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

Distance Functions Two point sets: A={a i } i=1…n B={b j } j=1…m Pairwise Correspondence: (a k 1,b t 1 ) (a k 2,b t 2 )… (a k N,b t N ) (1) Exact Matching: ||a k i – b t i ||=0 (2) RMSD (Root Mean Square Distance) Sqrt( Σ||a k i – b t i || 2 /N) < ε Hausdorff distance: h(A,B)=max aєA min bєB ||a– b|| H(A,B)=max( h(A,B), h(B,A))

Exact Point Matching in R 2 1.Determine the centroids C A,C B (i.e. arithmetic means) of the sets A and B. 2. Determine the polar coordinates of all points in A using C A as the origin. Then sort A lexicographically with respect to these polar coordinates (angle,length) obtaining a sequence (φ 1,r 1 )…(φ n,r n ). Let S A =(ψ 1,r 1 )…(ψ n,r n ), where ψ i = φ i mode n – φ i-1. Compute in the same way the correspondence sequence S B of the set B. 3. Determine whether S B is a cyclic shift of S A (i.e. S B is a substring of S A S A ). O(n log n)

Approximate Matching in R 2, R 3 (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 T r` – translation that takes r to r`. 3.For each k` (k`!=r`) define R k` – rotation around r that makes r,k`,k collinear. 4.Let E r`k` = R k` T r`. Let E`, h(E`(A),B)=min r`k` h(E r`k` (A),B). h(E`(A),B) <= 4*h(E opt (A),B) O(n 2 mlog 2 (n)) R 3 : h(E`(A),B) <= 8*h(E opt (A),B) O(n 3 mlog 2 (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={p i }, Q={q i }, i=1,…,n; find a 3-D rotation R 0 and translation T 0, such that min R,T  i |Rp i + T - q i | 2 =  i |R 0 p i + T 0 - q i | 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 the largest 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(n 7 ). (O(n 3 ) 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 p1p1 p2p2 p3p3

Complexity of Geometric Hashing O(n 4 + n 4 * BinSize) ~ O(n 5 ) (Naive alg. O(n 7 ))

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

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

Protein Structural Comparison FeatureExtraction Verification and Scoring CC Backbone Secondary Structures H-bonds Geometric Hashing Flexible Geometric Hashing Least Square Analysis Transformation Clustering Sequence Dependent Weights PDB files Other Inputs Rotation and Translation Possibilities GeometricMatching

Problems Redundancy in representation Solution: clustering Numerical Stability Solution: add geometrical constraints Accuracy is not always “the best policy” Always compute in a give error threshold Consistency of Solution