A cta Physicae Superficierum ■ Vol II • 1990
D E C O N V O L U T IO N A L M E T H O D S IN T H E R E C O N ST R U C T IO N O F L O C A L M ATER IA L PA R A M ETER S
HORST SCHWETLICK
Institut für Allgemeine Elektrotechnik, Technische Universität Berlin, D -1000 Berlin 10, Germany
IN TR O D U C T IO N
D eeonvolutional m ethods are used for processing ultrasonic impulse echo signals in order to obtain m ore detailed quantitative inform ation about material properties and defects. Examples o f one-dim ensional processing to increase the axial resolution of a single А -scan are given in [ 1 ] and two-dim ensional deconvolution o f С -scan images are presented in [ 2 ] . In the m ethods described in this paper, deconvolution is part o f the reconstruction process. For band-limited and noise contam inated data, the deconvolution may provide inaccurate results. In such cases either estim ations can be used or procedures utilizing prior inform ation which require greater mathematical and numerical effort [3 ]. U tilizing the latter in com bination with the algorithm given in [ 4 ] , the im pedance profile o f a layered medium can be reconstructed in the direction of sound propagation. M ultiple reflections are included in the process of reconstruction. The spatial and temporal radiation pattern o f the transducer in pulse echo m ode is included in the m odel given in [ 5 ] . The reconstruction, based on three-dim ensional deconvolution, provides an estim ate o f the local reflectivity as a function o f space, independent o f the geometry o f the reflector.
D E C O N V O L U T IO N A N D A PR IO R I IN FO RM ATION
The ultrasonic echo can be represented as the convolution o f an incident wavelet and a reflectivity function o f the specimen being tested. D econvolution represents the inverse problem, which is to determine the reflectivity from known reflection and wavelet data. The com putation o f the deconvolution is performed numerically from the digitized data. With m ost available transducers the inform ation o f the reflection data is contained in a limited passband. As a result the deconvolution problem may not be well posed. In Fig. 1 the classification of
m ethods is given. An estim ate o f the impulse response can be com puted by considering only passband data as it is done by truncation o f data outside the passband, by Wiener filtering, and by inverting the convolution matrix. A robust estim ate for the impulse response with high resolution and accuracy can be achieved by including prior information in the com putation. The solution must fit the data and at the sam e time minimize or m axim ize a functional relation representing the a priori inform ation. The m inim ization o f the L j-norm and the m axim ization o f the inform ation entropy have been investigated in this context. D irect, straightforward solutions as well as spectral extrapolation m ethods are derived from these principles in [3 ] . All these m ethods force the solution to have a m inim um number o f nonzero com ponents and a high resolution. The result in one-dim ensional processing is a sparse spike train impulse response and in higher dim ensions an im age with a small number o f points.
R ECONSTRUCTION O F A LAYERED M EDIU M
High resolution deconvolution was applied to the reconstruction o f the im pedance profile o f a one-dim ensional layered m edium from plane wave im pulse echo data. The com putation was done in two steps. First, an estim ate of the im pulse response data was obtained by a high resolution deconvolution m ethod. Since the algorithm given in [ 4 ] is stable for a non-band limited impulse response, the impedance profile was com puted in the second step from that im pulse response. In the exam ple in Fig 2, ultrasonic reflection data from two acrylic plates in water were analysed. An extrapolation method based on the m inim ization o f the L j-norm was used to find an im pulse response from the reflection data which is non-bandlim ited up to the N yquist frequency. The first layer in the corresponding im pedance profile was com pletely recovered. Toward deeper regions, errors tend to accum ulate and reduce the accuracy.
TH R EE-DIM EN SION AL D EC O N V O LU TIO N
Three-dim ensional deconvolution based on Wiener filtering was used to obtain qualitative inform ation on a reflecting region. In this approach the echo is considered to be the convolution o f the local reflectivity with the spatial and temporal characteristics o f the transducer. The reconstruction o f the reflectivity can be looked on as the elim ination of transducer characteristics by deconvolution which is com puted by division in the wavenumber frequency dom ain. In the exam ple in Fig. 3 the scanned reflection data from an agar (gelatine) surface and an embedded 0.5 mm glass sphere were used to reconstruct the local reflectivity. The result clearly shows the increased amplitude for the solid point reflector.
1 .0 0 2 . 0 0 3 . 0 0 4 .0 0 Z a l t / ; i a
F ra q u a n x /M H i
c)
S c h a lla u f x a lf c a 2 / p a
Fig. 2. Reconstruction o f a layered medium from impulse echo data: a) measured echo data and corresponding amplitude spectrum; b) estimated impulse response as a result of deconvolution and
corresponding spectrum; c) reconstructed impedance profile.
ACKNOW LED GEM ENT
The author w ould like to thank Prof. J. Kapelewski for initiating this article. The support o f the German Alexander von H um boldt-Stiftung, the Japan Society for the Prom otion o f Science and the FIM Project at T U -B erlin is also gratefully acknowledged.
e c h o d a t a T ran sdu ce r d a t a F ig . 3. Rec ons tru cti on of a p la n e an d a p o in t re fl ec to r fr om sc an n ed im p u ls e ec h o da ta : a) n u m e r ic a l p rocedure; b) en v el o p e of th e m ea su re d im p u ls e ec h o d a ta ; c) re co n st ru cte d r ef le c tiv it y .
REFERENCES
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[2 ] Hundt, E.E, Trautenberg, R A , Digital Processing of Ultrasonic D ata by Deconvolution IEEE Trans. Sonics and Ultrason. 27: 249 (1980).
[ 3 ] Schwetlick, H., Miyashita, T , Kessel, W , Schätzung einer Impulsantwort hoher Auflösung aus band begrenzten Daten, A E Ü 42: 160 (1988).
[ 4 ] Schwetlick, H., Inverse Methods in the Reconstruction o f Acoustical Impedance Profiles, J. Acoust. Soc. Am. 73: 1179 (1983).
[5 ] Schwetlick, H , Ueda, M , Tabei, M , Reconstruction o f Local Reflectivity with 3-D De- convolution, 17th Int. Symp. Acoustical Imaging, Sendai, Japan (1988).
