Homework 4 Notes

1
Homework 4 Notes

• Connelly Barnes
• COS 323

2
MATLAB

• Semicolon at endline suppresses output.
• gtgt 1 2
• ans
• 1 2
• gtgt 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.
• Adouble(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 ?) gt
• Transformed image with same size as A.

5
Image Transform

• gtgt imshow(A)

6
Image Transform

• gtgtimshow(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) gt 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
Mask

• If A, B are pair of images, then
• imtransformSimple(B, x y ?) 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, ?) (F R3 ? R).
• Find vector x y ? 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 i1, 2,
  3, then repeat gt 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, ? 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 ?) to get merged
  image. Use color.

16
Caveats

• Minimization finds a local minimum.
• Not necessarily the global minimum.

F
Local
Global
x
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).