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Learning a Scale-Invariant Model for Curvilinear Continuity

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The Quest of Boundary Detection

- Widely used for mid/high-level vision tasks
- Huge literature on edge detection

[Canny 86]

- Typically measuring local contrast
- Approaching human performance?

[Martin, Fowlkes & Malik 02]

[Fowlkes, Martin & Malik 03]

2

Continuity in Human Vision

- [Wertheimer 23]
- [Kanizsa 55]
- [von der Heydt et al 84]
- evidence in V2
- [Kellman & Shipley 91]
- geometric conditions of completion
- [Field, Hayes & Hess 93]
- quantitative analysis of factors
- [Kapadia, Westheimer & Gilbert 00]
- evidence in V1
- [Geisler et al 01]
- evidence from ecological statistics

… … … …

5

Extensive literature on curvilinear continuity

[Shashua & Ullman 88], [Parent & Zucker 89], [Heitger & von der Heydt 93], [Mumford 94], [Williams & Jacobs 95], [Elder & Zucker 96], [Williams & Thornber 99], [Jermyn & Ishikawa 99], [Mahamud et al 03], …, …

Problems with most of the previous approaches

no support from any groundtruth data

usually demonstrated on a few simple/synthetic images

no quantitative evaluation

Continuity in Computer Vision6

Outline

- Ecological Statistics of Contours
- A Scale-Invariant Representation
- Learning Models of Curvilinear Continuity
- Quantitative Evaluation
- Discussion and Future Work

7

Outline

- Ecological Statistics of Contours
- Groundtruth boundary contours
- Power law in contours
- A multi-scale Markov model
- A Scale-Invariant Representation
- Learning Models of Curvilinear Continuity
- Quantitative Evaluation
- Discussion and Future Work

8

t(s+1)

s+1

t(s)

s

Contour Geometry- First-Order Markov Model

[Mumford 94, Williams & Jacobs 95]

- Curvature: white noise ( independent from position to position )
- Tangent t(s): random walk
- Markov assumption: the tangent at the next position, t(s+1), only depends on the current tangent t(s)

10

Prediction: Exponential Distribution

If the first-order Markov assumption holds…

- At every step, there is a constant probability p that a high curvature event will occur
- High curvature events are independent from step to step

Let L be the length of a segment between high-curvature points

- P( L>=k ) = (1-p)k
- P( L=k ) = p(1-p)k

L has an exponential distribution

13

Power Laws in Nature

- Power laws widely exist in nature
- Brightness of stars
- Magnitude of earthquakes
- Population of cities
- Word frequency in natural languages
- Revenue of commercial corporations
- Connectivity in Internet topology

… …

- Usually characterized by self-similarity and scale-invariant phenomena

15

t(1)(s+1)

s+1

- Coarse-to-fine contour completion

- [Ren & Malik 02]

Multi-scale Markov Models

t(s+1)

- Assume knowledge of contour orientation at coarser scales

s+1

2nd Order Markov:

P( t(s+1) | t(s) , t(1)(s+1) )

Higher Order Models:

P( t(s+1) | t(s) , t(1)(s+1), t(2)(s+1), … )

t(s)

s

16

Contour Synthesis

First-Order Markov:

P( t(s+1) | t(s) )

Multi-scale Markov:

P( t(s+1) | t(s) , t(1)(s+1), t(2)(s+1), … )

[Ren & Malik 02]

17

Outline

- Ecological Statistics of Contours
- A Scale-Invariant Representation
- Piecewise linear approximation
- Constrained Delaunay Triangulation
- Learning Models of Curvilinear Continuity
- Quantitative Evaluation
- Discussion and Future Work

18

Use Pb (probability of boundary) as input

Combining local brightness, texture and color cues

Trained from human-marked segmentation boundaries

Outperform existing local boundary detectors including Canny

Local “Probability of Boundary”

- [Martin, Fowlkes & Malik 02]

19

Threshold Pb and find connected boundary pixels

Recursively split the boundaries until each piece is approximately straight

b

b

a

c

a

c

Split at C

Piecewise Linear Approximation

minimize

20

Standard in computational geometry

Dual of the Voronoi Diagram

Unique triangulation that maximizes the minimum angle

avoiding long skinny triangles

Efficient and simple randomized algorithm

Delaunay Triangulation

21

A variant of the standard Delaunay Triangulation

Keeps a given set of edges in the triangulation

[Chew 87]

[Shewchuk 96]

- Still maximizes the minimum angle
- Widely used in geometric modeling and finite elements

Constrained Delaunay Triangulation

22

A typical scenario of contour completion

high contrast

high contrast

low contrast

- CDT picks the “right” edge, completing the gap

The “Gap-filling” Property of CDT

23

Outline

- Ecological Statistics of Contours
- A Scale-Invariant Representation
- Learning Models of Curvilinear Continuity
- Transferring Groundtruth to CDT
- A simple model of local continuity
- A global model w/ Conditional Random Fields
- Quantitative Evaluation
- Discussion and Future Work

26

Transferring Groundtruth to CDT

- Human-marked boundaries are given on the pixel-grid
- Label the CDT edges by bipartite matching

d

distance threshold d in matching

CDT edges

Phuman: percentage of pixels

matched to groundtruth

human-marked

boundaries

27

pb1, G1

pb0, G0

Model for Continuity

- Goal: define a continuity-enhanced Pb on CDT edges
- Consider a pair of adjacent edges in CDT:
- Each edge has an associated set of features
- average Pb over the pixels belonging to this edge
- indicator G, gradient edge or completed edge?
- Continuity: angle

“bi-gram”

28

Binary Classification

- Assuming contours are always closed: each vertex in the CDT graph is adjacent to either zero or two true boundary edges

- A binary classification problem: (0,0) or (1,1)

“bi-gram”

29

pb1, G1

pb0, G0

Learning Local Continuity

- Binary classification: (0,0) or (1,1)
- Transferred Groundtruth labels on CDT edges
- Features:
- average Pb
- (G0*G1): both are gradient edges?
- angle
- Logistic regression

30

PbL: Pb + Local Continuity

Evidence of continuity comes from both ends

pb1, G1

pb2, G2

1

2

pb0, G0

take max. over all possible pairs

L

L

=

PbL

31

Variants of the Local Model

- More variants of the local model
- alternative classifiers ( SVM, HME, … )
- 4-way classification
- additional features
- learning a 3-edge (tri-gram) model
- learning how to combine evidence from both ends
- No significant improvement in performance

32

Local inference

Xi+1

Xi

A Global Model of Continuity?

X={X1,X2,…,Xm}

Global inference incorporating all

local continuity information?

33

For each edge i, define a set of features

{g1,g2,…,gh}

Potential function exp(i)at edge i

For each junction j, define a set of features

{f1,f2,…,fk}

Potential function exp(j)at juncion j

Conditional Random Fields

X={X1,X2,…,Xm}

[Pietra, Pietra & Lafferty 97]

[Lafferty, McCallum & Pereira 01]

34

Conditional Random Fields

Potential function on edges {exp(i)}

Potential function on junctions {exp(j)}

This defines a probability distribution over X:

X={X1,X2,…,Xm}

where

Estimate P(Xi|)

35

Buliding a CRF Model

- What are the features?
- edge features are easy: Pb, G
- junction features: type and continuity
- How to make inference?
- How to learn the parameters?

X={X1,X2,…,Xm}

Estimate P(Xi|)

36

degg=0,degc=2

degg=0,degc=2

Junction Features in CRF

- Junction types (degg,degc):

degg=1,degc=0

degg=0,degc=2

degg=1,degc=2

- Continuity term for degree-2 junctions

37

Inference w/ Belief Propagation

Fr

- Belief Propagation
- Xi: state of the node (edge) i
- Fq: state of the factor (junction) q
- potentials on Xi,Xj,Xk, Fq={Xi, Xj, Xk}
- want to compute PbG=P(Xi)
- mqi: “belief” about Xi from Fq

Xj

mjq

mir

mqi

Xi

Fq

mkq

Xk

- The CDT graph has many loops in it

38

Inference w/ Loopy Belief Propagation

- Loopy Belief Propagation
- just like belief propagation
- iterates message passing until convergence
- lack of theoretical foundations and known to have convergence issues
- however becoming popular in practice
- typically applied on pixel-grid
- Works well on CDT graphs
- converges fast
- produces empirically sound results

[Berrou 93], [Freeman 98], [Murphy 99], [Weiss 97,99,01]

39

Learning the Parameters

- Maximum-likelihood estimation in CRF

Let denote the groundtruth labeling on the CDT graph

- Many possible optimization techniques
- gradient descent, iterative scaling, conjugate gradient, …
- Gradient descent works well

40

there are more non-boundary edges than boundary edges

a continuation is better than a line-ending

junctions are rare

G-edges are better for continuation than C-edges

Interpreting the Parameters

- The junction parameters (degg,degc) on the horse dataset:

(0,0)= 2.8318

(1,0)= 1.1279

(2,0)= 1.3774

(3,0)= 0.0342

(2,0)= 1.3774

(1,1)= -0.6106

(0,2)= -0.9773

41

Outline

- Ecological Statistics of Contours
- A Scale-Invariant Representation
- Learning Models of Curvilinear Continuity
- Quantitative Evaluation
- The precision-recall framework
- Experimental results on three datasets
- Discussion and Future Work

42

Datasets

- Baseball player dataset [Mori et al 04]
- 30 news photos of baseball players in various poses, 15 training and 15 testing
- Horse dataset [Borenstein & Ullman 02]
- 350 images of standing horses facing left, 175 training and 175 testing
- Berkeley Segmentation Dataset [Martin et al 01]
- 300 Corel images of various natural scenes and ~2500 segmentations, 200 training and 100 testing

43

Evaluating Boundary Operators

- Precision-Recall Curves [Martin, Fowlkes & Malik 02]
- threshold the output boundary map
- bipartite matching with the groundtruth

m pixels on human-marked boundaries

k matched pairs

n detected pixels above a given threshold

Precision = k/n, percentage of true positives

Recall = k/m, percentage of groundtruth being detected

- Project CDT edges back to the pixel-grid

44

Use Phuman the soft groundtruth

label defined on CDT graphs:

precision close to 100%

Pb averaged over CDT edges: no worse than the orignal Pb

No Loss of Structure in CDT

45

Continuity improves boundary detection in both low-recall and high-recall ranges

Global inference helps; mostly in low-recall/high-precision

Roughly speaking,

CRF>Local>CDT only>Pb

46

In Conclusion…

- Constrained Delaunay Triangulation is a scale-invariant discretization of images with little loss of structure;

- Boundary contours are scale-invariant in nature;

- Moving from 100,000 pixels to <1000 edges, CDT achieves great statistical and computational efficiency;

- Curvilinear Continuity improves boundary detection;
- the local model of continuity is simple yet very effective
- global inference of continuity further improves performance
- Conditional Random Fields w/ loopy belief propagation works well on CDT graphs

- Mid-level vision is useful.

53

Future Work

- To add more features into CRF
- region-based features
- avoiding spurious completions
- tri-gram model
- To train CRF w/ different criteria
- e.g., area under the precision-recall curve
- Max-margin Markov networks
- To use CRF for feature selection
- To apply CDT+CRF to other mid-level vision problems, e.g., figure/ground organization

54

Figure/Ground Organization

- A classical problem in Gestalt psychology

[Rubin 1921]

- “Perceptual organization after grouping”
- Gestalt principles for figure/ground
- surroundedness, size, convexity, parallelism, symmetry, lower-region, common fate, familiar configuration, …
- Very few computational studies

[Hinton 86], [von der Heydt 93]

55

Shape context [Belongie, Malik & Punicha 01]

- Clustering shape context into prototypical shape configurations or “shapemes”
- Local figure/ground discrimination with shapemes

Using Shapemes for Figure/Ground

- To capture mid-level information:

“local” shape configuration

56

Shapemes

57

One feature for each junction type

F

G

G

G

F

F

- Add a continuity term

CRF for Figure/Ground

F={F1,F2,…,Fm}

Fi{Left,Right}

59

Preliminary Results on Figure/Ground

- Chance error rate
- Local operator w/ shapemes
- Using human segmentations:
- Averaging local cues on human-marked boundaries
- CRF w/ junction type
- CRF w/ junction type and continuity
- To use CDT graphs

50%

39%

29%

28%

21%

60

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