Bayesian segmentation for damage image using MRF prior

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1 Bayeian egmentation for damage image uing MRF prior G. Li 1, F.G. Yuan 1, R. Haftka and N. H. Kim 1 Department of Mechanical and Aeropace Engineering, North arolina State Univerity, Raleigh, N, , USA Department of Mechanical and Aeropace Engineering, Univerity of Florida, Gaineville, FL 3611, USA ABSTRAT Image egmentation for quantifying damage baed on Bayeian updating cheme i propoed for diagnoi and prognoi in tructural health monitoring. Thi cheme enable taking into account the prior information of the tate of the tructure, uch a patial contraint and image moothne. Baye law i employed to update the egmentation with the patial contraint decribed a Markov Random Field and the current oberved image acting a a likelihood function. Segmentation reult demontrate that the propoed algorithm hold promie of earching a crack area in the SHM image and focuing on the real damage area by eliminating the peudo-hadow area. Thu more precie crack etimation can be obtained than the conventional K-mean egmentation by hrinking the fuzzy tail which often exit on both ide of the crack tip. Keyword: Damage imaging, Damage etimation, Image egmentation, Markov random field, Gibb random field. 1. INTRODUTION Aircraft plate-like tructure are prone to develop crack due to cyclic load and everely corroive ervice environment. Diagnoi and prognoi of damage are eential iue in tructural health monitoring for preventing catatrophic failure and predicting the remaining life of tructure. Ultraonic wave imaging with variou enor array form ha potential to detect the damage or abnormity on large-cale, complex plate-like tructure. Damage location, hape and everity evaluation i one of the mot important topic in tructural health monitoring. The Lamb wave imaging method ha been recently hown to be effective in detecting damage uch a crack or delamination in tructure [1][][3]. Damage imaging method have focued on time-reveral baed migration technique in both time-pace domain and frequency-wave number domain [4][5][6][7]. Thee method can generate image indicating the damage condition of tructure by interpolating the data meaured from different form of enor array. The paper aim at quantifying the damage baed on the image generated from exiting imaging technique. Since the ize of the crack provide the eential information for tructural health prognoi to etimate reidual life of the tructural component according to crack propagation law uch a Pari law [8] in metal, which relate tre intenity factor and crack growth under fatigue. Image egmentation i an image proceing technique, which eparate the whole detected image into everal meaningful region with homogeneou attribute, e.g., damaged region/non-damaged region, according to the image itelf and ome prior knowledge. Figure 1 how a typical intenity image and it egmentation. Although human diagnoi with experience can identify the egmented region from the oberved image, but it i not an eay job for a computer to automatically divide an oberved image (epecially noie contaminated image with fuzzy information) into meaningful region. Image egmentation baed on tatitical model ha been developed in medical image analyi [9][1] and computer viion [11] [1] [13]. Bayeian updating cheme ued for image egmentation i introduced to merge the oberved image data with the prior knowledge, including patial contraint, trength and ditribution of diturbance to achieve more precie egmentation reult than thoe obtained uing traditional image egmentation method like K-mean method [14]. The ervice environment and pecific aircraft tructure lead to the following three main propertie of the detected image uing ultraonic wave when compared to medical imaging technique for human bodie. Firt, the large cale of aircraft tructure and online diagnotic requirement promote relative low denity of enor ditribution. Second, evere variation of temperature and humidity during the in-flight ervice alter the dynamic repone of detected tructure. Third, due to noie, electromagnetic diturbance from the environment, the monitoring ytem itelf and tructural Senor and Smart Structure Technologie for ivil, Mechanical, and Aeropace Sytem 9 edited by Maayohi Tomizuka, hung-bang Yun, Victor Giurgiutiu, Proc. of SPIE Vol J 9 SPIE code: X/9/$18 doi: / Proc. of SPIE Vol J-1 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

2 vibration, the experimental image from SHM ultraonic wave diagnoi are alway of relatively low quality. It i difficult for the traditional image egmentation like the threhold method, K-mean clutering to effectively eparate the damage region from the background. A reliable image egmentation cheme, which can be relatively immune to noie and fuzzy tail, i imperative tbtain a meaningful egmentation and precie damage quantification. la la I / /- Boundary Figure 1 An Image of intenity and it binary egmentation In thi paper, image egmentation baed on Bayeian updating with Markov Random Field i invetigated for plate-like tructural health monitoring uing ultraonic guided Lamb wave. Thi paper i organized a follow. Section introduce the Bayeian updating concept in image egmentation. Section 3 introduce Markov Random Field and Gibb Random Field a prior information in the Bayeian updating framework. Section 4 dicue how to model the image egmentation problem with Bayeian updating framework, and to quantify the image egmentation a a problem of maximizing a poterior (MAP). An illutrative example i alo given in thi ection. Section 5 preent the application of the egmentation method for revere-time migration image in frequency-wavenumber domain. Section 6 carrie out ome dicuion on the reult. Section 7 provide concluding remark.. IMAGE SEGMENTATION BASED ON BAYESIAN UPDATING Image egmentation refer to the proce of partitioning an image into multiple region. The goal of egmentation i to implify the repreentation of an image into a cla of certain repreentation, each of which hold ditinct characteritic and i eaier to ditinguih among them. Reearcher have developed many method baed on traditional clutering method uch a fuzzy mathematic, neural network, and tatitical model. Bayeian updating i one viable approach, which take advantage of a tatitical model to combine the current oberved data with prior knowledge, uch a patial contraint and moothne level. Bayeian updating i tatitical inference in which evidence or obervation are ued to update or to newly infer the probability that a hypothei may be true. It ue apect of the cientific method, which involve collecting evidence that i meant to be conitent or inconitent with a given hypothei. A evidence accumulate, the degree of belief in a hypothei hould change. With enough evidence, it hould become very high or very low. Thoma Baye firtly introduced Bayeian theorem [15], which relate the conditional and marginal probabilitie of event x and y by Where, py ( xpx ) ( ) px ( y) = (1) py ( ) px ( ) i the prior probability or marginal probability of x before the obervation. It i prior in the ene that it doe not take into account any information about y. px ( y) i the conditional probability of x after the obervation, given y. It i alo called the poterior probability becaue it i derived from or depend upon the pecified value of y. Proc. of SPIE Vol J- Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

3 py ( x ) i the likelihood function, which i the conditional probability of y given x. p( y ) i marginal probability of y, and act a a normalizing contant. In image egmentation, the prior function px ( ) can repreent prior aumption about the image for monitoring the tructural health. For intance, the aumption can include how the pixel contrain each other in the pace, and how evere the environmental noie i. In addition, a previou monitoring image and prior knowledge about the tructure can alo be conidered a prior information. The likelihood function relate to the oberved information, which i the detected intenity image in diagnoi from guided wave imaging with finite enor array, uch a linear enor array [16] or ditributed enor array [17]. 3. MARKOV RANDOM FIELD AND GIBBS RANDOM FIELD Markov Random Field (MRF) image modeling ha been ued uccefully in many image proceing technique [18]. The ucce of Markov Random Field modeling mainly arie from it ytematic and flexible treatment of the contextual information in the image. Prior knowledge about the image egmentation can be eaily quantified by Markov Random Field model parameter. Image egmentation procee the property of contextual moothne of the cla label in the image pace o that a pixel with a particular cla label i likely to hare the label with it immediate neighbor. Moreover a Bayeian framework uing MRF provide feaible optimal olution. The optimization proce uing patial local interaction make parallel and local computation poible. Following are ome baic definition [19] and derivation for the image egmentation baed on Bayeian updating cheme. Neighborhood A neighborhood ytem η aociated with the whole image Ω i a collection of neighborhood η = { η Ω }, where eachη i a neighborhood of the pixel at the ite of atifying (1) The pixel itelf doe not belong to it own neighborhood, or η ; () The pixel belonging to the neighborhood of pixel t implie that pixel t belong to the neighborhood of pixel, or η t implie t η lique A clique i a ubet of the whole image Ω if two different element of are neighbor. Figure give the nd order neighborhood and all the available clique in thi 8-element neighboring ytem. Figure All poible clique for the econd-order neighborhood ytem Markov Random Field A Markov Random Field on ( Ω, η) i a random field with it probability property of each ite in the whole field atifying the Markovian property decribed a following, Proc. of SPIE Vol J-3 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

4 P( X = x X = x, t Ω, t ) = P( X = x X = x, t η ), Ω. t t t t Gibb Random Field A random field X with (Ω, η, ), i a Gibb Random Field (GRF, or a random field with a Gibb ditribution) if it joint ditribution ha the form 1 P( x) = exp Vc ( x) Z () c where i the et of all clique in Ω, Z i the normalizing contant, and V c (x) i the clique potential aociated with cliquec. There i no particular retriction on the clique potential definition. A long a the reulting Gibb ditribution atifie the definition of the probability, the aociated clique potential are valid. The Gibb potential can be defined uch that ome pecific feature of the image can be identified and emphaized. In addition, it i not neceary to ue all type of clique for a given neighborhood ytem; that i, any pecific et of clique type can be electively ued. A Gibb Random Field i defined by a joint probability. On the contrary, the MRF i defined baed on a conditional probability. Prior Knowledge about the problem uch a the moothne contraint on the cla label can be incorporated into the Gibb ditribution by the choice of pecific clique type and their potential. For example, the moothne of the cla label in image pace can be meaured by defining the clique potential uch that a high poitive clique potential i aigned only when all cla label in the clique are identical. Hammerley-lifford Theorem On (Ω, η, ), a random field X i a Markov random field with repect to η if and only if P(x) ha a Gibb ditribution with repect toη. Hammerley-lifford Theorem build a bridge between MRF and GRF. The joint ditribution (i.e., Gibb ditribution) can be contructed from the local conditional probability (i.e. MRF). A a equence of the theorem, it i now poible to expre the conditional probability of a Markov Random Field in term of clique potential. Thi i ueful in practice becaue it i eay chooe the clique type and their potential to decribe the deired local behavior. For example, local patial relationhip uch a moothne and continuity of the neighboring pixel can be pecified by iotropic pair clique potential β. It i crucial for the theorem to hare the ame neighborhood ytem η and the aociate clique c for both MRF and GRF. 4. PROBLEM MODELING WITH MAXIMUM A POSTERIOR 4.1 Bayeian updating with MRF/GRF The image contructed by the ignal from the enor array for tructural health diagnoi i uually noiy and with fuzzy tail a dicued in Section 1. In Bayeian updating cheme, the current image, which can be conidered a a likelihood function, i updated by introducing prior aumption. MRF/GRF dicued in Section 3 i employed to eliminate the effect of noie in the image by deignating thee imilar neighbor with high probability. Thu the egmentation procedure depend on not only the image intenity but alo the pace retriction property of the neighboring ytem. For computational efficiency, the nd -order clique i ued, which contain an 8-element neighborhood. According to the MRF-GRF equivalence decribed by Hammerley-lifford theorem, the clique potential i given in the form of Gibb denity a Eq.() The Gibb potential value i defined a [1], if x xq and, q V ( x) = β = + β if x xq and, q where β i poitive. The choice of β will affect the patial contraint in the GRF. Larger β reult in tronger patial contraint; that i, neighbor are more likely to have the ame label. (3) Proc. of SPIE Vol J-4 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

5 The image egmentation problem i ill-poed imply becaue it olution pace i too large. The olution pace can be reduced by incorporating prior knowledge about the image, uch a the moothne in the problem formulation. The Maximum a poteriori (MAP) criterion [] ha frequently been ued to characterize the prior knowledge. According to the MAP criterion, given a realization of a random field, the goal i to find an optimal realization of x ), which maximize the a poteriori probability P( x y ) for all poible realization of x. By exploiting the chain rule and taking the monotonically increaing logarithmic function, the maximization can be equivalently expreed a: ) x = arg max P( x y) x Py ( xpx ) ( ) = arg max x Py ( ) = arg max Py ( xpx ) ( ) x = arg max[ln Py ( x) + ln Px ( )] x The oberved image can be modeled with a Gauian ditribution [1], and the prior function i the Gibb Random Field, which hold the probability in Eq. (5). Thu the MAP problem can be decribed a: x ( ) exp p x y y μ V x σ (4) ( ) (5) In thi equation, x i the egmentation label ranging from 1 to k. S i the poition of the etimated grid. y i the oberved x value (obtained image from SHM ytem). And μ i the mean value in the window with a certain ize centered at poition. px ( y) i the probability of the evaluated grid belonging to the cluter label x, given the intenity of ome grid. For the poterior probability, if px ( = y) > px ( = 1 y), then the etimated x hould be ; and if px ( = y) < px ( = 1 y), then the etimated x hould be 1; thu the etimation of the egmentation label ˆx i achieved. Thi poterior probability comprie of two part. The firt part i Gauian ditribution likelihood, and the econd part i the prior denity function -- Gibb potential. The firt part i eentially K-mean method, which calculate the ditance between the evaluated value and the clutering center. For the firt likelihood part, if the evaluated pixel value i cloe to the cluter center, the poterior energy function will be enhanced. The econd part i the adjutable part according to the neighboring environment. For the prior Gibb potential part, if the neighbor clique member i the ame label (from initialization or the previou egmentation), the Vc(x) hould be negative, then the poterior energy denity will be enhanced. On the other hand, if the clique member are at different label, the poterior energy denity will be weakened. For a given cae, neglecting the prior ditribution by etting β =, thi algorithm degenerate to a K-mean clutering method, which only count the ditance between each point with the clutering center. 4. An illutrative example To illutrate the procedure and the power of the Bayeian baed algorithm for SHM image egmentation, a mall ize grey-level matrix, which repreent a imple image of intenity caling from to 1, i hown in Figure 3. Proc. of SPIE Vol J-5 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

6 A B. ci ci c ci ci ci o. k 8. if F o a o a oa I '.; H I j o pi Figure 3 Matrix repreentation of original image for illutrative example The image ha three patche of high-level area at the firt glance marked a the darker color. The left-ide areaa coniting of element (.9,.7,.75,.8 and.85) i a relative large area, and the two right-ide one, which are element.777 and.8, are mall. In image analyi, thee mall area are uually caued by environment noie, thu they hould be eliminated from the image in the egmentation procedure. Figure 4 i the correponding noie contaminated matrix, whoe intenity image repreentation can be een in Figure 3(a). A 14 O (:.1.3. B. 3. t'7 '. II:11F3 IIII. 1.3 ' ': Lf 811 Lf n 91 1fi Ill I 8 T ' ui (I 9 'III' I L 9 ' ' ' ' 111 6T H I S '. 8. cii IIII. ' 'lilies O.j.O...O3 3.(, 1 6. a ) ri 1)6 i LL L T.' E3 Tu 9 I 1 ' ' '. 6 L 9 Ou" 1' I.. 9 L Figure 4 Matrix repreentation of the noie contaminated image A traditional K-mean clutering algorithm i applied tbtain the initial egmentation. The algorithm attempt to categorize the etimated ite to the cloet cluter center by the meaurement in ditance. Matrix in Figure 5 and it intenity image repreentation in Figure 7(b) are the initial egmentation by K-mean indicating three higher level area with region label 1 and other lower level area with region label. Three typical cell are highlighted to illutrate the Bayeian updating procedure. ell D3 i a high grey value pixel with mainly high grey value neighbor; ell F35 i a typical low grey value pixel with all lower grey value neighbor; and cell H9 i a typical high grey value pixel with lower grey value neighbor. When updating the egmentation label, the immediate neighboring information hould be introduced by the formulation of GRF. Proc. of SPIE Vol J-6 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

7 All the pixel in the image may be roughly claified into three categorie mentioned above. The factor howing how they behave in the updating proceduree i the different number of ame label pixel in it neighboringg ytem. For example, the.77 the cell D19 ha 4 high level neighbor, while it right neighbor.8 only ha two high level neighbor. Therefore.77 i more likely to belong to be cluter 1 than it neighbor.8. B? o a 8 o a o i ' ' i F F D 3 Ii I 3 3! n n I j n Figure 5 Initial egmentation with traditional K-mean clutering To etimate the poterior probability, the mean value of each region need to be calculated a follow: For the cla (lower grey level cluter/ light area): μ =.69 1 For the cla 1 (higher grey level cluter/ dark area): μ =.844 Pixel in all the image may be etimated with Eq. (5). Table1 lit three type of typical cell with index D19, H16, and F with the Gibb potential part given on the right ide, indicating the adjutment proceduree for each cae. The final egmentation reult i given in Figure 7(c) which keep the dominated high level zone and eliminate the iolated malll one. The reult in matrix form are hown in Figure 6. ell Poition D19 Table 1 Poteriorr calculation with Gibb potential Poterior probability with Gibb potential p= exp y σ = exp [ σ p1= exp y σ = exp [ σ μ V ( x = ) ] 1 μ V ( x = 1) ] V V = ( x ) = ( x 1) Gibb potential part V = V = Proc. of SPIE Vol J-7 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

8 : a : a H16 F p= exp y σ = exp [ σ p1= exp y σ = exp [ σ p= exp y σ = exp [.. 69 σ p1= exp y σ = exp [ σ 1 μ V ( x = ) ] 1 μ V ( x = 1) ] μ V ( x = ) ] 1 μ V ( x = 1) ] V V V V = ( x ) = ( x 1) = ( x ) = ( x 1) V = V = V = V = 8β = 8β 8β = 8β B t) F F a U1 a I I I I I J ' ' ' 48 () o ' 49 Figure 6 Bayeian egmentation with MRF prior Proc. of SPIE Vol J-8 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

9 .. (a) (b) (c) Figure 7 omparion of (a) the original image and (b) K-mean reult and (c) Bayeian updated reult 5. SEGMENTATION FOR F-K MIGRATION IMAGES Imaging algorithm for diagnoing damage in plate-like tructure have drawn continual interet. The time reveral cheme developed in time-pace domain and frequency-wave number domain ha been uccefully verified it effectivene and efficiency for damage detection. The migration imaging technique, baed on Mindlin plate theory, i one of the mot promiing method for multi-damage identification in plate-like tructure uing cattered Lamb wave in combination with the time-coincidence imaging condition. The migration technique can effectively interpret the enor data recorded by a ditributed linear array enor ytem and make it poible to etablih an active, in-ervice, and intelligent monitoring ytem. The image for egmentation i obtained from an active diagnotic linear array of actuator/enor, which i ued to excite/receive the flexural wave. The wave field cattered from the damage and enor array data are yntheized uing a two-dimenional explicit finite difference cheme to model wave propagation in the plate baed on the Mindlin plate theory. The damage image can be repreented by the cro-correlation in frequency domain a: e I( x, y) = w ( x, y, ω) w ( x, y, ω) ω (6) e in which I( xy, ) i the pixel value at ( x, y ), w ( x, y, ω) and w ( x, y, ω ) are the extrapolated excitation and cattered wave-field in frequency domain, repectively. Senor array Senor array.., S S Damage Noie Fuzzy tail Etimated damage boundary (a) (b) Figure 8 (a) f-k migration image and (b) the Bayeian baed egmentation When K-mean egmentation i applied to the f-k migration image, ome of the medium grey-level in the fuzzy tail will be egmented a the damaged area. Markov Random Field feature the property of emphaizing the local contraint by Proc. of SPIE Vol J-9 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

10 introducing the Gibb potential in the poterior probability. Thu the damaged area from the egmentation hrink regardle of ghot image. Figure 8(b) give the egmentation reult and it region reulting from Bayeian updating cheme. A a comparion, for a mm crack cae, the crack length obtained from the K-mean (without noie) egmentation i 3 mm, while the etimated crack ize obtained from Bayeian updating algorithm i 5 mm, providing more precie crack ize etimation. Furthermore, the crack ize etimated i larger than the true crack ize, enuring the conervative meaure of the damage. The parameter β in the Gibb potential play an important role of the egmentation procedure. It determine how the neighboring ytem affect the label of the etimated ite. When β =, the Bayeian egmentation reduce to the K- mean clutering, which egment ome of the noie area in the ame region of the damaged area and the reult i given in Figure 9(b). It i difficult to determine the crack length with uch egmentation. With the increae in β, the noie i uppreed, and the damage area hrink to the real damaged area. Reult how that the egmentation doe not change when β varie from.9 to 1, which how the tability of the egmentation. The reaon for the convergence phenomenon i that the Gibb potential provide play only an adjutment role in the Bayeian updating procedure, but not a dominating role. (a) Original (b) β= (c) β=.1 (d) β=.5 Figure 9 Bayeian baed image egmentation with different Gibb potential The reolution of the egmentation firt depend on the quality of the image. For the imaging procedure uing f-k migration technique, the center frequency of the excitation i 15 khz a hown in Figure 1. The maximum frequency over 3% frequency magnitude i about 44 khz, and the thickne of the monitored aluminum plate in the imulation i 3. mm. The mainly excited mode in the imulation i the firt-order aymmetric mode, A mode. From the diperive curve the velocity of A mode for Lamb wave i 3.5 m/m. Through Eq. (7), the wavelength for the highet frequency in the excitation ignal can be obtained. v λ = (7) f The hortet wavelength in the excitation frequency band i 1.4 mm. Therefore, the etimated error of 5 mm in crack ize i maller than half of the hortet wavelength. To improve the accuracy of the crack ize etimation, higher frequency excitation may be ued. However there will be a tradeoff between the complexity of multi-mode in high frequencie and the image reolution. Proc. of SPIE Vol J-1 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

11 1 8.6 r r.4 44 khz.1 L OL 5 i':': '1' 4 Frequency (khz) Figure 1 Excitation waveform in time-domain and frequency-domain 6. ONLUSIONS The propoed Bayeian baed egmentation algorithm for etimating damage ize ha been hown to poe the following advantage: (1) Spatial contraint uing the neighboring ytem i better than K-mean clutering, which only conider the grey-level ditance between the pixel and the clutering center. A a reult, the damage area i focued uppreing the fuzzy tail that may appear on both crack tip. () Bayeian baed egmentation hold the promie of reducing noie by applying the Gibb Random field, which aume that neighboring pixel more likely belong to be the ame cla. (3) The egmentation reult give damage hape and region etimation of the damaged area. In other word, the Bayeian updating with the MRF a a prior can efficiently egment an image into multiple region, which ditinguihe damaged region from undamaged region. For future work, the egmentation reult can alo be ued a prior information for the next Bayeian updating cheme, thereby enhancing the current image. Furthermore, different diagnotic technique that form the image can contribute to Bayeian updating and make the egmentation more precie and reliable. 7. AKNOWLEDGEMENT The author gratefully acknowledge the upport of the reearch by the U.S. Air Force under Grant FA and by the NASA under Grant NNX8A334. REFERENES 1. L. Wang and F. G. Yuan, Active Damage Localization Technique baed on Energy Propagation of Lamb Wave, Smart Structure and Sytem, 3(), 1-17(7).. V. Giurgiutiu, L. Yu, K. Jame and R. J. hritopher, In Situ Imaging of rack Growth with Piezoelectric-Wafer Active Senor, AIAA Journal, 45(11), (7). 3. Y. Gómez-Ullate and F. Montero de Epinoa, Non Detructive Evaluation of arbon Fiber Reinforced Plate Uing Lamb Wave: A omparion between Pitch-atch Air oupled Technique and Sector Image Obtained with Embedded Piezoelectric Linear Array, Ultraonic Sympoium, IEEE, (7). Proc. of SPIE Vol J-11 Downloaded from SPIE Digital Library on 8 Oct 9 to Term of Ue:

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