# Image De-Quantization Using Plate Bending Model
## 6 Discussion
### Cross-Terms  
There are previous higher-order polynomial approximations for this problem, but to our knowledge, the axes are handled individually. For instance, a fourth-order generalization of the Laplace equation without cross-terms could be an \"almost biharmonic\" equation where the cross-terms are downweighted, and while they give some contribution, their effect is much more subtle.  
Not connecting the two axis has both advantages and disadvantages. In case of noise, it will not propagate into the other dimension, and will not generate wave-like patterns, but also smoothness will be limited. There are at least two ways to find balance between the approaches.  
First, a lower-order stretching could be applied. In this approach, not only the plate bending energy should be minimized, which leads to the biharmonic equation, but an extra energy term which is proportional with the surface area. Applying this second energy term would lead to the Laplace equation. However, the sum of the two energies lead to non-trivial solutions, because the surface minimization tends to generate sharp edges at the border LMM regions. The weighting between the two energy terms could be chosen based on the application.  
Second, anomalous points could be detected. The main idea is as follows: If the reconstructed function has extrema at a given point, and this point also lies at the border of two regions in the low-bit-depth image, then this originates from noise. The reason is simple. If a large smooth region has extrema, it is unlikely that it will be exactly on the edge of the borders. Therefore, this point should be masked. However, if the point was in fact a real extrema, removing it would not significantly change the image, because the rest of the points should still produce continuous smooth surface, and this process would regenerate the extrema, even without the control point. This process can be iteratively performed until all the anomalous points are taken care of. Actually, this approach can applied axis-by-axis, which can take care of possible ringing at edges. The main disadvantage of this approach is its time-consuming nature due to the iterations.  
Third, alternative elasticity models could also be used. Referring to the literature, solving the biharmonic equation minimizes the mean curvature (\(B\)) ([PERSON] et al., 2010) of a surface. Using principal curvatures (\(\kappa_{1}\), \(\kappa_{2}\)), the mean curvature is as follows:  
\[B=(\kappa_{1}+\kappa_{2})^{2}. \tag{10}\]  
An alternative approach could be minimizing the total curvature (\(T\)) ([PERSON] et al., 2010):  
\[T=\kappa_{1}^{2}+\kappa_{2}^{2}. \tag{11}\]  
This is equivalent to minimizing the thin plate energy functional. The biharmonic model is permissive towards saddle points, while the thin plate model prefers surfaces which can be approximated locally with a spherical surface.  
The benefits of having higher-order plate-like approximation, having more traditional spline approximation, or a combined Laplace-biharmonic model depends on the content--especially on the structuredness, noise, and presence of saddle-like areas. From an application point of view, if similar high-resolution content exists, a simulated degradation-reconstruction process could be used to select the best solution from the aforementioned options in order to maximize a chosen metric, or to maximize observers' opinion score.