A 3-D reference frame can be uniquely defined by the ordered vertices of a non-degenerate triangle - PowerPoint PPT Presentation

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A 3-D reference frame can be uniquely defined by the ordered vertices of a non-degenerate triangle

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Title: A 3-D reference frame can be uniquely defined by the ordered vertices of a non-degenerate triangle


1
A 3-D reference frame can be uniquely defined by
the ordered vertices of a non-degenerate triangle
p1
p2
p3
2
Pattern MatchingNaive algorithm
  • For each pair of triplets, one from each molecule
    which define almost congruent triangles compute
    the rigid motion that superimposes them.
  • Count the number of point pairs, which are
    almost superimposed and sort the hypotheses by
    this number.

3
Naive algorithm (continued )
  • For the highest ranking hypotheses improve the
    transformation by replacing it by the best RMSD
    transformation for all the matching pairs.
  • Complexity assuming order of n points in both
    molecules - O(n7) .
  • (O(n3) if one exploits protein backbone
    geometry.)

4
Object Recognition Techniques
  • Pose Clustering
  • Geometric Hashing
  • (and more )

5
Pose Clustering
  • Clustering of transformations.
  • Match each triplet from the first object with
    triplet from the second object.
  • A triplet defines 3D transformation. Store it
    using appropriate data structure.
  • High scoring alignments will result in dense
    clusters of transformations.

Time Complexity O(n3m3)Clustering
6
Pose Clustering (2)
  • Problem How to compare 2 transformations?
  • Solutions
  • Straightforward
  • Based on transformed points

7
Geometric Hashing
  • Inavriant geometric relations
  • Store in fast look-up table

8
Geometric Hashing - Preprocessing
  • Pick a reference frame satisfying pre-specified
    constraints.
  • Compute the coordinates of all the other points
    (in a pre-specified neighborhood) in this
    reference frame.
  • Use each coordinate as an address to the hash
    (look-up) table and record in that entry the
    (ref. frame, shape sign.,point).
  • Repeat above steps for each reference frame.

9
Geometric Hashing - Recognition 1
  • For the target protein do
  • Pick a reference frame satisfying pre-specified
    constraints.
  • Compute the coordinates of all other points in
    the current reference frame .
  • Use each coordinate to access the hash-table to
    retrieve all the records (ref.fr., shape sign.,
    pt.).

10
Geometric Hashing - Recognition 2
  • For records with matching shape sign. vote for
    the (ref.fr.).
  • Compute the transformations of the high scoring
    hypotheses.
  • Repeat the above steps for each ref.fr.
  • Cluster similar transformation.
  • Extend best matches.

11
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12
Complexity of Geometric Hashing
O(n4 n4 BinSize) O(n5 )
(Naive alg. O(n7))
13
Advantages
  • Sequence order independent.
  • Can match partial disconnected substructures.
  • Pattern detection and recognition.
  • Highly efficient.
  • Can be applied to protein-protein interfaces,
    surface motif detection, docking.
  • Database Object Recognition a trivial extension
    to the method
  • Parallel Implementation straight forward

14
Structural Comparison Algorithms
  • Ca backbone matching.
  • Secondary structure configuration matching.
  • Molecular surface matching.
  • Multiple Structure Alignment.
  • Flexible (Hinge - based) structural alignment.

15
Protein Structural Comparison
PDB files
Feature Extraction
Geometric Matching
Verification and Scoring
Rotation and Translation Possibilities
Least Square Analysis
Ca
Other Inputs
Geometric Hashing
Backbone
Secondary Structures
Transformation Clustering
Flexible Geometric Hashing
H-bonds
Sequence Dependent Weights
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