Quadratic Form Distance (QFD)

QMap model

› QFD to L2 Space Transformation

QMap and MAMs

› Experimental evaluation

Conclusion

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Content-based similarity searching

› Pair-wise similarity

› High-dimensional (feature) vectors

Fast query processing

› Efficiency

› Effectiveness

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Similarity measuring:

QFDA (u,v) = √ (u – v)A(u – v)T

u, v feature vectors (1×n)

A similarity matrix/QFD matrix (n×n)

› positive definite (zAzT > 0)

› static / dynamic correlations

› data independent

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Applications

› QBIC project (Querying Images by content)

› 2D & 3D shapes

› Protein structures

› MindReader

Advanced

› SQFD (Signature QFD)

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Transformation approaches

› QBIC system

Lower-bounding (e.g. Faloutsos et. al 1994)

› Contractive reduction techniques

› SVD / KLT decompositions

Combination (e.g. Hafner et. al 1995)

› Transformation to k-dimensional Lp space

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Metric Access Methods (MAMs)

› Effective/efficient similarity searching

› Reduce distance computations

› Complexity depends on distance function

QFD is considered as expensive – O(n2)

› Indexing needed

We show the transformation of QFD

› Obtain cheaper distance function – O(n)

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QFDEuclidean (L2)

distance

Correlated

dimensions

Independent

dimensions

Expensive – O(n2) Cheap – O(n)

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Transform QFD space

› L2 instead of QFD

› Preserving distances (homeomorphism)

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Transform QFD space

› L2 instead of QFD

› Preserving distances (homeomorphism)

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Rotate (weighted L2 space)

Scale (L2 space)

Transformation matrix B

› obtained by Cholesky decomposition:

A = BBT

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Linear

transformation

1. QFDA (u,v) = √ (u – v)A(u – v)T

2. Cholesky decomposition: BBT = A

3. QFD (u,v) = √ (u – v)BBT(u – v)T

4. QFD (u,v) = √ [(u – v)B][(u – v)B]T

5. QFD (u,v) = √ (uB – vB)(uB – vB)T

6. L2 (u’,v’) = √ (u’ – v’)(u’ – v’)T

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Application of QMap in MAM› Sequential (SEQ) file

› Pivot Table

› M-tree

1,000,000 images (512 dimensional RGB histogram)

Actions› Indexing

› Querying

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Method (model) “Winner”

SEQ file (QFD)

SEQ file (QMap) QFD

Pivot Table (QFD)

Pivot Table(QMap) QMap

M-tree (QFD)

M-tree (QMap) QMap

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Method (model) “Winner”

SEQ file (QFD)

SEQ file (QMap) QMap

Pivot Table (QFD)

Pivot Table(QMap) QMap

M-tree (QFD)

M-tree (QMap) QMap

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QMap model

› Space transformation: QFD → L2

› Distance-preserving (homeomorphic)

› Data-independent

› Output is explicitly formulated

QMap model is separated from the

usage of any access methods

› Superior performance

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