  In signal quantization, it is well-known that introducing adaptivity to
quantization schemes can improve their stability and accuracy in quantizing
bandlimited signals. However, adaptive quantization has only been designed for
one-dimensional signals. The contribution of this paper is two-fold: i). we
propose the first family of two-dimensional adaptive quantization schemes that
maintain the same mathematical and practical merits as their one-dimensional
counterparts, and ii). we show that both the traditional 1-dimensional and the
new 2-dimensional quantization schemes can effectively quantize signals with
jump discontinuities. These results immediately enable the usage of adaptive
quantization on images. Under mild conditions, we show that the adaptivity is
able to reduce the reconstruction error of images from the presently best
$O(\sqrt P)$ to the much smaller $O(\sqrt s)$, where $s$ is the number of jump
discontinuities in the image and $P$ ($P\gg s$) is the total number of samples.
This $\sqrt{P/s}$-fold error reduction is achieved via applying a total
variation norm regularized decoder, whose formulation is inspired by the
mathematical super-resolution theory in the field of compressed sensing.
Compared to the super-resolution setting, our error reduction is achieved
without requiring adjacent spikes/discontinuities to be well-separated, which
ensures its broad scope of application.
  We numerically demonstrate the efficacy of the new scheme on medical and
natural images. We observe that for images with small pixel intensity values,
the new method can significantly increase image quality over the
state-of-the-art method.
