Lecture 7 Problems

The following two arrays contain the descriptors of features extracted from two images. The descriptors are 2-dimensional (much lower than we would usually use in pracice); each column of the matrix is the descriptor for a feature. In all of the following, use matrix-style (i, j) indexing with indices starting at 1. For example, feature 4 in image 1 has descriptor $\begin{bmatrix}3 & 1\end{bmatrix}^T$.

$$F_1 = \begin{bmatrix} 0 & 1 & 4 & 3 \\ 1 & 0 & 4 & 1 \end{bmatrix}\\ $$

$$F_2 = \begin{bmatrix} 2 & 5 & 1\\ 1 & 5 & 2 \end{bmatrix}$$

1. Create a table with 4 rows and 3 columns in which the $(i,j)$th cell contains the SSD distance between feature $i$ in image 1 and image $j$ in image 2.

2. For each feature in image 1, give the index of the closest feature match in image 2 using to the SSD metric.

3. For each feature in image 2, give the index of the closest feature match in image 1 and the ratio distance between each feature and its closest match.

Suppose you’ve aligned two images using feature matches using a translational motion model; that is, you have a vector $\mathbf{t} = \left[t_x, t_y\right]$​ that specifies the offset of corresponding pixels in image 2 from their coordinates in image 1. We’d like to warp image 2 into image 1’s coordinates and combine the two together using some blending scheme (maybe we’ll average them or something).

4. Give a 3x3 affine transformation matrix that can be used to warp image 2 into image 1’s coordinates.
5. If image 1’s origin is at its top left and $t_x$ and $t_y$​ are both positive, what’s the size of the destination image that can contain the combined image?