The preliminary results from attempts to derive depth and diameter information from digitized stereo images of impact features on the LDEF are reported. Contrary to our prior assumption, we find that impact craters in the T6 A1 alloy are not paraboloid in cross section, but rather are better described by a 6th-order polynomial curve. We explore the implications of this discovery

Sapp, Clyde A.

See, Thomas H.

Zolensky, Michael E.

English

NASA Technical Reports Server

N93-29359
3-D CRATER ANALYSIS OF LDEF IMPACT FEATURES
FROM STEREO IMAGERY
Clyde A. Sapp and Thomas H. See
Lockheed Engineering & Sciences Co., Houston, TX 77058
Phone: 713/483-5141, FAX: 713-483-5347
Michael E. Zolensky
SN2/NASA, Johnson Space Center, Houston, TX 77058
Phone: 713/483-5128, FAX: 713-483-5347
SUMMARY
We report here preliminary results from attempts to derive depth and diameter
information from digitized stereo images of impact features on LDEF. Contrary to our prior
assumption, we find that impact craters in the T6 A1 alloy are not paraboloid in cross section, but
rather are better described by a 6th-order polynomial curve. We explore the implications of this
discovery.
INTRODUCTION
In expectation of the LDEF return, the requirement for a system to analyze the hyper-
velocity impact craters on the space-exposed surfaces of the spacecraft was determined. Ideally,
this analysis system would be able to define in three dimensions the surface structure of each
crater to a high degree of precision. As a minimum, the system should be able to determine the
true depth and diameter of each crater. The 'true' depth is defined as the deepest point in the
crater as measured from the level of the ambient surface, and the 'true' diameter is the inside
diameter of the crater when measured at the level of the ambient surface (see Figure 1).
A number of constraints were placed upon this system design. The budgetary limitations
were fairly severe, and the time frame for technique investigations was short. It was essential
that the analysis system use a technique that was non-destructive and remote (i.e., no contact
with the material surface permitted). In addition, the system must use a technique that could be
incorporated into a portable system to be used at Kennedy Space Center during the deintegration
of the LDEF spacecraft.
It was decided to use binocular imagery to analyze the crater morphologies. It was fairly
inexpensive to achieve, and made use of some existing hardware to collect the information. A
portable system configuration consisted of a portable PC equipped with a color video digitizing
board and a color video multiplexer, a binocular microscope, a pair of video cameras, and a pair
of optical disk drives with removable media. This system configuration would collect pairs of
color digital images and store them to the optical media for later analysis. It was also decided to
write software that would automatically register the image pairs on a pixel by pixel basis using a
traditional cross-correlation technique. The parallax information in each pixel registration
would provide depth data for each pixel, and thereby provide a full three-dimensional
representation of the crater surface.
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During thethreemonthdeintegrationof LDEF,theMeteoroidandDebrisSpecial
InvestigationGroup(M&D SIG)generatedapproximately5000digital color stereoimagepairs
of impact-relatedfeaturesfrom all space-exposedsurfaces.An earlierpaper(1) describesthe
theoryandpracticeof determiningthis 3-dimensionalfeatureinformationfrom stereoimagery.
RECENTWORK
Theattemptsto analyzetheKSCimageryusingtraditionalcross-correlationwere
unsuccessfuldueto severalproblemsinherentin thedata. Therewasasignificantdifferencein
thephotometricresponsesbetweenthetwo camerasdueto alackof photometriccalibration. In
ananalogworld, thisproblemcouldbeeasilyrectifiedby compensatingfor thedifferentgains
andoffsets. In thedigitizedimages,however,thedatahasalreadybeenquantizedandtruncated
makingit impossibleto recovermuchof the information. Therewasalsoaproblemwith a lack
of detail in manyof theimagesdueto depthof field limitationsandlack of focal calibrations
betweencameras.Mostof thecratersdigitizeddisplayedahigh degreeof specularreflectivity,
which is incompatiblewith cross-correlationtechniques.Speculareflectionsarestrongly
viewing angledependent,which meansthathighcontrastdetailsseenfrom onecameraarelikely
to bevery low contrast,or eveninvisible from theothercamera.Theseproblems,combined
with a poorinitial understandingof thetaskcomplexity,causedtheplannedapproachof
automatedregistrationvia cross-correlationto beunsuccessful.
Dueto theproblemsencounteredin attemptingto implementthefully automated
software,thedecisionwasmadeto getan interactive(man-hourintensive)methodworking, and
thencomebacklaterandcontinuethedevelopmentof afully automatedcapabilityastime
permitted.The interactiveapproachwasto allowananalystto selectaseriesof tie-pointsfrom
animagepair, andusethethree-dimensionalinformationof thetie-pointsto performaleast-
squaresparametricfit to definethecrater'sgeometry.(A tie-point is apair of points, onefrom
eachof thetwo images,whichrepresentthesamepoint onasurface,i.e., a tie-point 'ties'the two
imagestogetherat asinglepoint.) Theinitial approachrequiredthata few basicassumphonsbe
made.Theassumptionswerethat 1) thecratersarebasicallyparaboloid,2) thecratersare
central-symmetricto anaxiswhich is perpendicularto theambientplane,and3) therewasliable
to besomeinherenterror in thetie-pointselections.
The interactivedatacollectionsoftwarewassetup sothattheanalystwouldselecttie-
pointsin threesets,oneeachfor theambientsurface,thecraterinterior, andthelip of thecrater.(Note: the lip tie-pointswerecollectedjust for statisticalinformation. No attemptwasmadeto
parametricallydefinethelip geometry.) The first step in the analysis was to first ca_!cu!_ate=the
distance from the focal plane for each tie-point in all three data sets. The analysis softwar_
would then calculate a least-squares fit for the ambient plane and compensate for rotations and
offsets of the crater surface with respect to the camera's focal plane in all three data sets. A least
squares fit of a paraboloid to the interior crater points was then performed. The intersection of
the ambient plane with the paraboloid then determined the ideal crater depth, and the width of
the paraboloid at the intersection with the ambient plane defined the ideal crater diameter.
i-n order to test tile accurac_c-of the-interactlve analys_s software, three craters were
selected which were large enough to perform fairly accurate manual depth and _meter ....
measurements. The manual measurements were performed resulting in measure_depths of i47,
455, and 933 microns and diameters of 279, 1254, and 2426 microns. Binocular images of each
crater were digitized, and the interactive data collection of tie-points was completed. The tie-
point data was analyzed using the parametric fit software, and the outputs were compared to the
manual measurements. There was an expected error in the manual measurements of
340
approximately2%,but for this accuracytest,themanualmeasurementswereacceptedas'true.'
Theresultsweresomewhatdisappointing.Theaverageerror in theestimatedcraterdepthwas
15.5%,andtheaverageerrorin thediameterestimatewas-11.9%. Bothof theseerrorswere
significantly largerthanwasconsideredacceptable.
In attemptingto determinethesourceof theencounterederrors,threepossiblecauses
wereisolatedandtested.Thepossibilitiesweredefinedas1)software'bugs,' 2) analgorithm
which wasoverly sensitiveto inputerrors,and3) afalsebasicassumptionregardingparaboloid
cratergeometry.Theerror sourcesweretestedin theorderlistedabove,asthis wasdeemedto
be theorderof their likelihood.
In orderto testthesoftwareintegrity,several'perfect'paraboloidcraterswerecomputer
generated,andcorrespondingtie-pointsweregenerated.Theanalysisresultedin 0% error. The
!ie-pointcollection,andinitial depthcalculationswerealsotestedseparatelyusingholesdrilled
m aluminumastestcases.Thesetwo testscombineddemonstratedthatthesoftwarewas
performingasexpected.
To testthealgorithm'ssensitivityto inputerrors,thepreviouslygenerated'perfect' craters
were again used. A Monte Carlo technique was applied to generate randomized errors in the
tie-point data prior to being input to the analysis software. The magnitude of the induced errors
was greater than or equal to the maximum expected input errors. Numerous runs were
.performed in batch mode with statistical analysis of the resulting outputs. This analysis resulted
m an average error in the depth estimate of 2.5%, and an average error in the diameter estimate
of 7.5%. These errors were larger than desired, but still not large enough to account for the
errors encountered in the analysis of the real craters.
It was decided then to test the basic assumption of the crater geometry. In order to test
this assumption, five impact craters were generated in T6 aluminum alloy, which is the most
common exposed material on the LDEF. These craters were large enough to be easily cross-
sectioned; the sizes ranged from 3.1 to 7.0 mm in depth and 7.0 to 19.8 mm in diameter. Each
crater was then carefully cross-sectioned through its center. Digital monocular images of the
cross section of each crater were generated, and a high resolution two-dimensional digitization
of the interior surface structure of each crater was then performed. The digitizations contained
66 to 111 data points each to attempt to minimize the errors. A series of two-dimensional
polynomials were then fitted to the digitized points. Second order (Eq. 1), fourth order (Eq. 2),
and sixth order (Eq. 3) polynomial curve fits were each performed. No odd order polynomials
were used because the assumption that a crater is central symmetric was still in effect. A first
order term (bx) was left in the fit equations in order to compensate for any axial rotations
incurred during the initial digitization.
y = a + bx + cx2
y=a+ bx + cx2 + dx4
y = a + bx + cx2 + dx4 + ex6
(Equation 1)
(Equation 2)
(Equation 3)
The results of these curve fits compared to the raw data were somewhat surprising. The
2nd order curve fits were consistently deeper and wider than the actual craters. The 4th order
curve fits were consistently shallower and wider than the actual craters. The 6th order curve
fits, however, resulted in inconsistent errors in depth and diameter. Figure 2 shows images of
the three cross-sectioned craters with the superimposed 2-D curve fits. Figure 3 depicts the total
amount of error encountered in the curve fits, and Table 1 summarizes the percent errors in
depth and diameter estimates for each of the five test shots from each type of curve fit. The
consistency of the magnitude and especially sign of the errors in the second and fourth order
curve fits suggests that these errors are not due to random factors, but instead are due to the
unsuitability of these equations for defining the crater geometry.
341
Idealized Crater Geometry
Paraboloid
(x.i)2 + (y.j)2 = a(z-k)
Lip
Plane: z = 0
'True' Diameter
Ambient
Surface
(i, j, k)
Cross _tionai View
Figure 1 -Initial idealized crater geometry assumed for this investigation, employing a
paraboloid cross-section. True depth and diameter are indicated.
342
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Table 1
Shot#
Errors in Depth and Diameter Estimates For the Five Test Shots From Each Type of
Curve Fit
Depth Error (%) Diameter Error (%)
2nd Order 4th Order 6th Order 2n_l Order 4th Order 6th Order
34 5.57 -3.92 -2.43 9.80 2.07 1. I 1
148 7.14 -3.63 -1.89 6.29 0.70 -0.06
159 6.79 -3.97 -1.33 9.30 1.88 0.76
160 6.57 -3.39 1.44 8.46 1.47 -0.10
163 8.70 -3.61 1.57 7.69 1.26 0.04
| ==
Comparison of Polynomial Curve Fits
16
159
148
Shot Number 34
_4_000 °8°°°
600O
. 5000
30oo
2ooo
0
e- (N
0 _
Sum( err 2 )
Figure 3 Summary of polynomial curve fits for all 5 large test
craters. The vertical axis indicates the sum of square of the
errors for diameter and depth.
When comparing the results of the cross-section study to the 3-D paraboloid fits on the
three original test craters, a rather puzzling discrepancy arises: the 2-D parabolas were
consistently deeper and wider than the actual craters, but the 3-D paraboloids were consistently
deeper and narrower than the actual craters. The explanation for this discrepancy has not yet
been determined. It is possible that findings of the cross-section study hold true only for the size
range which was tested (7 - 20 mm, much larger than the LDEF impact craters), or perhaps the
overall crater geometry is more a factor of the particle velocity upon impact. What seems more
likely though is that the majority of the problem is due to input errors. The initial three (small)
344
test craters had only 12, 9, and 7 tiepoints, while the five cross-sectioned craters had 66 to 111
data points. The effect that a single pixel of uncertainty has in the vertical is inversely
proportional to two times the tangent of the angle of separation between the two cameras. In the
cases being looked at here, that proportion is approximately four to one. This means that a
single data point has about four times the uncertainty in the 3-D paraboloid analysis as a similar
point in the 2-D polynomial curve fits. Also, the initial test crater images suffered from the
same focus and depth of field problems, which increases the amount of error in the data point
selection.
CONCLUSIONS AND FUTURE WORK
The conclusions based on the work done thus far is that the assumption that impact
craters are basically paraboloids is false, at least for the size ranges tested by cross-section. The
current algorithm's sensitivity to input errors is also a major concern. Future testing needs to
address the issue of algorithm sensitivity versus the number of input data points, as this was not
addressed during the initial sensitivity testing. Further cross-section tests are planned for smaller
craters to determine if the initial results of the cross-section tests are size related. Investigations
will be made into methods for minimizing the effect of input errors to the 3-D analysis. It may
also be possible to derive a correction factor from the data which would enable the continued
usage of the paraboloid fit to determine a crater's depth and diameter. We also hope to write a
semi-automated tie-point selection routine which will use existing manually selected tie-points
as 'seeds' to enable a much greater number of input data points. If this venture is successful, we
may be able to perform a 6th order 3-D polynomial curve fit to the craters for a much more
reliable crater definition.
REFERENCE
(1) See T.H., Allbrooks M.K., Atkinson D.R., Sapp C.A., Simon C.G. and Zolensky M.E.
(1992) Meteoroid and Debris Special Investigation Group: Data Acquisition Procedures.
LDEF - 69 Months in Space, NASA CP-3134, p.459-475.
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Three-D crater analysis of LDEF impact features from stereo imagery

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