1. Homework 4 Notes Connelly Barnes COS 323

2. MATLAB Semicolon at endline suppresses output. >> [1 2] ans = 1 2 >> [1 2];

3. Images imread(filename) reads image as: –h x w (greyscale) or –h x w x 3 (color) –uint8, values in [0, 255]. Do alignment in greyscale, doubles. A=double(imread(filename)); if size(A,3)==3 A = rgb2gray(A); end imshow(A) with double A assumes colors are in [0, 1]. So divide by 255 before imshow() at some point.

4. Image Transform imtransformSimple(A, [x y θ]) => Transformed image with same size as A.

5. Image Transform >> imshow(A);

6. Image Transform >>imshow(imtransformSimple(A,[100 0 pi/4]));

7. Parts of Assignment Paradigm: Write an error function, then minimize that function. Part 1: Image difference –F(x, y, theta) => Error –Want to minimize error. Part 2: Minimization –Minimizing a function of n variables is a well-studied problem, general solutions are known.

8. Image Difference Find sum squared difference on region where images overlap.

9. Image Difference Overlap region image 1's region. So in image 1's coordinates, can represent overlap region with a mask, 1 = overlap, 0 = no overlap.

10. Mask

11. If A, B are pair of images, then imtransformSimple(B, [x y θ]) transforms image B. To find mask, transform array of all ones with same parameters.

12. Image Difference Now follow directions of assignment: use mask to find sum squared difference. Use.* (element-wise product) operator, not * (matrix product) when multiplying mask against an image.

13. Minimization Given F(x, y, θ) (F: R 3 → R). Find vector [x y θ] minimizing F. If smooth, can find local minimum by general algorithm. So treat F as a black box and solve minimization independently of the previous part. Golden Section search finds local minimum of 1D function. Do Golden Section search for dimensions i=1, 2, 3, then repeat => taxicab minimization.

14. Minimization For convenience, loop over the dimension i that you're minimizing via Golden Section search (write F as a vector function). Extra credit: plot [x, y, θ] points that the algorithm searches over in 3D (use plot3() ), observe zig-zagging of taxicab minimization.

15. Merge Results Use mergeImages(A, B, [x y θ]) to get merged image. Use color.

16. Caveats Minimization finds a local minimum. Not necessarily the global minimum. x F Local Global

17. Solution Start with a good guess (gets full credit on this assignment). If F is quadratic, can solve exactly via a linear system. Not so in our case. General solution is to start with many random guesses (extra credit).