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The Pyramid Match Kernel and Its Improvement. Guo LiJun. PMK: Kristen Grauman,Trevor Darrell ICCV’05, Oral Projects that use LIBPMK Multiple Kernel Learning from Sets of Partially Matching Image Features, UKACC Control 2008.

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PMK: Kristen Grauman,Trevor Darrell ICCV’05, Oral
  • Projects that use LIBPMK
    • Multiple Kernel Learning from Sets of Partially Matching Image Features, UKACC Control 2008.
    • Distributed Image Search in Camera Sensor Networks, ACM SenSys 2008.
    • Automated Annotation of Drosophila Gene Expression Patterns Using a Controlled Vocabulary, Bioinformatics 2008.
    • Photo-based Question Answering, ACM Multimedia 2008.
    • Unsupervised Feature Selection via Distributed Coding for Multi-view Object Recognition, CVPR 2008.
    • Combining Brain Computer Interfaces with Vision for Object Categorization, CVPR 2008.
    • Scalable Classifiers for Internet Vision Tasks, CVPR Internet Vision Workshop 2008.
    • Object Category Recognition Using Probabilistic Fusion of Speech and Image Classifiers, MLMI 2007.
    • Envisioning Sketch Recognition: A Local Feature Based Approach to Recognizing Informal Sketches, Ph.D. thesis (2007).
    • Adaptive Vocabulary Forests for Dynamic Indexing and Categry Learning, ICCV 2007.
Related Work
    • Local Features
    • Kernel Methods
    • Earth Mover’s Distance
    • Multiresolution Histograms
sets of features1
invariant region descriptors

local shape features

examples under varying conditions

Sets of features

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
other solution
Other solution

Measuring Similarity Between Sets of Features


2)Bags of Prototypical Features

3)Computing Correspondences

existing set kernels
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

partial matching for sets of features
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 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!

pyramid match kernel
Number of newly matched pairs at level i

Measure of difficulty of a match at level i

Pyramid match kernel

Approximate partial match similarity

feature extraction

Histogram pyramid: level i has bins of size 2i

Feature extraction
counting matches
Counting matches

Histogram intersection

counting new matches
matches at this level

matches at previous level

Difference in histogram intersections across levels counts number ofnew pairs matched

Counting new matches

Histogram intersection

pyramid match kernel1
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

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

Pyramid match kernel:

Existing set kernel approaches:


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

Eichhorn and Chapelle 2004

object recognition results1
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
  • Inspect intersections to obtain correspondences between features
  • Higher confidence correspondences at finer resolution levels



pyramid match regression
Pyramid match regression
  • Pose estimation from contour features
  • Train SVR with CG data
  • Features: shape context histograms
summary pyramid match kernel
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 kernel1
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

Disregard all information about the spatial layout of the features


Spatial Pyramid Matching

Spatial Pyramid Matching

Learn Method: PMK or EMD-NN