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L08_MDS

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					Computer Science
Department              Technion-Israel Institute of Technology




          Flattening via Multi-
          Dimensional Scaling
                      Ron Kimmel

               www.cs.technion.ac.il/~ron



               Geometric Image Processing Lab
                   Outline

 On isometric surfaces and bending invariant
  signatures.
 Multi-dimensional scaling techniques:
   Classical.
   Least squares.
   Fast.
 Surface classification: experimental results.
                    Matching Surfaces
 Problem:
 Given 2D surfaces, define a measure of their similarity.
 Classical techniques:
   Find a rigid transformation that maximizes some
     measure.
   Match key points on the surface.
   Compare local or semi-differential invariants, e.g.
     matching graphs.
                         Bending Invariant Signatures

 Isometric surface matching via bending invariant signatures:
    Map the surface into a small Euclidean space, in which
     isometric surfaces transform to similar (rigid) surfaces.
 Advantages:
    Handle (somewhat) non-rigid objects.
    A global operation, does not rely on selected
     key points or local invariants.
                               Bending invariant signatures
                               Basic Concept
  Input – a surface in 3D


 Compute geodesic distance matrix [D]
   between each pair of vertices
    δ ij= geodesic_distance(vertex i ,vertex j )
    [D] ij = δ ij


  Extract from [D] coordinates in an m
  dimensional Euclidean space via
       Multi-Dimensional Scaling (MDS).

Output – A 2D surface embedded in
             m
         in R (for some small m)
Geodesic
           Fast Marching on Surfaces
Distance
                          Euclidean
                          Distance



           Source point
                       Multi-Dimensional Scaling
 MDS is a family of methods that map similarity
  measurements among objects, to points in a small
  dimensional Euclidean space.
 The graphic display of the similarity measurements
  provided by MDS enables to explore the geometric
  structure of the data.      x
                              i




Dissimilarity          MDS           coordinates in m-dimensional
 measures                                  Euclidean Space

                                                 2
                                    Σ(δ ij - dij )
             Stress
            Function     Stress =
                                         Σδ ij
                                            A simple example
                    1       2         3          4           5          6         7   8          9           10
Y                                                                               Stockholm
1. London               0
          Dublin
2. Stockholm        569         0

3. Lisbon           667     1212
                                     Amsterdam
                                       0                                                                           Rotation
                                                                 Berlin
4. Madrid           London
                    530 1043           201           0

5. Paris            141     617        596       431             0Prague                                          Reflection
                                    Paris
6. Amsterdam        140     446        768       608          177           0
7. Berlin           357     325        923       740          340       218           0
8. Prague           396
                 Madrid     423        882       690         Rome 272
                                                              337                 114       0
9. Rome
Lisbon              569     787        714       516          436       519       472      364       0
10. Dublin          190     648        714       622          320       302       514      573   755          x
                                                                                                              0


    1        2          3       4           5            6        7              8         9         10
12.7         0      32.8    29.4          14.6       8.8          4.5           6.4       16.4   13.8
    4.1    13.9     13.2    14.9          9.0        8.2         16.6           19.6      25.4           0
                              Flattening via MDS
 Compute geodesic distances between pairs
  of points.
 Construct a square distance matrix of
  geodesic distances^2.
                                                     0
 Find the coordinates in the plane via multi-           0
  dimensional scaling.                                       0
  The simplest is `classical scaling’.
 Use the flattened coordinates for
   texturing the surface, while preserving
      the texture features.
        Zigelman, Kimmel, Kiryati,
        IEEE TVCG 2002
       Grossmann, Kiryati, Kimmel, IEEE TPAMI 2002


    Bending invariant surface matching
       Elad (Elbaz), Kimmel CVPR 2001
Flattening
Flattening
Distances - comparison
Texture Mapping
Texture Mapping
                    Fattening via MDS: 3D

  Original                   Original
             Fast                       Fast




 Least       Classical        Least     Classical
Squares                      Squares
                  Fattening via MDS: 3D

Original                    Original
               Fast                       Fast




 Least     Classical       Least       Classical
Squares                   Squares
Input Surfaces
                          Bending Invariant Signatures



                    ?




Elad, Kimmel, CVPR’2001
                          Bending Invariant Signatures



                    ?




Elad, Kimmel, CVPR’2001
                          Bending Invariant Signatures



                    ?




Elad, Kimmel, CVPR’2001
                          Bending Invariant Signatures




Elad, Kimmel, CVPR’2001
                          Bending Invariant Signatures




Elad, Kimmel, CVPR’2001
                          Bending Invariant Signatures
Original surfaces                 Canonical surfaces in R3




Elad, Kimmel, CVPR’2001
                                                                 Bending Invariant Clustering

 2nd moments based MDS for clustering
           Original surfaces                                                                                 Canonical forms
                                                         D
                                  D     D
     0.8                           A                                                           0.8
                        A                                            E
                                                                     E
     0.7                                                                 A                     0.7
                                                                         D                                                           E
     0.6                                                                 E B                   0.6                              EE
                                                                                                                                                                          C
     0.5                                                                   C                   0.5
                                                                           C                                                                                              C
                                                                                                                                                         D                 C
     0.4                                                         B             C               0.4                                                        DD
                                                                               E                                                                                D
     0.3                                                                                       0.3                                                  BB
                  F                                          A                                                                                     B B
     0.2          F                              B                                             0.2
                                                         B       C                                                                                               A A
                                                                                                                                                                AA
     0.1                     F                                                                 0.1
                                             F
                                                                                           1                                                                              1
      0                                                                              0.8        0                                                                   0.8
      1                                                                                         1
            0.8
                  0.6
                                                                               0.6
                                                                                     *A=human body
                                                                                                     0.8
                                                                                                           0.6         FFF                                0.6
                                                                         0.4                                            F                          0.4
                            0.4
                                       0.2                       0.2                 *B=hand                     0.4
                                                                                                                          0.2                0.2
                                                     0   0                           *C=paper                                        0   0
                                                                                     *D=hat
                                                                                     *E=dog
                                                                                     *F=giraffe



Elad, Kimmel, CVPR’2001
                          Classical Scaling                                   Young et al. 1930


Given n points in        R   k
                               , denote            pi  [ xi1 , xi2 ,..., xik ]T

Define coordinates vector                          P  [p1 , p 2 ,..., p n ]                 T


The Euclidean distance between 2 points:

                           x            x    x                             x 
                              k                       k
                     2                       l 2                l 2                    l 2
      d  pi  p j
        2
       ij
                                      l
                                      i      j                  i      2x x
                                                                          l
                                                                          i
                                                                               l
                                                                               j       j
                             l 1                    l 1
                                                            2
                          p i  2p i p  p j
                                  2           T
                                              j
                   Classical Scaling

Define the `centering’ matrix J  I  n 11T
                                       1

where 1  [1,1,...,1]T     p1 2 p1 2 ... p1 2 
            
               n           2        2          2

Let the n  n matrix Q  
                           p2   p2      ... p 2 
                             .     .     ...   . 
                           2        2          2
                           pn
                                pn      ... p n 
                                                 
We have that QJ  0 and also    QJ T    JQT  0

Thus,    JDJ  J(Q  Q T  2PP T )J
                                            ~~ T
            JQJ  JQ J  2JPP J  0  0  2PP
                       T          T
                   Classical Scaling
                ~                              1 n l
The coordinates P are related to P by ~i  xi   xm
                                      x l   l
                             ~~ T              n m1
Thus the operation  2 JDJ  PP
                     1

               is also called `double centering’.
Applying SVD, we can compute  1 JDJ  UU
                                                  T
                                   2
        where UU T  I
                                           ~
and the coordinates can be extracted as P  U
                                                    12

If we choose to take only part of the eignstructure,
   then, our approximation minimizes the Frobenius
   norm
                       ˆ
                    U  U    T
                                  F
                     MDS

Matlab code for 2D flattening

           J  eye ( n )  ones ( n ). / n;
           B  0.5 * J * D * J ;
          [Q , L]  eigs ( B,2, ' LM ' );
           newy  sqrt ( L(1,1)). * Q (:,1);
           newx  sqrt ( L( 2,2)). * Q (:, 2);



                                   Zigelman, Kimmel, Kiryati IEEE TVCG 2002
                          Classical Scaling

The eigenvalues are the 2nd order moments of the
  flattened surface, since by definition

      ~i 2  U[i, :]1 2 U[i, :] 1 2 T   U[i, :] U[i, :] T  
    xm
    n

                      i             i         i                       i
   m 1


all the cross 2nd order moments vanish by the unitarity
    of U, thus `flattening’.
                                                                                  Conclusions
       A method for bending invariant signatures
       Based on:
         Fast marching on surfaces
         MDS LS/Classical/Fast
       Results:
         Texture mapping
         Bending invariant signatures
         Classification of isometric surfaces.
                                               D
                           D     D                                                              0.8
0.8               A         A
                                                        E
                                                        E                                       0.7
0.7                                                         A                                                                      E
                                                            D                                   0.6                             EE
0.6                                                         E B                                                                                                     C
                                                                                                0.5
0.5                                                           C                                                                                                      C
                                                                                                                                                                     C
                                                              C                                                                                      DD
                                                                                                0.4                                                   D
0.4                                                 B          C                                                                                          D
                                                               E                                0.3                                             BB
0.3                                             A                                                                                              B B
            F
                                                                                                0.2
0.2         F                             B
                                               B    C                                                                                                      AA A
                                                                                                                                                            A
0.1                    F                                                                        0.1
                                      F                                                                                                                             1
 0                                                                            1                  0
 1                                                                      0.8                      1                                                            0.8
      0.8                                                         0.6                                 0.8
                                                                                                            0.6         FFF
                                                                                                                         F                     0.4
                                                                                                                                                     0.6
            0.6                                             0.4                                                   0.4
                      0.4                           0.2                                                                   0.2            0.2
                                0.2
                                              0 0                                                                                  0 0
                       Least Squares MDS

 Standard optimization approach to solve the
  minimization problem of the stress cost function.
 Solved via ‘scaling by maximizing convex function’
  (SAMCOF) algorithm.
 Starting with a random solution and iteratively
  minimizing another stress function, which satisfies.

                        ƒ (x,z) ≥ ƒ (x) for x ≠ z
                          and ƒ (z,z) = ƒ (z)


 The complexity is O(n 2 ).
 Converges to the optimal solution.
                          Fast MDS

 The fast MDS: heuristic efficient technique O(mn).
 Works recursively by generating a new dimension at each step,
   Providing m-dimensional coordinates after m recursion steps.
 Project the vertices on a selected ‘line’.
    First, the algorithm selects the
      Farthest two vertices.
    Next, all other vertices are projected
      On that line using the cosine law.
    Next step is to project all items to
      An (n-1) hyper plane (H) that is
      Perpendicular to the line that
      Connects those vertices.
    Generate a new distance matrix.
    Repeat the last three steps m times.

				
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posted:3/31/2012
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