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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.