Daniel Gill1

1
Learning Shape Distances for Classification
Is Pinocchios Nose Long or His Head Small ?

• Daniel Gill1
• A joint work with Y. Ritov1 and G. Dror2
• 1Department of Statistics, The Hebrew University
• 2Department of Computer Science, The Academic
  College of Tel-Aviv Yaffo

2
Outline

• Shape Shape Metric
• Problems With the Standard Shape Metric
• General Quadratic Metric Shape Classification
  Learning
• Kernel Machines for Shape Classification
• Applications
• Conclusions

3
What Is Shape?

• An equivalence class under certain type of group
  of transformations (e.g., translation, scaling,
  and rotation).

4
Shape Representation by Landmarks

• A finite set of particularly meaningful and
  salient points which can be identified by
  computer and humans.
• The landmarks correspondences are known.
• A useful representation for planar shapes is by
  complex vectors.

i
-1
1
-i
5
The Full Procrustes Distance Between Shapes

• The minimal Euclidean distance between two
  configurations achieved by translation, scaling,
  and rotation of x towards y
• This distance is symmetric !
• (y is the transpose of the complex conjugate of
  y).
  

6
The Minimizing Parameters Translation
i
-1
1
-i
7
The Minimizing Parameters Scaling
The Minimizing Parameters Rotation
The orientation of y is not important.
i
-1
1
-i
8
The Full Procrustes Mean Shape

• Minimizes the sum of square FP distances to each
  configuration in the set
• Can be solved as an eigenvalue problem.

• It is the eigenvector corresponding to the
  largest eigenvalue of

9
The Generalized Full Procrustes Analysis

• Several configurations are pairwise fitted to a
  single common consensus.
• Provides a useful visualization.

Females (blue) and males (green).
10
The Pinocchio Effect (Chapman 1990)

• The equal treatment of the least-squares
  superimposition provides a poor discrimination
  when landmarks are not equal in their
  variability.
• Lets have a look at a simple example.

11
The Pinocchio Effect

• Trying to align the two configurations
• by minimization of the sum-of-square
  differences
• affects all landmarks.

• Two heads which differ only by the tip of their
  nose.

• The longed-nose head is diminished and tilted.
• Moreover, the superimposition is highly
• depended on the landmarks choice.

12
Generalizing the Full Procrustes Metric

• By a symmetric positive semi-definite matrix
  .
• We use the decomposition.

• Normalizing position
• Normalizing scale
• Normalizing orientation

13
The General Quadratic Full Procrustes Mean Shape

• Is the one that minimizes the sum of general
  quadratic full-Procrustes distances to each
  configuration in the set

• It is the eigenvector corresponding to the
  largest eigenvalue of

14

• But how should the matrix Q be chosen ?

15
Metric Learning

• Understanding the input features and their
  importance for the task may lead to an
  appropriate metric.
• Estimating the metric from the data itself might
  result in a better performance than that achieved
  by off the shelf metrics.
• If for example we deal with a classification
  task, we will seek a metric that provides a
  good separation between classes.

16
Fisher Linear Discriminant (FLD)

• A linear projection of the data that maximizes
  the ratio of the between-class scatter and the
  within-class scatter of the transformed data.

Discriminative Projection
Non-Discriminative Projection
17
FLD-Like Metric Learning for Shapes

• The desired metric maximizes the ratio
  of the between-class scatter and within-class
  scatter
• This optimization problem cannot be solved as the
  regular FLD, and only a local maximum is
  guaranteed.

nk is the no. of samples in class k.
The distance between a shape and the mean shape
of its class.
18
So Whats Really The Difference Between Men
Women

• Superimpositions of mean facial configurations
  
• females (solid line) and males (dashed line).

Full Procrustes metric
Learnt Procrustes metric
19
Plugging The Learnt Metric Into A Kernel SVM

• The learnt metric can improve the performance of
  classifiers.
• SVMs use kernel function k(, ) that can be
  thought of as a similarity measure between the
  input vectors, and must be an inner-product in
  some space.

?(x)

• The kernel is a dot product of the transformed
  input vectors

Input space
Feature space
20
Plugging The Learnt Metric Into A Kernel SVM

• Replacing the predefined kernels with ones that
  are designed for the task at hand is likely to
  improve the performance of the classifier.
• An essential contribution when training examples
  are scarce.
• The following function is an inner-product kernel
  
• is a positive constant.

21
Experimental Results

• Task Gender classification tasks (except for the
  
• schizophrenia MRI dataset
  Mouse Vertebrae).
• Leave-One-Out error rates

• All datasets except for the facial images were
  taken from
• http//www.maths.nott.ac.uk/personal/ild/s
  hapes/

22
Conclusions

• The Procrustes metric is invariant under
  similarity transformations.
• This metric has disadvantages when dealing with
  classification tasks where the variability of
  different landmarks has different meaning.
• Labeled samples provide some information about
  the inter\intra variability of the samples
  classes.
• The learnt metric uncovers discriminative
  features, and allows useful visualization.

23
Conclusions (cont.)

• The Procrustes kernel ( QI ) is preferable over
  the standard Euclidean-based kernel (with
  pre-Procrustes analysis of the data)
• The learnt metric - based kernel improves the
  classifier performance even more.

24
Thank You !
25
The End