The Pyramid Match Kernel: Discriminative Classification with Sets of Image Features

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The Pyramid Match Kernel: Discriminative Classification with Sets of Image Features. Kristen Grauman Trevor Darrell MIT. Sets of features. invariant region descriptors. local shape features. examples under varying conditions. Sets of features. Problem.

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### The Pyramid Match Kernel: Discriminative Classification with Sets of Image Features

Kristen Grauman

Trevor Darrell

MIT

invariant region descriptors

local shape features

examples under varying conditions

Sets of features
Problem

How to build a discriminative classifier using the set representation?

• Kernel-based methods (e.g. SVM) are appealing for efficiency and generalization power…
• But what is an appropriate kernel?
• Each instance is unordered set of vectors
• Varying number of vectors per instance

Compute pair-wise similarity between all vectors in each set

• Wallraven et al., Lyu, Boughhorbel et al.
• General family of algebraic functions combining local (vector) kernels
• Shashua & Hazan

High complexity

Existing set kernels
• Fit(parametric) model to each set, compare with distance over models

Kondor & Jebara, Moreno et al., Lafferty & Lebanon, Cuturi & Vert,

Wolf & Shashua

Restrictive assumptions

Ignoring set statistics

Partial matching for sets of features

Compare sets by computing a partialmatching between their features.

Robust to clutter, segmentation errors, occlusion…

Pyramid match overview

Pyramid match kernel measures similarity of a partial matching between two sets:

• Place multi-dimensional, multi-resolution grid over point sets
• Consider points matched at finest resolution where they fall into same grid cell
• Approximate similarity between matched points with worst case similarity at given level

No explicit search for matches!

Number of newly matched pairs at level i

Measure of difficulty of a match at level i

Pyramid match kernel

Approximate partial match similarity

,

Histogram pyramid: level i has bins of size 2i

Feature extraction
Counting matches

Histogram intersection

matches at this level

matches at previous level

Difference in histogram intersections across levels counts number ofnew pairs matched

Counting new matches

Histogram intersection

histogram pyramids

number of newly matched pairs at level i

measure of difficulty of a match at level i

Pyramid match kernel
• Weights inversely proportional to bin size
• Normalize kernel values to avoid favoring large sets
Efficiency

For sets with m features of dimension d, and pyramids with L levels, computational complexity of

Pyramid match kernel:

Existing set kernel approaches:

or

Example pyramid match

pyramid match

optimal match

Approximation of the optimal partial matching

[Indyk & Thaper]

Matching output

Trial number (sorted by optimal distance)

100 sets with 2D points, cardinalities vary between 5 and 100

Building a classifier
• Train SVM by computing kernel values between all labeled training examples
• Classify novel examples by computing kernel values against support vectors
• One-versus-all for multi-class classification

Convergence is guaranteed since pyramid match kernel is positive-definite.

Object recognition results
• ETH-80 database 8 object classes
• Features:
• Harris detector
• PCA-SIFT descriptor, d=10

Eichhorn and Chapelle 2004

Object recognition results
• Caltech objects database 101 object classes
• Features:
• SIFT detector
• PCA-SIFT descriptor, d=10
• 30 training images / class
• 43% recognition rate

(1% chance performance)

• 0.002 seconds per match
Localization
• Inspect intersections to obtain correspondences between features
• Higher confidence correspondences at finer resolution levels

target

observation

Pyramid match regression
• Pose estimation from contour features
• Train SVR with CG data
• Features: shape context histograms
Summary: Pyramid match kernel

optimal partial matching between sets of features

difficulty of a match at level i

number of new matches at level i

Summary: Pyramid match kernel
• A new similarity measure based on implicit correspondences that approximates the optimal partial matching
• linear time complexity
• no independence assumption
• model-free
• insensitive to clutter
• positive-definite function
• fast, effective object recognition
Future work
• Geometric constraints
• Fast search of large databases with the pyramid match for image retrieval
• Use as a filter for a slower, explicit correspondence method
• Alternative feature types and classification domains