Maximum Likelihood Estimation

For a change from working on the motion estimation side of the project, i decided to take a look at replacing the end of the super-resolution pipeline. The end of the pipeline is the part responsible for reversing the effects of the forward model. Up until know i have implemented this inversion using the pseudo-inverse function, whilst this is an effective approach for small images, and images which do not contain noise it will become infeasable to use this method once i start using larger images, and once i move ontoconsidering real world low resolution examples that contain noise.

The other motivation for implementing this part of the project now asides from the change from looking at motion estimation is that the maximum likelihood estimation approach is similar to work that i have been doing in my Neural networks class (ECE 173).



The X that satisfies the above equation is the X vector that is most likely to have been the original high resolution image. By finding the derivative of the above equation and setting it equal to zero we can then calculate the X vector that corresponds to the minimum of the function. The equation below shows the derivative:

Rearranging this formula and then applying the gradient descent method allows the X vector corresponding to the minimum error to the iteratively obtained.

Where

and


The graph below shows the error between the reconstruction and the original high resolution images against the number of iterations that were performed in order to acquire the reconstruction.

The reconstructed image can be seen below: