# Novel Deep Level Image State Ensemble Enhancement Method for M87 Imaging
## 2 Mathematics of Image State Decomposition and Enhancement
### ISE Reconstruction Matrices  
For levels of decomposition, \(\tau_{m,n}\) in (9) represents the Ln ISEE image reconstruction. S\({}_{m,63}\) is used because the 63 rd state Ln ISED image has the _most information removed_ when compared to all other ISED images states. If you refer to Table 1, you can see that all the \(\psi\)-ensemble states are activated (on). This indicates that the maximum amount of information is filtered out of the original image, and the resulting ISED image contains the least amount of information when compared to the original image. Incidentally, it would have the poorest full-reference quality metrics as well. The logic behind this is so that there is no overlapping of image data in the reconstruction and oversaturation in the image for a finite range of colors. The restoration will produce an image closer to the original image but with a narrower band of colors removed from the image as opposed to the previous wideband ISED method. This causes the SSIM to increase and improves PSNR and IMMSE.  
With an increase in Ln (further levels of decomposition) used in conjunction with the reconstruction for the five selected ISED states (S\({}_{m,30}\), S\({}_{m,44}\), S\({}_{m,45}\), S\({}_{m,46}\), and S\({}_{m,47}\)), the image begins to evolve to a convergence condition resulting in closer similarities to the original image. As one increases Ln for ISEE reconstruction, the greater the similarity for the convergent case. Not all ISEE reconstructions converge, some remain divergent or are partly convergent. This is due to either the intrinsic properties of the image and the lack of information contained within the ISED filter.  
ISED is not a single filter but a class of filters, each filter with its own unique attribute that selectively removes a finite bandwidth of information from the image. ISED has the characteristics of a wideband filter. However, since we designed the algorithms to generate many possible filter states,it has the advantage of producing the right tool for the right application. When you take the ISED image with the least amount of information (S\({}_{\text{m},63}\)) and combine the image with the lower state filter images (S\({}^{-}\)\({}_{\text{m},\text{n}}\), where \(n=1\)\(\sim\)62) it will produce a narrow filter that remains a subset of the original image. This improves the full-reference quality metrics. The equation for ISEE in (9) was derived from a deep understanding of how ISED images and filters operate. You could choose other filters of other ISED images other than S\({}_{\text{m},63}\), but this would cause an oversaturation in regions of the image. The generative image would be outside of the domain of the original image. It was not our desire to have this property, although, in some cases, it may enhance features. Additionally, it would produce a generative image further from the ground base truth. In ISED, S\({}_{\text{m},63}\) is like the skeleton of the image, and the combined dissimilar ISED filter restores a great deal of information to produce the ISEE reconstruction.  
\[\tau_{m,n}=S_{m,63}+S_{m,n}^{-}\text{ where, }\text{n}=1\sim 62\text{ and }\text{m}=1 \sim\text{Ln} \tag{9}\]  
For this study, 378 generated Ln ISED images, 378 generated Ln ISED filters, and 372 generated SEE images were produced from the SST test image.