Initial Conditions:

The range of scale is very little. Scaling will give good results in the range from approximately 0.95 to 1.15. Results can be seen in the figure below.

The range of rotations is from approximately -10 to 10 degrees. Although the metric is above 90% but the ellipse is not fitted well in my opinion.

The range of translations on the X-axis is was very large. It ranged from approximately 40 to 160. Samples between x=35 and x=61 were also tested but are not shown here.

The range of translations on the Y-axis is from approximately 52 to 185.

Two sample outputs of the ones tested can be seen here .One is close to the upper boundaries and the other is out of lower boundaries.

Why the eclipse stopped? The reason that the eclipse gets stuck is that it falls in local minima (or maxima in this case). As can be seen from the figures below, the eclipse is correctly fitted on one corner (or a region) of the brain and skull. This is a local maxima. In order to get out of it requires that it's metric decrease a significant amount in order to "jump out" of the local maxima. In other word the number of dark pixels will decrease and the white pixels will increase, which will cause the metric value to decrease. But the program will not do because it's maximizing, that that's why it gets stuck. 

Optimizers:

I used the OnePlusOneEvolutionary-Optimizer. This Optimizer stimulates the biological evolution of a set of samples in the search space.

 Did the algorithm reach convergence faster?  No it didn't. It took a lot more time for the metric value to reach a stable condition. I had to limit the number of iterations of the optimizer for the program to stop at some point. I am defining its performance as its computation time and the final result together. If the inputs initial conditions were within a good range .

Did the optimizer perform the same? No, it performed differently in the range of translations bounded before with the other optimizer. The range became narrower. Two sample outputs can be seen below:

Bandwidth:

 What happens to the range of transformation parameters? I did not see an increase in the range of initial transformations. This can be seen in the table below. I tested a range of initial conditions near the boundaries of the original condition and the results did not look good. Below is a sample of the tested ranges.

As the bandwidth increases the number of iterations increases and decreases (this can be seen in the graph below).  But in general it has a decreasing trend.

Also, the metric value of the changes in bandwidth can also be seen below. As in the # of iterations, this also increases and decreases. But it won't increase back to the high metric value as in bandwidths 10 and 15.

This amount is not that significant in ideal initial conditions. But in a wider range of transformations and rotations, a major increase can be seen in the number of iterations. The graph above shows a comparison of the original transformation parameters compared to a wider range sample as the bandwidth increase. Also, as the bandwidth increases the number of iterations also increases. If the bandwidth is small the eclipse latches on to the dark pixels at first. In other words the priority, is latching to the darker pixels then to the white ones. This can be seen in the images provided for this section.

Part 2:

The program first thresholds the image at intensity 100 using the ThresholdImageFilter. The output of this image is another image which basically keeps the pixels with intensities higher than 100 and throws the rest out. Then the program finds the bounding box around the image. By having the bounding box, it is easy to compute the scale and translation of the image, having the data of the original image ( I compute everything relative to the center of the bounding box) The output of the program (on the last line) states the initial estimates as:

[scale,rotation,x_translation,y_translatin]

Giving these initial conditions to the reg.exe program will cause the eclipse to initial start at a very good position and scale. The source code can be found in the directory named Part2. The program is called estimate.cxx and the relative CMakeLists.txt is provided for convenience. The program prints out correct initial conditions as far as I have tested and should cause no problem. Below are the results of the program on the images provided. 

Note: The resulting image for the translated image would suddenly disappear at the last iteration, but I could see the eclipse fitted correctly through the run. I couldn't get a snapshot while the program was running.

Part 3:

The program starts off the hierarchical fitting with sigma=15. With sigma=25, my computer executed very very slowly towards the end and I couldn't see the actual images. So my program starts with sigma=15 and at each level decreases sigma by step=2. Note that the brain images on the display screen are not seen as smoothed differently. I did not change the display image but the registration does it's computation on the newly less blurred image and all computations are right. Also, while running the program in order to go to the next level you must press "q" in order to finish the previous level.