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Closely inspired by the total variation (TV) model of Rudin, Osher and Fatemi
[Physica D, 60:259-268,1992], we propose the quantized or quantum TV model
(either with a preassigned quanta set $Q$ or without), and study the associated
mathematical properties and computational algorithms. An algorithm based on stochastic or
Markovian gradient descent is proposed to handle the discrete programming nature of the
quantum TV model, which further leads to a two-step iterative algorithm for the
computationally more challenging free quantum TV model. We also demonstrate several
major applications of the proposed models and algorithms in bar code scanning, image
quantization, and image segmentation.