Supper Resolution is the field of improving the resolution of pictures. There are different ways of doing so, some using additional information (such as multiple images)
Abstract
Supper Resolution is the field of improving the resolution of pictures. There are different ways of doing so, some using additional information (such as multiple images) and other using image processing methods, such as forward and backward diffusion.
All methods give good but not perfect results, thus another improvement was needed.
This project implements the Ricci flow, which uses the Ricci curvature to operate the heat equation on the metrics of an image.
In the process, the project gives new solution for the problem of integrating an image from the gradient field.
Image Enlargement-Interpolation
- There are different interpolation methods used for picture enlargement:
- Bilinear
- Bicubic
- Interpolation kernel (lanczos2 in matlab)
- All suffer from edge blurring (unwanted effect)
- Thus further improvement is needed
Image Interpolation-Demonstration
Forward and Backward Diffusion
- Improves image interpolation results
- Uses the heat equation –
- Forward diffusion- noise cleaning
- Backward diffusion – edge sharpening
- Treats the image as a surface and operates on it
- The results are not perfect
Forward and Backward Diffusion-Demonstration
Picture Enlargement – Ricci flow
- When we enlarge an image, we stretch the metric of the image
- Therefore, working on the metric of the image seems like a more natural way to enlarge images, and might give better results
- The Ricci curvature is the laplacian of the metric, so we will use it instead of the normal laplacian
- This gives us the Ricci flow
Ricci Flow Diagram
Poisson Solver Method
- The linear Poisson equation can be described as:
- Where:
– K is the second-difference matrix
–U is the required vector (Function)
–F is a given vector (Function)
- Eigen values of K are:
- Eigen vectors of K are discrete sines
- Expand F as a combination of eigen vectors:
- Divide each
by
. - U is obtained by :
Improved Reconstruction
- Identifying outliers as the main problem enabled us to improved the reconstruction results
- An “Outliers Filter” was built:
–For each pixel, let AVG be the average of its adjacent pixels
–If the distance of each pixel from AVG is larger than a certain value, the pixel’s value is changed to AVG
Conclusions
- Surface Reconstruction from its gradient field
–Integration is an important part of the Ricci flow
–Existing solutions for the integration problem give mediocre results
–Using image processing methods we improved the results of existing integration methods
- Ricci flow
–We achieved success in edge sharpening, however witnessed unwanted side effects
–Calculating Ricci curvature from the interpolated image is problematic
–The way the Ricci curvature is calculated (geometric, combinational) has great influence on the metric in the Ricci flow
Future Works
- Improve the unwanted side effects seen in the results of the Ricci flow
- Use the Ricci flow with Ricci curvature calculated from the interpolated image
- Investigate whether there is a better approximation for the Ricci curvature
–For instance: without neglecting the non diagonal members of the metric