Description

(30%) Euler’s method: https://en.wikipedia.org/wiki/Euler_method For ^{dy}_{dt} + 2y = 2 e ^{4t}; y(0) = 1,


Derive its closedform solution.



Use Euler’s Method to nd the approximation to the solution at t = f1; 2; 3; 4; 5g, and compare to the exact solution in (a).



Use di erent step size h = f0:1; 0:05; 0:01; 0:005; 0:001g and plot out the approximated function value.


(70%) Geodesic shooting. Implement geodesic shooting by the following two strategies and compare the di erences between the nal transformations _{1} at time point t = 1.
(a)
^{dv}_{dt}^{t} = K[(Dv_{t})^{T} v_{t} + div(v_{t}v_{t}^{T} )];
^{d }^{t}
= v_{t} _{t}:
(b)
^{dv}_{dt}^{t} = K[(Dv_{t})^{T} v_{t} + div(v_{t}v_{t}^{T} )];
^{d }^{t}
= D _{t} v_{t}:
Note: Use your code of frequency smoothing in PS1 to implement the smoothing operator K (set the truncated number of frequency as 16^{2}).

Deform a given source image by using the transformations _{1} obtained from (a) and (b). * Use Euler integration to solve the above ordinary di erential equations.
IMPORTANT NOTES:

Interpolation function: MATLAB function interp2 with the option `spline’.

Initial velocity eld v_{0} and source image are included in the data folder. The initial transformation _{0} is an image coordinate, which can be easily generated from MATLAB.

All results should be clearly reported and discussed in the report.