Transformations

1
Transformations

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

2
Inexact Alignment (slides by Prof. Haim Wolfson).
Simple case two closely related proteins with
the same number of amino acids.
Assume transformation T is given
Question how to measure an alignment error?
3
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)
4
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.
5
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
6
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

7
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.

8
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.

9
GEOMETRIC PATTERN MATCHING UNDER RIGID MOTION(C.
Guerra, V. Pascucci, 1999)

• Problem 1. Find a transformation T, if it exists
  that brings A to within a given distance, say e,
  of B, i.e. H(T(A),B)
• Problem 2. Find the minimum Hausdorff distance
  under a rigid motion
• D(A, B) min t (t(A), B)
• where t is a rigid motion

10
Hausdorff Distance

• 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.

11
Exact solution in 2D

• 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