Docstoc

mds

Document Sample
mds Powered By Docstoc
					Texture Mapping using Surface Flattening via
         Multi-Dimensional Scaling

        G.Zigelman, R.Kimmel, N.Kiryati
    IEEE Transactions on Visualization and
             Computer Graphics
                     2002
Multidimensional scaling (MDS)

   The idea: compute the pairwise geodesic distances
    between the vertices of the mesh:

               M   dist (xi , x j ) 
                            2
                                             nn

                            2
   Now, find n points in R , so that their distance matrix is
    as close as possible to M.
                                   q2


                                        q1


                                                                 2
MDS – the math details

We look for X’,
                          |          | 
                                        
                   X    x1       xn   R d n
                          |
                                     | 
                                        
such that || M’ – M || is as small as possible, where


                                             
                                               2
         M   dist (xi , x j )   xi  x j   R nn
                     2

                                                
M’ is the Euclidean distances matrix for points xi’.


                                                            3
        MDS – the math details

                                 
        Ideally, we want: M    xi  x j 
                                           2

                                                                                                                  M
                                            

                                                   x   x , x   x  
                                                              i       j             i       M
                                                                                            j



                                                  || x  ||  || x  ||  2  x , x     M
                                                          i
                                                                  2
                                                                               j
                                                                                        2
                                                                                                    i         j




 || x  || || x  ||
 1
                        || x1 || 
                                   
                                        || x  || || x  ||
                                        1
                                                                      || xn || 
                                                                                
                                                                                                       x1        |           | 
                                                                                                                                  
                1                                      2

 || x  || || x  ||   || x 2 ||     || x  || || x  ||          || xn || 
 2

                2
                                   
                                   
                                        1
                                       
                                                       2
                                                                                
                                                                                
                                                                                                                  x1        x n 
                                                                                            
                                                                                                                  |
                                                                                                                               |  
                                                                                                        xn
 || x  || || x  ||   || xn ||      || x  || || x  ||          || xn || 
 n                                
                n                       1             2                                                                         

                  want to get rid of these                                                              X T               X
                                                                                                                                    4
MDS – the math details

Trick: use the “magic matrix” J :
                       1        1
                                  n           1
                                               n
                       1                        
                   J  n       1    1
                                       n      1
                                               n
                                                
                       1
                                               
                       n             1
                                       n      1  n n
                                                 

                      a     a         a  J  0

                                    b
                                     
                                 J b  0
                                     
                                     
                                    b
                                     
                                                         5
MDS – the math details

Cleaning the system:
       || x  || || x  ||   || x1 ||   || x1 || || x 2 ||    || xn || 
       1             1
                                                                            
       || x  || || x  ||   || x 2 ||   || x1 || || x 2 ||   || xn || 
J     2             2
                                                                             2X  X   M
                                                                                      T
                                                                                                 J
                                                                           
      
       || x  || || x  ||                                                 
       n             n       || xn ||   || x1 || || x 2 ||
                                                                  || xn || 
                                                                              



                              2 X T X   JMJ
                              X T X    1 JMJ : B
                                           2


                                         X T X   B


                                                                                                      6
How to find X’

We will use the spectral decomposition of B:

                                             |  1
                                                                                            T
                       |                                       |                   | 
       X T X   B   v1                      
                                            v n 
                                                                                      
                                                               v1                 vn 
                       |                    |            n  |                   | 
                                                                                   
                                                                          T
                         |   1                    1                                            T
            |     |                                                           |       |        | 
                                                                                            
            |     |     |                                                |       |        | 
X T X    v1   vd    vn           d           
                                                             d               v1     vd       vn 
                                                                                            
                         |                                            
            |
            |
                    |
                         | 
                               d d                                    
                                                                               |
                                                                               |
                                                                                       |         | 
                                                                                                 | 
                   |                          n 
                                                                     n 
                                                                                     |           

              nd

                            X T                                       X
                                                                                                  7
How to find X’

So we find X’ by throwing away the last nd eigenvalues

                              1 v1          
                                             
                  X                        
                      
                                             
                                              
                              d vd           d n




                 X   arg min X T X   B
                          X                           L2



                     A L2          Aij 2
                                       i, j


                                                            8
Flattening results (Zigelman et al.)




                                       9
Flattening results (Zigelman et al.)




                                       10
Flattening results (Zigelman et al.)




                                       11
The end

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:2/17/2013
language:Unknown
pages:12