1. Part 2: part-based models
by Rob Fergus (MIT)

2. Problem with bag-of-words
Motivation slide. Clearly location has to help, but bag-of-words doesn’t have it…
All have equal probability for bag-of-words methods
Location information is important

3. Overview of section Representation Recognition
Computational complexity
Location
Appearance
Occlusion, Background clutter
Recognition
Demos

4. Representation

5. Model: Parts and Structure
Globally different, but have portions in common

6. Representation Object as set of parts Model:
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]

7. 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

8. Sparse representation
+ Computationally tractable (105 pixels  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

9. Region operators Local maxima of interest operator function
Can give scale/orientation invariance
Figures from [Kadir, Zisserman and Brady 04]

10. The correspondence problem
Model with P parts
Image with N possible assignments for each part
Consider mapping to be 1-1
Key problem with approach & motivates design choices. Assume for the moment that we have a set of regions.
NP combinations!!!

11. The correspondence problem
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
Many – 1
- Quattoni, Collins and Darrell, 04

12. Location

13. 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.
Shape Model
Factor graph
Variables L 2 3 L 2 3
Factors
S(L)
S(L,2)
S(L,3)
A(L)
A(2)
A(3)
Shape
Appearance

14. Different connectivity structures
Felzenszwalb & Huttenlocher ‘00
Fergus et al. ’03
Fei-Fei et al. ‘03
Crandall et al. ‘05
Fergus et al. ’05
Crandall et al. ‘05
O(N2)
O(N6)
O(N2)
O(N3)
Csurka ’04
Vasconcelos ‘00
Bouchard & Triggs ‘05
Carneiro & Lowe ‘06
from Sparse Flexible Models of Local Features Gustavo Carneiro and David Lowe, ECCV 2006

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

16. Hierarchical representations
Pixels  Pixel groupings  Parts  Object
Multi-scale approach increases number of low-level features
Amit and Geman ‘98
Bouchard & Triggs ‘05
Images from [Amit98,Bouchard05]

17. Some class-specific graphs
Articulated motion
People
Animals
Special parameterisations
Limb angles
Images from [Kumar, Torr and Zisserman 05, Felzenszwalb & Huttenlocher 05]

18. Dense layout of parts Layout CRF: Winn & Shotton, CVPR ‘06
Part labels (color-coded)
Each pixel is labelled – regions of pixels have the same part label. 18

19. How to model location? Explicit: Probability density functions
Implicit: Voting scheme
Invariance
Translation
Scaling
Similarity/affine
Viewpoint
Affine transformation
Similarity transformation
Translation and Scaling
Translation

20. Explicit shape model Cartesian Polar 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
Mikolajczyk et al., CVPR ‘06

21. Spatial occurrence distributions Matched Codebook Entries
Implicit shape model
Use Hough space voting to find object
Leibe and Schiele ’03,’05
Spatial occurrence distributions x y s
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
Recognition
Matched Codebook Entries
Probabilistic Voting
Interest Points

22. Deformable Template Matching
Berg, Berg and Malik CVPR 2005
Template
Query
Formulate problem as Integer Quadratic Programming
O(NP) in general
Use approximations that allow P=50 and N=2550 in <2 secs
% The idea of deformable templates is not new. In
% the early 1970's at least three groups were
% working on similar ideas:
% Grenander in statistical pattern theory
% von der Malsburg in neural networks
% and Fischler and Eschlager in computer vision
On the left we have the model template of a helicopter. We characterize the template by a set of feature points sampled from edges in the image. These are shown as the approximately 40 colored dots in the left image. On the right we show a query image. Here feature points are extracted along edges in the entire image. Many of these will be on background or clutter, but some are on the object of interest. Our problem is to find a correspondence between the model points at left and the correct subset of feature points on the right.

23. 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)
invariance of the characteristic scale
Figures from Mikolajczyk & Schmid

24. Multiple views Mixture of 2-D models Frontal Profile
Weber, Welling and Perona CVPR ‘00 20 40 60 80
100 50 55 65 70 75 85 90 95
Orientation Tuning
angle in degrees
% Correct
Component 1
Component 2
Frontal
Profile

25. 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

26. Appearance

27. Representation of appearance
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)
Dependency structure
Often assume each part’s appearance is independent
Common to assume independence with location

28. 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

29. Appearance representation
SIFT
Decision trees
[Lepetit and Fua CVPR 2005]
PCA
Figure from Winn & Shotton, CVPR ‘06

30. Occlusion Explicit Implicit
Additional match of each part to missing state
Implicit
Truncated minimum probability of appearance
µpart
Appearance space
Log probability

31. Background clutter Explicit model Use a sub-window
Generative model for clutter as well as foreground object
Use a sub-window
At correct position, no clutter is present

32. Recognition

33. What task? Classification Localization / Detection
Object present/absent in image
Background may be correlated with object
Localization / Detection
Localize object within the frame
Bounding box or pixel-level segmentation

34. Efficient search methods
Interpretation tree (Grimson ’87)
Condition on assigned parts to give search regions for remaining ones
Branch & bound, A*

35. Distance transforms Felzenszwalb and Huttenlocher ’00 & ’052
Model
Felzenszwalb and Huttenlocher ’00 & ’05
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 xi
Image pixel
Appearance log probability at xi for part 2 = A2(xi)
Log probability
f(d) = -d2

36. 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 xg xh xi xj xk xl
Image pixel
A2(xi)
A2(xl)
A2(xj)
A2(xg)
A2(xh)
A2(xk)
Log probability

37. 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.

38. Demo images
Sub-set of Caltech face dataset
Caltech background images

39. Demo Web Page

40. Demo (2)

41. Demo (3)

42. Demo (4)

43. Demo: efficient methods

44. Stochastic Grammar of Images S.C. Zhu et al. and D. Mumford

45. Context and Hierarchy in a Probabilistic Image Model Jin & Geman (2006)
animal head instantiated by bear head
e.g. animals, trees, rocks
e.g. contours, intermediate objects
e.g. linelets, curvelets, T-junctions
e.g. discontinuities, gradient
animal head instantiated by tiger head

46. Parts and Structure models Summary
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

48. References
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55. Quest for A Stochastic Grammar of Images Song-Chun Zhu and David Mumford

60. 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

61. Connectivity of parts
To find best match in image, we want most probable state of L,
Run max-product message passing L 2 3 md ma mb mc
S(L)
S(L,2)
S(L,3)
A(L)
A(2)
A(3)
Take O(N2) to compute:
For each of the N values of L, need to find max over N states

62. Different graph structures6 1 3 5 2 3 2 3 1 2 1 4 5 4 6 6 4 5
Fully
connected
Star structure
Tree structure
O(N6)
O(N2)
O(N2)
Sparser graphs cannot capture all interactions between parts

63. 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
Translation Invariant shape
Euclidean shape
Affine shape
Figures from [Leung98]

64. Translation-invariant shape
Figure space density:
Translation-invariant form
e.g. P=3, move 1st part to origin
Shape space density is still Gaussian

65. Affine Shape Density Affine Shape density (Dryden-Mardia):
Euclidean Shape density is of similar form
Can learnt parameters of DM density with EM!
[Leung98],[Welling05]

66. 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] Y V X U
Figure Space
Shape Space
Figures from [Leung98]

67. Learning

68. 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

70. Learning using EM Task: Estimation of model parameters
Chicken and Egg type problem, since we initially know neither:
Model parameters
- Assignment of regions to parts
Let the assignments be a hidden variable and use EM algorithm to learn them and the model parameters

71. Learning procedure Find regions & their location & appearance
Initialize model parameters
Use EM and iterate to convergence:
E-step: Compute assignments for which regions belong to which part
M-step: Update model parameters
Trying to maximize likelihood – consistency in shape & appearance

72. Example scheme, using EM for maximum likelihood learning
1. Current estimate of 
2. Assign probabilities to constellations
Large P
...
pdf
Image 1
Image 2
Image i
Small P
3. Use probabilities as weights to re-estimate parameters. Example: 
Large P x +
Small P x
+ … =
new estimate of 

73. Priors Implicit Explicit model () space 1 2 n
Structure of dependencies in model
Parameterisation of model
Feature detectors
Explicit
p()
MAP / Bayesian learning
Fei-Fei ‘03
model () space 1 2 n
p(n )
p(2 )
p(1 )

74. Learning Shape & Appearance simultaneously
Fergus et al. ‘03

75. Learn appearance then shape
Weber et al. ‘00
Model 1
Choice 1
Parameter
Estimation
Model 2
Choice 2
Parameter
Estimation
Preselected Parts (100)
Predict / measure model performance
(validation set or directly from model)

76. 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

77. Number of training images
More supervision, fewer images needed
Few unknown parameters
Less supervision, more images.
Lots of unknown parameters
Over-fitting problems

78. Number of training examples
6 part Motorbike model
Priors