Inverse Scattering Study

By Roberto Lavarello, Graduate Student and Professor Michael L. Oelze

Acoustical inverse scattering techniques allow obtaining quantitative information about the distribution of acoustical properties such as speed of sound and attenuation inside an object. Such information is obtained by measuring the scattered field using several transmission-reception configurations, and using the wave equation to relate these measurements with the wavenumber function inside the object. Unlike some other tomographic techniques such as X-ray tomography and MRI, acoustical inverse scattering has yet to be widely accepted. This is due to the lack of both applications that can exploit its unique features, and robust but efficient inversion methods.

The most successful inversion techniques consist of discretizing the integral form of the wave equation using the method of moments, and using Newton-type iterative methods to solve the resulting equations. Because of computational efficiency issues, marching methods have started to gain some attention lately. Frequency domain methods suffer from phase wrapping, so in order to reconstruct regions of large contrast one can resort to multiresolution reconstructions using frequency hopping.

Figure 1 shows inverse scattering reconstructions using the distorted Born iterative method in the frequency domain and simulated data. The ideal speed of sound contrast profile is shown in Figure 1(a). The effects of phase wrapping are shown in Figure 1(b), where a frequency high enough so that divergence occurs is used. Even though some of the features of the object are reconstructed, the reconstruction lacks quantitative value. The frequency hopping approach, i.e., the use of multiple frequency information to perform the reconstruction- is shown in Figures 1(c) and (d). First, the object is reconstructed with a low frequency so that convergence is achieved and the coarse features of the object are captured. Then, the coarse reconstruction is used as initial guess for a higher frequency reconstruction, which can better resolve the finer features of the object.

Figure 1: Reconstruction of speed of sound profile using the distorted Born iterative method. a) Ideal distribution of the contrast in speed of sound. b) Single frequency reconstruction at 2 MHz. c) Single frequency reconstruction at 1 MHz. d) Frequency hopping reconstruction using scattered data at 1 and 2 MHz.

The two dimensional inverse scattering simulations were validated experimentally using unfocused circular piston transducers as transmitters and receivers. A phantom was manufactured using a thin (0.23 mm thick) rubber balloon filled with a saline solution (saline concentration of approximately 25 grams/100 milliliters). The speed of sound of the saline solution was measured using time of flight measurements. The scattered data was collected for receiving angles between -60º and 60º. Figure 2 shows the scanning configuration and equipment used in the experiments. Figure 3 shows the measured scattered fields at both 0.64 MHz and 1.2 MHz. Frequency hopping was used in this experiment to avoid convergence issues due to frequency wrapping.

Figure 2: Setup for experimental configuration. (a) Picture of the equipment used, showing the fixed transmitter transducer (left), the yellow balloon phantom (center), and the rotating arm with the attached receiver transducer (right). (b) Schematic of the scanning configuration showing values for relevant distances and sizes.





Figure 3: Scattered fields as a function of the receiving angle. Magnitude in dB (left) and phase in radians (right) of the scattered field at (a) 0.64 MHz and (b) 1.2 MHz. Both the measured (blue) and ideal (red) scattered data are shown.

The reconstructed speed of sound is shown in Figure 4. Profiles for the speed of sound contrast are shown in Figures 4(c) and (d). The blue line shows the ideal profile. The red line shows the profile reconstructing using ideal scattered data generated with a method of moment solver. The black line shows the profile reconstructed using the measured scattered data. The difference in the reconstruction error of the speed of sound contrast between ideal and measured data is around 1.4 percent. It can be observed that the profile reconstructed using a higher frequency has sharper edges and reduced amplitude of the oscillation around the mean speed of sound value of the phantom.

Figure 4: DBIM reconstruction of the balloon phantom using experimental data. a) Ideal speed of sound distribution. b) Reconstructed speed of sound distribution at 1.2 MHz. c) Reconstructed speed of sound contrast profile through the center of the cylinder at 0.64 MHz. d) Reconstructed speed of sound contrast profile through the center of the cylinder using 1.2 MHz.

As in many other tomographic techniques, acoustical inverse scattering performs better when the region of interest can be illuminated from every direction. This poses a constraint on the type of targets that can be imaged. The application that has received most of the attention is the imaging of the female breast for early detection of cancer. We propose an original application which consists of obtaining specific values of acoustical properties of tumor cells in order to develop improved cell models for the Quantitative Ultrasonic Imaging (QUS) project. We believe that this project is perfectly suited to take advantage of the current state of the art inverse scattering methods, and at the same time expand the applicability of these types of techniques.

References:


W. C. Chew and Y. M. Wang. Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method. IEEE Transactions on Medical Imaging,9, 218-225, 1990.


O. Haddadin and E. Ebbini. Imaging strongly scattering media using a multiple frequency distorted Born iterative method. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control,45, 1485-1496, 1998.


D. Borup, S. Johnson, W. Kim, and M. Berggren. Nonperturbative diffraction tomography via Gauss-Newton iteration applied to the scattering integral equation. Ultrasonic Imaging,14, 69-85,1992.


F. Natterer and F. Wubelling. Marching schemes for inverse acoustic scattering problems. Numerische Mathematik,100, 697-710, 2005.

This work is supported in part by a 3M Non-tenured Faculty Grant