CS-544 Optimization in Computer Vision

1. Use the augmented Lagrangian method to find a surface that minimises smoothness cost while interpolating points given below;You should represent the surface as a height map on a 256 x 256 grid, representing the unit square. You should represent the surface as a height map. You should measure the smoothness as the L2 norm of the gradient of the height map. The points to interpolate are: (0, 0, 1); (0, 1/2, 0); (0, 1, 1); (1/2, 0, 0); (1/2, 1/2, 1); (1/2, 1, 0); (1, 0, 1); (1, 1/2, 0); (1, 1, 1). You should use a second order method to do the inner optimization (fminunc) but be careful - you don't want to build a (256x256) x (256x256) Hessian!
2. Of course, the solution to this problem can be obtained by solving a linear system. Compare this solution with the solution obtained above.
3. Now construct a solution where smoothness is measured using the L1 norm. This is much trickier.
4. Now construct a surface of minimum area that interpolates these points and the line segments on the grid joining them (I'll draw this in class). Use the augmented lagrangian method, and plot the surface you obtain.
5. Now minimize the surface area of the interpolating surface that interpolates only the vertices - you should get an odd surface. Plot this.

This is a broad MP intended to be educational.