- difficulties?
- questions?
- Did you try other averaging methods?
- Bilateral Filter?

- Recap
- Panoramas,Mosaics
- Some cool papers
- Warping
- Homographies
- Lab 3

take 2 or more images and combine them to get a "better one"

idea: capture multiple low-res (LR) images and fuse them into a single super-resolved (SR) image

use highest gradient

Confocal Stereo [Hasinoff and Kutulakos 2007]

- Compact Camera FOV = 50 x 35°
- Human FOV = 200 x 135°
- Panoramic Mosaic = 360 x 180°

- 4 correspondances
- reproject one on the other

- Automatic correspondances: Feature detector
- Projection in a different space
- Blending

same center of projection!

do we need geometry of the two planes in respect to the eye?

but what kind of transformation?

but what kind of transformation?

Projective – mapping between any two projection planes with the same center of projection

- rectangle should map to arbitrary quadrilateral
- parallel lines aren’t
- but must preserve straight lines
- same as: project, rotate, reproject

called Homography

$\begin{bmatrix}wx'\\wy'\\w\end{bmatrix} = \begin{bmatrix}* & * & *\\* & * & *\\* & * & *\end{bmatrix} \begin{bmatrix}x\\y\\1\end{bmatrix}$

$p' = Hp$

Initialize with sift images, compute Wrap between images

as 2D matrices (translation cannot be expressed as a 2D matrix)

These transformations are a nested set of groups Closed under composition and inverse is a member

Given a coordinate transform (x’,y’) = T(x,y) and a source image f(x,y), how do we compute a transformed image g(x’,y’) = f(T(x,y))?

Backward Mapping eliminate holes

Needs a invertible wrap function: Not always possible

- Input correspondences at key feature points
- Define a triangular mesh over the points
- Same mesh in both images!
- Warp each triangle separately from source to destination (affine)

- Create an intermediate shape (by interpolation)
- Warp both images towards it
- Cross-dissolve the colors in the newly warped images
- $Image_{halfway} = (1-t)*Image_1 + t*image_2$

same center of projection!

$\begin{bmatrix}wx'\\wy'\\w\end{bmatrix} = \begin{bmatrix}* & * & *\\* & * & *\\* & * & *\end{bmatrix} \begin{bmatrix}x\\y\\1\end{bmatrix}$

$p' = Hp$

PP3 is a projection plane of both centers of projection, so we are OK

- Find the homography H given a set of p and p’ pairs
- How many correspondences are needed?
- Tricky to write H analytically, but we can solve for it!

$\begin{bmatrix}wx'\\wy'\\w\end{bmatrix} = \begin{bmatrix}a & b & c\\d & e & f\\g & h & i\end{bmatrix} \begin{bmatrix}x\\y\\1\end{bmatrix}$

9 unknowns and $w'$

$w'$ is easy: $w' = gy + hx + i$

Set up a system of linear equations:

$Ah = b$

where vector of unknowns $h = [a,b,c,d,e,f,g,h]^T$

Need at least 8 eqs, (setting $i=1$)

Solve for h. If overconstrained, solve using least-squares

$min\|Ah-b\|^2$

Due: 10 November

- Warping with Homographies
- Compute homographies
- Manual Mosaic
- Blending?

These slides have been prepared with materials, slides, and discussions from the following.

Fredo Durand, Alexei Efros, Videos and papers from the authors