Geometric Pattern Matching in 3D space

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Geometric Pattern Matching in 3D space

• Concettina Guerra

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Geometric Matching ofPoint Clouds
Two input sets
Two sets after superimposition
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Matching objects in Computer Vision
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Protein basics
From Sequence
To Structure
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A protein chain

• Input protein sequence
• Output protein structure

Unfolded protein Denaturate state
Folded protein Native state
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Protein Structure Alignment
Human Hemoglobin alpha-chain pdb1jebA
Human Myoglobin pdb2mm1
Another example G-Proteins 1c1yA,
1kk1A6-200 Sequence id 18 Structural id 72
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Continuity Constraint

• Given two objects A and B
• If point P and Q of A are matched to point P abd
  Q of B then
• If P lt P then it must also be Q lt Q
• ( lt stands for preceds)

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Transformations

• Translation
• Translation and Rotation
• Rigid Motion (Euclidian Trans.)
• Translation, Rotation Scaling

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Inexact Matching Simple case two closely sets
with the same number of points .
Assume transformation T is given
Question how to measure an alignment error?
( by Prof. Haim Wolfson).
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Distance Functions

• Two point sets Aai i1n
• Bbj j1m
• Pairwise Correspondence
• (ak1,bt1) (ak2,bt2) (akN,btN)

(1) Exact Matching aki bti0
(2) Bottleneck max aki bti (3) RMSD
(Root Mean Square Distance) Sqrt(
Saki bti2/N)
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Hausdorff DistanceAnother distance function

• Let A a1, a2, ..., am B b1, b 2, ..., bn
  be sets of either points or segments.
• Definition. (Hausdorff Distance)
• H(A, B) max (h(A, B), h(B, A))
• where the one-way Hausdorff distance is
• h(A, B) maxa minb r (a, b)
• where a (b) is a point of A (B) and r (a, b), is
  a metric.

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Correspondence is Unknown
Given two configurations of points in the
three dimensional space,
find those rotations and translations of one
of the point sets which produce large
superimpositions of corresponding 3-D
points.
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Largest Common Point Set (LCP) problem
Given egt0 and two point sets A and B find a
transformation T and equally sized subsets A (a
subset of A) and B (a subset of B) of maximal
cardinality such that dist(A,T(B)) e.
Bottleneck metric optimal solution in O(n32.5)
C. Ambuhl et al. 2000
RMSD metric open problem
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Exact solution in 2Dusing Hausdorff Distance

• This problem is generally solved as a problem of
  intersection of unions of disks in the
  transformation space.
• Time complexity O( m3 n3 log2nm) in R2

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Two instances of the problem

• Similarity of the two sets of atoms with known
  correspondences
• Aai , Bbi , i1,,n
• ai ?? bi
• Similarity of the two sets of atoms with unknown
  correspondences
• Aai , Bbj , i1,,n j1,,m
• ai(k) ?? bj(k) k1,,Kltn,m

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Superposition RMSD

• Given two sets of 3-D points with known
  correspondences
• Aai , Bbi , i1,,n
• find a 3-D rotation R and translation T that
  minimizes
• D2minR,T Si Rai T - bi 2
• RMSDD / sqrt(n)
• A closed form solution exists for this task.

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Orthogonal Procrustes problem

• The Solution is based on Singular Value
  Decomposition (SVD) of the correlation matrix A
  of the points
• Aij Sk ak ibk j
• where ak i is the ith component of the vector
  ak
• The solution involves eigenvalue analysis of a
  correlation matrix of the points.

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Finding Correspondences

• Geometric Hashing, Indexing (Wolfson et al., 1998)

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Hashing function

• From an Object
• To invariant Features
• To t-ples of numbers
• To indeces
• Use the t indeces to access a t-dimensional hash
  table

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Indexing Methodsfor Fast retrieval of 3D
patterns

• Select a set of target objects
• Create and store a hash table indexed by
  invariant geometric properties of the selected
  objects
• Update the databases as new objects are found
• Use the table to identify the most similar
  object for a target object.

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Reference Frame

• A 3-D reference frame (r. f.) can be defined by
  three non collinear points
• Invariant
• the coordinates of any other point in the r.f.

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Geometric Hashing PreprocessingTable
Construction

• Pick a reference frame.
• Compute the coordinates of all the other points
  in this reference frame.
• Use the triplet of coordinate as an index to
  the hash (look-up) table and record in that entry
  the record
• (object, ref. Frame)
• Repeat above steps for each reference frame.

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Geometric Hashing - Recognition

• For the target object 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 (prot., r.f., shape
• sign., pt.).

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Geometric Hashing - Recognition

• For records with matching shape sign. vote for
  the (object, r.f.).
• Compute the transformations of the high
  scoring hypotheses.
• Repeat the above steps for each r.f.

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Complexity of Geometric Hashing

• Preprocessing
• O(Nn4)
• Match Detection/Recognition
• O(Rns).
• N- number of objetcs.
• O(n)- no. of features in an object.
• s - size of a hash-table entry.
• R size of the hash table

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Shape histograms
Based on a partition of the space in which the
objects/molecules reside, i.e. a decomposition
into cells corresponding to the histogram bins.
Shell bins
Sector bins Combined bins
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Euclidean Distance

• Given two N-dimensional vectors p and q
• deuclid(p, q) S i1,N (pi qi)2 (pq)
  (pq)T

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Shortcomings of Euclidean distance
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Example Proteins
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Quadratic distance

• If AI then the quadratic distance coincides with
  the Euclidean distance

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Similarity Weights

• Setting the elements of A
• d(i,j) represents the distance of bins i and j
• The higher s the more similar A to I

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Generalization of geometric histogram Shape
Distribution

• build shape distributions from 3D polygonal
  models and compute a measure of their
  dissimilarities.
• Shape matching problem is reduced to sampling,
  normalization, and comparison of probability
  distributions,

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Shape DistributionSelecting a Shape Function

• A3 Measures the angle between three random
  points on the surface of a 3D model.
• D1 Measures the distance between a fixed point
  and one random point on the surface. We use the
• centroid of the boundary of the model as the
  fixed point.
• D2 Measures the distance between two random
  points on the surface.
• D3 Measures the square root of the area of the
  triangle between three random points on the
  surface.
• D4 Measures the cube root of the volume of the
  tetrahedron between four random points on the
• surface.

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Three steps

• 1) to select discriminating shape functions
• 2) to efficiently construct shape functions for
  each 3D model,
• 3) to compute a dissimilarity measure for pairs
  of distributions.

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Experimental Results
Plot of D2 values scaled to align their mean
values