Taylor-Series-Based Derivation of the Resolution of Null Subtraction Imaging for a Uniform Linear Array

Null subtraction imaging (NSI) is a non-linear beamformer that aims to improve the spatial resolution of ultrasound images. NSI incoherently combines three delay-and-sum (DAS) outputs from the same RF data using three related apodizations on receive. NSI has been advocated to have many advantages in different domains such as B-mode imaging, plane wave imaging, power Doppler imaging, and for large-pitch arrays. However, despite its increasing popularity, an explicit relationship between NSI resolution (interpreted as the mainlobe width) and various parameters (such as the DC offset value c, array aperture, and wavelength) is not known, making system design and intuitive reasoning about the method difficult. Therefore, in the current work, we derive the theoretical NSI array pattern and give an approximate expression for the −6dB mainlobe width. Our derivation is based on a Taylor series-expansion of the analytical NSI array pattern, which is valid over the mainlobe region for the range of c values typically seen in the literature. The results show that the NSI mainlobe width is proportional to $c \lambda /D$ , which is the DC offset value multiplied by the wavelength and divided by the aperture size, and therefore has a similar wavelengh and aperture dependency as the classical DAS mainlobe. The work is validated numerically, also showing that the NSI mainlobe width approaches the DAS mainlobe width as c approaches infinity.