Eddy current-based material loss imaging system prototype
Brian Brewington, George Cybenko
Thayer School of Engineering, Dartmouth College
Hanover, NH 03755-8000
ABSTRACT
Systems that associate positions with eddy current probe readings present inspection data more directly
and usefully than the more common impedance plane display. We have economically constructed such an
integration of an eddy current testing device with a position feedback device which gives 0.005”
achievable resolution. The position and probe readings are fed to a PC and plotted in real-time as a
projection into a user-defined rectangle in 3-space. All spatial coordinates and all inspection data are
retained for model building and record keeping. Unsampled eddy-current readings are interpolated using
a spatial average, consistent with the observed averaging effect of the probe. An empirically-determined
convolution kernel permits an estimate of the surface profile on the opposite side of the sample by
deconvolution. This can then be combined with the position to provide a model of the sampled area.
Keywords: eddy current, corrosion imaging, corrosion detection, non-destructive evaluation (NDE)
1. INTRODUCTION
Military and commercial sectors have an ongoing need for a non-destructive evaluation (NDE) tool that is capable of not
only detecting corrosion, but also quantifying the extent of the damage. This is particularly important to the U.S. Air Force,
which plans to keep their fleet of KC-135, B-52, and E-3 aircraft in service for perhaps another 40 years1. Since corrosion
prevention, maintenance and control were not at all well understood when these aircraft were designed and built in the
1950’s, their designs are not particularly resistant to corrosion. The aircraft are in need of routine inspection to determine if
corrosion damage has occurred, and further down time to replace corroded parts. The commercial sector, while less
concerned about the difficulty of replacing aging aircraft in the fleet, is certainly concerned with the negative effects of
corrosion on their older aircraft, how much corrosion is tolerable, and how best to detect it.
Recent corrosion research2 regarding the KC-135 has brought several needs of the military to attention, stating that existing
NDE technologies are inadequate for corrosion detection. The majority of the difficulties stem from the fact that there is no
quantitative standard by which to grade the severity of corrosion; maintenance decisions are currently made on only
qualitative information. Worse still, inadequate knowledge of corrosion rates in different environments makes it difficult to
schedule maintenance intelligently. This is part of a larger problem: there are currently no models which predict potential
future corrosion with any degree of certainty. Even more problematic is a lack of engineering data on how corrosion affects
structural integrity, and where on the aircraft corrosion is most likely to cause a structural failure. Records of past aircraft
maintenance, which could be used to help develop an understanding of the above, are not well enough maintained and
integrated into current decision-making to be of much use. There is clearly a great deal of work to be done in the
determination of damage tolerance, prescription of maintenance, record keeping, and detection and quantification of
corrosion.
With all this in mind, we focus our efforts on the development of a more useful tool for gauging the severity of material
loss, presenting results in an easy to interpret fashion, then storing them for future access. Recent advances in computing
and positioning technology have made such corrosion imaging systems affordable, an accomplishment which might have
been infeasible just a few years ago. The availability of such new technologies have allowed us to construct an economical
integration of an eddy current testing device with a positioning arm, at about a fifth of the price of a comparably equipped
commercially available system. We collect impedance plane data using an ordinary eddy current probe, while tracking its
location using the positioning arm. Both peripherals report data to a PC, which plots the data vs. position and records it on
disk. Following the initial data collection, interpolation and reconstruction give an estimate of the severity of material loss
in the area. Our software includes full spatial positioning, database integration, and network capability for remote analysis
of test results. We view the use of such computerized data collection systems as a necessary step towards more prudent
scheduling of maintenance and the development of predictive models of corrosion growth.
2. BACKGROUND
2.1 Position determination
2.1.1 Overview and hardware specifications
A variety of instruments are available which provide position feedback in real time. Our criteria for selection of a
mechanism were (1) resolution, (2) accuracy, and (3) compatibility with a magnetic device (an eddy current probe). Several
categories of devices exist, ranging from video monitoring and processing of the inspection, to wireless computer “mice,” to
a wide variety of mechanical sensors. The wireless mice were ruled out due to the interference with the electromagnetics of
the probe itself, in addition to lack of spatial precision. We also considered a gyroscopic mouse (a cordless mouse
containing a gyroscope which allows inference of position from acceleration and initial position), but ruled it out because of
the propagation of errors and the lack of resolution. Video was judged to be potentially too difficult to implement outside of
a laboratory setting. We were left with an astonishing array of passive mechanical positioning devices from which to
choose.
For its resolution, accuracy, and ease of integration, the MicroScribe-3D arm (figure 1), developed by Immersion Corp.
(San Jose, CA) was an excellent choice. The product was developed primarily for digitizing 3-dimensional objects and
rendering virtual reality environments, but is quite general and well-suited to our application. The arm’s interface with the
PC uses RS-232 serial port communications at up to 115 Kbps. Encoder readings are sampled at 1 kHz, with 1 ms latency
from encoder read to transmission3. A 50” spherical workspace and five degrees of freedom allow the user to easily cover a
large area with excellent resolution (0.005” average)3 and good absolute accuracy (0.017” mean value). Unfortunately,
resolution and dexterity are not as good when operating near the boundary of the workspace as the manipulator loses
degrees of freedom. The problem
is similar to a human trying to
perform some fine motor control
task (such as writing one’s name)
with arm fully extended, allowing
only rotation at the shoulder or
twisting of the elbow and wrist. As
a result, the useful workspace is
somewhat smaller, near a 40”
sphere, but still adequate for most
tasks. A further hindrance was the
existence
of
manipulator
singularities within the workspace,
though these can generally be
avoided by reaching the same point
in an alternate orientation. To
adapt the device for eddy current
inspections, we attached the eddy
current probe to the last link of the
Figure 1: The MicroScribe-3D arm (left) and probe attachment close-up.
arm with a custom-made clamp.
2.1.2 Denavit-Hartenberg transformation conventions
Calculation of probe position from encoder values at the joints is accomplished by application of the Denavit-Hartenberg
parameterization4,5 for homogeneous coordinate transformations. This convention expresses any rigid-body spatial
transformation as a sequence of four motion primitives: (i) a rotation of α radians about the x-axis of the current frame, (ii)
a translation of a units along that same x-axis, (iii) a rotation of θ radians about the (now transformed) z-axis of the frame,
and (iv) a translation of d units along that same axis. Each of these four is completely described by a single parameter,
which may be dynamic or static depending on the mechanical joint type involved (prismatic or revolute). Expressing each
primitive as a 4×4 homogeneous transformation matrix, the matrix product of each of the above transformations (in order)
gives a general transformation parameterization, typically called an “A-matrix,”4 which describes the transformation as a
function of the joint variable(s) in question:
 cos θ
sin θ cos α
A=
 sin θ sin α

0

− sin θ
0
cosθ cos α
cos θ sin α
0
− sin α
cos α
0

− d sin α 
d cos α 

1

a
(1)
For the MicroScribe arm, all of the joints are rotary, making a, d, and α physical constants (the values of which are
hardwired into the device by the manufacturer*), while θ is the (variable) joint angle. Premultiplying a point and
orientation by the transformation above will give the point’s new coordinates and orientation with respect to the original
frame of reference; premultiplying by A-1 (if A is nonsingular) reverses the transformation. The upper left 3×3 represents
the x, y, and z unit vectors in the transformed frame, and the last column is the location of the origin of the transformed
frame in terms of the original coordinates. Applying one such transformation at each joint, we transform the origin through
each of the five joints until we reach the tip of the probe. The overall transformation is the matrix product of the individual
transformations applied at each joint:
T5 = A1 (θ1 )A 2 (θ2 )A 3 (θ3 )A 4 (θ4 )A 5 (θ5 )
0
(2)
A completely general expression for this matrix product is quite unruly; it is left in this form for compactness. When this
net transformation is applied to the origin, we find endpoint coordinates in the original reference frame. Generally in our
work we were not interested in the orientation of the final frame, just in the location of its origin; this allows us to transform
just a point (4-vector) rather than a point and an orientation (4×4 matrix). In software, we implement transformations by
reading encoder values, scaling these readings by the range of the encoder (maximum number of counts), and multiplying
by the angular range. For every position sampling, the A-matrices were constructed and multiplied together to give a
transformation from the base through all of the joints, thereby allowing the end of the probe to be located.
2.2 Eddy current inspection fundamentals
2.2.1 Overview
Eddy current inspection technology works by generating an alternating current in a coil in the vicinity of the material being
inspected and measuring some portion of the material’s response. Our instrumentation allows us to infer defects in the
material due to a change in coil impedance compared to a reference value set by balancing the instrument. A number of
features of the defect will affect how this impedance behaves, among which are straightforward thickness reduction, lattice
defects, chemical impurities, and flaw geometry.6 With most variables held constant, an examination of impedance change
allows one to infer some unknowns, but a completely general inference is perhaps not a reasonable goal due to the myriad
factors involved. Truly precise interpretations of eddy current tests can only be made in carefully controlled environments.
*
To accommodate the probe attachment, it was necessary to modify the final transformation based on the physical offsets of the probe from the manufacturer’s
stylus tip (see figure 1 for detail).
2.2.2. Electromagnetics of eddy current inspection
Relative motion between a magnetic field and a conducting medium will induce a current in the material. Lenz’s Law7
states that these induced currents will flow in a direction which opposes the change in magnetic flux. General eddy current
devices use this property for flaw detection by generating alternating current in a loop (or a series of coils encircling a
ferrite core), thereby generating a magnetic field, which in turn induces a current in the material being tested. This induced
current generates its own magnetic field, which opposes the change in the field which created it. The secondary field can
then be sensed by a pickup coil on either side of the specimen. The change in the impedance in the pickup coil can be
measured directly. In our particular application, the pickup coil is built into the probe, concentric with the first, or driver,
coil, in an arrangement called a reflection probe. This particular probe (Zetec number 927-8655, pictured in figure 1) is
recommended by Zetec, Inc. (Issaquah, Washington) for use in detection of corrosion in thin skins.
The strength of the induced currents falls off exponentially as one moves along the central axis of the inducing field away
from the source, to a greater or lesser degree depending on the driver frequency, conductivity of the material, and relative
permeability†. As a general rule, high driving frequency or high conductivity implies a smaller depth of penetration. The
(1 − 1e ) current strength depth of penetration is given (in inches) by8
 ρ 
. 
δ = 198
µf 

(3)
where f is the driving frequency (in Hz), ρ is the material’s resistivity (in µΩ⋅cm), and µ is the relative permeability
(dimensionless). Sixty-three percent of the induced current is enclosed between this depth and the material surface; Zetec
recommends9 using 2.6δ as a minimum thickness for a test sample. We found 15-20 kHz to be the optimal scanning
frequency range for detecting defects on the opposite face of our standards. Eddy current scanners generally allow an
inspector to tune the driving frequency to a desired value, chosen to match the frequency response of the probe as well as
the item being scanned.
Clearly, the choice of inspection parameters such as frequency and probe type to a large extent determines what the
inspection is capable of revealing. Since the impedance measurement is (by nature) a spatially-averaging process applied
only in the general vicinity of the tip of the probe, the resolution of an eddy current probe will be on the order of the
diameter of the driver coils.8 A ferrite core can serve to concentrate the flux, thereby reducing the effective coil diameter
and improving resolution. This determines the resolution of the probe itself. Our use of “resolution” here is somewhat of a
misnomer; certain flaws (such as cracking) have effects which extend much further than the actual, physical size of the
defect in the material.
Figure 2: The typical impedance plane signatures of a defect (left) and a probe liftoff (right).
†
Permeability can have a dramatic effect on the depth of current penetration. Our inspections are limited to aluminum alloys, having unit permeability. Please
see Bray and McBride1 regarding the special requirements for eddy current testing of ferromagnetic materials.
In our optimal scanning frequency range, it was observed that defects generally had a relative phase of 0.5π radians, making
the imaginary part of the reading an excellent indicator of thickness loss. The image of the circular defect in figure 4 (in a
later section) shows the magnitude of the impedance reading, almost all of which is imaginary component. However, using
this as a primary indicator does pose some problems, as the more general defect signature is the impedance magnitude
increase. With this in mind, the data collection routine was designed to be quite general, and will read all quantities and
allow the inspector to make technical decisions.
2.2.3 Interpretation of eddy current inspections
In general, in an area of reduced thickness, one should be able to observe a phase lag relative to the reference impedance, as
well as an increased magnitude8. The greater the thickness reduction, the greater the (theoretical) increase in impedance
magnitude and phase lag. The primary exception to this rule is the “liftoff” error, occurring whenever the probe is (for
example) rocked from side to side, momentarily losing close contact with the surface under inspection. This problem is
generally avoided by adding a phase shift so that the lift-off error is very nearly along the negative real axis, making it
easily identifiable in the data as having phase of approximately ±π radians, or as having imaginary part of essentially zero
(see figure 2). A liftoff error can also occur whenever the distance from the probe tip to the surface of the work changes,
due to variations in coating thickness, differences in operator methods, or differences in probe construction‡. Perhaps the
most vexing aspect of the problem is the difficulty it poses when one attempts to compare inspection data gathered at
different distances from the sample: there is nothing ostensibly “wrong” with the data, but no meaningful comparison can
be drawn unless the data was collected at the same separation distance.
3. SOFTWARE IMPLEMENTATION
3.1 Motivation and design objectives
The traditional eddy current inspection has two major drawbacks which limit its effectiveness from a management
perspective. First, for a given test, there are no hard and fast quantitative rules an inspector can follow, resulting in large
variations in inspector opinions of percent material loss (PML) and necessitating a great deal of inspector training. The
effectiveness of corrosion imaging systems in reducing variations in inspector opinions has been documented by Howard
and Mitchell.10 Second, there are only very limited facilities for recording the results of individual inspections, and access
to past inspection data is difficult or impossible. Current Air Force practice is to treat every plane which leaves the
inspection and repair facility as a “new” aircraft.1 Although there is certainly a practical limit to how much data could be
retained from each inspection, doing so is a necessary first step in developing statistical models of susceptibility to and
growth of corrosion in aging aircraft. Access to past inspections would immediately help identify those designs which
effectively inhibit (or promote) corrosion. Furthermore, even a basic understanding of the repair history of a few aircraft
would allow more intelligent scheduling of future maintenance, an idea known as “condition-based maintenance,” discussed
in greater detail in section 5.
3.2 Description of operation
3.2.1 User interface
Due to the large amount of visual content and the need for data acquisition capability, we chose LabVIEW software,
developed by National Instruments Corp. of Austin, Texas, as the development platform. LabVIEW is a visual
programming environment that uses a proprietary programming language, G, dedicated to the creation of “virtual
instrumentation” (VI’s) emulating electronic devices such as multimeters or oscilloscopes. The general programming
paradigm consists of dropping graphical controls and indicators (such as buttons, lights, sliders, charts, 2D plots, etc.) on a
screen to form the front-end, and programming by wiring a block diagram symbolizing the flow of data between the widgets
and other functional units. A small sample of the block diagram is shown in the right half of figure 2. LabVIEW allows for
‡
Most eddy current probes have their coil in a protective casing, inevitably creating a small separation between the end of the coils and the surface. The
manufacturer must maintain tight tolerances on the separation between the windings and the physical tip of the probe, or results obtained using different probes
may not be comparable.
the creation of an attractive and user-friendly interface with minimal programming effort, with extensive analysis tools on
the back end.§
Our main interface (fig. 2, left side) is deceptively simple from a user’s perspective, consisting of about 70 LabVIEW
procedures working in tandem to produce a single screen of output. Program operation consists of establishing
communication with the arm, setting the desired spatial resolution of the inspection, choosing an inspection area, collecting
data, saving the data, logging the inspection, and processing the data. Resolution can be set to any value between 0.005”
and 0.5”, and defines the block size in the projection plot on the right side of the display. This plot is a projection into a
rectangle in space which the user selects by defining two corners, and approximating a third. The third corner is found by
choosing the closest point on the sphere defined by taking the first two corners as endpoints of a diameter (the fourth corner
is then implicit from the other three). All positions are projected into the plane so defined, and those that lie in the
selection region are plotted. The z-coordinates relative to the projection plane are retained to allow for eventual 3dimensional reconstruction of the inspection area, even for surfaces with substantial curvature. Extreme curvature could
present a problem since the projection mapping is not injective, though problems of this nature could generally be avoided
by clever selection of the inspection area, or at worst, partitioning the inspection area.
Figure 3: The eddy current inspection program desktop (left), showing selected inspection data as well as a projection plot of the inspection area. A sample
of the LabVIEW programming language “G” is shown on the right.
The program will store any desired portion of the impedance dataset, such as magnitude, phase, real part, imaginary part,
as well as the coordinates of the probe in 3-space. All these are collected in memory, and the user has the option of
selecting which of these should be stored on disk following the inspection. Data is collected whenever the probe tip is in
range, no liftoff error is discernible, and the user has toggled into “add points” mode. The program screens out liftoff errors
according to the criteria that impedance readings with magnitude greater than a user-defined threshold, and with absolute
value of phase in the range π±0.02 are the result of a probe liftoff. Work is currently underway to implement a more
sophisticated filter based on a calibration provided by the inspector.
3.2.2 Interpolation of eddy current readings
For a medium to high resolution inspection, it is simply not practical to sample the workspace at every point in the
collection area, a task much like coloring the entire area with a ball point pen. Rather, we can sample at a number of points
within the area and interpolate, giving an approximation of the readings at unsampled points. Interpolation by averaging is
particularly attractive because the defect-to-probe system is lowpass (see section 4), retaining almost none of the fine
structure of the defect surfaces.
§
Conceivably, an eddy current instrument could be programmed into the software if coupled with a sufficiently powerful A/D card (and processor) capable of the
high driving frequencies found in current analog instrumentation. This could eventually obviate the need for a bulky, expensive eddy current scanner, but for the

moment, we are still using a Zetec MIZ -22 Eddy Current Instrument for our primary data collection.
The interpolation algorithm is a two-dimensional, iterated weighted average. The unsampled point in question is the center
element in an n×n grid of elements (where n is odd), some of which have been sampled. The known samples are used in the
average, weighted by some monotonically decreasing function of their distance (in pixels) from the point in question. The
choice of the weighting function is somewhat arbitrary at first, motivated only by the expected resolution and influence of
the probe compared with the resolution of the inspection. For the moment note that we use a Gaussian weighting function
with approximately 0.15” standard deviation, and we approximate using points inside of this distance from the point in
question. If the interpolation still leaves points unestimated due to overly sparse sampling, the routine is repeated with
interpolated values now treated as actual samples. This becomes particularly unreliable upon multiple iterations; it is
recommended that the initial sampling be as dense as possible or that resolution be decreased, leaving fewer intermediate
points.
To select n, we note that it would not be reasonable to expect a point in the graph to have an influence beyond that which
the physical system would allow. As a result, the weighting function and n are chosen by discretizing the empirically
determined point-spread function of the surface defect-to-probe system. The details of finding this function are treated in
the next section. It is admittedly circular to find the system’s response using interpolated data when the interpolations are
to be based on the anticipated system response, but for a good initial guess, the resulting errors are negligible.
Figure 4: Eddy current inspection of a manufactured sample. The plane of the defect, about 1” in diameter, is not parallel to the surface of the plate, hence
the gradient in the eddy current contour image. At the upper left, there is approximately 30% thickness reduction, while the PML at lower right is about 5%.
4. THE INVERSE PROBLEM
Once eddy current data is collected for a sample, the next problem is to determine what surface profile produced the
observed relative impedances. This is a rather daunting system identification task; as mentioned previously, a truly precise
interpretation is perhaps not a reasonable goal of a general purpose system. At the very least, though, we should obtain an
estimate of percent material loss. Towards this end, we model using linear system theory, whereby any input and output
imply the transfer function (or point spread function) which relates them. Unfortunately, finding this function is severely
ill-posedfor this system; a great deal of fine structure is permanently lost to averaging effects (due to the lowpass action of
the system; see section 4.1 and figure 7). As a result, fundamentally different effects can look quite similar to the eddy
current scanner. Limitations notwithstanding, if we specify the defect profile input precisely, then we can deconvolve the
output to find a model system transfer function. Once we have the system function, a number of signal restoration
algorithms exist which allow us to return an estimate of the PML at positions in any eddy current scan.
4.1 Impulse response
4.1.1 One-dimensional case
We begin by approximating the one-dimensional impulse response h(x) of a surface defect on the side opposite the probe.
The test system is found by scanning perpendicular to a long (manufactured) slot with a known surface profile. If we
sample the slot at a sufficient distance from its ends, any cross section of this scan is an output from a one-dimensional
system. We can average the responses to find a single input and a single output related by some transfer function. To study
the effects of different slot depths and widths, four different slot profiles were used. All test data are presented in figure 5,
showing modeled responses, slot profiles, eddy current scans, and sample photographs.
Preliminary work by Santosa11 indicates that a Gaussian point spread function should approximate system behavior, making
the estimation problem a matter of choosing optimal variance and a vertical scaling factor which minimize a measure of the
prediction error.12 The system function takes the form
h( x ) =
 x2 
a
exp − 2 
 2σ 
2πσ
(4)
where a and σ are parameters to be determined in the minimization of the sum of squared errors at positions in the onedimensional test system z(n) → y(n):
2

min a ,σ  ∑ ∆x( z (n)∗ h(n)) − y(n) 

 n
[
]
(5)
As the discrete convolution sum in (5) approximates a convolution integral, we must approximate the differential element
by the resolution of the inspection. This appears as ∆x in the equation above; the convolution operator, “∗”, indicates the
linear convolution of the discrete sequences z and h. Likewise, an approximation to a two-dimensional convolution integral
would require a scale factor of (∆x)2, and so forth. We found that much of the nonlinearity in the system could be removed
by using the square of the defect profile as the input z(n). This imposes a positivity constraint on the input, which aids the
convergence of the reconstruction algorithm presented in section 4.2.
Figure 5: Test inputs and outputs for the one dimensional system identification routine. Photos of the slots are at upper left; the average depth across the
slots (in 0.040” 7075 aluminum plate stock) is shown at lower left. Eddy current scan results (top right) show magnitude at 20 kHz across the slots. The
plots on the lower right show the average relative impedance magnitude across the slots.
In pursuit of the values of a and σ which optimize (5), analytic optimization methods are of little use since the objective
function is not differentiable. We employ a variant of the steepest descent algorithm12. The directional derivative at a point
is approximated by evaluating the objective function on evenly spaced points about a small ellipse centered at the point in
question. A line search is performed in the direction of steepest descent, from which a new minimum is found for the
objective function. The location of this new minimum is then fed back into the optimization, and we repeat. The algorithm
is stopped (somewhat arbitrarily) when there was no more than a 1×10-6 % change in the value of the objective function.
The one-dimensional convolution kernel, plotted below in figure 5, was therefore determined to be
f ( x) =
(
2
 1
x  
 
exp − 
 
.
)  2  014984
2π ( 014984
.
102.985
)
(6)
where x is in inches. The impedance magnitude is a relative measurement, so it has no dimensions and its scale is entirely
arbitrary. As f (x) is an impulse response, it must also be dimensionless.** From (6), the standard deviation σ is
approximately 0.15 inches, so 95% of the system’s response to an impulse is contained within (roughly) 0.3 inches to either
side.
Figure 6: Impulse responses in both one and two dimensions, appropriately normalized. Axes are labeled by distance from center in inches.
Figure 7: Power spectral density of the impulse response.
4.1.2 Generalizing to two dimensions
It is reasonable to assume that if the material is isotropic, then the magnetic field in the material (generated by the probe)
will be symmetric about the probe axis.†† In this case, the one-dimensional impulse response can be revolved about its axis
of symmetry and renormalized to find the two-dimensional response. In rescaling the function, we preserve the variance σ2
and the vertical scale factor a. We normalize such that the total area under the one-dimensional kernel is equal to the total
volume under the two-dimensional kernel, thereby forcing the two systems to have the same DC gain:
f (x, y) =
 x2 + y2 
a
exp


2πσ 2
 2σ 2 
(7)
Note that by including the differential ∆x (which has dimensions of inches) in the convolution sum, the response is made dimensionless.
The isotropy condition, unfortunately, will not hold in the neighborhood of an edge. Edge effects are quite strong, causing a substantial increase in impedance
magnitude in the pickup coil. How to properly account for edge effects is an active area of research.
**
††
Again, x and y are in inches, and a and σ are the same as in the one-dimensional case. When f (x, y) is discretized and
convolved with an input, a scale factor is necessary for each differential approximated, so we multiply the result of the
convolution by (∆x)2 for a scan of resolution ∆x.
4.2 Surface reconstruction using iterative deconvolution
The final problem we address is that of attempting to partially recover a surface profile from eddy current data. In reversing
the system, it is tempting to simply deconvolve by termwise division of discrete Fourier transforms, though this operation is
particularly susceptible to high-frequency noise‡‡,13. An alternative approach14,15 allows us to deconvolve the impulse
response out of the output without actually realizing the inverse system. The process is based on the fact that a good
estimator of the input of the system will have a response very similar to the observed output. The residuals of the estimation
are then scaled (for numerical stability) and subtracted from the input. To infer the input x(n1,n2) from the output y(n1,n2)
and the transfer function h(n1,n2), we use the following input and recursion:
x0 (n1 , n2 ) = λ0 y(n1 , n2 )
[
(8)
(
)]
xk +1 (n1 , n2 ) = x k (n1 , n2 ) + λk y (n1 , n2 ) − ∆x 2 h(n1 , n2 )∗∗ C x k (n1 , n2 )
(9)
The subscript on x denotes the iteration number, the double asterisk is two-dimensional convolution (much faster if
implemented by FFT14). The convolution is scaled by the square of the block size, ∆x2. The operator C constrains the input
found by the previous iteration; in our case, we impose a positivity constraint on the possible inputs x. The parameter λk is
chosen to assure eventual convergence while assuring stability. After Conchello and Hansen 15, we use a value of λk which
decreases geometrically with each iteration, for a fast start and a stable finish:
λk +1 = cλk ; c < 1 .
(10)
The iterative step is repeated as often as desired, with the exact number depending largely on the initial value of λ and the
constant c. As with all deconvolution problems, this one is particularly susceptible to measurement noise, so the output is
filtered with an anti-noise filter prior to deconvolution. The cutoff frequency of the filter was chosen to eliminate noise at
frequencies in which the system function’s PSD had fallen more than 50 dB below DC gain. In figure 7, we show the result
of running the deconvolution algorithm to determine the input which produced the eddy current scan on the far left. This
simple example shows how difficult it can be to recover an exact profile, although the algorithm does an excellent job of
recovering the defect depth in the center. Direct measurement of the depth in the center shows it to be 0.007”, while the
reconstruction calculates this depth as 0.008”.
Figure 7 (above): Demonstration of the deconvolution algorithm. The eddy current scan is shown at far left. The actual defect, roughly 1” square, is
pictured in the center photograph; iterative deconvolution gives a comparable result at far right. Flaw depths match well at the center of the reconstruction,
but the peaks at the corners of the square are anomalous. Lowpass filtering before deconvolution limits the influence of measurement noise.
‡‡
This method of deconvolution can be used if we threshold the discrete Fourier transforms of the input and output to remain above some magnitude; this is
accomplished by guaranteeing that the magnitude not fall below some small quantity, without changing the phase in those frequency bins. See Lim13 for details.
5. FUTURE POSSIBILITIES
A major challenge in the maintenance of complex mechanical systems, such as aircraft, automobiles, and ships for example,
is the use of service records to build models of how systems degrade with time and use. At present, it is possible in principle
to generate and archive computerized maintenance data obtained from vehicles during servicing. That data can be stored in
centralized databases together with detailed vehicle usage history (that is, how the vehicle has been used, where it has been
stationed and so on). One of our main research goals is to prototype a distributed database and modeling system which
would be capable of integrating maintenance and service data records. Using such a system, a maintenance inspector could
access previous inspection results and vehicle service data for the vehicle currently under inspection as well as for other
vehicles. That information could be processed by a modeling system resulting in specific suggestions for closer inspection
and/or servicing. For example, archived information about the state of corrosion at numerous locations on different KC135's could serve as an excellent prediction tool, from which maintenance requirements and schedules could be set.
We call this possibility "condition-based maintenance," whereby the service and maintenance history of the aircraft would be
considered in planning its inspection and repair. Using information from a database, we believe it possible to model the
system stochastically by isolating those factors which most accurately predict the state of disrepair of an element in a
distributed system. The conditions under which an aircraft or other vehicle is in service may largely define its state of
corrosion, in type, extent, and location. Neural networks, statistical inference and other machine learning methods would be
necessary to implement such a system. We will be designing and prototyping such a condition-based maintenance system for
aircraft corrosion as the next step of our research in this area.
6. SUMMARY
In response to the demonstrated need for improved corrosion detection equipment, we have developed a software system
which integrates an eddy current scanner with a positioning device, gives an image of the scan, and estimates the severity of
the defect. This is accomplished by locating the probe’s endpoint using homogenous coordinate transformations,
associating an eddy current reading with that position, and storing the data. Inspection data is displayed in real time, and
the user can choose from a plot of magnitude, phase, imaginary part, or real part of the pickup coil impedance. Any
unsampled points are estimated using a weighted spatial average, a lowpass system which approximates probe behavior as
accurately as possible. Estimation of the material loss is accomplished using post-sampling deconvolution of the system’s
impulse response from observed output. Integrated with a database, the software permits future access of test results and the
potential for building models of corrosion growth rates. Our ultimate goal is to develop a predictive tool that will render
maintenance scheduling more efficient.
7. ACKNOWLEDGMENTS
This effort was supported by Air Force Office of Scientific Research grant number F49620-95-1-0305. The U.S.
Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright
notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as
necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of
Scientific Research or the U.S. Government. The authors also wish to thank Dartmouth College for facilities and support,
as well as Eric Hansen for his valuable input on multidimensional digital systems.
8. REFERENCES
1. Rennel, Robert, from Proceedings of Aging Aircraft Workshop, Boston, MA, September 1995 (unpublished).
2. Neiser, Donald, OC-ALC Aging Aircraft Disassembly and Hidden Corrosion Detection Program, Proceedings of the
AFOSR Logistics Conference: “Basic Research for Logistics,” Dayton, Ohio, October 31 to November 1, 1995.
3. Immersion Corporation, Programmer’s Technical Reference Manual: MicroScribe-3D and MicroScribe-3DX,
Immersion Corp., San Jose, CA, 1995.
4. Andeen, Gerry B., Robot Design Handbook, McGraw-Hill, New York, 1988.
5. Craig, John J., Introduction to Robotics: Mechanics and Control, 2nd ed., Addison-Wesley, Reading, MA, 1989.
6. Boeing Commercial Airlines, Structures and Corrosion, D6-54769 (rev. 1), The Boeing Company, Seattle, WA, 1989.
7. Purcell, Edward M., Electricity and Magnetism, 2nd edition, McGraw-Hill Inc., New York, 1985.
8. Bray, D. E. and McBride, D., Nondestructive Testing Techniques, Wiley-Interscience, New York, 1992.
9. Zetec Corporation, MIZ-22 Operating Guide, Zetec, Issaquah, WA, 1995.
10. Howard, Michael A. and Mitchell, Geoffrey O., “Nondestructive inspection for hidden corrosion in U.S. Air Force
aircraft lap joints: test and evaluation of inspection procedures,” presented by ARINC corporation at NDE Conference in
San Diego, October, 1995.
11. Santosa, Fadil, personal communication, June 1996.
12. Luenberger, David G., Linear and Nonlinear Programming, 2nd edition, Addison-Wesley, Reading, Massachusetts,
1989.
13. Lim, Jae S., Two-dimensional Signal and Image Processing, Prentice Hall, Englewood Cliffs, New Jersey, 1990.
14. Dudgeon, Dan E. and Mersereau, Russell M., Multidimensional Digital Signal Processing, Prentice-Hall, Inc.,
Englewood Cliffs, NJ, 1984.
15. Conchello, José-Angel and Hansen, Eric W., Enhanced 3-D reconstruction from confocal scanning microscope images.
1: Deterministic and maximum likelihood reconstructions, Applied Optics, Vol. 29, No. 26, September 1990.