K-Space analysis for plane-wave ultrasound imaging

Ultrasound imaging can be considered as an inverse scattering problem in which the unknown distribution of the material parameters mass density and compressibility is reconstructed from measurements of scattered waves on the borders around the object. Unfortunately, this inverse problem is mathematically ill-posed and nonlinear: While the wave propagation is considered linear, the mapping of material parameters to measurements is nonlinear because the superposition principle is not valid: The sum of the wavefield solutions for two individual scatterers is not the correct solution when both scatterers are present because of multiple scattering. The usual way to deal with this nonlinearity is to use the linear Born approximation neglecting multiple scattering. Based on this approximation the analysis of ultrasound imaging is possible and pulse-echo ultrasound and transmission ultrasound using plane waves will both be described in the spatial frequency domain (k-space). One result of this analysis is the fundamental half-wavelength resolution limit. Also, the characteristics of reconstruction results in transmission and reflection imaging can be explained as k-space-filtered versions of the material parameters. For reflection imaging low spatial frequencies cannot be not reconstructed, explaining the main image characteristics observed in clinical imaging, e.g. the speckle pattern and missing absolute values. However, all limits resulting from the linear Born approximation are not necessarily valid when nonlinear reconstruction methods are applied. Simulations will be presented that demonstrate that absolute material parameter values can be reconstructed even in reflection imaging. In theory, also the half-wavelength resolution limit must not be valid. When the computational complexity and convergence issues of these methods can be overcome, ultrasound image quality would be drastically improved. Left: Linear reconstruction (not demodulated) at 100 kHz. Right: non-linear full wavefield inversion of a numerical breast phantom using 100 kHz – 200 kHz ultrasound. Session: Ultrasound Velocity Imaging and Diagnostics Fast imaging and vector flow estimation, an engineer's perspective.