Part 2: part-based models

by Rob Fergus (MIT)

Problem with bag-of-words

• All have equal probability for bag-of-words methods

• Location information is important

Overview of section

• Representation– Computational complexity– Location – Appearance– Occlusion, Background clutter

• Recognition– Demos

Representation

Model: Parts and Structure

Representation• Object as set of parts

– Generative representation

• Model:– Relative locations between parts– Appearance of part

• Issues:– How to model location– How to represent appearance– Sparse or dense (pixels or regions)– How to handle occlusion/clutter

Figure from [Fischler & Elschlager 73]

History of Parts and Structure approaches

• Fischler & Elschlager 1973

• Yuille ‘91• Brunelli & Poggio ‘93• Lades, v.d. Malsburg et al. ‘93• Cootes, Lanitis, Taylor et al. ‘95• Amit & Geman ‘95, ‘99 • Perona et al. ‘95, ‘96, ’98, ’00, ’03, ‘04, ‘05• Felzenszwalb & Huttenlocher ’00, ’04 • Crandall & Huttenlocher ’05, ’06• Leibe & Schiele ’03, ’04

• Many papers since 2000

Sparse representation+ Computationally tractable (105 pixels 101 -- 102 parts)

+ Generative representation of class

+ Avoid modeling global variability

+ Success in specific object recognition

- Throw away most image information

- Parts need to be distinctive to separate from other classes

Region operators– Local maxima of

interest operator function

– Can give scale/orientation invariance

Figures from [Kadir, Zisserman and Brady 04]

The correspondence problem• Model with P parts• Image with N possible assignments for each part• Consider mapping to be 1-1

• NP combinations!!!

• 1 – 1 mapping– Each part assigned to unique feature

As opposed to:

• 1 – Many– Bag of words approaches – Sudderth, Torralba, Freeman ’05– Loeff, Sorokin, Arora and Forsyth ‘05

The correspondence problem

• Many – 1- Quattoni, Collins and Darrell, 04

Location

Connectivity of parts• Complexity is given by size of maximal clique in graph• Consider a 3 part model

– Each part has set of N possible locations in image– Location of parts 2 & 3 is independent, given location of L– Each part has an appearance term, independent between parts.

L

32

Shape Model

S(L,2) S(L,3) A(L) A(2) A(3)

L 32Variables

Factors

Shape Appearance

Factor graph

S(L)

from Sparse Flexible Models of Local FeaturesGustavo Carneiro and David Lowe, ECCV 2006

Different connectivity structures

O(N6) O(N2) O(N3)

O(N2)

Fergus et al. ’03Fei-Fei et al. ‘03

Crandall et al. ‘05Fergus et al. ’05

Crandall et al. ‘05Felzenszwalb & Huttenlocher ‘00

Bouchard & Triggs ‘05 Carneiro & Lowe ‘06Csurka ’04Vasconcelos ‘00

How much does shape help?• Crandall, Felzenszwalb, Huttenlocher CVPR’05• Shape variance increases with increasing model complexity• Do get some benefit from shape

Hierarchical representations

• Pixels Pixel groupings Parts Object

Images from [Amit98,Bouchard05]

• Multi-scale approach increases number of low-level features

• Amit and Geman ‘98• Bouchard & Triggs ‘05

Some class-specific graphs• Articulated motion

– People– Animals

• Special parameterisations– Limb angles

Images from [Kumar, Torr and Zisserman 05, Felzenszwalb & Huttenlocher 05]

Dense layout of parts

Layout CRF: Winn & Shotton, CVPR ‘06

Part labels (color-coded)

How to model location?

• Explicit: Probability density functions

• Implicit: Voting scheme

• Invariance– Translation– Scaling– Similarity/affine– Viewpoint

Similarity transformationTranslation and ScalingTranslationAffine transformation

• Cartesian – E.g. Gaussian distribution– Parameters of model, and – Independence corresponds to zeros in – Burl et al. ’96, Weber et al. ‘00, Fergus et al. ’03

• Polar – Convenient for

invariance to rotation

Explicit shape model

Mikolajczyk et al., CVPR ‘06

Implicit shape model

Spatial occurrence distributionsx

y

s

x

y

sx

y

s

x

y

s

Probabilistic Voting

Interest PointsMatched Codebook Entries

Recognition

Learning• Learn appearance codebook

– Cluster over interest points on training images

• Learn spatial distributions– Match codebook to training images– Record matching positions on object– Centroid is given

• Use Hough space voting to find object • Leibe and Schiele ’03,’05

Deformable Template Matching

QueryTemplate

Berg, Berg and Malik CVPR 2005

• Formulate problem as Integer Quadratic Programming• O(NP) in general• Use approximations that allow P=50 and N=2550 in <2 secs

Other invariance methods• Search over transformations

– Large space (# pixels x # scales ….)

– Closed form solution for translation and scale (Helmer and Lowe ’04)

• Features give information– Characteristic scale

– Characteristic orientation (noisy)

Figures from Mikolajczyk & Schmid

Multiple views

Component 1

Component 2

• Mixture of 2-D models– Weber, Welling and Perona CVPR ‘00

Frontal Profile

0 20 40 60 80 10050

55

60

65

70

75

80

85

90

95

100Orientation Tuning

angle in degrees

% C

orre

ct

Multiple view points

Thomas, Ferrari, Leibe, Tuytelaars, Schiele, and L. Van Gool. Towards Multi-View Object Class Detection, CVPR 06

Hoiem, Rother, Winn, 3D LayoutCRF for Multi-View Object Class Recognition and Segmentation, CVPR ‘07

Appearance

Representation of appearance

• Dependency structure– Often assume each part’s

appearance is independent – Common to assume

independence with location

• Needs to handle intra-class variation– Task is no longer matching of descriptors– Implicit variation (VQ to get discrete appearance)– Explicit model of appearance (e.g. Gaussians in SIFT space)

Representation of appearance• Invariance needs to match that of

shape model

• Insensitive to small shifts in translation/scale– Compensate for jitter of features– e.g. SIFT

• Illumination invariance– Normalize out

Appearance representation

• Decision trees

Figure from Winn & Shotton, CVPR ‘06

• SIFT

• PCA

[Lepetit and Fua CVPR 2005]

Occlusion• Explicit

– Additional match of each part to missing state

• Implicit– Truncated minimum probability of appearance

Log

prob

abili

ty

Appearance spaceµpart

Background clutter

• Explicit model– Generative model for clutter as well as foreground

object

• Use a sub-window– At correct position,

no clutter is present

Recognition

What task?

• Classification– Object present/absent in image– Background may be correlated with object

• Localization / Detection– Localize object

within the frame– Bounding box or

pixel-level segmentation

Efficient search methods• Interpretation tree (Grimson ’87)

– Condition on assigned parts to give search regions for remaining ones

– Branch & bound, A*

Distance transforms

• Distance transforms– O(N2P) O(NP) for tree structured models

• How it works– Assume location model is Gaussian (i.e. e-d2 )– Consider a two part model with µ=0, σ=1 on a 1-D image

f(d) = -d2

Appearance log probability at xi for part 2 = A2(xi)

xi

Image pixel

Log

prob

abili

ty

L

2

Model• Felzenszwalb and Huttenlocher ’00 & ’05

Distance transforms 2• For each position of landmark part, find best position for part 2

– Finding most probable xi is equivalent finding maximum over set of offset parabolas

– Upper envelope computed in O(N) rather than obvious O(N2) via distance transform (see Felzenszwalb and Huttenlocher ’05).

• Add AL(x) to upper envelope (offset by µ) to get overall probability map

xixg xj xlxh

A2(xi)

A2(xl)

A2(xj)A2(xg)

A2(xh)A2(xk)

xk

Log

prob

abili

ty

Image pixel

Parts and Structure demo

• Gaussian location model – star configuration• Translation invariant only

– Use 1st part as landmark

• Appearance model is template matching• Manual training

– User identifies correspondence on training images

• Recognition– Run template for each part over image– Get local maxima set of possible locations for each part– Impose shape model - O(N2P) cost– Score of each match is combination of shape model and

template responses.

Demo images• Sub-set of Caltech face dataset• Caltech background images

Demo Web Page

Demo (2)

Demo (3)

Demo (4)

Demo: efficient methods

Stochastic Grammar of ImagesS.C. Zhu et al. and D. Mumford

animal head instantiated by tiger head

animal head instantiated by bear head

e.g. discontinuities, gradient

e.g. linelets, curvelets, T-junctions

e.g. contours, intermediate objects

e.g. animals, trees, rocks

Context and Hierarchy in a Probabilistic Image ModelJin & Geman (2006)

Parts and Structure modelsSummary

• Correspondence problem• Efficient methods for large # parts and # positions in image• Challenge to get representation with desired invariance

Future directions:• Multiple views• Approaches to learning • Multiple category training

References

2. Parts and Structure

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[FeiFei03] L. Fei-Fei, R. Fergus, and P. Perona. A Bayesian approach to unsupervised one-shot learningof object categories. In Proceedings of the 9th International Conference on ComputerVision, Nice, France, pages 1134-1141, October 2003.[FeiFei04] L. Fei-Fei, R. Fergus, and P. Perona. Learning generative visual models from few trainingexamples: an incremental bayesian approach tested on 101 object categories. In Workshopon Generative-Model Based Vision, 2004.[FeiFei05] L. Fei-Fei and P. Perona. A Bayesian hierarchical model for learning natural scene categories.In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,San Diego, CA, volume 2, pages 524-531, June 2005.[Felzenszwalb00] P. Felzenszwalb and D. Huttenlocher. Pictorial structures for object recognition. InProceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages2066-2073, 2000.[Felzenszwalb05] P. Felzenszwalb and D. Huttenlocher. Pictorial structures for object recognition. InternationalJournal of Computer Vision, 61:55-79, January 2005.[Fergus_Datasets] R. Fergus and P. Perona. Caltech Object Category datasets. http://www.vision.caltech.edu/html-files/archive.html, 2003.[Fergus03] R. Fergus, P. Perona, and P. Zisserman. Object class recognition by unsupervised scaleinvariantlearning. In Proceedings of the IEEE Conference on Computer Vision and PatternRecognition, volume 2, pages 264-271, 2003.[Fergus04] R. Fergus, P. Perona, and A. Zisserman. A visual category lter for google images. InProceedings of the 8th European Conference on Computer Vision, Prague, Czech Republic,pages 242-256. Springer-Verlag, May 2004.[Fergus05 R. Fergus, P. Perona, and A. Zisserman. A sparse object category model for ecientlearning and exhaustive recognition. In Proceedings of the IEEE Conference on ComputerVision and Pattern Recognition, San Diego, volume 1, pages 380-387, 2005.[Fergus_Technote] R. Fergus, M. Weber, and P. Perona. Ecient methods for object recognition using theconstellation model. Technical report, California Institute of Technology, 2001. [Fischler73] M.A. Fischler and R.A. Elschlager. The representation and matching of pictorial structures.IEEE Transactions on Computer, c-22(1):67-92, Jan. 1973.

[Grimson87] W. E. L. Grimson and T. Lozano-Perez. Localizing overlapping parts by searching theinterpretation tree. IEEE Transactions on Pattern Analysis and Machine Intelligence,9(4):469-482, 1987.[Harris98] C. J. Harris and M. Stephens. A combined corner and edge detector. In Proceedings ofthe 4th Alvey Vision Conference, Manchester, pages 147-151, 1988.[Hart68] P.E. Hart, N.J. Nilsson, and B. Raphael. A formal basis for the determination of minimumcost paths. IEEE Transactions on SSC, 4:100-107, 1968.[Helmer04] S. Helmer and D. Lowe. Object recognition with many local features. In Workshop onGenerative Model Based Vision 2004 (GMBV), Washington, D.C., July 2004.[Hofmann99] T. Hofmann. Probabilistic latent semantic indexing. In SIGIR '99: Proceedings of the22nd Annual International ACM SIGIR Conference on Research and Development inInformation Retrieval, August 15-19, 1999, Berkeley, CA, USA, pages 50-57. ACM, 1999.[Holub05] A. Holub and P. Perona. A discriminative framework for modeling object classes. InProceedings of the IEEE Conference on Computer Vision and Pattern Recognition, SanDiego, volume 1, pages 664-671, 2005.[Kadir01] T. Kadir and M. Brady. Scale, saliency and image description. International Journal ofComputer Vision, 45(2):83-105, 2001.[Kadir_Code] T. Kadir and M. Brady. Scale Scaliency Operator. http://www.robots.ox.ac.uk/~timork/salscale.html, 2003.[Kumar05] M. P. Kumar, P. H. S. Torr, and A. Zisserman. Obj cut. In Proceedings of IEEE Conferenceon Computer Vision and Pattern Recognition, San Diego, pages 18-25, 2005.[Leibe04] B. Leibe, A. Leonardis, and B. Schiele. Combined object categorization and segmentationwith an implicit shape model. In Workshop on Statistical Learning in Computer Vision,ECCV, 2004.[Leung98] T. Leung and J. Malik. Contour continuity and region based image segmentation. InProceedings of the 5th European Conference on Computer Vision, Freiburg, Germany,LNCS 1406, pages 544-559. Springer-Verlag, 1998.[Leung95] T.K. Leung, M.C. Burl, and P. Perona. Finding faces in cluttered scenes using randomlabeled graph matching. Proceedings of the 5th International Conference on ComputerVision, Boston, pages 637-644, June 1995.

[Leung98] T.K. Leung, M.C. Burl, and P. Perona. Probabilistic ane invariants for recognition. InProceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages678-684, 1998.[Lindeberg98] T. Lindeberg. Feature detection with automatic scale selection. International Journal ofComputer Vision, 30(2):77-116, 1998.[Lowe99] D. Lowe. Object recognition from local scale-invariant features. In Proceedings of the7th International Conference on Computer Vision, Kerkyra, Greece, pages 1150-1157,September 1999.[Lowe01] D. Lowe. Local feature view clustering for 3D object recognition. In Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition, Kauai, Hawaii, pages682-688. Springer, December 2001.[Lowe04] D. Lowe. Distinctive image features from scale-invariant keypoints. International Journalof Computer Vision, 60(2):91-110, 2004.[Mardia89] K.V. Mardia and I.L. Dryden. \Shape Distributions for Landmark Data". Advances inApplied Probability, 21:742-755, 1989.[Sivic05] J. Sivic, B. Russell, A. Efros, A. Zisserman, and W. Freeman. Discovering object categoriesin image collections. Technical Report A. I. Memo 2005-005, Massachusetts Institute ofTechnology, 2005.[Sivic03] J. Sivic and A. Zisserman. Video Google: A text retrieval approach to object matchingin videos. In Proceedings of the International Conference on Computer Vision, pages1470-1477, October 2003.[Sudderth05] E. Sudderth, A. Torralba, W. Freeman, and A. Willsky. Learning hierarchical modelsof scenes, objects, and parts. In Proceedings of the IEEE International Conference onComputer Vision, Beijing, page To appear, 2005.[Torralba04] A. Torralba, K. P. Murphy, and W. T. Freeman. Sharing features: ecient boostingprocedures for multiclass object detection. In Proceedings of the IEEE Conference onComputer Vision and Pattern Recognition, Washington, DC, pages 762-769, 2004.

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Quest for A Stochastic Grammar of ImagesSong-Chun Zhu and David Mumford

Example scheme

• Model shape using Gaussian distribution on location between parts

• Model appearance as pixel templates• Represent image as collection of

regions– Extracted by template matching:

normalized-cross correlation

• Manually trained model– Click on training images

Connectivity of parts• To find best match in image, we want most probable

state of L, • Run max-product message passing

S(L,2) S(L,3) A(L) A(2) A(3)

L 32

mbmc

md

Take O(N2) to compute:

For each of the N values of L, need to find max over N states

S(L)

ma

Different graph structures

1

3

4 5

6

2

1

3

4 5

6

2

Fully connected Star structure

1

3

4

5

6

2

Tree structure

O(N6) O(N2) O(N2)

• Sparser graphs cannot capture all interactions between parts

Euclidean & Affine Shape• Translation, rotation and scaling Euclidean Shape

• Removal of camera foreshortenings Affine Shape

• Assume Gaussian density in figure space

• What is the probability density for the shape variables in each of the different spaces?

Feature space

TNN yyxx ],,,,,[ 11

Euclidean shape

TNN vvuu ],,,,,[ 33

Affine shape

TNN vvuu ],,,,,[ 44

Translation Invariant shape

TNN vvuu ],,,,,[ 22

Figures from [Leung98]

Translation-invariant shape• Figure space density:

• Translation-invariant form

• Shape space density is still Gaussian

e.g. P=3, move 1st part to origin

Affine Shape Density• Affine Shape density (Dryden-Mardia):

• Euclidean Shape density is of similar form

}2

{||

||)!3()(

2

},,{

4

1)3(

2/

4,321

ik

kkkk i

kiN

g

i

ieN

p

LC

Uu

[Leung98],[Welling05]• Can learnt parameters of DM density with EM!

Shape• Shape is “what remains after differences due to translation,

rotation, and scale have been factored out”. [Kendall84]

• Statistical theory of shape [Kendall, Bookstein, Mardia & Dryden]

X

Y

Figure SpaceU

V

Shape Space

N

N

y

yx

x

X

1

1

N

N

v

vx

u

U

3

3

Figures from [Leung98]

Learning

Learning situations

• Varying levels of supervision– Unsupervised– Image labels– Object centroid/bounding box– Segmented object– Manual correspondence

(typically sub-optimal)

• Generative models naturally incorporate labelling information (or lack of it)

• Discriminative schemes require labels for all data points

Contains a motorbike

• Task: Estimation of model parameters

Learning using EM

• Let the assignments be a hidden variable and use EM algorithm to learn them and the model parameters

• Chicken and Egg type problem, since we initially know neither:

- Model parameters

- Assignment of regions to parts

Example scheme, using EM for maximum likelihood learning

1. Current estimate of

...

Image 1 Image 2 Image i

2. Assign probabilities to constellations

Large P

Small P

3. Use probabilities as weights to re-estimate parameters. Example:

Large P x + Small P x

pdf

new estimate of

+ … =

Priors• Implicit

– Structure of dependencies in model– Parameterisation of model– Feature detectors

• Explicit– p()– MAP / Bayesian learning– Fei-Fei ‘03

model () space

1

2n

p( n

)

p(2 )

p(1 )

Learning Shape & Appearance simultaneously Fergus et al. ‘03

Learn appearance then shape

ParameterEstimation

Choice 1

Choice 2ParameterEstimation

Model 1

Model 2

Predict / measure model performance(validation set or directly from model)

Preselected Parts (100)

Weber et al. ‘00

Discriminative training• Sparse so parts need to be distinctive of class

• Boosted parts and structure models– Amores et al. CVPR 2005– Bar Hillel et al. CVPR 2005

• Discriminative features– Weber et al. 2000– Ullman et al.

• Train discriminatively on parameters of generative model– Holub, Welling,

Perona ICCV 2005

Number of training images

• More supervision, fewer images needed– Few unknown parameters

• Less supervision, more images.– Lots of unknown parameters

• Over-fitting problems

Number of training examples

Priors

6 part Motorbike model