# Image De-Quantization Using Plate Bending Model
## 2 Related Work  
Consider an image that is discretized using \(n\) bits. Any value between two discrete levels is rounded. The rounding strategy can be the nearest integer, or the nearest \"smaller/larger than the signal\" integer. In both cases, the original signal (\(f_{original}\)) is rounded to a discretized signal (\(g\)). The difference (\(f_{original}-g\)) is the lost information which is the target of the recovery operation. Using the \(\mathcal{R}\) rounding operator and \(x,y\) spatial coordinates, the problem is as follows:  
\[g(x,y)=\mathcal{R}\left\{f_{original}(x,y).\right\} \tag{1}\]  
The lost signal must remain within certain limits. If the rounding operator is \"rounding to the nearest integer\", then the limit is \(\pm 0.5\) unit, but it could also be \"rounding down\" or other operators. The general formulation is:  
\[\text{lower limit}\leq f_{original}(x,y)-g(x,y).\leq\text{upper limit} \tag{2}\]  
What is the \(f(x,y)\) approximation that makes the recovered signal the most plausible, most appealing?  
\[f_{original}(x,y)\approx f(x,y). \tag{3}\]  
Early strategies utilized blurring for false contour removal. For instance, [PERSON] and [PERSON] [2] used random blurring. Blurring and exploitation of the human visual system's properties remained a research interest, and led to low- and high-frequency separation techniques and adaptive cancellation, as reviewed by [PERSON] and [PERSON] [3]. The most challenging issue with this approach is the fact that  
Figure 1: Example of non-continuous regions. The sky exhibits strong false contours, and the foreground ship partitions the sky into several regions. The photo was taken by the first author.  
blurring might also affect real edges. The separation of false contours from real edges is a very difficult problem on its own.  
The most recent approaches describe the false contour removal as a pure mathematical de-quantization challenge, and they do not aim to model the human visual system. [PERSON]'s algorithm [PERSON] (2016) uses flooding-based downward/upward distance maps to calculate step ratio and linearly interpolate. This method achieves very nice results for gradients, but fails to recover the local maximal/minimal (LMM) regions.  
[PERSON] et al. (2015) developed a content-adaptive method using a virtual skeleton marking (VSM) model which transforms the 2D extrapolation problem into a 1D interpolation, but it is still based on linear interpolation. In a follow-up paper [PERSON] (2016), a spatially varying sparsity-based de-quantization method was published, based on the observation that the second derivatives are positive/negative in the LMM regions and approximately zero in the other regions. The derived model is a Thikonov regularization-based anisotropic total variation minimization problem.  
A similar approach was used by [PERSON] (2016), who developed a bounded interval regularization method, where the recovered signal must remain within limits, which is enforced with a penalty term outside limits. The algorithm in the allowed interval minimizes an energy function, which is quadratic in absolute value of the gradient (\(|\
abla f|^{2}\)). This formulation is equivalent to a bounded Laplace problem, where internal points (not exceeding the limits) satisfy the Laplace equation:  
\[\Delta f(x,y)=0. \tag{4}\]  
This partial differential equation approach has several advantages, including the possibility of handling non-uniform quantization levels. However, it fails to reconstruct LMM regions, and perceived smoothness of the image might be low when the original signal's second derivative is significantly different from zero.  
[PERSON] et al. (2015) developed an AC/DC separation model with maximum a posteriori AC estimation, where DC denotes the mean of the signal, and AC is the difference of the original signal and the DC component. They viewed the problem from a minimum mean square error perspective, and used a graph smoothness prior. The method works on blocks, where overlapping blocks are used to minimize artifacts at the edges.