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Epipolar lines. epipolar plane. epipolar lines. epipolar lines. Baseline. O’. O. Rectification.

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Rectification

- Rectification: rotation and scaling of each camera’s coordinate frame to make the epipolar lines horizontal and equi-height,by bringing the two image planes to be parallel to the baseline
- Rectification is achieved by applying homography to each of the two images

Cyclopean coordinates

- In a rectified stereo rig with baseline of length , we place the origin at the midpoint between the camera centers.
- a point is projected to:
- Left image: ,
- Right image: ,
- Cyclopean coordinates:

Disparity

- Disparity is inverse proportional to depth
- Constant disparity constant depth
- Larger baseline, more stable reconstruction of depth (but more occlusions, correspondence is harder)

(Note that disparity is defined in a rectified rig in a cyclopean coordinate frame)

Random dot stereogram

- Depth can be perceived from a random dot pair of images (Julesz)
- Stereo perception is based solely on local information (low level)

Compared elements

- Pixel intensities
- Pixel color
- Small window (e.g. or ), often using normalized correlation to offset gain
- Features and edges (less common)
- Mini segments

Dynamic programming

- Each pair of epipolar lines is compared independently
- Local cost, sum of unary term and binary term
- Unary term: cost of a single match
- Binary term: cost of change of disparity (occlusion)
- Analogous to string matching (‘diff’ in Unix)

Dynamic Programming

- Shortest path in a grid
- Diagonals: constant disparity
- Moving along the diagonal – pay unary cost (cost of pixel match)
- Move sideways – pay binary cost, i.e. disparity change (occlusion, right or left)
- Cost prefers fronto-parallel planes. Penalty is paid for tilted planes

Probability interpretation: Viterbi algorithm

- Markov chain
- States: discrete set of disparity
- Log probabilities: product sum

Probability interpretation: Viterbi algorithm

- Markov chain
- States: discrete set of disparity
- Maximum likelihood: minimize sum of negative logs
- Viterbi algorithm: equivalent to shortest path

Dynamic Programming: Pros and Cons

- Advantages:
- Simple, efficient
- Achieves global optimum
- Generally works well
- Disadvantages:

Dynamic Programming: Pros and Cons

- Advantages:
- Simple, efficient
- Achieves global optimum
- Generally works well
- Disadvantages:
- Works separately on each epipolar line, does not enforce smoothness across epipolars
- Prefers fronto-parallel planes
- Too local? (considers only immediate neighbors)

Markov Random Field

- Graph In our case: graph isa 4-connected gridrepresenting one image
- States: disparity
- Minimize energy of the form
- Interpreted as negative log probabilities

Iterated Conditional Modes (ICM)

- Initialize states (= disparities) for every pixel
- Update repeatedly each pixel by the most likely disparity given the values assigned to its neighbors:
- Markov blanket: the state of a pixel only depends on the states of its immediate neighbors
- Similar to Gauss-Seidel iterations
- Slow convergence to (often bad) local minimum

Graph cuts: expansion moves

- Assume is non-negative and is metric:
- We can apply more semi-global moves using minimal s-t cuts
- Converges faster to a better (local) minimum

α-Expansion

- In any one round, expansion move allows each pixel to either
- change its state to α, or
- maintain its previous state

Each round is implemented via max flow/min cut

- One iteration: apply expansion moves sequentially with all possible disparity values
- Repeat till convergence

α-Expansion

- Every round achieves a globally optimal solution over one expansion move
- Energy decreases (non-increasing) monotonically between rounds
- At convergence energy is optimal with respect to all expansion moves, and within a scale factor from the global optimum:

where

α-Expansion (1D example)

- But what about?

α-Expansion (1D example)

- Such a cut cannot be obtained due to triangle inequality:

Common Metrics

- Potts model:
- Truncated :
- Truncated squared difference is not a metric

Reconstruction with graph-cuts

Original Result Ground truth

A different application: detect skyline

- Input: one image, oriented with sky above
- Objective: find the skyline in the image
- Graph: grid
- Two states: sky, ground
- Unary (data) term:
- State = sky, low if blue, otherwise high
- State = ground, high if blue, otherwise low
- Binary term for vertical connections:
- If the state of a node is sky, the node above should also be sky (set to infinity if not)
- If the state of a node is ground, the node below should also be ground
- Solve with expansion move. This is a binary (two state) problem, and so graph cut can find the global optimum in one expansion move

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