  Reconstructing 2D freehand Ultrasound (US) frames into 3D space without using
a tracker has recently seen advances with deep learning. Predicting good
frame-to-frame rigid transformations is often accepted as the learning
objective, especially when the ground-truth labels from spatial tracking
devices are inherently rigid transformations. Motivated by a) the observed
nonrigid deformation due to soft tissue motion during scanning, and b) the
highly sensitive prediction of rigid transformation, this study investigates
the methods and their benefits in predicting nonrigid transformations for
reconstructing 3D US. We propose a novel co-optimisation algorithm for
simultaneously estimating rigid transformations among US frames, supervised by
ground-truth from a tracker, and a nonrigid deformation, optimised by a
regularised registration network. We show that these two objectives can be
either optimised using meta-learning or combined by weighting. A fast scattered
data interpolation is also developed for enabling frequent reconstruction and
registration of non-parallel US frames, during training. With a new data set
containing over 357,000 frames in 720 scans, acquired from 60 subjects, the
experiments demonstrate that, due to an expanded thus easier-to-optimise
solution space, the generalisation is improved with the added deformation
estimation, with respect to the rigid ground-truth. The global pixel
reconstruction error (assessing accumulative prediction) is lowered from 18.48
to 16.51 mm, compared with baseline rigid-transformation-predicting methods.
Using manually identified landmarks, the proposed co-optimisation also shows
potentials in compensating nonrigid tissue motion at inference, which is not
measurable by tracker-provided ground-truth. The code and data used in this
paper are made publicly available at https://github.com/QiLi111/NR-Rec-FUS.
