  Heatmap regression has become the mainstream methodology for deep
learning-based semantic landmark localization, including in facial landmark
localization and human pose estimation. Though heatmap regression is robust to
large variations in pose, illumination, and occlusion in unconstrained
settings, it usually suffers from a sub-pixel localization problem.
Specifically, considering that the activation point indices in heatmaps are
always integers, quantization error thus appears when using heatmaps as the
representation of numerical coordinates. Previous methods to overcome the
sub-pixel localization problem usually rely on high-resolution heatmaps. As a
result, there is always a trade-off between achieving localization accuracy and
computational cost, where the computational complexity of heatmap regression
depends on the heatmap resolution in a quadratic manner. In this paper, we
formally analyze the quantization error of vanilla heatmap regression and
propose a simple yet effective quantization system to address the sub-pixel
localization problem. The proposed quantization system induced by the
randomized rounding operation 1) encodes the fractional part of numerical
coordinates into the ground truth heatmap using a probabilistic approach during
training; and 2) decodes the predicted numerical coordinates from a set of
activation points during testing. We prove that the proposed quantization
system for heatmap regression is unbiased and lossless. Experimental results on
popular facial landmark localization datasets (WFLW, 300W, COFW, and AFLW) and
human pose estimation datasets (MPII and COCO) demonstrate the effectiveness of
the proposed method for efficient and accurate semantic landmark localization.
Code is available at http://github.com/baoshengyu/H3R.
