a new algorithm for generating quadrilateral meshes and its

7 downloads 0 Views 1MB Size Report
Keywords: Quadrilateral mesh, mesh generation, finite element method, image registration. 1. ... ages using algorithms for meshing polygonal domains.
A NEW ALGORITHM FOR GENERATING QUADRILATERAL MESHES AND ITS APPLICATION TO FE-BASED IMAGE REGISTRATION S. Ramaswami

M. Siqueira 

T. Sundaram

J. Gallier

J. Gee

½ Rutgers University, Camden, NJ 08102, USA, [email protected] ¾ University of Pennsylvania, Philadelphia, PA 19104, USA, marcelos,tessa@seas.upenn.edu ¿ Universidade Federal de Mato Grosso do Sul, Campo Grande, MS 79070-900, Brazil.

ABSTRACT

                               

                                                       !                       "   #                  $       !                    %       $                     $        $                              #     

                   1.

INTRODUCTION

                        &  $    

                           $                       "                    $                  '      Æ                   

                                        ()*                                                          (+* ,   

      $                     $       ! (- . / 0*      (.*             (1 2*       3    #        $                              $     4  $   $                              (5*                            ()6* "                                                        !                    

                                             "                   $        7    $                             8      $          

      $        

 )26           $                   #                       $          !                "                                      $          9 +      $    %                                                           $          9 - $                                   #     ! ,           8     $                                  :;          

                       !    " 8  +               #       # " 8  -                  $          " 8  .                           " 8  /         7                         $      " 8  0    !         # 2. BACKGROUND AND RELATED WORK

       $                                 $ 

             

    $         

                                 ())* "      

     $  $                 $                          $   <                           $    (- )+ / 1 )- 2*             $  $                

           =                                     Ê             Ì    Ê  Ì    '     #    > + 9+  +                                                               $         ? (-* 3            - $         / 9 /  /  +        "           !     $                    

   (-*                        $           $     3   !     $                                   ?'               @         

         

          $          ()+*          

         A        $    3     

     ()+*                $    8     (1*          $     #   

                                      #              $   !                     :            $                                         $    =  B      (2*

                   $                $    ?           $                      

   !      3                                     

         #                  $                                     $    ,    ()-*     $         #              "                $                  

   C                                    =                             ,         

              (1 2*       ()-*     (1 2*            !             $  $                  (- )+ )-* #              (1 2* #                  :

                      (1 2*                         3       $

                            

    ().* ?                                      $                                3. THE ALGORITHM

"             

                   $       !                              D                       8      E           $ 

                   $                              $         9 + $          9+ 8               8      

                D                                $                   $         9-         9+ 8                            3.1 Polygonal Regions with Holes

          $                

          $                                       1                         $         ?       

           

  

                                      =        8  + =    

                                                                              

                    ?8                          D                                     6 )                      6 D                             =

                                  D        

                )                   

  %               =

           % 

                                           

                               

                 . / 0  1         

             $      8              "         $   

                  "  $                                        

                                                                         %                     $             8      "           F    G $                                   

                               $   ?                                           >                                             ) 8              8                  

        + Ú

Ú





!



  



   

  



 





   

Figure 1: The non-empty triangle H.

"                            H          D         

     H        ) "           

         $                 8            

           %         <                   "#$   

                   %

     

                       

    

          

     

      

                                                   

+

-







   



   

!  

 











Figure 2: (a) Degenerate quadrilateral. (b) Degenerate pentagon.

   

        

     

   !             "    #   

     !  

  !                " #   

 "  #              

!         

   

        

                  !                           $     !    

              





%            

                          $                      8  -- = )            %                    $                D 

             )   +              %              

                =

   8                  $         

  %             $                    +            %                        =     

 8                       

  "                   $          H      + D              > H D 

       $    

          3  ,  -)+  #              $         $                   %                 =             %         

       -    .
                 

            "             

                               "            $                 <   8                

                 A     . ? =  -++  $           $  

 8      ;         3              $          8      "    

      

        $     

  

     

   

 

   !

&    '                    

                  

  



 ! 



   

 



                ,  -).        ?8

                       

!  %    

   =           >   I   8              $        8     ¼                  $      H A            

      /                  H       $     #      $        

         ? =  -+)

               $          8                     

                 







   





   







   

 





  

¼



&   

















   

Figure 6: Cases i and ii of Step 4c.



   

   



 



  

&   





  









!



   

   



  

 









   



 

    



   







   



Figure 5: Cases 4a and 4b of Step 4.

! %    

        

              >        / "                                                                   

   ) "              =  -+-                $          H A            

       8      ;               H !               

  

 =            >   "            $            

                    ,       

       > 

               0   1    2<                D  

 >          ?8  I   8             $         



 



  

         ,    >                  %   =  -+-            

         F  G                          $      H A           8      8               $     8                 "                      H "                  

        $                

 



  







  







  



Ú





  







  

  

Ú 

Ú

Figure 7: Case iii of Step 4c (dashed edges are cross-edges).



  

 "  >                  =  -+) "  >   

        $                       =  -++ "        $                 

    

8              ;              

  

     

   &    

             ?8           2< J K                        =  -+. "  K   

                $         

      =  -++    

             =  -+/ "                   $           

 8          +                                "          8  )K-  8  .       "

                            

                             $                      "     $        8            $ "        8       

                            $   !      8              







 ½

 ¾ ¿









½

 ¾ ¿





Figure 8:  for case iv of Step 4c (dashed edges are cross-edges).

"  (       

!                      ! ! 

                 !                          !    )     

   

9+

9+



"                  8         *

                  8       A   

                    8          9 +                       $                 $          

      9 +           $      $                                                        )

  + "                                  $             3.2 Small Polygonal Regions

"                    $                     . / 0  1                                %          I  

   #    

   (0*  ()/*

#  +,- &     !  

          !      

       

#  +,- &         

    !      .       !      

       

                                                                 

 % $        8                       8                   $            $                A                        


  

  !   '                                '    0      '

      !                   '                         !   

#  +'1- /   !        !

      )                    2 '      '               !              0     !                              !   

          0     !                             !          !            

   

Ì >

# ! +'1- /   !     

        !       )         !                 0     !             

"                                                          

               

             

# % +'1- /   !

     

                  /         

      !     )     !              0     !           

Ì >

3.3 Constrained Quadrilateral Meshes

,       $                         #                                     8  /       $                                )6 ?        

             

            

           

 $  

=              

   =                                               " 

     $     

             

 >                            A                      =                   -))       $      

   9 + $         9 + 8      D        8                A                       "     

            $      $        9 - 

       9 + 8      3.4 Implementation and Results

%           J99     3   (    & /!  JL =         5         --.0        )                    $       ()0*   5   $      )10+ $          5               !   06M        

    5                     (- )-* ,                               $     %      $                   !   5                5         ()1*       ()2* 4.

MESHES FROM IMAGES

I                              <                

             

  ().*     #      

                             #                           

                                                                     ()5*

Figure 10: Contours of a human brain image.







Figure 9: (a) Triangular mesh of “Lake Superior”. (b) Quadrilateral mesh of Lake Superior. (c) Mesh in (a) after post-processing.

                         

                                                                                      )6                             

5. AN APPLICATION                                    (+6*                                   

                    &    

     !               "                         !        ?  (+)*                         

              =  L  ?A (++*           #                      

      ? '      

   L  ?A'  #     

   "  8     ;     # " N       D  =   :   D=:     

  #                        ¼                    ))        

         " N 3  

 " N       L  ?A'        #      $   $           8  -     $                                   %                  +               :;       $  +/0  +/0                 

    D       8  .              

           

    $            %     ()0*                  +6  +/  -6   --     $                 8  - D        $                      %           " N          2  2 .  . +  +  )  )        )          

            : )K.             +6  +/  -6   --     : /K2  $        )K. : 5K)+         /K2 : )-K)0        2  2 .  . +  +  )  )        









Figure 11: (a) Source image. (b) Target image and its associated mesh. (c) Image resulting from warping image in (a). (d) Subtraction of (c) from image in (b).

            +6                        ) : )K.        /K)0           $     -  .

                $                                    )  -  

          +6     IJ    "  I  """   +/0:?  ; :   %  52 %                       $ ;:8  &                          ¼      +   !         +6    

& '(  ')  '( ) + . / 0 1 2 5 )6 )) )+ )). )/ )0

+5+) -/.5 .5). 2+/. )0./ )5.) +/2) .-)2 )11+61+1.1 ..55 )6+. .650 )0-2. 0//-0

).1+ )156 +.2) .)1)0/1 )5/1 +06/ .-0. )12/ +625 +11) ././ )625 .++/ )00.) 006.5

.-5+ /--2 1-5. )+.+0 --6) -251 /)2/ 202) -//1 .)0) //)1 56.+))+ 2-+6 --6+. )-)/2.

Table 1: Size of the meshes 1–16.

          )  +           !                 ;:8 "   $    /K2    06M                    )K.        $

     5K)+   /K2         )6M  $          $      5K )+     0)M             )K.          !  ;:8       $                ;:8       $              ;:8                    )  -          D 

     $   

 $           

 $         

              

              

   . $       

           -    

& *  +   ,- +& ) ) + + . . / 0 1 2 5 )6 )) )+ )). )/ )0

) ) ) ) . . . . . . . . . . . .

)6 )1 )+ +) )1 -6 -+ /+ +) +. -/5 ++1 -/ 0+ )+ .0 +6/ )66)

)15/ )1+/ )1-/ )052 )1+6 )02) )012 )00+ )101 )1)+ )05)002 )1.. )1.1 )05+ )000 )2/0 )055 )0)) )/5-

Table 2: Summary of the registration results.

C    1  ))   1/M   $             ).          ;:8                                   

               

            ))            

     )+      -   )-  )-   $    1    -       8  -  $     ))                 1    6. CONCLUSIONS

%                                  $   ,                        & 

          (-*           $       !                 

 $                          

 (1 )- 2* %       $                 

                             ,        

 $                                     !    

               

           #           !           $  

        $         $                $           %             8  /          #      $         (1 )- 2*                                                     ACKNOWLEDGEMENTS

%   #  #      

        '         D8 L D JJ;6+6.+5-             JDI$ ?!  References

()* 8# @ F%   L =     O "    J    C  : G * )) 45  ))/K)+0 +66+ (+* ? :     F: L   ,

    G  3 P   

 3  %

 (   %  8   )55+ (-* 

3 = % ,  : 8   Q  @ F8    J C   !

  I G * . 3  3 3 (   11K2+ )55+ (.* ? :     FC   :   J   I# G * 0 45  1K)5 )551 (/* ;  8 ;  I   L FJ        C  G 3   (  2 )   &  

   5 +/1K+10 )552 (0* ?   3   ;  8 8   R B F8  J C    I  8 G *    &  3   &&3 =DJ8 +++-  0+-K0-/ 8 B  +66)

(1* 8  N =  @3 "   FC   :        J    I#   8$ J G * 1 45  0)K 10 )552 (2* B   D 8  N "   FC    :            J    J  I#   ;   J G

* 5 45  +)1K++/ +666 (5* :    L  % FJ   8   Q   : :      C   G * 0 45  .-1K..1 )551 ()6* 3  Q N  F     L  =   -      

:  :  ;   G * %%% $6 &  )2.K)5+ )552 ())* =   F          $   G * &34  3 (   51K)60 )52/ ()+* @  ? 8  @ N #  F     J         :  C   :G  7  8 4   %   -)  ) 01K2. )55) ()-* , 8 8   : J 8 8  8 FC :<  "       C : G  7  8 4    %   5  .. )-)1K)-.6 )555 ().* L! ! ; % ; 9    *   %  +    +66+ ()/* ;  8 8 $  : 8  L   @ L @ FC   :   ? "  ;   G  ; :8J"8 6-)0    J    "   8  Q    I   I     I  Q8  +66()0* 8# @ F   <     + C  : L        G '  $:   & 3 (   )+.K)-- I    I  Q8  : )550 ()1* P  8  N F   ? 

    : 8   G * 5

45  -1-K-2. +666 ()2* N  I FJ Q< "   C         :G * 0 45  ..5K .0) )551 ()5* 3 @ 8 # @ F8  Q   I# = 8      G * 1       9  ;     )  )-.K).- )55+

(+6* 3A @ 3#  3  4     5     J;J I +66) (+)* ? J