CS559:  Computer  Graphics  

Lecture  8:  Warping,  Morphing,  3D  Transforma?on  

Li  Zhang  Spring  2010  

Most  slides  borrowed  from  Yungyu  Chuang  

Last  ?me:  Forward  warping  fwarp(I, I’, T) { for (y=0; y<I.height; y++) for (x=0; x<I.width; x++) { (x’,y’)=T(x,y); I’(x’,y’)=I(x,y); } } I I’

x

x’

T

Last  ?me:  Inverse  warping  iwarp(I, I’, T) { for (y=0; y<I’.height; y++) for (x=0; x<I’.width; x++) { (x,y)=T-1(x’,y’); I’(x’,y’)=I(x,y); } } I I’

x x’

T-1

Non-­‐parametric  image  warping  

•  Mappings  implied  by  correspondences  

•  Inverse  warping  

P’ ?

Non-­‐parametric  image  warping  

Warping  between  two  triangles  

•  Idea:  find  an  affine  that  transforms  ABC  to  A’B’C’    

A  

C  B  

A’  

C’  

B’  

Barycentric  coordinates  

•  Idea:  represent  P  using  A1,A2,A3  

Non-­‐parametric  image  warping  

P’

Barycentric coordinate

P

Turns  out  to  be  equivalent  to  affine  transform,  why?    

Non-­‐parametric  image  warping  

Gaussian

View this problem as a signal reconstruction problem

Demo  

•  hWp://www.colonize.com/warp/warp04-­‐2.php  

•  Warping  is  a  useful  opera?on  for  mosaics,  video  matching,  view  interpola?on  and  so  on.  

Image  morphing  

Image  morphing  •  The  goal  is  to  synthesize  a  fluid  transforma?on  from  one  image  to  another.  

image #1 image #2 dissolving

•  Cross dissolving is a common transition between cuts, but it is not good for morphing because of the ghosting effects.

(1-t) · Image1 + t · Image2

Image  morphing  •  Why  ghos?ng?  

•  Morphing  =  warping  +  cross-­‐dissolving  

shape (geometric)

color (photometric)

morphing

cross-dissolving

Image  morphing  image #1 image #2

warp warp

Morphing  sequence  

Image  morphing  

create  a  morphing  sequence:  for  each  ?me  t  1.  Create  an  intermediate  warping  field  (by  

interpola?on)  

t=0 t=1 t=0.33

A(0) A(1) A(0.33)

B(0) B(1) B(0.33) C(0) C(1) C(0.33)

Image  morphing  

create  a  morphing  sequence:  for  each  ?me  t  1.  Create  an  intermediate  warping  field  (by  

interpola?on)  2.  Warp  both  images  towards  it  

t=0 t=1 t=0.33

A(0) A(1) A(0.33)

B(0) B(1) B(0.33) C(0) C(1) C(0.33)

Image  morphing  

create  a  morphing  sequence:  for  each  ?me  t  1.  Create  an  intermediate  warping  field  (by  

interpola?on)  2.  Warp  both  images  towards  it  

t=0 t=1 t=0.33

A(0) A(1) A(0.33)

B(0) B(1) B(0.33) C(0) C(1) C(0.33)

Image  morphing  

create  a  morphing  sequence:  for  each  ?me  t  1.  Create  an  intermediate  warping  field  (by  

interpola?on)  2.  Warp  both  images  towards  it  

3.  Cross-­‐dissolve  the  colors  in  the  newly  warped  images  

t=0 t=1 t=0.33

A(0) A(1) A(0.33)

B(0) B(1) B(0.33) C(0) C(1) C(0.33)

More  complex  morph  

•  Triangular  Mesh  

Results  

Michael Jackson’s MTV “Black or White” http://www.youtube.com/watch?v=YVoJ6OO6lR4 (5:20 later)

Mul?-­‐source  morphing  

Mul?-­‐source  morphing  

The  average  face  

•  hWp://www.uni-­‐regensburg.de/Fakultaeten/phil_Fak_II/Psychologie/Psy_II/beautycheck/english/index.htm  

3D  Face  morphing  

http://www.youtube.com/watch?v=nice6NYb_WA Blanz and Vetter, SIGGRAPH 1998

Where  to  now…  

•  We  are  now  done  with  images  

 3D  Graphics  Pipeline  

Rendering (Creating, shading images from geometry, lighting, materials)

Modeling (Creating 3D Geometry)

3D  Graphics  Pipeline  

Rendering (Creating, shading images from geometry, lighting, materials)

Modeling (Creating 3D Geometry)

Want to place it at correct location in the world Want to view it from different angles Want to scale it to make it bigger or smaller Need transformation between coordinate systems -- Represent transformations using matrices and matrix-vector multiplications.

Next  …  

•  We  will  spend  several  weeks  on  the  mechanics  of  3D  graphics  – 3D  Transform  

– Coordinate  systems  and  Viewing  

– Drawing  lines  and  polygons  –  Ligh?ng  and  shading  

•  More  advanced:  mathema?cal  representa?on  of  lines  and  surfaces,  texture  mapping,  anima?on  

Recall:  All  2D  Linear  Transforma?ons  •  Linear  transforma?ons  are  combina?ons  of  …  

–  Scale,  – Rota?on,  –  Shear,  and  – Mirror  

2D  Rota?on  

•  Rotate  counter-­‐clockwise  about  the  origin  by  an  angle  θ  

x

y

x

y

θ

Rota?ng  About  An  Arbitrary  Point  

•  What  happens  when  you  apply  a  rota?on  transforma?on  to  an  object  that  is  not  at  the  origin?  

x

y

?

Rota?ng  About  An  Arbitrary  Point  

•  What  happens  when  you  apply  a  rota?on  transforma?on  to  an  object  that  is  not  at  the  origin?  –  It  translates  as  well  

x

y

x

How  Do  We  Fix  it?  

•  How  do  we  rotate  an  about  an  arbitrary  point?  – Hint:  we  know  how  to  rotate  about  the  origin  of  a  coordinate  system  

Rotating About An Arbitrary Point

x

y

x

y

x

y

x

y

Back  to  Rota?on  About  a  Pt  

•  Say  R  is  the  rota?on  matrix  to  apply,  and  p  is  the  point  about  which  to  rotate  

•  Transla?on  to  Origin:  •  Rota?on:  •  Translate  back:  •  How  to  express  all  the  transforma?on  using  matrix  mul?plica?on?    

Scaling  an  Object  not  at  the  Origin  

•  What  happens  if  you  apply  the  scaling  transforma?on  to  an  object  not  at  the  origin?  

Composing  rota?ons,  scales  

Rotation and scaling are not commutative.

Inver?ng  Composite  Transforms  

•  Say  I  want  to  invert  a  combina?on  of  3  transforms  

•  Op?on  1:  Find  composite  matrix,  invert  

•  Op?on  2:  Invert  each  transform  and  swap  order  

Inver?ng  Composite  Transforms  

•  Say  I  want  to  invert  a  combina?on  of  3  transforms  

•  Op?on  1:  Find  composite  matrix,  invert  

•  Op?on  2:  Invert  each  transform  and  swap  order  

•  Obvious  from  proper?es  of  matrices  

Homogeneous  Transform  Advantages  

•  Unified  view  of  transforma?on  as  matrix  mul?plica?on  – Easier  in  hardware  and  solware  

•  To  compose  transforma?ons,  simply  mul?ply  matrices  – Order  maWers:  BA  vs  AB  

•  Allows  for  transforming  direc?onal  vectors  

•  Allows  for  non-­‐affine  transforma?ons  when  last  row  is  not  (0,0,1)  

Direc?ons  vs.  Points  

•  We  have  been  talking  about  transforming  points  

•  Direc?ons  are  also  important  in  graphics  – Viewing  direc?ons  – Normal  vectors  

– Ray  direc?ons  •  Direc?ons  are  represented  by  vectors,  like  points,  and  can  be  transformed,  but  not  like  points  

(1,1) (-2,-1)

x

y

Transforming  Direc?ons  

•  Say  I  define  a  direc?on  as  the  difference  of  two  points:  d=a–b  – This  represents  the  direc)on  of  the  line  between  two  points  

•  Now  I  transform  the  points  by  the  same  amount:  a’=M*a+t,  b’=M*b+t  

•  d’=a’–b’=M*(a-­‐b)=M*d  

•  Transla?on  does  not  maWer  

Homogeneous  Direc?ons  •  Transla?on  does  not  affect  direc?ons!  •  Homogeneous  coordinates  give  us  a  very  clean  way  of  handling  this  

•  The  direc?on  (x,y)  becomes  the  homogeneous  direc?on  (x,y,0)  

•  M  can  be  any  linear  transforma?on:  rota?on,  scaling,  etc  

Homogeneous  Direc?ons  •  Transla?on  does  not  affect  direc?ons!  •  Homogeneous  coordinates  give  us  a  very  clean  way  of  handling  this  

•  The  direc?on  (x,y)  becomes  the  homogeneous  direc?on  (x,y,0)  

•  M  can  be  any  linear  transforma?on:  rota?on,  scaling,  etc  –  Uniform  scaling  changes  the  length  of  the  vector,  but  not  the  direc?on  

–  How  to  represent  this  equivalence    

Homogeneous  Coordinates  •  In  general,  homogeneous  coordinates  (x,  y,  w)    

•  How  to  interpret  the  case  for  w  =  0?  •  Point  at  infinity:  direc?onal  vector  

•  (1,2,1)    (2,4,2)  •  (2,3,0)    (6,9,0)  

Transforming  Normal  Vector  

•  Why  normal  vectors  are  special?  

Transforming  normal  vectors    

tangent normal

tangent’

normal’

M

If M is a rotation,