Advanced Array Imaging for Breast and Prostate Sonography
Vaidyanathan, Ravi Shankar
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[Abstract from Thesis “Introduction.”] In conventional medical ultrasound such as B-mode imaging, the amplitude of the backscattered ultrasound pulse is used to image tissues along a fixed beam direction1. This imaging technique works best in static organs, and it is difficult to image moving organs like the heart. The M-mode imaging technique is better for cardiac applications. For better image resolution, ultrasound tomography systems were developed in which ultrasound data were acquired by transducers placed in a circle around the object2. This task of deriving the structure of the object from scattered radiation is known as the inverse scattering problem. The inverse scattering problem is known by several names like reflectivity tomography3 and diffraction tomography5, 6, 7 etc. Scattering refers to the effects on wave propagation due to an inhomogeneous medium. Since the inhomogenieties are unknown, the goal is to determine their properties – the spatial variation in density, compressibility, geometrical distribution etc. With the scattered wave field, determining the scatterer is called the inverse problem. As for the geometry of the scattering theory, the scatterer is assumed to be present in a homogeneous reference medium with known properties. Following the notations used in Lehman8, the acoustic pressure, p, in this medium satisfies the Helmholtz equation (2 + k2) p(r) = 0 where the pressure field is given by p(r,t)=p0+p1(r,t) The ambient pressure, p0 is constant. Since the scatterer is present in the reference homogeneous medium, the pressure field can be written as p0(r) = pinc(r) + psc(r) where pinc refers to the incident field and psc is the scattered field. In an ideal situation the incident pressure field is taken as a plane wave pinc(r) = p0 eikz where k is the complex wave number which is given by k=(/c) (1- iM) where M is the compressional viscosity. Now, we are in a position to introduce the integral representations of the scattered field. In the region exterior to the scatterer, the pressure field is given by (2 + k2) p0(r) = 0 Introducing the Green’s function G(r – r’) = eik|r-r’|/|r-r’| that will satisfy the inhomogeneous impulse equation (2 + k2) G(r – r’) = -4(r-r’) Using one of the most frequently used approximations, the Rayleigh-Born approximation we can modify equation (7). At large distance the Green‟s function can be approximated by G(r – r’) ~ eikr/r e-ikr.r’ which holds true for k0r‟2/r <<1. A Fourier diffraction theorem based reconstruction technique using the Born approximation is derived in Radial Reflection Diffraction Tomography (RRDT) 8. Though my work is concerned with time-domain reconstruction techniques, I will discuss some existing frequency domain reconstruction techniques.