On ray tracing for sharp changing media

Abstract

Ray tracing is an integral part of modern imaging and inversion techniques. Several methods have been proposed depending on the requirements of the application. Some algorithms are best applied to fast changing material properties, like an interface between two differing media, while others are well suited to media with gradually changing properties, like composite materials. In this paper, an enhanced numerical algorithm for ray tracing is presented. Focus is given to solutions involving ordinary differential equations with initial-value conditions. The proposed algorithm is the result of a combination of two classical implementations and the authors show that it is well suited for media with both sharp and gradual changes in the index of refraction. Additionally, the authors present an application of ray path computation by using the technique known as the shooting method.

Refraction-corrected ray-based inversion for three-dimensional ultrasound tomography of the breast

“…(Here, we set n o = 1 and a = 1.) [9,32,38]. This phantom has two interesting and useful properties which will be used below to test the ray tracing algorithms' accuracy.…”

“…Early approaches to UST used an assumption that the acoustic waves propagate along straight lines [6], thus neglecting diffraction, refraction and scattering [37]. Refraction (bending) of the rays can be incorporated by basing the calculation of each ray trajectory on Fermat's principle, which states that the energy tends to propagate along the path with minimal acoustic length [32,36,37,38]. However, because rays inherently assume a high-frequency approximation, diffraction and scattering effects are still neglected [37].…”

“…Another class of ray-tracing approaches, two-point ray tracing, treats the emitters and receivers as ray-emission and ray-reception points and computes only the rays between them (for all emitterreceiver pairs). The numerical methods for calculating the trajectory of a ray that links a pair of emission and reception points can be categorised into bending [48,57,49,41,42,47] and shooting [32,33,38] approaches. In the bending method, the problem is framed as a two-point boundary value problem.…”

“…This is known as the ray-linking problem [32,58,59]. Shooting methods have been widely used for 2D UST imaging [32,33,38], but little work has been done on their extension to 3D.…”

Refraction-corrected ray-based inversion for three-dimensional ultrasound tomography of the breast

“…This is known as the ray-linking problem [4,60,61]. Shooting methods have been widely used for 2D UST imaging [4,5,53], but little work has been done on their extension to 3D.…”