Construction of Orthogonal Transmit Sequences Using the Nearest Orthogonal Matrix

Abstract

A wide variety of transmit sequences can be employed in medical ultrasound, including plane waves, diverging waves, and focused beams. The choice of sequence often involves trade-offs between resolution, signal-to-noise ratio (SNR), frame rate, and harmonic imaging capabilities. However, the desirable mathematical property of orthogonality (i.e., absence of cross-talk) between transmits has generally received less attention. This property, often lacking, becomes particularly relevant for the recent REFoCUS (retrospective encoding for conventional ultrasound sequences) technique, which we in this work connect to the array signal processing technique called beamspace processing. Given an arbitrary transmit sequence, REFoCUS enables the recovery of signals from single-element transmissions (known as the multistatic dataset) thereby enhancing beamforming flexibility. In this context, the choice of transmit sequence influences the recovery process when using the intuitively appealing and computationally efficient adjoint-based method, which must be replaced by a regularized pseudoinverse for general applicability. In the current work, we derive the “closest” alternative to any chosen transmit sequence that makes the regularized and adjoint methods yield equal estimates of the multistatic dataset, and show via numerical experiments a reduction in beam and/or element cross-talk. The derivation is based on a matrix nearness problem of finding the nearest orthogonal (or unitary) matrix to the encoding matrix using singular value decomposition (SVD). The resulting transmit sequences offer a time-domain equivalent understanding of the regularized REFoCUS method, as well as a solution for optimizing the invertibility of ultrasound sequences.