Computer Vision I - Algorithms and Applications: Discrete ...ds24/lehre/cv1_ws_2013/VL10.pdf · Computer Vision I - Algorithms and Applications: Discrete Labelling Carsten Rother

Computer Vision I -Algorithms and Applications:

Discrete Labelling

Carsten Rother

Computer Vision I: Discrete Labelling03/02/2014

Roadmap this lecture (chapter B.4-5 in book)

• Define: Markov Random Fields

• Formulate applications as discrete labeling problems

• Discrete Optimization: • Pixels-based: Iterative Conditional Mode (ICM),

Simulated Annealing• Line-based: Dynamic Programming (DP)• Field-based: Graph Cut and Alpha-Expansion

03/02/2014 Computer Vision I: Discrete Labelling 2







Machine Learning: Big Picture


”Normal” Machine Learning:

f : Z N (classification) f : Z R (regression)

Input: Image, textOutput: real number(s)

f : Z X

Input: Image , textOutput: complex structure object

(labelling, parse tree) Parse tree of a sentence

Image labellingChemical structure

Structured Output Prediction:

Structured Output Prediction


Ad hoc definition (from [Nowozin et al. 2011])

Data that consists of several parts, and not only the parts themselves contain information, but also the way in which the parts belong together.

Graphical models to capture structured problems


Write probability distribution as a Graphical model:

• Directed graphical model (also called Bayesian Networks)• Undirected graphical model (also called Markov Random Field)• Factor graphs (which we will use predominately)

• A visualization to represent a family of distributions• Key concept is conditional independency• You can convert any concrete distribution from one

visualization to another

Basic idea:

References:

- Pattern Recognition and Machine Learning [Bishop ‘08, chapter 8]

- several lectures at the Machine Learning Summer School 2009 (see video lectures)- Computer Vision 2 and Intelligente Systeme

Factor Graphs

• 𝑃(𝒙) is defined by a factor graph as follows:


𝑓: partition function

𝐹: Factor

𝔽: Set of all factors

𝑁(𝐹):Neighbourhood of a factor

𝜓𝐹: function (not distribution) depending on 𝒙𝑁(𝐹)

𝑃 𝒙 =1

𝑓

𝐹∈𝔽

𝜓𝐹(𝒙𝑁 𝐹 ) where 𝑓 = 𝒙 𝐹∈𝔽𝜓𝐹(𝒙𝑁 𝐹 )

Factor Graphs - example


Defines an unobserved variable

means that these variables are in one factor

Defines an observed variable

𝑃 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5 =1

𝑓𝜓 𝑥1, 𝑥2, 𝑥4 𝜓 𝑥2, 𝑥3 𝜓 𝑥3, 𝑥4 𝜓 𝑥4, 𝑥5 𝜓 𝑥4

𝑥1 𝑥2

𝑥3𝑥4

𝑥5

Introducing energies


𝑃 𝒙 =1

𝑓 𝐹∈𝔽𝜓𝐹(𝒙𝑁 𝐹 ) =

1

𝑓 𝐹∈𝔽 exp{−𝜃𝐹(𝒙𝑁 𝐹 )} =

1

𝑓exp{ −

𝐹∈𝔽𝜃𝐹 𝑥𝑁 𝐹 } = 1

𝑓exp{ −𝐸(𝒙) }

The energy 𝐸 𝒙 is just a sum of factors:

E 𝒙 =

𝐹∈𝔽

𝜃𝐹 𝑥𝑁 𝐹

The most likely solution 𝑥∗ is reached by minimizing the energy:𝑥∗ = 𝑎𝑟𝑔𝑚𝑎𝑥𝑥 𝑃 𝑥𝑥∗ = 𝑎𝑟𝑔𝑚𝑖𝑛𝑥 𝐸(𝑥)

(since it is: − log𝑃 𝑥 = log 𝑓 + 𝐸 𝒙 = constant + 𝐸(𝒙) )

Gibbs Distribution


𝑃 𝑥 =1


E 𝒙 =

𝐹∈𝔽

𝜃𝐹 𝑥𝑁 𝐹

Is a so-called Gibbs distribution or Boltzmann Distribution

with energy 𝐸


Definition: Order

arity 3 arity 2

Definitions:

• Order: the arity (number of variables) of the largest factor

• Markov Random Field: Random Field with low-order factors

• Note, “cliques” are defined for undirected graphical models, which we don’t use here, we use the name “factors”. (cliques and factors are slightly different)

𝑥1 𝑥2

𝑥3𝑥4

𝑥5

Factor graph with order 3

𝐸 𝑥 = 𝜃 𝑥1, 𝑥2, 𝑥4 + 𝜃 𝑥2, 𝑥3 + 𝜃 𝑥3, 𝑥4 + 𝜃 𝑥5, 𝑥4 + 𝜃(𝑥4)

arity 1

Two examples


𝑃1 𝑥1, 𝑥2, 𝑥3 =1

𝑓exp{ 𝑥1𝑥2 + 𝑥2𝑥3 + 𝑥1𝑥3}

𝑃2 𝑥1, 𝑥2, 𝑥3 =1

𝑓exp{𝑥1𝑥2𝑥3}

We always tray to write distributions as “factorized” as possible

Note 𝑃2 cannot be written as a sum of pairwise energy terms

The family view of distributions


Family of all distributions

Family of all distributions with same 𝑈𝑛𝑑𝑖𝑟𝑒𝑐𝑡 𝐺𝑟𝑎𝑝ℎ𝑖𝑐𝑎𝑙 𝑀𝑜𝑑𝑒𝑙𝑠

Realization of a distribution

𝑃1 𝑥1, 𝑥2, 𝑥3 =1

𝑓exp{ 𝑥1𝑥2 + 𝑥2𝑥3 + 𝑥1𝑥3}

𝑃1𝐹 𝑥1, 𝑥2, 𝑥3 =

1

𝑓( 𝜓 𝑥1, 𝑥2 𝜓 𝑥2, 𝑥3 𝜓 𝑥1, 𝑥3 )

𝑃12𝑈 𝑥1, 𝑥2, 𝑥3 =

1

𝑓𝜓 𝑥1, 𝑥2, 𝑥3

𝑃2 𝑥1, 𝑥2, 𝑥3 =1

𝑓exp{𝑥1𝑥2𝑥3}

𝑃2𝐹 𝑥1, 𝑥2, 𝑥3 =

1

𝑓𝜓 𝑥1, 𝑥2, 𝑥3

Family of all distributions with same 𝑓𝑎𝑐𝑡𝑜𝑟 𝑔𝑟𝑎𝑝ℎ

this is known as a triple clique

𝑃1 𝑃2

𝑃1𝐹 𝑃2

𝐹𝑃12𝑈

What to infer?

• MAP inference (Maximum a posterior state):

𝑥∗ = 𝑎𝑟𝑔𝑚𝑎𝑥𝑥 𝑃 𝑥 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑥𝐸 𝑥

• Probabilistic Inference, so-called marginals:𝑥𝑖∗ = 𝑎𝑟𝑔𝑚𝑎𝑥𝑥𝑖 𝑃 𝑥𝑖

(where 𝑃 𝑥𝑖 = 𝑘 = 𝒙 | 𝑥𝑖=𝑘 𝑃(𝑥1, … 𝑥𝑖 = 𝑘,… , 𝑥𝑛)


Example: MAP versus Marginals


Input Image Ground Truth Labeling

MAP solution (each pixel has 0,1 label)

marginals solution (each pixel has a probability

between 0 and 1)

This lecture: Discrete Inference in Order 2 models


𝑃 𝑥 =1


E 𝒙 =

𝑖

𝜃𝑖 𝑥𝑖 +

𝑖,𝑗

𝜃𝑖𝑗 𝑥𝑖 , 𝑥𝑗 +

𝑖,𝑗,𝑘

𝜃𝑖,𝑗,𝑘 𝑥𝑖 , 𝑥𝑗 , 𝑥𝑘 + …

Gibbs distribution

Unary terms Pairwise terms Higher-order terms

MAP inference: 𝑥∗ = 𝑎𝑟𝑔𝑚𝑎𝑥𝑥 𝑃 𝑥 = 𝑎𝑟𝑔𝑚𝑖𝑛𝑥𝐸 𝑥

Label space: binary 𝑥𝑖 ∈ 0,1 or multi-label 𝑥𝑖 ∈ 0,… , 𝐾

We only look at energies with unary and pairwise factors







Examples: Order


4-connected; pairwise MRF

Higher-order RF

𝐸(𝑥) = 𝜃𝑖𝑗 (𝑥𝑖, 𝑥𝑗)𝑖, 𝑗 Є𝑁4

higher(8)-connected; pairwise MRF

Order 2 Order 2 Order n

𝐸(𝑥) = 𝜃𝑖𝑗 (𝑥𝑖, 𝑥𝑗)

“Pairwise energy” “higher-order energy”

𝐸(𝑥) = 𝜃𝑖𝑗 (𝑥𝑖, 𝑥𝑗)𝑖, 𝑗 Є𝑁8 𝑖, 𝑗 Є𝑁4

+𝜃(𝑥1, … , 𝑥𝑛)

Stereo Matching


d=4

d=0

Ground truth depthImage – left(a) Image – right(b)

• Images rectified• Ignore occlusion

(see last lecture / exercise for occlusion handling)

𝐸(𝒅): {0, … , 𝐷}𝑛 → 𝑅

Energy:

Labels: 𝑑𝑖 disparity (shift) of pixel 𝑖

𝑑𝑖

Label only left image

Stereo Matching - Energy


• Unary terms (see last lecture)Patch-Cost for a pixel i with disparty 𝑑𝑖is:

𝜃𝑖 𝑑𝑖 =

𝑗∈𝑁𝑖

𝐼𝑗𝑙 − 𝐼𝑗−𝑑𝑖𝑟2

Left: 𝐼𝑙

E 𝒅 =

𝑖

𝜃𝑖 𝑑𝑖 +

𝑖,𝑗

𝜃𝑖𝑗 𝑑𝑖 , 𝑑𝑗

Right: 𝐼𝑟

𝑑𝑖

𝜃𝑖 (𝑑𝑖)

𝜃𝑖𝑗 (𝑑𝑖, 𝑑𝑗)

Stereo Matching Energy - Smoothness


[Olga Veksler PhD thesis, Daniel Cremers et al.]

|𝑑𝑖 − 𝑑𝑗|co

stNo truncation(global min.)

𝜃𝑖𝑗 𝑑𝑖 , 𝑑𝑗 = |𝑑𝑖 − 𝑑𝑗|

𝑑𝑖

𝜃𝑖 (𝑑𝑖)

𝜃𝑖𝑗 (𝑑𝑖, 𝑑𝑗)

Stereo Matching Energy - Smoothness


discontinuity preserving potentials[Blake&Zisserman’83,’87]

cost

No truncation(global min.)

with truncation(NP-hard optimization)

|𝑑𝑖 − 𝑑𝑗|

𝜃𝑖𝑗 𝑑𝑖 , 𝑑𝑗 = min( 𝑑𝑖 − 𝑑𝑗 , 𝜏)

Stereo Matching: Simplified Random Fields


No MRF; Block MatchingPixel independent (WTA)

No horizontal links Efficient since independent chains

Ground truthPairwise MRF[Boykov et al. ‘01]

Image Segmentation – Reminder first lecture

03/02/2014 24

𝜃𝑖𝑗(𝑥𝑖 , 𝑥𝑗)

𝑥𝑗

𝜃𝑖(𝑥𝑖)𝑥𝑖

E 𝒙 =

𝑖

𝜃𝑖 𝑥𝑖 +

𝑖,𝑗

𝜃𝑖𝑗 𝑥𝑖 , 𝑥𝑗

Binary Label: 𝑥𝑖 ∈ {0,1}

Computer Vision I: Discrete Labelling

Unary term - Reminder

03/02/2014 25

Optimum with unary terms onlyDark means likely

backgroundDark means likely

foreground

𝜃𝑖(𝑥𝑖 = 0)𝜃𝑖(𝑥𝑖 = 1)

New query image 𝑧𝑖


Pairwise term - Reminder

03/02/2014 26

Most likely Most likely Intermediate likely

“Ising Prior”

most unlikely

This models the assumption that the object is spatially coherent

𝜃𝑖𝑗 𝑥𝑖 , 𝑥𝑗 = |𝑥𝑖 − 𝑥𝑗|

When is 𝜃𝑖𝑗(𝑥𝑖 , 𝑥𝑗) small, i.e. likely configuration ?


Texture Synthesis


Input

Output

[Kwatra et. al. Siggraph ‘03 ]

b

a

O

1

E 𝒙 =

𝑖

𝜃𝑖 𝑥𝑖 +

𝑖,𝑗




𝜃𝑖 𝑥𝑖

𝜃𝑖𝑗 𝑥𝑖 , 𝑥𝑗 = 0 𝑖𝑓 𝑥𝑖 = 𝑥𝑗

𝜃𝑖 𝑥𝑖 = ∞ if an image does not exist at pixel 𝑖, otherwise 0

Texture Synthesis


Input

Output

[Kwatra et. al. Siggraph ‘03 ]

b

a

O

1

E 𝒙 =

𝑖

𝜃𝑖 𝑥𝑖 +

𝑖,𝑗




𝜃𝑖 𝑥𝑖

a

b

a

b

𝑖 𝑗 𝑖 𝑗

Good case: Bad case:

𝐸(𝒙) = |𝑥𝑖 − 𝑥𝑗| [ |𝑎𝑖 − 𝑏𝑖| + |𝑎𝑗 − 𝑏𝑗| ]𝑖, 𝑗 Є𝑁4

Face Freeman Image Quilting


You can choose this as well as exercise

[A. Efros and W. T Freeman, Image quilting for texture synthesis and transfer, SIGGRAPH 2001]

• Unary term matches “dark” rice pixels to dark face pixels

• Place source image at random positions on the output canvas

• You can also use dynamic program (see article)

source image

Output canvas

Panoramic Stitching


Use identical energy

Panoramic Stitching


Use identical energy

Interactive Digital Photomontage


[Agarwala et al., Siggraph 2004]

Interactive Digital Photomontage


[Agarwala et al., Siggraph 2004]

Video Synthesis


OutputInput

Video

Video (duplicated)

A 3D labeling problem

Same pairwise terms but now in 𝑥, 𝑦, and 𝑡(time)-direction

Image Retargeting

http://swieskowski.net/carve/


http://swieskowski.net/carve/

Image Retargeting


Image Retargeting


E 𝒙 =

𝑖

𝜃𝑖 𝑥𝑖 +

𝑖,𝑗




𝜃𝑖 𝑥𝑖

Force label 0

Force label 1

Cut (sketched)

Label 0 Label 1

𝜃𝑖𝑗 𝑥𝑖 , 𝑥𝑗 = Ii − Ij simple gradient measure

Extension – Scene Carving


[Mansfield, Gehler, Van Gool, Rother, ECCV ‘10]

Image Hand drawn depth-ordering

Normal seam carving Scene carving

Examples: Order



Higher-order RF







+𝜃(𝑥1, … , 𝑥𝑛)

Avoid Discretization artefacts


[Boykov et al. ‘03, ‘05]

Larger connectivity can model true Euclidean length (also other metric possible)

Eucl.

Length of the paths:

4-con.

5.65

8

1

8-con.

6.28

6.28

5.08

6.75

Assume 4-connected grid.Can you choose edge weights in such a way that the dark yellow and blue segmentation have different length?

Avoid Discretization artefacts

[Boykov et al. ‘03, ‘05]

4-connected Euclidean

8-connected Euclidean (MRF)

8-connected geodesic (CRF)


Examples: Order



Higher-order RF







+𝜃(𝑥1, … , 𝑥𝑛)

Advanced Object recognition


• Many other examples: ObjCut Kumar et. al. ’05; Deformable Part Model Felzenszwalb et al.; CVPR ’08; PoseCut Bray et al. ’06, LayoutCRF Winn et al. ’06

• Maximizing / Marginalzation over hidden variables

“parts”

“instancelabel”

“instance”

[LayoutCRF Winn et al. ’06]







Inference – Big Picture … for CV2


• Combinatorial Optimization

• Binary, pairwise MRF: Graph cut, BHS (QPBO)

• Multiple label, pairwise: move-making; transformation

• Binary, higher-order factors: transformation

• Multi-label, higher-order factors: move-making + transformation

• Dual/Problem Decomposition

• Decompose (NP-)hard problem into tractable once.Solve with e.g. sub-gradient technique

• Local search / Genetic algorithms

• ICM, simulated annealing


Inference – Big Picture … for CV2

• Message Passing Techniques

• Methods can be applied to any model in theory(higher order, multi-label, etc.)

• DP, BP, TRW, TRW-S

• LP-relaxation

• Relax original problem (e.g. {0,1} to [0,1]) and solve with existing techniques (e.g. sub-gradient)

• Can be applied any model (dep. on solver used)

• Connections to message passing (TRW) and combinatorial optimization (QPBO)

Also in CV2: continuous variables (we have seen PatchMatch from last lecture), continuous domain; Gaussian conditional random fields

Function Minimization: The Problems


• Which functions are exactly solvable?

• Approximate solutions of NP-hard problems

Function Minimization: The Problems


• Which functions are exactly solvable?Boros Hammer [1965], Kolmogorov Zabih [ECCV 2002, PAMI 2004] , Ishikawa [PAMI 2003], Schlesinger [EMMCVPR 2007], Kohli Kumar Torr [CVPR2007, PAMI 2008] , Ramalingam Kohli Alahari Torr [CVPR 2008] , Kohli Ladicky Torr [CVPR 2008, IJCV 2009] , Zivny Jeavons [CP 2008]

• Approximate solutions of NP-hard problemsSchlesinger [1976 ], Kleinberg and Tardos [FOCS 99], Chekuri et al. [2001], Boykov et al. [PAMI 2001], Wainwright et al. [NIPS 2001], Werner [PAMI 2007], Komodakis [PAMI 2005], Lempitsky et al. [ICCV 2007], Kumar et al. [NIPS 2007], Kumar et al. [ICML 2008], Sontag and Jakkola [NIPS 2007], Kohli et al. [ICML 2008], Kohli et al. [CVPR 2008, IJCV 2009], Rother et al. [2009]

ICM - Iterated conditional mode


Gibbs Energy:

𝑥2

𝑥1𝑥3 𝑥4

𝑥5

𝐸 𝒙 = 𝜃12 𝑥1, 𝑥2 + 𝜃13 𝑥1, 𝑥3 +𝜃14 𝑥1, 𝑥4 + 𝜃15 𝑥1, 𝑥5 +⋯



𝑥2

𝑥1𝑥3 𝑥4

𝑥5

Idea: fix all variable but one and optimize for this one

Selected 𝑥1 and optimize:

Gibbs Energy:

• Can get stuck in local minima• Depends on initialization

ICM Global min

𝐸 𝒙 = 𝜃12 𝑥1, 𝑥2 + 𝜃13 𝑥1, 𝑥3 +𝜃14 𝑥1, 𝑥4 + 𝜃15 𝑥1, 𝑥5 +⋯

𝐸′ 𝒙 = 𝜃12 𝑥1, 𝑥2 + 𝜃13 𝑥1, 𝑥3 +𝜃14 𝑥1, 𝑥4 + 𝜃15 𝑥1, 𝑥5

ICM - parallelization


• The schedule is a more complex task in graphs which are not 4-connected

Normal procedure:

Step 1 Step 2 Step 3 Step 4

Parallel procedure:

Step 1 Step 2 Step 3 Step 4


Extension / related techniques


• Simulated annealing (next)• Block ICM (see exercise)• Gibbs sampling• Lazy Flipper: MAP Inference in Higher-Order Graphical Models by Depth-limited

Exhaustive Search [Bjoern Andres, Joerg H. Kappes, Ullrich Koethe, Fred A. Hamprecht]

Reminder: Simulated annealing (Lecture 10)


Gibbs Distribution with a temperature 𝑇: 𝑃𝑇 𝑥 =1

𝑓exp{ −𝐸(𝒙)/𝑇 }

Basic Idea: Start optimization with high temperature and find good initial solution and then go to low temperatures.

Function: –𝐸

Label 𝑥(for illustration: One

Pixel with Label ranges from 0 to 40)

(original distribution)Label 𝑥 Label 𝑥

Label 𝑥 Label 𝑥

Simulated annealing (correct)


Algorithm.Choose an annealing schedule, e.g. (𝑡1, … , 𝑡𝑚) = (100,100,…,1,1,1)Initialize all variables 𝒙

Compute current highest probability: 𝑃𝑏𝑒𝑠𝑡 = 𝑃𝑡1(𝒙)

For 𝑗 = 1…𝑚{

For a random selection of variables 𝑥𝑖{

select a new sample 𝑘 for 𝑥𝑖 // simple sample procedure next

if 𝑃𝑡𝑗 𝒙 ≥ 𝑃𝑏𝑒𝑠𝑡

{accept 𝑥𝑖 = 𝑘

𝑃𝑏𝑒𝑠𝑡 = 𝑃𝑡𝑗 𝒙

}}

}

Gibbs Distribution with a temperature 𝑇: 𝑃𝑇 𝑥 =1

𝑓exp{ −𝐸(𝒙)/𝑇 }

How to sample a label for one pixel?

03/02/2014 55

How to sample from a general discrete probability distribution of one variable 𝑝(𝑥), 𝑥 ∈ {0,1,… , 𝑛}?

1. Define “intervals” whose lengths are proportional to 𝑝(𝑥)

2. Concatenate these intervals

3. Sample into the composed interval uniformly

4. Check, in which interval the sampled value falls in.

Below is an example for 𝑝 𝑥 ∝ {1,2,3} (three values).


1 2 3

Can sometimes work surprisingly well !


Ground Truth Input (20x50 pixels)

0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2

0.2 0.2 0.2 0.2 0.2

MRF: 80 connectivity - illustrationMRF has 20x50 nodes (pixels)

5x5 blur kernel

Binary image de-convolution:

Can sometimes work surprisingly well !


Ground Truth QPBO 80% unlab. (0.1sec)Input

ICM E=6 (0.03sec)QPBOP 80% unlab. (0.9sec) GC E=999 (0sec)

BP E=71 (0.9sec) QPBOP+BP+I, E=8.1 (31sec) Simulated annealingE=0 (1.3sec)







What is dynamic programming?

dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It will examine all possible ways to solve the problem and will find the optimal solution.


[From Wikipedia]

Dynamic Programming on chains


No MRF; Block MatchingPixel independent (WTA)

No horizontal links Efficient since independent chains

Ground truthPairwise MRF[Boykov et al. ‘01]



qp r

• Pass messages from left to right • Message is a vector with 𝐾 entries (𝐾 is the amount of labels)• Read out the solution from final message and final unary term• globally exact solution• Other name: min-sum algorithm

𝑀𝑞→𝑟

E 𝒙 =

𝑖

𝜃𝑖 𝑥𝑖 +

𝑖,𝑗∈𝑁2


Unary terms pairwise terms in a row

𝑀𝑝→𝑞 𝑀𝑟→𝑠𝑀𝑜→𝑝



qp r

𝑀𝑞→𝑟

Define the message:

𝑀𝑞→𝑟 k = minx{Mp→q x + 𝜃𝑞 𝑥 + 𝜃𝑞,𝑟(𝑥, 𝑘)}

Information from previous nodes

Local information

Connection to next node

Message stores the energy up to this point for 𝑥𝑟 = 𝑘:

𝑀𝑞→𝑟 k = min𝑥1…𝑞,𝑥𝑟=𝑘

E 𝑥1, … , 𝑥𝑞 , 𝑥𝑟 = 𝑘

Dynamic Programming on chains - example


Dynamic Programming on trees


Treestructure

root

Example: Part-based object recognition

[Felzenschwalb, Huttenlocher ‘01]

Mes

sage

s

Extensions (see CV2)

• Can be done with marginal: sum-product algorithm

• Can be done on fields: Belief Propagation

• Can be done in higher-order factor graphs

• Speed up trick with distance transforms

• Shortest path in a graph can also be done with dynamic programming


Chain as shortest path General shortest path Variables x

Label

Dynamic Programming in vision – 2 scenarios

• The two dimensions are pixels (on chain) and labels:

• Stereo

• Many other applications

• The two dimensions are pixel (x-direction) and pixels (y-direction)• Segmentation with Intelligent Scissors

[Mortenson et al. Siggraph 95] in Gimp, Adobe Photoshop, etc.

• Image Retargeting

• Image stitching also possible but rarely done

• Border matting [Rother et. al. Siggraph ’04]


Image Retargeting


E 𝒙 =

𝑖

𝜃𝑖 𝑥𝑖 +

𝑖,𝑗




𝜃𝑖 𝑥𝑖

Force label 0

Force label 1

labeling (sketched)

Label 0 Label 1

𝜃𝑖𝑗 𝑥𝑖 , 𝑥𝑗 = Ii − Ij simple gradient measure

Path from top to bottom

Binary labeling

In this case the problem can be represented in 2 ways: labeling or path finfing

You can do that as an exercise, please see details in: http://www.merl.com/reports/docs/TR2008-064.pdf







Move Making Algorithms


Search Neighbourhood

Current Solution

Optimal Move

Solution Space

Ener

gy

Solution space 𝐾 𝒙 , where 𝐾 is number of labels and 𝒙 number of pixels

Move-Making Algorithms

• ICM: only local (pixels), move space: 𝐾

• DP: only optimal on chains, move space 𝐾|𝒙𝑙𝑖𝑛𝑒|

• Graph Cut (next), move space: 2|𝒙|


Space of Solutions (x) : 𝐾𝑛

Move Space: 2nSearch Neighbourhood

Current Solution

𝑛 Number of Variables

𝐾 Number of Labels

A series of globally optimal large moves

Combinatorial Optimization for pairwise MRFs

• GraphCut:

• Energy has unary and pairwise terms only

• global optimal solution in polynomial time (𝑂(𝑛3) and better)

• It must hold for all pairwise terms:𝜃𝑖𝑗 0,1 + 𝜃𝑖𝑗 1,0 ≥ 𝜃𝑖𝑗 0,0 + 𝜃𝑖𝑗 1,1

(known as submodularity condition)

• Used in many applications

• Many other techniques (LP-relaxations, message passing)- See CV 2


E 𝒙 =

𝑖

𝜃𝑖 𝑥𝑖 +

𝑖,𝑗


Unary terms Pairwise terms

𝜃𝑖𝑗 𝑥𝑖 , 𝑥𝑗 = |𝑥𝑖 − 𝑥𝑗|

Important example (Ising prior):

Graph cut in a nutshell

• Details in CV2


Converting the energy minimization problem into a problem called “finding maximum flow in a network”

10

1

Sink

Source

5

6 8

0

𝑥1 𝑥2 E 𝑥1, 𝑥2 = 𝜃1 𝑥1 + 𝜃2 𝑥2 + 𝜃12 𝑥1, 𝑥2

Alpha-Expansion


[Boykov , Veksler and Zabih 2001]

Sky

House

Tree

Ground

Initialize with TreeStatus: Expand GroundExpand HouseExpand Sky

• Variables take label a or retain current label


Move energy is submodular

• How to construct move energy (see exercise)

• Move energy is submodular if:• Unary Potentials: Arbitrary

• Pairwise potentials: Metric

[Boykov, Veksler, Zabih]

𝜃𝑖𝑗 (𝑙𝑎, 𝑙𝑏) = 0 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑙𝑎 = 𝑙𝑏

Examples: Truncated linear (not truncated quadratic)

[Boykov , Veksler and Zabih 2001]

• Other moves: alpha-beta swap, range move, fusion move etc.

𝜃𝑖𝑗 (𝑙𝑎, 𝑙𝑏) + 𝜃𝑖𝑗 (𝑙𝑏, 𝑙𝑐) ≥ 𝜃𝑖𝑗 (𝑙𝑎, 𝑙𝑐)

𝜃𝑖𝑗 (𝑙𝑎, 𝑙𝑏) = 𝜃𝑖𝑗 (𝑙𝑏, 𝑙𝑎) ≥ 0

Reminder truncated linear


discontinuity preserving potentials[Blake&Zisserman’83,’87]

𝜃𝑖𝑗 (𝑑𝑖, 𝑑𝑗) = min(|𝑑𝑖 − 𝑑𝑗|, 𝜏)

cost

No truncation(global min.)

with truncation(NP-hard optimization)

|𝑑𝑖 − 𝑑𝑗|

Example – Panoramic Stitching


Example – Panoramic Stitching


Summary this lecture


• Formulation applications as discrete Labeling problems




Computer Vision I - Algorithms and Applications: Discrete ...ds24/lehre/cv1_ws_2013/VL10.pdf · Computer Vision I - Algorithms and Applications: Discrete Labelling Carsten Rother

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