Michael J. Berggren

Inverse scattering solutions by a sinc basis, multiple source, moment method — Part III: Fast algorithms

olving the inverse scattering problem for the Helmholtz wave equation without employing the Born or Rytov approximations is a challenging problem, but some slow iterative methods have been proposed. One such method suggested by us is based on solving systems of nonlinear algebraic equations that are derived by applying the method of moments to a sinc basis function expansion of the fields and scattering potential. In the past, we have solved these equations for a 2-D object of n by n pixels in a time proportional to n5. In the present paper, we demonstrate a new method based on FFT convolution and the concept of backprojection which solves these equations in time proportional to n3 X log(n). Several numerical examples are given for images up to 7 by 7 pixels in size. Analogous algorithms to solve the Riccati wave equation in n3 X log(n) time are also suggested, but not verified. A method is suggested for interpolating measurements from one detector geometry to a new perturbed detector geometry whose measurement points fall on a FFT accessible, rectangular grid and thereby render many detector geometrics compatible for use by our fast methods.

Wave Equations and Inverse Solutions for Soft Tissue

Ultrasound has been used in the echo mode as a diagnostic tool for many years. The images produced by the echo mode are not qualitative and do not measure absolute tissue properties. Rather, echo-mode images display the changes in acoustic impedance. Thus, the boundaries of tissues are imaged. Such images are of value for study of anatomy and tissue morphology. Tumors are detected by their shape and not by their tissue type. Thus, classification of malignant or benign tumors is difficult by use of echo-mode imaging alone. The recently developed straight-line ultrasound transmission tomographic methods provide images which are qualitative and absolute (not relative), but have inferior resolution compared to the echo methods [Greenleaf, et al., 1978].

Ultrasound Inverse Scattering Solutions From Transmission And/Or Reflection Data

Although historically the Born or Rytov linear approximations have received a great deal of attention, it is now more apparent that only a full nonlinear formulation of the inverse scattering problem, such as those we have developed, provide the accuracy for quantitative clinical ultrasound imaging. Our inverse scattering solutions have been developed to reconstruct quantitative images of speed of sound, density, and absorption using the exact Helmholtz wave equation without perturbation approximations. We have developed fast algorithms which are based upon Galerkin or moment discretizations and use various iterative solution techniques such as back propagation and descent methods. In order to reconstruct images with reflection-only scanner geometries we have extended our algorithms to include multiple frequency data. We have demonstrated a procedure for imaging inhomogeneous density distributions. We also discuss the significance and potential applications of these new methods.

Analysis of Inverse Scattering Solutions from Single Frequency, Combined Transmission and Reflection Data for the Helmholtz and Riccati Exact Wave Equations

Various numerical methods to solve the exact inverse scattering problem are presented here. These methods consist of the following steps: first, modeling the scattering of acoustic waves by an accurate wave equation; second, discretizing this equation; and third, numerically solving the discretized equations. The fixed-point method and the nonlinear Newton-Raphson method are applied to both the Helmholtz and Riccati exact wave equations after discretizations by the moment method or by the discrete Fourier transform methods. Validity of the proposed methods is verified by computer simulation, using exact scattering data from the analytical solution for scattering from right circular cylindrical objects. (Acoustical Imaging 15, Halifax, Nova Scotia, July, 1986)

Performance of Fast Inverse Scattering Solutions for the Exact Helmholtz Equation Using Multiple Frequencies and Limited Views

We have previously reported fast algorithms for imaging by acoustical inverse scattering using the exact (not linearized) Helmholtz wave equation [1]. We now report numerical implementations of these algorithms which allow the reconstruction of quantitative images of speed of sound, density, and absorption from either transmission or reflection data. We also demonstrate the application of our results to larger grids (up to 64 × 64 pixels) and compare our results with analytically derived data, which are known to be highly accurate, for scattering from right circular cylindrical objects. We report on the performance of our algorithms for both transmission and reflection data and for the simultaneous solution of scattering components corresponding to speed of sound and absorption. We have further examined the performance of our methods with various amounts of random noise added to the simulated data. We also report on the performance of one technique we have devised to extract quantitative density images from our algorithms.

Implementation of scatterer size imaging on an ultrasonic breast tomography scanner

Quantitative ultrasound (QUS) techniques make use of frequency-dependent information from backscattered echoes normally discarded in conventional B-mode imaging. Using scattering models and spectral fit methods, properties of tissue microstructure can be estimated. The use of full angular spatial compounding has been proposed as a means of improving the variance of scatterer property estimates and spatial resolution of QUS imaging. In this work, preliminary experimental results from a QUS implementation on an ultrasonic breast tomography scanner from TechniScan, Inc. are presented. The imaging target consisted of a cylindrical gelatin phantom of 7.8 cm diameter. The phantom contained uniformly distributed glass bead inclusions of 85 m mean diameter. The scanner provided reflection-mode data using arrays with 6 MHz nominal center frequency for 17 different angles of view distributed between 0° and 360°. Tomographic images of speed of sound were also generated by the scanner and used for refraction-compensation and registration of the effective scatterer diameter (ESD) estimates corresponding to ROIs at different angles of view. Only data from the surface of the array to the center of the tomography gantry were analyzed for each angle of view, which resulted in 8.5 effective angles of view per ROI. ESD estimates were obtained using ROIs of size 4 mm by 4 mm with a 50% overlap. The average mean and standard deviation of the single angle of view estimates considering the 17 data sets were 85.4 µm and 12.2 µm, respectively. The resulting ESD mean and standard deviation of the compounded image were 85.2 µm and 4.1 µm, respectively. The preliminary experimental results presented here represent the first implementation of QUS on an ultrasonic breast tomography scanner and demonstrate some of the benefits of integrating these technologies, i.e., the availability of full angular spatial compounding and integration with tomographic speed of sound images.

Ultrasonic characterization of mycelial morphology as a fractal structure

Preliminary measurements of scattering of ultrasound from filamentous microorganisms in aqueous-medium suspensions are reported. The data are used to characterize the structuring (morphology) between the organisms quantitatively by modelling the morphology in terms of fractal structures and obtaining the fractal dimension as the characteristic parameter. The fractal dimension is calculated from the slope of a log-log plot of scattered intensity versus the scattering vector magnitude. The intensity is measured over a range of magnitudes of the scattering vector by varying the scattering angle as well as the ultrasonic frequency. Results from microbial suspensions of different species and different morphologies indicate that the fractal dimension can be used to quantify the morphological state of a suspension. The fractal dimension ranges from around 1.5 for suspensions with pelletous morphology to around 2.5 for suspensions with filamentous morphology.

Fast iterative algorithms for inverse scattering solutions of the Helmholtz and Riccati wave equations

Solving the inverse scattering problem for the Helmholtz wave equation without employing the Born or Rytov approximations is a challenging problem, but some slow iterative methods have been proposed [1, 2], One such method suggested and demonstrated by us is based on solving systems of nonlinear algebraic equations that are derived by applying the method of moments to a sine basis function expansion of the fields and scattering potential [2, 3]. In the past, we have solved these equations for a 2-D object of n by n pixels in a time proportional to n5 [1–3], We now describe further progress in the development of new methods based on FFT convolution and the concept of backprojection [4, 5], which solves these equations in time proportional to n3 • log(n).

Full inverse scattering vs. Born-like approximation for imaging in a stratified ocean

This Oceans 93 paper describes full nonlinear inversion that gives quantitative reconstructions of submerged and/or buried objects with the aid of a layered Greens function. The computational speed is preserved by use of Fast Fourier Transforms and stabilized biconjugate gradients, for both the homogeneous background and layered background case.<<ETX>>

Acoustic Inverse Scattering Images from Simulated Higher Contrast Objects and from Laboratory Test Objects

Our previously reported acoustic inverse scattering imaging algorithms, based on the exact (not linearized) Helmholtz wave equation, are accurate and robust when applied to scattering data from small or low contrast objects. In order to compare the relative efficacy of our inverse scattering methods with linear approximations such as the Born or Rytov approximations, we have reviewed the limitations of those methods with both theoretical and experimental data. We have augmented our original alternating variable algorithms with Newton-Raphson-like methods which give improved convergence with higher contrast objects (up to 20% contrast in the scattering potential) andSLASHor larger grid sizes. We have also developed a new, fast algorithm for computing forward scattering solutions from two non overlapping right circular cylinderical objects using a Bessel function series expansion. When using our inverse scattering algorithms to reconstruct images from this independently generated data, we find excellent agreement. We demonstrate the ability of our methods to reconstruct separate images of compressibility, absorption, and density for simulated data from a simple breast phantom. We also report on our progress in reconstructing images of larger size, as measured in wavelengths, and in using more realistic tissue simulating models as test objects.. (Acoustical Imaging 16, Chicago, Illinois, June, 1987).