Fast Thresholding of SVD Clutter Filter Using the Spatial Similarity Matrix and a Sum-Table Algorithm

Singular value decomposition (SVD) has become a standard for clutter filtering of ultrafast ultrasound datasets. Its implementation requires the choice of appropriate thresholds to discriminate the singular value subspaces associated with tissue, blood, and noise signals. Comparing the similarity of the spatial singular vectors was shown to be a robust and efficient method to estimate the SVD thresholds. The correlation of the spatial singular vector envelopes gives the spatial similarity matrix (SSM), which usually exhibits two square-like domains juxtaposed along the diagonal of the SSM, representing the tissue and the blood subspaces. Up to now, the proposed methods to automatically segment these two subspaces on the SSM were of high computational complexity and had a long processing time. Here, we propose an optimized algorithm using a sum-table approach that decreases the complexity by two orders of magnitude:
$O(n^{4}) to O(n^{2})$
. The proposed method resulted in processing times lower than 0.08 s for datasets of 2000 frames, whereas previous algorithms took more than 26 h, so an improvement by a factor of 106. We illustrated this adaptive square-fitting on the SSM in the in vivo case of human neonate brain imaging and carotid imaging with various conditions of clutter. This optimization of SVD thresholding is essential to develop the use of adaptive clutter filtering, especially for real-time applications or block-wise processing.