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A proposal for the reconstruction of buried defects from
photothermal images
U. Seidel, H. Walther, J. Burt
To cite this version:
U. Seidel, H. Walther, J. Burt. A proposal for the reconstruction of buried defects from photother
mal images. Journal de Physique IV Proceedings, EDP Sciences, 1994, 04 (C7), pp.C7551C7554.
ďż˝10.1051/jp4:19947129ďż˝. ďż˝jpa00253183ďż˝
JOURNAL
DE
PHYSIQUE
IV
Colloque C7,'supplCment au Journal de Physique
111,
Volume
4,
juillet
1994
A
proposal for the reconstruction of buried defects from photothermal
images
U.
Seidel, H.G. Walthe~ and
J.A.
Burt*
Institut fur Optik und Quantenelektronik,
FriedrichSchilkrUniversitat,
MaxWienPlatz
1,
07743 Jena, Germany
*
YorkUniversity, Department of Physics and Astronomy, 4700 Keele Street, North York, Canada,
M3J
I
P3
The object reconstruction from synthesized and noisy photothermal signals was carried out by
deconvoluting contrast scans by means of photothermal point spread function. The unknown
defect depth parameter was estimated by minimizing the "imaginary content" of the
deconvol
uted real valued defect distribution. The algorithm shows remarkable accuracy even in the
presence of 10%noise in finding the true defect depth. Essential for its convergence is an
effective noise suppression which we achieved by a modified Wiener filter.
1 .Introduction
Photothermal techniques make it possible "to see" under the surface of opaque samples. Due to the
diffusive nature of thermal waves and their mutual interference the images of buried objects are blurred.
Resemblance between image and object fades out with increasing depth of the object.
At photothermal imaging there arises the necessity to reconstruct buried defects from photothermal
images. Recently some efforts are reported to solve this problem
eg.[l]. Here we present a first attempt
to make use of the photothermal point spread function as defined in [2] for object reconstruction. The
method is based on the property of the photothermal image which can be expressed mathematically as
a convolution integral of the point spread function G and a distribution function D of thermal
inh
omogeneities. Then the measurable photothermal contrast as defined in [3] can be written as
We have to keep in mind that G depends on both the defect depth and the kind of thermal
inhomoge
neity (thermal conductivity and/or thermal density).
Below we restrict our considerations on twodimensional thermal patterns whose extension in depth are
small compared with their distance to sample surface or with thermal diffusion length.
2. Reconstruction of buried thermal inhomogeneities
2.1 Generation of the signals
In the following section we will consider the generation of "synthetic" photothermal signals by the
photothermal PSF. We generate those signals by spatially convolving the object distribution
D(x,y,q)
with the PSF G(x,y,z,) given in [2] in order to simulate a scan across an infinitely thin stripe shaped
defect of width
2R, infinitely extended in x direction and buried at depth
z,
[3]. So, the defect is only
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:19947129
C7552 JOURNAL DE PHYSIQUE
IV
a 2D rather than a 3D object although it exists in a 3D domain.
where
@
is the Heaviside function. We generate the photothermal signal in terms of the amplitude
contrast
K, and phase contrast K,, In order to meet real conditions, under which photothermal experi
ments are carried out, one has to involve a certain amount of noise in the synthesized signal. We did
this by adding an independently, uncorraletedly and uniformly distributed noise to amplitude and phase.
Its value was set to a certain percentage of the contrast value just above the center of the defect. Figure
1
shows a typical contrast signal in amplitude and phase, obtained by the spatial convolution of a defect
distribution
D(y,z,=l,R=2) with an overlaid noise of
10%.
These kinds of "synthesized" signals are
the starting point for the proposed procedure of deconvolution.
2.2
Deconvolution of photothermal signals
by
means of the photothermal
PSF
As the "measured" complex valued contrast
K
=
KA
+
iK, is thought of being the result of a convolu
tion, in principle, we can obtain the object distribution by appropriate deconvolution. By Fourier
transforming all mathematical expressions (are labeled by indexes
FT)
DFT can be obtained as ratio
KFT/GFT. But, in reality, there are at least 2 problems:
1)
K
results from noisy measurements.
2) Both defect depth
z,
and defect type are usually unknown and have to be estimated by the recon
struction procedure.
To overcome these difficulties we write
The summand WF in the denominator represents a modified Wiener filter, its significance for the
deconvolution process will be discussed in section
2.4.
Obviously, we only obtain the true defect distribution D(x,y,z,) as the Fourier back transform of
DFT(k,,&,%) if we deconvolute it with the correct GFT(k,,kY,zk), taken at the correct depth z,
=
z,
assuming the same thermal type of defect in numerator and denominator. Usually, these values are
unknown and we have to determine the correct
GFT(kx,k,,z& from an independent criterion.
2.3
The deconvolution criterion
In order to obtain the correct defect distribution as the result of the deconvolution process one has to use
such a
G,(kx,ky,z3 in the denominator of equ.(3) which is as close as possible to the one corresponding
to the actual inhomogeneity under test. One could repeat the deconvolution process with varying
parameters in the denominator and so try to look for the best match. For simplicity, we restrict our
efforts to the search of the unknown defect depth
z,
and deconvolute the "measured" contrast K(x,y,q)
with a number of PSF's taken at guessed depths
2,.
Scanning across the subsurface inhomogeneity
measuring at
N
sample points yi, assuming a certain depth zk for each scan, we monitor the imaginary
content of deconvoluted defect by a function
Because
D(x,y,q) has
to
be a real valued function we will find the correct depth z,
=
z,
if rms(zk)
=
0
or, in the presence of noise, rms(zk) reaches a minimum among the various depths investigated.
Numerically, this means to find the global minimum of the pointwise function rms
=
rms(z3.
2.4
Noise
suppression
The proposed deconvolution procedure is very sensitive to noise and strongly needs a mechanism
to
suppress spikes in the deconvoluted result, which lead to a total loss in information after
D,(k,,k,,z,)
was Fourier transformed into D(x,y,z,). We follow the approach in
[4]
using the modified Wiener Filter
in the denominator of equation
3.
Adding a summand WF to the denominator prevents it from being too
small. This kind of noise suppression can be derived from optimal Wiener filtering providing the noise
has a white frequency dependence
[5].
The constant WF reflects the power spectrum of the noise
involved and has to be chosen appropriately. Figures
2b,c show the result of under and overfiltering.
Choosing the filter constant WF too small (underfiltering) results in a increase of noise in the
deconvol
ution result making it impossible to recognize the shape of the defect and leading to a decrease in
accuracy in determining the depth
G.
Overfiltering (WF too large) leads to smoothing out the defect
distribution, it is not longer possible to determine the shape or size of the defect, although its depth
z,
is found with reasonable accuracy.
Additional investigations also showed, that the deconvolution results only weakly depend on WF over
a wide range,
i.e. the choice of WF matching the power spectrum of the noise is not critical for the
performance of the deconvolution algorithm. We
could achieve reasonable results even for noise levels
of about
10%20%, a range which certainly covers the common experimental conditions.
amplitude connast
ja.u.]
phase contrast [rad]
Fig.
1
Calculated photother 0.05
i
0.2
ma1 contrast of a stripe
shaped
kinhomogene~ty of
o
width
2R/p
buried at depth
%Ip.
1;
IJ
Abscissa is scan path. 0.05
F

0.2
t
0.1
10.4
1
4
c

0.6
i
0.2
F
:
j
0.8
i
0.25
'1
10
8
6
4
2
0
2
4
6
8
10
scan
parh [a.u.]
scan
path [a.u.]
defect
distribution
1a.u.l
Fig.2
Object reconstruction from data of Fig.
1 using different values WF. (a: WF=O.Ol, b: WF=
0.0001, c: WF=
1
).
1,"
,
V
c
I
4
2
0
2
4
4
2
0
2
4
4
2
0
2
4
C7554 JOURNAL
DE
PHYSIQUE IV
2.5
Selected examples
Object reconstructions from buried stripe shaped inhomogeneities are reported. If the true type of
thermal distortion is assumed then the true defect depth can be estimated with %accuracy. If the type
of thermal distortion is unknown an increase in uncertainty of reconstruction the size (as shown in
Fig.3) and depth results.
deconvoluted defect distribution
la.u.1
deconVOluted defecr disnbutlon ia.u.1
0.4
4.5
1
10
8
6
4
2
0
2
4
6
8
10
10
8 6
4
2
0
2
4
6
10
scan
path
[au.]
scan
path 1a.u.l
Fig.3 Reconstructed kinhomogeneity in case of correct (left) and incorrect (right) assumption of the
type of thermal distortion.
3.
Limits and further development of the suggested method
Certainly the major drawback of the proposed method of inverting photothermal images is that one has
to assume the general shape of the defect before carrying out the deconvolution. That means in the
presented case of an indefinitely extended strip in xdirection to assume this geometry in order to use
the
IDPSF. Also, so far we have developed the method only for very thin defects (in terms of
p).
The
extension to real 3D defects is assumed to involve mathematical difficulties in deriving a 3DPSF.
Furthermore, it seems to be difficult to distinguish clearly between k and
bc)defects at this level of
theory. For further improvements one has to make use of the knowledge of the frequency dependence
of
G(x,y,q) and the signal.
The next step will be the extension of the algorithm to higher dimensions. The presented case is a
semi
2D problem, but even the application to disk shaped defects would require a 2DPSF and therefore 2D
Fourier transformation.
Additional difficulties will be encountered if we proceed with real measurements instead of the synthe
sized signals described above. Problems will arise from calibrating the depth and frequency axes by
applying the deconvolution algorithm on real measurements.
The authors wish to acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG)
and the Deutscher Akademischer Austauschdienst (DAAD).
[I]
D. J.Crowther, L.D.Favro, P.K.Kuo, R.L.Thomas, "Inversion of Pulsed Thermal Wave Images
for Recovery of the Shape of the Object",
Proc.7th.Int.Top.Meeting
on Photoacoustic and
Photothermal Phenomena
111, Doorwerth, Aug.2630, 1991
[2]
K.Friedrich, K.Haupt,U.Seidel,H.G. Walther, J.Appl.Phys.
72
(1992), 37593764
[3]
U.Seide1, K.Haupt, H.G. Walther, J.Burt, B.K.Bein, J.Appl.Phys.
75
(1994), No. 8
[4]
W.Braun, I.Price, B.C.Cadoff and N.C. Peterson, International Journal of Chemical Kinetics
21
(1989), 1029
[5]
W.H.Press, B.P.Flannery, S.A.Teukolsky, W.T.Vetterlein, Numerical Recipes in Pascal,
Cambridge University Press, Cambridge (UK), 1989