Problem Set II Solution

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  1. (30%) Euler’s method: For dydt + 2y = 2 e 4t; y(0) = 1,

    1. Derive its closed-form solution.

    1. 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).

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

  1. (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.


dvdtt = K[(Dvt)T vt + div(vtvtT )];

d t

= vt t:


dvdtt = K[(Dvt)T vt + div(vtvtT )];

d t

= D t vt:

Note: Use your code of frequency smoothing in PS1 to implement the smoothing operator K (set the truncated number of frequency as 162).

  1. 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.


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

  • Initial velocity eld v0 and source image are included in the data folder. The initial trans-formation 0 is an image coordinate, which can be easily generated from MATLAB.

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