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Localization of defects in magnetic permeable bar by multiple signal classification approach using rotating dipole model
ARTICLE IN PRESS NDT&E International 42 (2009) 664–667
Contents lists available at ScienceDirect
NDT&E International journal homepage: www.elsevier.com/locate/ndteint
Localization of defects in magnetic permeable bar by multiple signal classification approach using rotating dipole model R. Baskaran , M.P. Janawadkar Condensed Matter Physics Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, Tamilnadu, India
a r t i c l e in f o
a b s t r a c t
Article history: Received 4 November 2008 Received in revised form 14 May 2009 Accepted 26 May 2009 Available online 2 June 2009
A simplified search algorithm based on multiple signal classification (MUSIC) approach with rotating dipole model has been proposed to identify the defect locations inside a magnetically permeable bar. This new algorithm identifies the defect locations and the moments of the modelled anti-dipoles without the necessity to decompose the single column lead vector by singular value decomposition (SVD). Here, we find the location of the first dipole by using the rotating dipole model of MUSIC approach and the rest of the defect locations are identified using the orthogonal projection method coupled with rotating dipole model of MUSIC. Three deeply buried defects were modelled as antidipoles and the leakage fields due to these three deeply buried defects were simulated by direct forward calculation. Using the calculated data as an input, it has been possible to identify the number of defects and their locations by this new approach with significant reduction in computations, even in the presence of reasonable levels of additive noise. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Magnetic flux leakage Inverse problems Singular value decomposition Generalized inverse Rotating dipole Multiple signal classification Orthogonal projection
1. Introduction Magnetic flux leakage (MFL) is widely used to detect and characterize defects in ferromagnetic oil and gas pipelines and other structures like wire ropes of suspension bridges, rail tracks, and pressure vessels manufactured from magnetic steels [1–3]. Different approaches to identify the defect locations and characterization of the defects have been reported [4–6]. A detailed description of defect modeling based on multiple signal classification (MUSIC) approach and its advantage in identifying the number of defects has been presented elsewhere [6]. The main objective is to determine the number of defects, their locations and the strengths of the anti-dipoles that represent the defects from the leakage fields. In this paper, we show that it is possible to identify all the defect locations by the MUSIC approach with rotating dipole model coupled with an orthogonal projection of the measured field data, provided the defects are well separated. This approach avoids singular value decomposition (SVD) of the single dipole lead vector at all search locations [6], thus reducing the computations for calculating the cost function at each scan coordinate. We also show that it is sufficient to scan the cost function in two dimensions of x and y, keeping the depth z as a constant for identification of defects by the MUSIC search algorithm. The two modifications together significantly reduce
Corresponding author.
E-mail address: [email protected] (R. Baskaran). 0963-8695/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2009.05.005
the total computations for finding the approximate locations of the defects by the MUSIC making the inverse calculations more efficient. To get a better estimate of the locations of the defects, based on the information of the number of defects and their approximate locations obtained by this new approach, pseudoinverse calculations are subsequently performed as detailed elsewhere [6].
2. Defect identification by MUSIC If the size of the defect is small like a cavity with a closed surface, then the magnetic flux leakage due to the defect in a magnetically permeable material can be modelled as arising due to an anti-dipole positioned at the defect location. The sampled magnetic field bz (x, y, z) can be written as a product of a nonlinear function of space variables M (x, y, z, x0, y0, z0) called lead matrix and the magnetic dipole intensity vector Q corresponding to the defect dipoles. To include the effect of noise, which is inevitably present in any experimental measurement, we add a noise term N which is a Gaussian white noise with a mean zero and a variance of approximately 10% of the maximum flux leakage field. bzn ¼ MQ þ N
(1)
If the locations of the defects and the strengths of the antidipoles representing them are known apriori, the magnetic field due to all the defects can be calculated using the above Eq. (1).
ARTICLE IN PRESS R. Baskaran, M.P. Janawadkar / NDT&E International 42 (2009) 664–667
In the MFL, with the knowledge of bzn, we should be able to find the locations of the defects. In general, there is no unique solution to the above problem. However, if the number of defects is known apriori, then it is possible to find an approximate solution using the pseudo-inverse approach [5,6] by minimizing kek ¼ kbzn M 0 M0
1
bzn kF
(2)
This minimum norm solution can be rewritten as follows kek ¼ kP ? M bzn kF
(3)
?
where PM is the orthogonal projector of M [7]. If bzn is decomposed into [U S VT] by the SVD [7–9], then PM? T can be approximated by Um1 Um1 , where Um1 is the matrix comprising of the remaining m1 columns of the decomposed matrix U, representing the noise subspace [6]. In stationary dipole model of the MUSIC, the location and orientation of each dipole are unknown quantities, making this a five-dimensional search problem in Eq. (3). To reduce search in three dimensions further, SVD of the single dipole vector is taken. In the MFL all the antidipoles are expected to align in the same—X direction, which implies that the direction cosines of the dipole moment vector are known. As all the anti-dipoles are aligned in same direction, this situation is analogous to the case of rotating dipoles with all the dipoles aligned at a particular instant. In the application of MUSIC algorithm to the case of rotating dipoles, the correlation between the signal subspace and noise subspace is minimized. In rotating dipole model of MUSIC, G1, the single dipole lead vector itself constitutes the signal subspace. The cost function Jls [7] is the minimum of the eigen-values of the product of signal subspace and noise subspace and is given by J ls ¼
kU Tm1 G1 k kG1 k
665
present in the bar is calculated based on the forward problem discussed above neglecting the magnetic flux density due to the magnetization of the sample [6]. A Gaussian noise with mean zero and standard deviation of 10% of the maximum of bz is added to the magnetic field to simulate the noise present in measured data. The three anti-dipoles are assumed to have a strength of 30, 7 and 17 nA m, respectively. The z-component of the magnetic field is sampled at 1681 locations in a plane 3 mm above the top surface with x ranging from 20 to +20 mm in steps of 1 mm and y ranging from 20 to +20 mm in steps of 1 mm forming the bzn vector. Fig. 1 shows the plot of the magnetic flux leakage field due to these three defects. The cost function Jls is calculated for all the 1681 locations using the rotating dipole model of MUSIC, with z taken as 10 mm. As indicated in Fig. 2, a two-dimensional plot of the inverse of the cost function versus scan coordinates shows a peak at the dominant defect location indicating the presence of a defect at the corresponding x and y coordinates. A better estimate of the position is then obtained by the generalized inverse procedure [6], for a single dipole using the measured bzn. Having obtained the first defect location, the gain vector for the first dipole G1 is constructed. The orthogonally projected data matrix
(4)
where JG1J is used as a normalization factor. The cost function Jls when evaluated at dipole locations will yield a near zero value for low-noise data, whereas at all other locations the cost function will be nearly unity. The functional dependence of G1 on z is weak compared to that on x and y. Hence, we have found it sufficient in practice to assume an average z and scan for x and y coordinates only. To identify the position of the dominant dipole in the presence of noise, the minimum eigenvalue cost function is evaluated in the region of interest (x, y coordinates only) with an average z value (half the thickness of the plate approximately) and the inverse of this cost function is plotted. Peak in this graph indicates the x and y coordinates of the position of dominant dipole [7]. A better estimate of the defect position is obtained by the generalized inverse procedure [5] for single dipole using the measured bzn. Having identified the location of the dominant dipole, the locations of other defects are then identified by repeatedly using the method of orthogonal projection away from all the dipoles identified in the previous step [6,8].
Fig. 1. Calculated X–Y scan of magnetic field bzn for three widely separated subsurface defects with 10% noise added. This has been used as input for inversion using orthogonally projected rotating dipole model of MUSIC.
3. Results of numerical simulations In order to demonstrate the effectiveness of the above procedure, the magnetic flux leakage field, due to three buried defects is calculated in a magnetically permeable bar with three distinct defects. The magnetically permeable bar of size 40 mm 40 mm 15 mm is assumed to be magnetized by an applied field along the positive X-direction and three distinct anti-dipoles are positioned at (12, 15, 8), (8, 11, 12) and (12, 14, 10) representing the defect locations. The x–y plane is taken as the top surface of the bar with the bar placed symmetrically with respect to the x–z plane. The flux leakage field due to all the three defects
Fig. 2. Plot of inverse of cost function calculated from bzn, with respect to the scan coordinates utilized for identification of the strongest dipole.
ARTICLE IN PRESS 666
R. Baskaran, M.P. Janawadkar / NDT&E International 42 (2009) 664–667
bzn?1 is then obtained. The algorithm is repeated with bzn?1 which yields the position of the second defect represented by second most intense anti-dipole [6]. A better estimate of the positions of the two defects is then obtained by the generalized inverse procedure for two dipoles using the measured bzn. The process is repeated to get the position of the third defect. After identifying all the three defect locations accurately using generalized inverse, the orthogonally projected data matrix for the three identified defects bzn?3 is obtained. The search with the above algorithm does not reveal any peaks within the region of interest as shown in Fig. 3, indicating that no further
Fig. 3. Plot of inverse of cost function calculated from bzn?3 after identification of three dipoles, with respect to the scan coordinates in the region of interest.
defects exist. The stray peaks in Fig. 3 have an amplitude less than 1.0008 (o1%) which is below the noise level in the data and are too weak to be identified with possible defect location. The inferred locations of the defects agree with the those assumed in the forward calculation to within 1 mm in x and y coordinates. The calculated magnetic field using the inferred locations and strengths of the dipoles is shown in Fig. 4. The results are presented in Table 1 for 10% additive noise. For a quantitative comparison of the computational burden between this algorithm based on rotating dipole model of MUSIC scanning only in two dimensions (x and y) with fixed z and the earlier stationary dipole model of MUSIC scanning in three dimensions (x, y and z) [6], the number of floating point operations required for computing has been obtained from MATLAB 5.3 for both the algorithms and is denoted as flops. The number of flops for searching the dipole in two dimensions with fixed value of z using the rotating dipole model (no SVD of Gi) is approximately 20% of the number of flops for searching dipole in two dimensions with fixed value of z using stationary dipole model (with SVD of Gi). Due to reduced search domain in the revised approach, the computational burden reduces to 1/15th of the earlier approach. The computational burden associated with other parts of the algorithm such as non-linear least square minimization are common for both versions. The total number of operations for running the stationary dipole model algorithm with a three-dimensional search space is estimated to be 2.88 1012 flops, while the total number of operations required for running the revised algorithm based on rotating dipole model of MUSIC with z fixed at 10 mm presented in this paper is estimated to be 1.02 1011 flops. This directly implies that the computational burden of the revised algorithm is only about 3.5% of that for the stationary dipole model of MUSIC for defect detection using MFL.
4. Conclusion
Fig. 4. Calculated magnetic field at the measurement plane due to the defects identified by the orthogonally projected MUSIC algorithm using the rotating dipole model.
We have developed an improved approach by which it has been possible to identify the number of defect locations and hence the positions and strengths of the dipole moments in flux leakage experiments using the orthogonally projected MUSIC with rotating dipole model. The results clearly indicate the capability of the orthogonally projected MUSIC with rotating dipole model algorithm based on the pseudo-inverse technique to identify the buried defects even in the presence of reasonable levels of noise. We estimate that by this new approach, it has been possible to reduce the total computational burden to about 3.5% of that required for searching the dipoles in three dimensions using the conventional stationary dipole model of MUSIC to evaluate the number of defects, their positions and strengths. The ideas presented in this paper could be utilized for identification of location of buried defects in magnetic steel plates when the measurement is done after demagnetization of the plate. For plates of larger thickness than assumed in this modeling, however, it may be necessary to scan with z in larger steps.
Table 1 Dipole locations recovered using the orthogonally projected dipole model of the MUSIC approach identifying the three distinct dipoles. First dipole
Actual Calc.
Second dipole
Third dipole
x (mm)
y (mm)
z (mm)
Moment (nA m)
x (mm)
y (mm)
z (mm)
Moment (nA m)
x (mm)
y (mm)
z (mm)
Moment (nA m)
12.0 11.9
15.0 15.0
8.0 8.0
30.0 29.9
8.0 8.0
11.0 11.1
12.0 10.7
7.0 5.5
12.0 11.8
14.0 14.2
10.0 10.1
17.0 17.6
ARTICLE IN PRESS R. Baskaran, M.P. Janawadkar / NDT&E International 42 (2009) 664–667
Acknowledgement The authors would like to place on record their sincere gratitude to Dr. C.S. Sundar for the continued support. References [1] Mandal K, Atherton DL. A study of magnetic flux leakage signals. J Phys D: Appl Phys 1998;31:3211–7. [2] Mandache Catalin, Clapham Lynann. A model for magnetic flux leakage signal predictions. J Phys D: Appl Phys 2003;36:2427–31. [3] Bray DE, Stanley RK. Nondestructive evaluation. New York: McGraw-Hill; 1989.
667
[4] Bruno AC. Imaging flaws in magnetically permeable structures using the truncated generalized inverse on leakage fields. J Appl Phys 1997;82(12):5899–906. [5] Baskaran R, Pattabiraman M, Janawadkar MP. Imaging defects in a threedimensional magnetically permeable medium using pseudoinverse technique. J Appl Phys 2006;100:064909–11. [6] Baskaran R, Janawadkar MP. Defect localization by orthogonally projected multiple signal classification approach for magnetic flux leakage fields. NDT&E Int 2008;41(6):416–9. [7] Mosher John C, Lewis Paul S, Leahy Richard M. Multiple dipole modelling and localization from spatio-temporal MEG data. IEEE Trans Biomed Eng 1992;39(6):541–57. [8] Mosher John C, Leahy Richard M. Source localization using recursively applied projected (RAP) MUSIC. IEEE Trans Signal Process 1999;47(2):332–40. [9] Meyer Carl D. Matrix analysis and applied linear algebra. Philadelphia: SIAM; 2000.
Contents lists available at ScienceDirect
NDT&E International journal homepage: www.elsevier.com/locate/ndteint
Localization of defects in magnetic permeable bar by multiple signal classification approach using rotating dipole model R. Baskaran , M.P. Janawadkar Condensed Matter Physics Division, Indira Gandhi Centre for Atomic Research, Kalpakkam 603102, Tamilnadu, India
a r t i c l e in f o
a b s t r a c t
Article history: Received 4 November 2008 Received in revised form 14 May 2009 Accepted 26 May 2009 Available online 2 June 2009
A simplified search algorithm based on multiple signal classification (MUSIC) approach with rotating dipole model has been proposed to identify the defect locations inside a magnetically permeable bar. This new algorithm identifies the defect locations and the moments of the modelled anti-dipoles without the necessity to decompose the single column lead vector by singular value decomposition (SVD). Here, we find the location of the first dipole by using the rotating dipole model of MUSIC approach and the rest of the defect locations are identified using the orthogonal projection method coupled with rotating dipole model of MUSIC. Three deeply buried defects were modelled as antidipoles and the leakage fields due to these three deeply buried defects were simulated by direct forward calculation. Using the calculated data as an input, it has been possible to identify the number of defects and their locations by this new approach with significant reduction in computations, even in the presence of reasonable levels of additive noise. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Magnetic flux leakage Inverse problems Singular value decomposition Generalized inverse Rotating dipole Multiple signal classification Orthogonal projection
1. Introduction Magnetic flux leakage (MFL) is widely used to detect and characterize defects in ferromagnetic oil and gas pipelines and other structures like wire ropes of suspension bridges, rail tracks, and pressure vessels manufactured from magnetic steels [1–3]. Different approaches to identify the defect locations and characterization of the defects have been reported [4–6]. A detailed description of defect modeling based on multiple signal classification (MUSIC) approach and its advantage in identifying the number of defects has been presented elsewhere [6]. The main objective is to determine the number of defects, their locations and the strengths of the anti-dipoles that represent the defects from the leakage fields. In this paper, we show that it is possible to identify all the defect locations by the MUSIC approach with rotating dipole model coupled with an orthogonal projection of the measured field data, provided the defects are well separated. This approach avoids singular value decomposition (SVD) of the single dipole lead vector at all search locations [6], thus reducing the computations for calculating the cost function at each scan coordinate. We also show that it is sufficient to scan the cost function in two dimensions of x and y, keeping the depth z as a constant for identification of defects by the MUSIC search algorithm. The two modifications together significantly reduce
Corresponding author.
E-mail address: [email protected] (R. Baskaran). 0963-8695/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ndteint.2009.05.005
the total computations for finding the approximate locations of the defects by the MUSIC making the inverse calculations more efficient. To get a better estimate of the locations of the defects, based on the information of the number of defects and their approximate locations obtained by this new approach, pseudoinverse calculations are subsequently performed as detailed elsewhere [6].
2. Defect identification by MUSIC If the size of the defect is small like a cavity with a closed surface, then the magnetic flux leakage due to the defect in a magnetically permeable material can be modelled as arising due to an anti-dipole positioned at the defect location. The sampled magnetic field bz (x, y, z) can be written as a product of a nonlinear function of space variables M (x, y, z, x0, y0, z0) called lead matrix and the magnetic dipole intensity vector Q corresponding to the defect dipoles. To include the effect of noise, which is inevitably present in any experimental measurement, we add a noise term N which is a Gaussian white noise with a mean zero and a variance of approximately 10% of the maximum flux leakage field. bzn ¼ MQ þ N
(1)
If the locations of the defects and the strengths of the antidipoles representing them are known apriori, the magnetic field due to all the defects can be calculated using the above Eq. (1).
ARTICLE IN PRESS R. Baskaran, M.P. Janawadkar / NDT&E International 42 (2009) 664–667
In the MFL, with the knowledge of bzn, we should be able to find the locations of the defects. In general, there is no unique solution to the above problem. However, if the number of defects is known apriori, then it is possible to find an approximate solution using the pseudo-inverse approach [5,6] by minimizing kek ¼ kbzn M 0 M0
1
bzn kF
(2)
This minimum norm solution can be rewritten as follows kek ¼ kP ? M bzn kF
(3)
?
where PM is the orthogonal projector of M [7]. If bzn is decomposed into [U S VT] by the SVD [7–9], then PM? T can be approximated by Um1 Um1 , where Um1 is the matrix comprising of the remaining m1 columns of the decomposed matrix U, representing the noise subspace [6]. In stationary dipole model of the MUSIC, the location and orientation of each dipole are unknown quantities, making this a five-dimensional search problem in Eq. (3). To reduce search in three dimensions further, SVD of the single dipole vector is taken. In the MFL all the antidipoles are expected to align in the same—X direction, which implies that the direction cosines of the dipole moment vector are known. As all the anti-dipoles are aligned in same direction, this situation is analogous to the case of rotating dipoles with all the dipoles aligned at a particular instant. In the application of MUSIC algorithm to the case of rotating dipoles, the correlation between the signal subspace and noise subspace is minimized. In rotating dipole model of MUSIC, G1, the single dipole lead vector itself constitutes the signal subspace. The cost function Jls [7] is the minimum of the eigen-values of the product of signal subspace and noise subspace and is given by J ls ¼
kU Tm1 G1 k kG1 k
665
present in the bar is calculated based on the forward problem discussed above neglecting the magnetic flux density due to the magnetization of the sample [6]. A Gaussian noise with mean zero and standard deviation of 10% of the maximum of bz is added to the magnetic field to simulate the noise present in measured data. The three anti-dipoles are assumed to have a strength of 30, 7 and 17 nA m, respectively. The z-component of the magnetic field is sampled at 1681 locations in a plane 3 mm above the top surface with x ranging from 20 to +20 mm in steps of 1 mm and y ranging from 20 to +20 mm in steps of 1 mm forming the bzn vector. Fig. 1 shows the plot of the magnetic flux leakage field due to these three defects. The cost function Jls is calculated for all the 1681 locations using the rotating dipole model of MUSIC, with z taken as 10 mm. As indicated in Fig. 2, a two-dimensional plot of the inverse of the cost function versus scan coordinates shows a peak at the dominant defect location indicating the presence of a defect at the corresponding x and y coordinates. A better estimate of the position is then obtained by the generalized inverse procedure [6], for a single dipole using the measured bzn. Having obtained the first defect location, the gain vector for the first dipole G1 is constructed. The orthogonally projected data matrix
(4)
where JG1J is used as a normalization factor. The cost function Jls when evaluated at dipole locations will yield a near zero value for low-noise data, whereas at all other locations the cost function will be nearly unity. The functional dependence of G1 on z is weak compared to that on x and y. Hence, we have found it sufficient in practice to assume an average z and scan for x and y coordinates only. To identify the position of the dominant dipole in the presence of noise, the minimum eigenvalue cost function is evaluated in the region of interest (x, y coordinates only) with an average z value (half the thickness of the plate approximately) and the inverse of this cost function is plotted. Peak in this graph indicates the x and y coordinates of the position of dominant dipole [7]. A better estimate of the defect position is obtained by the generalized inverse procedure [5] for single dipole using the measured bzn. Having identified the location of the dominant dipole, the locations of other defects are then identified by repeatedly using the method of orthogonal projection away from all the dipoles identified in the previous step [6,8].
Fig. 1. Calculated X–Y scan of magnetic field bzn for three widely separated subsurface defects with 10% noise added. This has been used as input for inversion using orthogonally projected rotating dipole model of MUSIC.
3. Results of numerical simulations In order to demonstrate the effectiveness of the above procedure, the magnetic flux leakage field, due to three buried defects is calculated in a magnetically permeable bar with three distinct defects. The magnetically permeable bar of size 40 mm 40 mm 15 mm is assumed to be magnetized by an applied field along the positive X-direction and three distinct anti-dipoles are positioned at (12, 15, 8), (8, 11, 12) and (12, 14, 10) representing the defect locations. The x–y plane is taken as the top surface of the bar with the bar placed symmetrically with respect to the x–z plane. The flux leakage field due to all the three defects
Fig. 2. Plot of inverse of cost function calculated from bzn, with respect to the scan coordinates utilized for identification of the strongest dipole.
ARTICLE IN PRESS 666
R. Baskaran, M.P. Janawadkar / NDT&E International 42 (2009) 664–667
bzn?1 is then obtained. The algorithm is repeated with bzn?1 which yields the position of the second defect represented by second most intense anti-dipole [6]. A better estimate of the positions of the two defects is then obtained by the generalized inverse procedure for two dipoles using the measured bzn. The process is repeated to get the position of the third defect. After identifying all the three defect locations accurately using generalized inverse, the orthogonally projected data matrix for the three identified defects bzn?3 is obtained. The search with the above algorithm does not reveal any peaks within the region of interest as shown in Fig. 3, indicating that no further
Fig. 3. Plot of inverse of cost function calculated from bzn?3 after identification of three dipoles, with respect to the scan coordinates in the region of interest.
defects exist. The stray peaks in Fig. 3 have an amplitude less than 1.0008 (o1%) which is below the noise level in the data and are too weak to be identified with possible defect location. The inferred locations of the defects agree with the those assumed in the forward calculation to within 1 mm in x and y coordinates. The calculated magnetic field using the inferred locations and strengths of the dipoles is shown in Fig. 4. The results are presented in Table 1 for 10% additive noise. For a quantitative comparison of the computational burden between this algorithm based on rotating dipole model of MUSIC scanning only in two dimensions (x and y) with fixed z and the earlier stationary dipole model of MUSIC scanning in three dimensions (x, y and z) [6], the number of floating point operations required for computing has been obtained from MATLAB 5.3 for both the algorithms and is denoted as flops. The number of flops for searching the dipole in two dimensions with fixed value of z using the rotating dipole model (no SVD of Gi) is approximately 20% of the number of flops for searching dipole in two dimensions with fixed value of z using stationary dipole model (with SVD of Gi). Due to reduced search domain in the revised approach, the computational burden reduces to 1/15th of the earlier approach. The computational burden associated with other parts of the algorithm such as non-linear least square minimization are common for both versions. The total number of operations for running the stationary dipole model algorithm with a three-dimensional search space is estimated to be 2.88 1012 flops, while the total number of operations required for running the revised algorithm based on rotating dipole model of MUSIC with z fixed at 10 mm presented in this paper is estimated to be 1.02 1011 flops. This directly implies that the computational burden of the revised algorithm is only about 3.5% of that for the stationary dipole model of MUSIC for defect detection using MFL.
4. Conclusion
Fig. 4. Calculated magnetic field at the measurement plane due to the defects identified by the orthogonally projected MUSIC algorithm using the rotating dipole model.
We have developed an improved approach by which it has been possible to identify the number of defect locations and hence the positions and strengths of the dipole moments in flux leakage experiments using the orthogonally projected MUSIC with rotating dipole model. The results clearly indicate the capability of the orthogonally projected MUSIC with rotating dipole model algorithm based on the pseudo-inverse technique to identify the buried defects even in the presence of reasonable levels of noise. We estimate that by this new approach, it has been possible to reduce the total computational burden to about 3.5% of that required for searching the dipoles in three dimensions using the conventional stationary dipole model of MUSIC to evaluate the number of defects, their positions and strengths. The ideas presented in this paper could be utilized for identification of location of buried defects in magnetic steel plates when the measurement is done after demagnetization of the plate. For plates of larger thickness than assumed in this modeling, however, it may be necessary to scan with z in larger steps.
Table 1 Dipole locations recovered using the orthogonally projected dipole model of the MUSIC approach identifying the three distinct dipoles. First dipole
Actual Calc.
Second dipole
Third dipole
x (mm)
y (mm)
z (mm)
Moment (nA m)
x (mm)
y (mm)
z (mm)
Moment (nA m)
x (mm)
y (mm)
z (mm)
Moment (nA m)
12.0 11.9
15.0 15.0
8.0 8.0
30.0 29.9
8.0 8.0
11.0 11.1
12.0 10.7
7.0 5.5
12.0 11.8
14.0 14.2
10.0 10.1
17.0 17.6
ARTICLE IN PRESS R. Baskaran, M.P. Janawadkar / NDT&E International 42 (2009) 664–667
Acknowledgement The authors would like to place on record their sincere gratitude to Dr. C.S. Sundar for the continued support. References [1] Mandal K, Atherton DL. A study of magnetic flux leakage signals. J Phys D: Appl Phys 1998;31:3211–7. [2] Mandache Catalin, Clapham Lynann. A model for magnetic flux leakage signal predictions. J Phys D: Appl Phys 2003;36:2427–31. [3] Bray DE, Stanley RK. Nondestructive evaluation. New York: McGraw-Hill; 1989.
667
[4] Bruno AC. Imaging flaws in magnetically permeable structures using the truncated generalized inverse on leakage fields. J Appl Phys 1997;82(12):5899–906. [5] Baskaran R, Pattabiraman M, Janawadkar MP. Imaging defects in a threedimensional magnetically permeable medium using pseudoinverse technique. J Appl Phys 2006;100:064909–11. [6] Baskaran R, Janawadkar MP. Defect localization by orthogonally projected multiple signal classification approach for magnetic flux leakage fields. NDT&E Int 2008;41(6):416–9. [7] Mosher John C, Lewis Paul S, Leahy Richard M. Multiple dipole modelling and localization from spatio-temporal MEG data. IEEE Trans Biomed Eng 1992;39(6):541–57. [8] Mosher John C, Leahy Richard M. Source localization using recursively applied projected (RAP) MUSIC. IEEE Trans Signal Process 1999;47(2):332–40. [9] Meyer Carl D. Matrix analysis and applied linear algebra. Philadelphia: SIAM; 2000.