Nonlinear Multifrequency Multiparameter Quantitative Ultrasound Inverse Scattering Tomography

Algorithms for multiparameter quantitative ultrasonic tomography using multifrequency information are presented along with examples of their numerical implementation. A fluid model is employed to characterize the interaction of ultrasound with biological tissues yielding a scalar theory for the scattering and diffraction of ultrasonic waves by biological tissues. The scattering and diffraction of ultrasonic plane waves propagating through an arbitrary inhomogeneous, nondispersive medium, f, may be described by the nonlinear integral equation u({ bf x, k}) = u^{i}({bf x, k}) + k^2int_{rm D} G({bf x}, xi, k)f(xi)u( xi,{bf k})dxiquad { bf x}in D.The unknown object function f and the total field u are to be determined from measurements of the scattered field u^{s } = u - u^{i} on partial D u^{s }({bf x, k}) = k^2int _{rm D}G({bf x}, xi, k)f(xi)u(xi,{bf k})d xiquad {bf x}in partial D.The wave propagation vector k contains information on both frequency and direction of propagation of incident plane waves; hence, both multiple frequencies and multiple angles of incidence information are used in the measurements of the scattered field. The algorithms presented for the solution of these integral equations employ the method of moments and Newton's method. Rectangular coordinates and polar coordinates are studied. An alternating variables method is used with the method of moments to solve the integral equations in rectangular coordinates. Newton's method is used to solve the integral equations in polar coordinates which are written as a nonlinear operator equation Tw = 0 in a Banach space. The solution of Tw = 0 requires the solution of the linear operator equation T_sp{(w)}{ '}W = -Tw where the derivative of the operator, T_sp{(w)}{ '}, is the Frechet derivative of T at w. Upon discretization both methods require the solution of linear systems of equations. The conjugate gradient algorithm and the QR algorithm are used to solve these systems of equations and the singular value decomposition is used to study the spectra of these linear systems.