Ultrasound Tomography Calibration Using Structured Matrix Completion

Calibration of ultrasound tomography devices is a challenging problem and of highly practical interest in medical and seismic imaging. This work addresses the position calibration problem in circular apertures where sensors are arranged on a circular ring and act both as transmitters and receivers.We introduce a new method of calibration based on the time-of-flight (ToF) measurements between sensors when the enclosed medium is homogeneous. Knowing all the pairwise ToFs, one can find the positions of the sensors using multi-dimensional scaling (MDS) method. In practice, however, we are facing two major sources of loss. One is due to the transitional behaviour of the sensors, which makes the ToF measurements for close-by sensors unavailable. The other is due to the random malfunctioning of the sensors, that leads to random missing ToF measurements. On top of the missing entries, since in practice the impulse response of the piezoelectric and the time origin in the measurement procedure are not present, a time mismatch is also added to the measurements. In this work, we first show that a matrix defined from all the ToF measurements is of rank at most four. In order to estimate the structured and random missing entries, utilizing the fact that the matrix in question is shown to be low-rank, we apply a state-of-the-art low-rank matrix completion algorithm. Then we use MDS in order to find the correct positions of the sensors. To confirm the functionality of our method in practice, simulations mimicking the measurements of an ultrasound tomography device are performed.