Computer Vision
Exercise 4 – Labelling Problems
Outline
1. Energy Minimization (example – segmentation) 2. Iterated Conditional Modes
3. Dynamic Programming 4. Block-wise ICM
5. MinCut
6. Equivalent transformations + 𝛼-expansion 7. Row-wise stereo (Cyclopean view)
8. Assignments:
a) Segmentation b) Stereo
Energy Minimization (Segmentation)
Energy Minimization
Iterated Conditional Modes
Idea: choose (locally) the best label for the fixed rest [Besag, 1986]
Repeat:
extremely simple, parallelizable
coordinate-wise optimization, does not converge to the global optimum even for very simple energies
Dynamic Programming
Suppose that the image is one pixel high → a chain
The goal is to compute
Dynamic Programming – example
Dynamic Programming
General idea – propagate Bellman functions by
The Bellman functions represent the quality of the best expansion onto the processed part.
Dynamic Programming (algorithm)
is the best predecessor for -th label in the -th node Time complexity –
Iterated Conditional Modes (again, but now 2D)
Fix labels in all nodes but for a chain (e.g. an image row) Before (simple)
→
The “auxiliary” task is solvable exactly and efficiently by DP
MinCut
MinCut for Binary Energy Minimization
Search techniques
𝛼-expansion
𝛼-expansion
After 𝛼-expansion we have but we need ↓
in order to transform it further to MinCut.
What to do ?
Equivalent Transformation (re-parameterization)
Two tasks and are equivalent if
holds for all labelings .
− equivalence class (all tasks equivalent to ).
Equivalent transformation:
Equivalent Transformation
Equivalent transformation can be seen as “vectors”, that satisfy certain conditions:
Back to 𝛼 -expansion
Remember out goal:
It can be done by equivalent transformations.
Row-wise stereo
Pixel of the left image should be labelled by disparity values:
Constraint: 𝑑 𝑖 + 1 ≤ 𝑑 𝑖 − 1
Row-wise stereo (Cyclopean view)
Symmetric definition – transform the “coordinates”
We are searching for a 4-connected path.
Assignments
1. Segmentation:
a) Binary Segmentation with the Ising Model – MinCut.
b) Possible extensions: multi-label segmentation (with ICM, DP, 𝛼-expansion [1]), more complex appearance models [2],
contrast dependent edge potentials [3].
2. Stereo:
a) Block Matching, row-wise stereo.
b) Possible extensions: row-wise Iterated Conditional Mode, more complex data-terms (e.g. Normalized Cross-
Correlation), global solutions by MinCut [4], approximate solutions with 𝛼-expansion [1], re-parameterization [4].
a) – implemented (can be used as a template), b) – assignments Deadline – 07.02.2014 per e-mail an Dmytro.Shlezinger@...
Literature
[1] Boykov, Veksler, Zabih: Fast Approximate Energy Minimization via Graph Cuts. 2001
[2] Rother, Kolmogorov, Blake: “GrabCut” – Interactive Foreground Extraction using Iterated Graph Cuts. 2002
[3] Boykov, Jolly: Interactive Graph Cuts for Optimal Boundary &
Region Segmentation of Objects in N-D Images. 2001 [4] Ask me …
