Waveform tomography with signal sparsity

Images (1)-(4) show different stages of an experiment [4] with a 15x15x15 cm plastic test object. From left to right: (1) cubic mould with three stones placed inside, (2) ultrasonic travel-time  measurements via a pair of 55 kHz transducers, (3) CT scan of the interior, and (4) three dimensional CT image of the stones (blue) compared to the ultrasonic reconstruction (green) obtained using six face-centric source positions (yellow three-branched symbols).

Images (1)-(4) show different stages of an experiment [4] with a
15x15x15 cm plastic test object. From left to right: (1) cubic mould with three
stones placed inside, (2) ultrasonic travel-time measurements via a pair of 55
kHz transducers, (3) CT scan of the interior, and (4) three dimensional CT image
of the stones (blue) compared to the ultrasonic reconstruction (green) obtained
using six face-centric source positions (yellow three-branched symbols).

The inverse problems group of the Department of Mathematics, Tampere University
of Technology does research that is closely related to technological
advancements in many applied fields of science including astronomical and
astrophysical modeling, environmental engineering and biomedical imaging. We
concentrate, among other things, on reconstructing the 3D shape of an asteroid
based on the intensity of the reflected light as well as estimating the total
volume of wood within a tree from laser point cloud data. Our projects are also
part of the activity of the Finnish Centre of Excellence in Inverse Problems
Research Finnish Centre of Excellence in Inverse Problems Research. For a book
on applied inverse problems, see [1].

One of our recent areas of focus has been waveform tomography with signal
sparsity. This scenario is relevant in several ultrasonic, radio and microwave
applications, when the number of signal transmission and/or receival points
needs to be low. Such situations occur, for example, non-destructive testing
(NDT) of materials and structures, biomedical microwave tomography and
ultrasonography, as well as in various atmospheric, pedospheric, geological, and
astrophysical investigations. To enable accurate forward (data) simulation for a
realistic domain structure, we use the finite element and finite-difference
time-domain method for spatial and temporal discretization, respectively. The
permittivity distribution can be recovered utilizing the linearized version of
the forward model together with classical or statistical inversion methodology
applied to waveform or compressed data, e.g., the travel-time of the signal. In
addition to the inversion methodology, we also currently develop compatible
hardware-level solutions for gathering the data.

In waveform tomography, our primary goal has been to advance design of future
planetary missions targeting at subsurface imaging of small planetary objects
(SPOs), which will be needed, for example, to explore mineral resources
contained by an asteroid. On such a mission, sparsity of the signal will be
fundamental due to the strict limitations in mission payload and energy supply.
As the present state-of-the-art, the comet nucleus sounding experiment by
radiowave transmission (CONSERT) [2,3] aims at tomography of the comet
67P/Churyumov-Gerasimenko, utilizing a radio signal transmitted between the
Rosetta spacecraft and its lander Philae. In CONSERT, the internal permittivity
distribution of the target comet can be estimated akin to the terrestrial
georadar applications, since the signal velocity is proportional to the inverse
square root of the permittivity. We have recently studied a slightly extended
lander-to-orbiter scenario relying on a sparse set of multiple source (lander)
positions. The baseline of our numerical and laboratory results is that the
source count should be at least one plus the dimensionality of the domain, i.e.,
three in 2D and four in 3D [4,5]. Our research on potential mission scenarios
will be continued with analysis of orbiter-to-orbiter schemes for multisatellite
formations, which are attractive as they do not necessitate landing within a
microgravity environment.

Our inversion results and experimental approaches also directly support the
ultrasonic NDT applications of civil engineering. With air-coupled ultrasound
and laser Doppler vibrometer technology, a signal penerating through an
investigated 3D structure can be scanned over a 2D surface, which allows
inversion under sparsity of signal sources. Optimizing the source count is
valuable regarding various on-site testing environments such as construction
sites, where the speed and ease of measurements can be crucial. In addition to
the astrophysical objectives, our future goal is to advance inversion
methodology regarding these aspects of NDT technology.
Sampsa Pursiainen

PhD Eng, Assistant Professor
Department of Mathematics
Tampere University of Technology
PO Box 553
33101 Tampere
Finland
E-mail: firstname.lastname@tut.fi
References

[1] J. P. Kaipio and E. Somersalo. “Statistical and Computational Methods for
Inverse Problems”. Berlin: Springer, 2004.

[2] W. Kofman et al. “The comet nucleus sounding experiment by radiowave
transmission (CONSERT): A short description of the instrument and of the
commissioning stages”. In: Space Science Reviews, 128:1-4 (2007).

[3] W. Kofman et al. “Properties of the 67P/Churyumov-Gerasimenko interior
revealed by CONSERT radar”. In: Science, 349:6247 (2015).

[4] S. Pursiainen and M. Kaasalainen. “Sparse source travel-time tomography of a
laboratory target: accuracy and robustness of anomaly detection”. In: Inverse
Problems, 30:114016 (2014).

[5] S Pursiainen and M Kaasalainen. “Detection of anomalies in radio tomography
of asteroids: source count and forward errors”, In: Planetary and Space Science,
99:36–47 (2014).

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