Slicing scheme of of thethe investigated (approximately0.60.6 × 0.6

2.2. 2D Algorithm Overview 2.2. 2D Algorithm Overview

2.2. 2D Algorithm Overview The percolation method proposed by Yamaguchi et al. [12] is similar to a scalable window The percolation method proposed by Yamaguchi et al. [12] is similar to a scalable window processing and takes into account the connectivity of the percolated pixels. The method requires The percolation method proposed by Yamaguchi et percolated al. [12] is pixels. similar tomethod a scalable window processing and takes into account the connectivity of the The requires setting a local investigation window containing the focal pixel, or seed pixel. A seed voxel is a voxel processing takes into account thecontaining connectivity of the percolated pixels. The method requires setting and a local investigation window the focal pixel, or seed pixel. A seed voxel is a voxel within a crack that is used as a starting point for the algorithm to fully investigate the crack setting a local investigation window containing thefor focal or seed pixel.investigate A seed voxel is a voxel within a crack that is used as a starting point the pixel, algorithm to fully the crack boundaries. This seed pixel is then percolated and some features of the percolated region are used to boundaries. This then percolated features theinvestigate percolated region are used to within a crack that is seed usedpixel as a is starting point forand the some algorithm to of fully the crack boundaries. determine if the resulting cluster is a crack or not. To speed up the method, which was otherwise determine if the resulting cluster is a crack or not. To speed up the method, which was otherwise This seed pixel then percolated and some features of [12] the percolated are used to determine slower thanisconventional methods, Yamagachi et al. introduced region two processes preventing thancluster conventional methods, Yamagachi et al. [12] introduced two processes preventing if theslower resulting is a crack or not. To speed up the method, which was otherwise useless percolation. First, if the seed pixel is located in the background, i.e., not in a crackslower or in a than useless percolation. First, if the seed pixel is located in the background, i.e., not in a crack or in a conventional methods, Yamagachi et al. [12] introduced two processes preventing useless percolation. pore, the percolation ends. The percolation also ends after reaching the boundaries of the local pore, the percolation ends. The percolation also ends after reaching the boundaries of the local First,window if the seed pixel is located in percolated the background, i.e., not in a crack orthreshold. in a pore, the percolation ends. if the circularity of the region is higher than a given window if the circularity of the percolated region is higher than a given threshold. Several modifications have been applied to the percolation process to improveifthe 2D percolation The percolation also ends after reaching the boundaries of the local window the2D circularity Several modifications have been applied to the percolation process to improve the percolationof the [7,12–14]. The scheme of the resulting algorithm and definition of all parameters can be found in percolated region higherofthan a given threshold. [7,12–14]. The is scheme the resulting algorithm and definition of all parameters can be found in Several modifications have been applied to the percolation process to improve the 2D percolation [7,12–14]. The scheme of the resulting algorithm and definition of all parameters can be found in Yamaguchi et al. [14]. The proposed algorithm is a two steps process applied to each

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pixel candidate (seed pixel), which are obtained using a pre-processing technique [7,24]. In the first Metals 2016, 6, 272 14 step, percolation is performed until it reaches the boundary of a predetermined investigation4 of window (N × N pixels). If the geometrical constraint is satisfied (i.e., circularity or roundness of the percolated Yamaguchi et al. [14]. The proposed algorithm is a two steps process applied to each pixel candidate region less than a specified threshold Fc), the percolation moves to the second step. The second phase (seed pixel), which are obtained using a pre-processing technique [7,24]. In the first step, percolation of theisalgorithm of increasing incrementally the investigation window size (N up×toNapixels). maximum performedconsists until it reaches the boundary of a predetermined investigation window size of or until the circularity reaches the threshold. In anyof case, only a percolated region If M thepixels geometrical constraint is satisfied (i.e., circularity or roundness the percolated region less with circularity less than the specified threshold moves will betoidentified asstep. a crack. than a specified threshold Fc), the percolation the second The second phase of the algorithm consists of increasing incrementally the investigation window size up to a maximum size

2.3. 3D of Algorithm M pixels or until the circularity reaches the threshold. In any case, only a percolated region with circularity less than the specified threshold will be identified as a crack.

The 3D algorithm developed in Matlab® is a direct extension of the 2D algorithm. To extend the 2D algorithm to the 3D case, three challenges had to be overcome. The first was to apply the percolation 2.3. 3D Algorithm by inspecting the 3D neighbor voxels (26 neighbors for each voxel) of all the voxels included in the Dp The 3D algorithm developed in Matlab® is a direct extension of the 2D algorithm. To extend the region of interest instead of the 2D neighbor pixels (8 neighbors for each pixel). This part was relatively 2D algorithm to the 3D case, three challenges had to be overcome. The first was to apply the straightforward. The second was to define an investigation box (3D) instead of a window (2D). The box percolation by inspecting the 3D neighbor voxels (26 neighbors for each voxel) of all the voxels faces included (or window sides) are used to define the boundary that will(8cause processing to pixel). stop when in the Dp region of interest instead of the 2D voxels neighbor pixels neighbors for each the percolation reaches those limits. As for the window, the box dimensions must be representative This part was relatively straightforward. The second was to define an investigation box (3D) instead of theoftypical crack length (approximately 5–15 cm are in the of carbon anodes)voxels embedded a window (2D). The box faces (or window sides) usedcase to define the boundary that willin the cause processing to stop when percolation reaches those to limits. As for the window, the box case, investigated sample [14]. The box the must also be able to adapt the case study. In the present dimensions must representative of theatypical crackoflength (approximately 5–1580 cm in the case of the samples consist ofbe anode slices having thickness 50 mm (approximately voxels). Moreover, carbon anodes) embedded in the investigated sample [14]. The box must also be able to adapt to the cracks are parallel to the thickness/width plane and are, in most cases, much longer than the slice case study. In the present case, thethrough samples 2consist of anode slices having a thickness of 50 mm thickness (i.e., most cracks propagate to 4 anode slices). (approximately 80 voxels). Moreover, cracks are parallel to the thickness/width plane and are, in most The investigation box thus proposed is shown in Figure 3. If the investigation box is fully cases, much longer than the slice thickness (i.e., most cracks propagate through 2 to 4 anode slices). embedded in the sample volume then the six faces of the box are considered boundary voxels. The investigation box thus proposed is shown in Figure 3. If the investigation box is fully Otherwise, if the investigation box then is notthecompletely the sample volume (i.e., box embedded in the sample volume six faces ofembedded the box arein considered boundary voxels. thickness is larger thaninvestigation the sample box volume then the box’s are modified to fit in Otherwise, if the is notthickness) completely embedded indimensions the sample volume (i.e., box the sample volume by removing boundary voxels. Thusthen onlythe four faces remains in list of boundary thickness is larger than the sample volume thickness) box’s dimensions arethe modified to fit in As the shown sample in volume Thus than only the fourinvestigation faces remainsbox, in the of of voxels. Figureby3,removing if the sliceboundary thicknessvoxels. is smaller thelist voxels boundary voxels. As shown in Figure 3, if the slice thickness is smaller than the investigation box, the dark faces of the box shown in Figure 3b are removed from the boundary voxels to avoid ending the voxels of the dark faces of the to boxthe shown in Figure 3b process, are removed the boundary to the process prematurely. Similarly 2D percolation thefrom investigation boxvoxels dimensions avoid ending the process prematurely. Similarly to the 2D percolation process, the investigation box increases iteratively in the second phase, e.g., N × N × N → (N + 2) × (N + 2) × (N + 2). The flowchart dimensions increases iteratively in the second phase, e.g., N × N × N → (N + 2) ×(N + 2) ×(N + 2). The shown in Figure 3 is also applied during the iteration process. flowchart shown in Figure 3 is also applied during the iteration process.

(a)

(b)

Figure 3. Flowchart for investigation box size determination: (a) flowchart; (b) investigation box.

Figure 3. Flowchart for investigation box size determination: (a) flowchart; (b) investigation box.

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The last challenge was to define a relevant 3D crack criterion (e.g., circularity in 2D). A three-dimensional object can be classified using several morphological features [25,26]. At this stage, it was however unnecessary to do so. Only a threshold value applied to a defined parameter is needed to distinguish a crack from a pore (or any other structures). The selected parameter was the sphericity Ψ (Equation (1)). It is defined as the ratio of the surface area of a sphere with the same volume Vp as the given region of interest to the surface area (Ap ) of the region of interest [27]. In the present case, anode macro cracks are relatively thin compared to their length and width. Hence, the sphericity was found to be a sufficient criterion for the 3D algorithm. π1/3 6Vp Ψ= Ap


2/3 (1)

3. Results and Discussion As mentioned previously, carbon anodes can exhibit different crack patterns [6]. In the present study cracks related to baking have been used to test the proposed 3D crack detection algorithm. Those cracks are generally correlated with significant density variation within the anode (slots and stub hole regions), as highlighted by Ben Boubaker et al. [28]. Variations in thermal (e.g., thermal expansion coefficient) and mechanical properties (Young, modulus, Poisson’s ratio, etc.) between zones of different densities could then induce stresses leading to anode cracking when the anodes are cooled down after baking. The number of visible cracks on an anode slice is generally limited (0–10) depending on the anode quality. In the present case, cracks with different morphologies were used to test the algorithm. 3.1. Pre-Processing In order to reduce computation time, it is preferable to define the candidate voxels or seed voxels (starting voxels for the percolation method) [14] instead of performing the percolation on each voxel in the 3D volume. In the present case, the same method used in the 2D analysis [24], i.e., the non-decimated wavelet transform (NDWT), was used to identify the seed voxels within the 3D volume (anode slice). Briefly, the NDWT method performs the convolution between a finite length wave (called a wavelet) of a given shape and the image signals in a given direction. Different frequency bands can be analysed by stretching the length of the wave and repeating the convolution process. Smaller waves capture the high frequency details (e.g., cracks). In this application, the Harr wavelet with one decomposition level was used. In a typical anode slice (70 × 468 × 1068 voxels with a resolution of 0.7 × 0.6 × 0.7 mm3 ), after filtering the NDWT results [24], approximately 10,000 voxels (0.03%) are tagged as seed voxels. However, not all seed voxels will initiate a percolation process since some of them may have been included in the region of interest (Dp) in a previous percolation analysis. The result of pre-processing for a typical 50 mm anode slice is shown in Figure 4. It is difficult to distinguish seed voxels located in cracks from those located in pores. For visualisation purposes, Figure 5 shows the seed voxels in the height/width plane located at half-thickness (thickness = 35 voxels).

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Figure 4. Seed voxels of a typical50 50 mmanode anode slice containing containing cracks and pores (10 voxels 7 mm). Figure 4. 4. Seed voxels cracksand andpores pores(10 (10voxels voxels≈≈ ≈ 7 mm). Figure Seed voxelsofofa atypical typical 50mm mm anode slice slice containing cracks 7 mm).

Figure 5. Seed voxels (white voxels, some are identified with black arrows) of a typical anode image Figure 5. 5. Seed voxels withblack blackarrows) arrows)ofofa atypical typical anode image Figure Seed voxels(white (whitevoxels, voxels,some some are are identified identified with anode image (center of the slice) containing cracks and pores: (a) full scale; (b) image zoomed in the region (center of the slice) containing cracks and pores: full (b) scale; (b) zoomed image zoomed in thecontaining region (center of the slice) containing cracks and pores: (a) full(a)scale; image in the region containing most cracks. containing most cracks. most cracks.

3.2. 3D 3D Percolation—Anode Percolation—Anode Slice Slice 3.2.3.2. 3D Percolation—Anode Slice Two seed seed voxels voxels located located in cracks cracks with with two two different different sizes sizes and and morphologies, morphologies, shown shown in in Figure Figure 6, 6, Two Two seed voxels located inincracks with two different sizes and morphologies, shown in Figure have been used to demonstrate the performance of the 3D percolation algorithm. The algorithm is 6, have been used to demonstrate the performance of the 3D percolation algorithm. The algorithm is have used to demonstrate the performance 3DHence percolation The algorithm notbeen yet optimized optimized from aa computation computation time point point of of the view. only the thealgorithm. crack detection detection efficiency is not yet from time of view. Hence only crack efficiency notisyet optimized from a computation time point of anode view. Hence only the crack detection efficiency analyzed. The first seed voxel is located near an slot and the second one close to an anode anode is is analyzed. The first seed voxel is located near an anode slot and the second one close to an analyzed. The first seed voxel is located near an anode slot and the second one close to an anode stub stub hole. hole. As As seen seen in in that that figure, figure, cracks cracks have have aa width width of of approximately approximately 22 to to 33 voxels voxels (1.4–2.1 (1.4–2.1 mm) mm) and and stub hole. As from seen in2D that figure, crackstohave a width of approximately to 3 voxelsprocess (1.4–2.1has mm) and seem, seem, point of view, view, be discontinuous. discontinuous. Also, the the 3D 3D 2percolation percolation been used seem, from aa 2D point of to be Also, process has been used from a 2D point of view, to be discontinuous. Also, the 3D percolation process has been used with and with and without the boundary voxels of the dark faces of the investigation box shown in Figure 4. with and without the boundary voxels of the dark faces of the investigation box shown in Figure 4. without the boundary voxels of the dark faces of the investigation box shown in Figure 4. The sphericity

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Metals 2016, 272 threshold 7 ofon 14 The sphericity been set to Ψ = 0.5. 7 and 8 show the crack detection results threshold has6,been set to Ψ has = 0.5. Figures 7 and 8 Figures show the crack detection results on both seed voxels The sphericity threshold hasan been set to Ψ = 0.5. Figures 7 and 8100 show the crack detection results on both seed voxels by using investigation box (100 × 100 × voxels) containing all boundary by using an investigation box (100 × 100 × 100 voxels) containing all boundary voxels (including the The sphericity threshold hasanbeen to Ψfaces = 0.5. 7 3b). and 8 show the crack detection results on both seed voxels by box Figures (100 × 100 × In 100 voxels) containing all boundary voxels (including theusing voxels ofinvestigation thesetdark Figure the first case (seed voxel #1), the voxels of the dark faces of Figure 3b). In the firstofcase (seed voxel #1), the percolation process leads both seed voxels by using an investigation box (100 × 100 × 100 voxels) containing all boundary voxels (including the voxels of the dark faces of Figure 3b). In the first case (seed voxel #1), the percolation process leads to a crack containing 2078 voxels. In the second case, 1367 voxels were to apercolation crack containing 2078 voxels. In the second case, 1367 voxels were identified. Compared to voxels (including voxels ofcrack the dark faces of Figure 3b).In Inthe thesecond firstwithout case (seed #1), the 2D processthe leads to apercolation containing 2078 voxels. case, 1367voxel voxels identified. Compared to 2D results [24] the 3D algorithm constraint onwere the percolation results [24] the 3D algorithm without constraint on the investigation box did not perform percolationCompared process to apercolation crack voxels. In the second case,are 1367 identified. to perform 2D results [24] the 3Dcrack algorithm without constraint onwere the investigation box didleads not wellcontaining (Figure 8),2078 i.e., visible continuations not voxels detected by wellthe (Figure 8), i.e., visible crack continuations are not detected by the algorithm. identified. Compared to 2D percolation results [24] the 3D algorithm without constraint on the investigation box did not perform well (Figure 8), i.e., visible crack continuations are not detected by algorithm.

investigation the algorithm.box did not perform well (Figure 8), i.e., visible crack continuations are not detected by the algorithm.

Figure 6. Selected seeds voxels (white voxels identified with black arrows) for 3D percolation

Figure 6. Selected seeds voxels (white voxels identified withwith black arrows) for 3D process: Figure 6.(a)Selected seeds voxels voxels identified black arrows) forpercolation 3D percolation process: seed voxel #1; (b) seed(white voxel #2. (a) seed voxel #1; (b) seed voxel #2. process: (a)Selected seed voxel #1; (b) seed (white voxel #2. Figure 6. seeds voxels voxels identified with black arrows) for 3D percolation process: (a) seed voxel #1; (b) seed voxel #2.

(a) (b) (a) (b) Figure 7. Cracks detection results (10 voxels ≈ 7 mm) without restriction on the investigation box (Ψ = (a) (b) Figure Cracks detection results (10 #2. voxels ≈ 7 mm) without restriction on the investigation box (Ψ = 0.5): (a)7.seed voxel #1; (b) seed voxel Figure 7. Cracks detection results (10 voxels ≈ 7 mm) without restriction on the investigation box 0.5): (a) 7. seed voxel #1; (b) seed voxel Figure Cracks detection results (10#2. voxels ≈ 7 mm) without restriction on the investigation box (Ψ = (Ψ = 0.5): (a) seed voxel #1; (b) seed voxel #2. 0.5): (a) seed voxel #1; (b) seed voxel #2.

Figure 8. Cracks detection results (white regions) without restriction on the investigation box. Image Figure 8. detection regions) restriction on crack: the investigation located atCracks mid width of theresults anode (white slice. Left crack:without near stub hole. Right near anodebox. slot.Image located at Cracks mid width of theresults anode slice. crack:without near stub hole. Right near anode box. slot. Image Figure 8. detection (whiteLeft regions) restriction on crack: the investigation Figure 8. Cracks detection results (white regions) without restriction on the investigation box. located at mid width of the anode slice. Left crack: near stub hole. Right crack: near anode slot.

Image located at mid width of the anode slice. Left crack: near stub hole. Right crack: near anode slot.

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of 14 The use of a constrained investigation box (i.e., using the flowchart of Figure 3) on the8selected The use of a constrained investigation box (i.e., using the flowchart of Figure 3) on the selected seed voxels led to athe results shown in Figures 9 and 10. the A total of 6243 voxels3)were identified as useled of to constrained investigation box9 (i.e., using of Figure on the selected seed The voxels the results shown in Figures and 10. A totalflowchart of 6243 voxels were identified as part partseed of the crack intothe caseshown (seedinvoxel #1) and in the second onewere (seed voxel #2). A third led thefirst results and17,832 10. of 6243 identified part of thevoxels crack in the first case (seed voxel Figures #1) and 917,832 in A thetotal second onevoxels (seed voxel #2). A thirdascrack crack seems to have been identified in Figure 10. However, the percolation process identified a volume of the crack in the first case (seed #1)10. and 17,832 in the the percolation second one (seed voxel #2). A third crack seems to have been identified invoxel Figure However, process identified a volume where all identified voxels are connected, as shown in Figure 9a. Hence, both cracks on the left seems to have been identified in Figure 10. However, the percolation process identified a volume where all identified voxels are connected, as shown in Figure 9a. Hence, both cracks on the left end end of of Figure Figure 10are arepart part the same 3Dcrack crack (percolated region related tocracks seed voxel The where all identified voxels aresame connected, as shown in Figure 9a. related Hence, both on#2). the#2). left 10 ofofthe 3D (percolated region to seed voxel Theend 3D 3D of Figure 10 are part of the same 3D crack (percolated region related to seed voxel #2). The 3Dhave algorithm has performed well in the second case (seed voxel #2), i.e., most voxels of the cracks algorithm has performed well in the second case (seed voxel #2), i.e., most voxels of the cracks have algorithm has performed well in the second case (seed voxel #2), i.e., most voxels of the cracks have beenbeen identified (Figure 9b). While also seems seemstotohave haveperformed performed well identified (Figure 9b). Whilethe thealgorithm algorithm also well for for the the firstfirst seedseed been identified (Figure 9b). While the algorithm to(right have well for the first voxels (Figure 9a),9a), some parts ofofthe cracks have been missed (rightperformed end Figure 10). This isseed mainly voxels (Figure some parts the cracks have also beenseems missed end ofof Figure 10). This is mainly voxels (Figure some partsresolution of thelimitation. cracks have been missed (right end of Figure Thishas is mainly related to image’s the 9a), image’s low limitation. Moreover, in cases, both cases, the10). percolation has related to the low resolution Moreover, in both the percolation identified related to some the image’s lowvoxels resolution limitation. Moreover, both This cases, the percolation has identified anode(atside Thickness as partsThis of in cracks. is directly related with some anode side voxels Thickness(at = 0) as parts =of0)cracks. is directly related with the constraint identified some side voxels (at Thickness = 0) as of cracks. This(Figure is directlyare related with the constraint onanode the investigation where onparts the(Figure box dark faces excluded on the investigation box, where voxelsbox, on the boxvoxels dark faces 3) are excluded 3) from the boundary the constraint on the investigation box, where voxels on the box dark faces (Figure 3) are excluded from theinitial boundary voxels (the investigation box is Hence, larger than the width).is Hence, voxels (the investigation boxinitial is larger than the width). the algorithm unable the to stop from the boundary (thepercolation initial investigation box during is larger the width). Hence, the algorithm is unable tovoxels stop the at this location thethan first phase of the percolation the percolation this location during the first of theduring percolation process. Reducing the initial algorithm is at unable toinitial stop the percolation at phase this location the first phase of the percolation process. Reducing the investigation box would decrease the crack detection performance [24]. investigation box would decrease the crack detection performance [24]. This could be avoided by process. Reducing the initial investigation box would the crack This could be avoided by increasing the anode volumedecrease investigated, e.g., detection an anode performance slice of 200 to[24]. 300 increasing the be anode volume investigated, e.g., an anodeinvestigated, slice of 200e.g., to 300 mm thick or This thick could avoided by theblock. anode volume anode slice of eventually 200 to(e.g., 300 on mm or eventually onincreasing a full anode Alternatively, decreasing theansphericity threshold a fullΨ anode block. Alternatively, decreasing the sphericity threshold (e.g., Ψ = 0.25) could have helped mm thick or eventually on a full anode block. Alternatively, decreasing the sphericity threshold (e.g., = 0.25) could have helped to eliminate the “side voxels” but this did not significantly improve the Ψ = 0.25) could have helped to eliminate the “side voxels” but this did not significantly improve the to eliminate the “side voxels” but this did not significantly improve the detection result. detection result.

detection result.

(a) (a)

(b) (b)

Figure 9. Crack detection results (10 voxels ≈ 7 mm) with restriction on the investigation box (Ψ = 0.5): (a)

Figure 9. Crack detection results (10 voxels ≈ 7 mm) with restriction on the investigation box (Ψ = 0.5): Figure 9. Crack detection results seed voxel #1; (b) seed voxel #2.(10 voxels ≈ 7 mm) with restriction on the investigation box (Ψ = 0.5): (a) (a) seed (b)seed seedvoxel voxel#2. #2. seedvoxel voxel #1; #1; (b)

Figure 10. Crack detection results (white regions) with restriction on the investigation box. Image Figure 10. Crack detection resultsslice. (white regions) withstub restriction on the investigation box. located at mid-width of the anode Left crack: near hole. crack: near anode slot.Image Figure 10. Crack detection results (white regions) with restriction onRight the investigation box. Image located located at mid-width of the anode slice. Left crack: near stub hole. Right crack: near anode slot.

at mid-width of the anode slice. Left crack: near stub hole. Right crack: near anode slot.

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3.3. 3D Percolation—Partial Anode Block 3.3. 3D Percolation—Partial Anode Block As pointed out in Section 3.2, voxels excluded from the investigation box (Figure 3) cannot be As in Section 3.2, voxels excluded from the investigation box (Figure 3) cannot be used to pointed stop the out percolation process in thin anode slices. A larger anode sample with similar crack used to stop the percolation process in thin anode slices. A larger anode sample with similar crack morphologies is needed to fully test the algorithm. Another anode has thus been cut with the same morphologies fully testa the Another anode hasofthus beenas cut with the same pattern shown is inneeded Figure 2tobut with slicealgorithm. thickness of 270 mm instead 50 mm, shown in Figure pattern shown in Figure 2 but with a slice thickness of 270 mm instead of 50 mm, as shown in Figure 11. 3 11. The voxel dimensions were kept the same, i.e., 0.7 × 0.6 × 0.7 mm . In that case, the crack is totally 3 . In that case, the crack is totally The voxel dimensions were kept the same, i.e., 0.7 × 0.6 × 0.7 mm embedded in the anode sample. Hence, all voxels of an investigation box can be used. A 2D view of embedded in the anodeis sample. all voxels of an as investigation box can beto used. 2Dpercolation view of the the crack investigated shownHence, in Figure 12 as well the seed voxel used startAthe crack investigated is shown in Figure 12 as well as the seed voxel used to start the percolation process. process.

Figure 11. Partial anode block. Figure 11. Partial anode block.

A representative percolation run with the same parameters as presented in Section 3.2 (initial box dimension of 100 × 100 × 100 voxels with a sphericity threshold of Ψ = 0.5) is presented in Figure 13. The voxel count at the end of the percolation is 19,319. In this representative run, the percolation ended by reaching the sphericity threshold. Hence, the investigation box did not affect the percolation process, which was the goal of using a thicker anode slice. Figure 14 shows 2D views of the percolated region superimposed on anode images at three different locations inside the anode sample. It can be seen that most of the central part of the crack has been detected. However, the algorithm still misses some parts of the crack as obtained on the thinner anode slices (Figures 8 and 10). Again, this is mainly related to the image low-resolution limitation. The sphericity threshold can be increased to include the missing voxels but the algorithm will flood the investigation box and cracks will not be detected. Hence, the easiest way to overcome this problem would be to increase image resolution, but it would also lead to a significant increase of the analysis cost.

the percolated region superimposed on anode images at three different locations inside the anode sample. It can be seen that most of the central part of the crack has been detected. However, the algorithm still misses some parts of the crack as obtained on the thinner anode slices (Figures 8 and 10). Again, this is mainly related to the image low-resolution limitation. The sphericity threshold can be increased to include the missing voxels but the algorithm will flood the investigation box and Metals cracks 2016, 6,will 272 not be detected. Hence, the easiest way to overcome this problem would be to increase 10 of 13 image resolution, but it would also lead to a significant increase of the analysis cost.

Figure 12. 2D view of the crack (top right) used to test the 3D percolation algorithm on thick anode

Figure 12. 2D view of the crack (top right) used to test the 3D percolation algorithm on thick anode slice: (a) full view; (b) zoom on the cracked zone investigated with the seed voxel (white dot slice: (a) full view; (b) zoom on the cracked zone investigated with the seed voxel (white dot identified identified with a black arrow). with black arrow). Metalsa2016, 6, 272 11 of 14

Figure 13. Crack detection results (10 voxels ≈ 7 mm) on the partial anode block with restriction on

Figure 13. Crack detection results (10 voxels ≈ 7 mm) on the partial anode block with restriction on the investigation box (Initial box dimensions of 100 × 100 × 100 voxels and Ψ = 0.5). the investigation box (Initial box dimensions of 100 × 100 × 100 voxels and Ψ = 0.5).

Metals 2016, 6, 272 Figure 13. Crack detection results (10 voxels ≈ 7 mm) on the partial anode block with restriction on 11 of 13

the investigation box (Initial box dimensions of 100 × 100 × 100 voxels and Ψ = 0.5).

Figure 14. 2D views of the percolated region (white) on partial anode block: (a) image located near Figure 14. 2D views of the percolated region (white) on partial anode block: (a) image located near the the left end of the partial anode block; (b) image located at the mid width (135 mm) of the partial left end of the partial anode block; (b) image located at the mid width (135 mm) of the partial anode anode block; (c) image located near the right end of the partial anode block. block; (c) image located near the right end of the partial anode block.

4. Conclusions 4. Conclusions A 3D crack detection tool, based on a percolation method, has been developed as a A 3D crack detection tool, based on a percolation has beenThe developed as aalgorithm non-destructive non-destructive quantification technique for bakedmethod, carbon anodes. proposed is a quantification technique baked carbon anodes. The proposed is a direct extension direct extension of a 2Dfor percolation algorithm previously adaptedalgorithm for low-resolution images. The of a 2D percolationofalgorithm previously adapted for low-resolution images. The development of the development the algorithm required the proposal of an adaptive investigation box taking into algorithm required the proposal of an adaptive investigation box taking into account the main features of the cracks of interest, and the percolated volume sphericity was used as the threshold criterion for crack identification. The performance of the proposed 3D algorithm was tested on 3D images of partial volumes of a baked carbon anode (anode slice) obtained using a CT scanner. The results have shown the necessity of an adaptive investigation box when cracks are not completely embedded within the anode sample. The percolation process is, however, most effective when investigated cracks are fully embedded in the anode sample. The proposed algorithm encountered, as with all other algorithms available in the literature, crack detection problems when applied to low resolution images. Nevertheless, unlike the Hessian-driven method, the 3D percolation proposed in this work was able to partially identify thin cracks (few voxels width). Hence, in its current state and without increasing the image resolution (0.7 × 0.6 × 0.7 mm3 ), the algorithm could be used by the industry to qualify, but not yet quantify, their anodes by highlighting the presence and spreading of cracks. In summary, the algorithm is very promising for quantifying cracks and could help enhance anode quality control. To improve the 3D crack detection, higher resolution images are required. This will be investigated in the upcoming months on the same anode samples investigated in this paper. This will be achieved by increasing the reconstruction time (so the analysis cost) in order to reach the maximum resolution (0.09 × 0.04 × 0.09 mm3 ) of the apparatus used in this study, i.e., the Siemens Somatom Sensation 64. Finally, the proposed algorithm will be challenged with the Hessian-driven method to see at which point the percolation algorithm proposed will lose its advantage (from a resolution point of view) over the former one. Acknowledgments: Authors would like to acknowledge the financial support of the Natural Sciences and Engineering Research Council (NSERC) and Alcoa. A part of the research presented in this paper was financed

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by the Fonds de Recherche du Québec-Nature et Technologie (FRQNT) by the intermediary of the Aluminium Research Centre-REGAL. Particular gratitude is dedicated to Hugues Ferland, from the REGAL group, at Laval University and to Louis-Frédéric Daigle, from INRS-ETE, for their technical support. Author Contributions: Donald Picard, Julien Lauzon-Gauthier and Carl Duchesne conceived and designed the experiments; Donald Picard performed the experiments; Houshang Alamdari, Mario Fafard and Donald P. Ziegler contributed in data analysis; Donald Picard wrote the paper and all co-authors commented/corrected it. Conflicts of Interest: The authors declare no conflict of interest.

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