# Image De-Quantization Using Plate Bending Model
## 5 Results  
One-dimensional comparison can be seen in Figure 3 for demonstration purposes. The Laplace solution cuts the top and bottom of the dynamic range, while a higher-order solution allows overshoot and reconstructs values in the LMM regions.  
Figure 4 shows two signals: a Gaussian and a Rosenbrock function [10]. The Gaussian does not have saddle points, while the Rosenbrock function is a frequently used test function in optimization due to its saddle-like shape with a relatively flat valley in the middle. The sub-figures show the high-bit-depth original, the reduced-bit-depth version, and the Laplace-reconstructed and biharmonic reconstructed solutions. The advantages of the higher-order approximation are obvious in the figure: the maximal region is smoothly recovered, and there is no second-order false contour.  
Table 1 compares peak signal-to-noise ratio (PSNR) [11], normalized root mean square error (NRMSE), and structural similarity index (SSIM) [12] for a few selected (2, 3, 5) bit depths of Gaussian and Rosenbrock images extended with zero padding, Laplacian method, and with the proposed method.  
Finally, an image of the night sky with the moon meant to demonstrate masking was used. When multiple objects were present in the scene, then the transitions led to overshoots. It is expected that the bended plate tries to keep its shape at the edges. However, if the image were segmented into  
Figure 3: Discretized signal samples with upper and lower limits for the possible original function values. The minimized squared gradient magnitude (Laplace problem) assumption leads to a piecewise linear solution (red), while minimizing the bending energy or mean curvature leads to a higher-order smooth solution (blue).  
parts, then reconstruction might be applied partially, ignoring highly textured areas. Similarly, if the neighboring pixels differ in more than one discretization unit, then it can be assumed that they belong to different image components, and the assumption about signal smoothness does not hold. For an example, see the moon in Figure 5. If these regions (moon and sky) were not handled separately, then due to the stiffness of the plate in the model, the moon would generate a halo around itself. The synthetic examples were generated and the photo was taken by the first author.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Function** & **Bit Depth** & **Method** & **NRMSE** & **PSNR** & **SSIM** \\ \hline Gaussian & 2 & zero pad & 0.3252 & 17.0239 & 0.4174 \\ [PERSON] & 2 & Laplace & 0.3362 & 16.7344 & 0.4661 \\ [PERSON] & 2 & biharmonic & 0.0713 & 30.2044 & 0.9462 \\ [PERSON] & 3 & zero pad & 0.1618 & 23.0882 & 0.7585 \\ [PERSON] & 3 & Laplace & 0.1535 & 23.5467 & 0.8187 \\ [PERSON] & 3 & biharmonic & 0.1169 & 25.9098 & 0.8877 \\ [PERSON] & 5 & zero pad & 0.0383 & 35.6112 & 0.9831 \\ [PERSON] & 5 & Laplace & 0.0308 & 37.4985 & 0.9918 \\ [PERSON] & 5 & biharmonic & 0.0371 & 35.8821 & 0.9885 \\ [PERSON] & 2 & zero pad & 0.4011 & 19.3146 & 0.3118 \\ [PERSON] & 2 & Laplace & 0.4457 & 18.3993 & 0.3855 \\ [PERSON] & 2 & biharmonic & 0.2315 & 24.0895 & 0.8097 \\ [PERSON] & 3 & zero pad & 0.2194 & 24.5547 & 0.5317 \\ [PERSON] & 3 & Laplace & 0.2646 & 22.9290 & 0.4988 \\ [PERSON] & 3 & biharmonic & 0.1520 & 27.7415 & 0.8251 \\ [PERSON] & 5 & zero pad & 0.0561 & 36.4029 & 0.8999 \\ [PERSON] & 5 & [PERSON] & 0.0721 & 34.2228 & 0.8555 \\ [PERSON] & 5 & biharmonic & 0.0681 & 34.7163 & 0.9319 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Reconstruction of [PERSON] and Rosenbrock functions evaluated with peak signal to noise ratio (PSNR), normalized root mean square error (NRMSE), and structural similarity index (SSIM) metrics. While at high bit depths both Laplace and biharmonic produced good results, at lower bit depth the biharmonic model had a significant advantage in all metrics.  
Figure 4: Top row: Gaussian function; Bottom row: Rosenbrock function. From left to right: 8-bit reference image, downscaled 2-bit image, Laplace reconstruction, and biharmonic reconstruction.