Object Recognition

Geometric Task:

find those rotations and translations of one of the point sets which produce “large” superimpositions of corresponding 3-D points.

Given two configurations of points in the three dimensional space,

T

Geometric Task (continued)

Aspects:

•Object representation (points, vectors, segments)

•Object resemblance (distance function)

•Transformation (translations, rotations, scaling)

Transformations

Translation

Translation and Rotation Rigid Motion (Euclidian Trans.)

Translation, Rotation + Scaling

txx

txUxRx

)( txUsxTx

Distance FunctionsTwo point sets: A={ai} i=1…n B={bj} j=1…m• Pairwise Correspondence: (ak1,bt1) (ak2,bt2)… (akN,btN)

(1) Exact Matching: ||aki – bti||=0

(2) RMSD (Root Mean Square Distance) Sqrt( Σ||aki – bti||2/N) < ε

• Hausdorff distance: h(A,B)=maxaєA minbєB ||a– b|| H(A,B)=max( h(A,B), h(B,A))

Exact Point Matching in R2

1. Determine the centroids CA,CB (i.e. arithmetic means) of the sets A and B.

2. Determine the polar coordinates of all points in A using CA as the origin. Then sort A lexicographically with respect to these polar coordinates (angle,length) obtaining a sequence (φ1,r1)…(φn,rn). Let SA=(ψ1,r1)…(ψn,rn), where ψi = φi mode n – φi-1 . Compute in the same way the correspondence sequence SB of the set B.

3. Determine whether SB is a cyclic shift of SA (i.e. SB is a substring of SASA).

O(n log n)

Approximate Matching in R2, R3 (Hausdorff distance)

E- Euclidian motion (translation and rotation), |A|=m, |B|=n

1. Select from A diametrically opposing points r and k. O(m log(m))

2. For each r` from B define Tr` – translation that takes r to r`.

3. For each k` (k`!=r`) define Rk` – rotation around r that makes r,k`,k collinear.

4. Let Er`k`= Rk` Tr` . Let E`, h(E`(A),B)=minr`k` h(Er`k`(A),B).

• h(E`(A),B) <= 4*h(Eopt(A),B)

• O(n2mlog2(n))

R3:• h(E`(A),B) <= 8*h(Eopt(A),B)

• O(n3mlog2(n))

M.T. Goodrich, J.S.B. Mitchell, M.W. Orletsky

Superposition - best least squares(RMSD) rigid alignment

Given two sets of 3-D points :P={pi}, Q={qi} , i=1,…,n;find a 3-D rotation R0 and translation T0, such that

minR,T i|Rpi + T - qi |2 = i|R0pi + T0- qi |2 .

A closed form solution exists for this task.It can be computed in O(n) time.

Model Database

Scene

Recognition

Lamdan, Schwartz, Wolfson, “Geometric Hashing”,1988.

Geometric Matching task = Geometric Pattern Discovery

Remarks:

• The superimposition pattern is not known a-priori – pattern discovery .

• The matching recovered can be inexact.

• We are looking not necessarily for thelargest superimposition, since other matchings may have biological meaning.

Straightforward 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.

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.)

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.

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.).

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.

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

p1

p2

p3

Complexity of Geometric Hashing

O(n4 + n4 * BinSize) ~ O(n5 )

(Naive alg. O(n7))

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 methodParallel Implementation – straight forward

Structural Comparison Algorithms

C backbone matching.

Secondary structure configuration matching.Molecular surface matching.Multiple Structure Alignment.Flexible (Hinge - based) structural alignment.

Protein Structural Comparison

FeatureFeatureExtractionExtraction

Verification Verification and Scoringand Scoring

CBackbone

SecondaryStructures

H-bonds

GeometricHashing

Flexible GeometricHashing

Least SquareAnalysis

TransformationClustering

Sequence Dependent Weights

PDB files

OtherInputs

Rotation andTranslationPossibilities

GeometricGeometricMatchingMatching

Problems

Redundancy in representation Solution: clustering

Numerical StabilitySolution: add geometrical constraints

Accuracy is not always “the best policy”Always compute in a give error threshold

Consistency of Solution