  Quantitative $T_1$ mapping by MRI is an increasingly important tool for
clinical assessment of cardiovascular diseases. The cardiac $T_1$ map is
derived by fitting a known signal model to a series of baseline images, while
the quality of this map can be deteriorated by involuntary respiratory and
cardiac motion. To correct motion, a template image is often needed to register
all baseline images, but the choice of template is nontrivial, leading to
inconsistent performance sensitive to image contrast. In this work, we propose
a novel deep-learning-based groupwise registration framework, which omits the
need for a template, and registers all baseline images simultaneously. We
design two groupwise losses for this registration framework: the first is a
linear principal component analysis (PCA) loss that enforces alignment of
baseline images irrespective of the intensity variation, and the second is an
auxiliary relaxometry loss that enforces adherence of intensity profile to the
signal model. We extensively evaluated our method, termed ``PCA-Relax'', and
other baseline methods on an in-house cardiac MRI dataset including both pre-
and post-contrast $T_1$ sequences. All methods were evaluated under three
distinct training-and-evaluation strategies, namely, standard, one-shot, and
test-time-adaptation. The proposed PCA-Relax showed further improved
performance of registration and mapping over well-established baselines. The
proposed groupwise framework is generic and can be adapted to applications
involving multiple images.
