Problem Set II Solution

Description

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

1. 1. Derive its closed-form solution.

1. 2. 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. 3. Use di erent step size h = f0:1; 0:05; 0:01; 0:005; 0:001g and plot out the approximated function value.

2. (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_tv_t^T )];

^{d t}

= v_{t t}:

(b)

^dv_dt^t = K[(Dv_t)^T v_t + div(v_tv_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 16²).

3. 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 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.