  Recent research highlights that the Directed Accumulator (DA), through its
parametrization of geometric priors into neural networks, has notably improved
the performance of medical image recognition, particularly with small and
imbalanced datasets. However, DA's potential in pixel-wise dense predictions is
unexplored. To bridge this gap, we present the Directed Accumulator Grid
(DAGrid), which allows geometric-preserving filtering in neural networks, thus
broadening the scope of DA's applications to include pixel-level dense
prediction tasks. DAGrid utilizes homogeneous data types in conjunction with
designed sampling grids to construct geometrically transformed representations,
retaining intricate geometric information and promoting long-range information
propagation within the neural networks. Contrary to its symmetric counterpart,
grid sampling, which might lose information in the sampling process, DAGrid
aggregates all pixels, ensuring a comprehensive representation in the
transformed space. The parallelization of DAGrid on modern GPUs is facilitated
using CUDA programming, and also back propagation is enabled for deep neural
network training. Empirical results show DAGrid-enhanced neural networks excel
in supervised skin lesion segmentation and unsupervised cardiac image
registration. Specifically, the network incorporating DAGrid has realized a
70.8% reduction in network parameter size and a 96.8% decrease in FLOPs, while
concurrently improving the Dice score for skin lesion segmentation by 1.0%
compared to state-of-the-art transformers. Furthermore, it has achieved
improvements of 4.4% and 8.2% in the average Dice score and Dice score of the
left ventricular mass, respectively, indicating an increase in registration
accuracy for cardiac images. The source code is available at
https://github.com/tinymilky/DeDA.
