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

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Epipolar lines
Epipolar lines

epipolar plane

epipolar lines

epipolar lines

Baseline

O’

O


Rectification
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


Rectification1
Rectification

Baseline

O’

O


Cyclopean coordinates
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

  • 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
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
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
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)


String matching
String matching

  • Swing → String

S t r i n g

Start

S w i n g

End


String matching1
String matching

  • Cost: #substitutions + #insertions + #deletions

S t r i n g

S w i n g


Dynamic programming1
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


Dynamic programming2
Dynamic Programming

Start

,

Complexity?


Probability interpretation viterbi algorithm
Probability interpretation: Viterbi algorithm

  • Markov chain

  • States: discrete set of disparity

  • Log probabilities: product sum


Probability interpretation viterbi algorithm1
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
Dynamic Programming: Pros and Cons

  • Advantages:

    • Simple, efficient

    • Achieves global optimum

    • Generally works well

  • Disadvantages:


Dynamic programming pros and cons1
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
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
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
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


Expansion
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


Expansion1
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






Expansion 1d example4
α-Expansion (1D example)

  • But what about?





Expansion 1d example8
α-Expansion (1D example)

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


Common metrics
Common Metrics

  • Potts model:

  • Truncated :

  • Truncated squared difference is not a metric


Reconstruction with graph cuts
Reconstruction with graph-cuts

Original Result Ground truth


A different application detect skyline
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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