  Continuous one-to-many mapping is a less investigated yet important task in
both low-level visions and neural image translation. In this paper, we present
a new formulation called MonoPix, an unsupervised and contrastive continuous
modulation model, and take a step further to enable a pixel-level spatial
control which is critical but can not be properly handled previously. The key
feature of this work is to model the monotonicity between controlling signals
and the domain discriminator with a novel contrastive modulation framework and
corresponding monotonicity constraints. We have also introduced a selective
inference strategy with logarithmic approximation complexity and support fast
domain adaptations. The state-of-the-art performance is validated on a variety
of continuous mapping tasks, including AFHQ cat-dog and Yosemite summer-winter
translation. The introduced approach also helps to provide a new solution for
many low-level tasks like low-light enhancement and natural noise generation,
which is beyond the long-established practice of one-to-one training and
inference. Code is available at https://github.com/lukun199/MonoPix.
