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# Mutual Information Algorithm applied to rigid registration

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### Mutual Information Algorithm applied to rigid registration

1. 1. Mutual  Informa-on  Algorithm   applied  to  Rigid  Registra-on   Vibha  Chaswal,  Ph.D.
2. 2. Registra-on     •  Image  registra-on   –  Deﬁne  geometric  transforma-ons  T  that  will  map  co-­‐ ordinats  between  one  image  onto  another  image  such  that   some  image  quality  criterion  is  maximized.   –  also  referred  to  as  image  fusion,  superimposi-on,   matching  or  merge
3. 3. Registra-on  algorithms     •  Used  to  ﬁnd  the  transforma-on     •  Rigid  &  aﬃne   –  Landmark  based   –  Edge  based   –  Voxel  intensity  based   –  Informa,on  theory  based   •  Non-­‐rigid   –  Registra-on  using  basis  func-ons   –  Registra-on  using  splines   –  Physics  based   •  Elas-c,  Fluid,  Op-cal  ﬂow,  etc.
4. 4. Rigid  Body  Registra-on  of  medical   images   •  The  anatomical  and  pathological  structures  do   not  deform  during  image  acquisi-ons   •  Tissue  deforma-ons  ignored  and  register   images  using  rigid  body  transforms   •  only  rota-ons  and  transla-ons   •  6  degrees  of  freedom:  3  transla-ons  and  3   rota-ons   •  Key  Characteris-c:  All  distances  are  preserved
5. 5. 3D  Rigid-­‐body  Transforma-ons   •  A  3D  rigid  body  transform  is  deﬁned  by:   –  3  transla-ons  -­‐  in  X,  Y  &  Z  direc-ons   –  3  rota-ons  -­‐  about  X,  Y  &  Z  axes   •  The  order  of  the  opera-ons  maZers   Transla-ons   Pitch   about  x  axis   Roll   about  y  axis   Yaw   about  z  axis
6. 6. Informa-on  theory  based  Rigid  body   Registra-on   •  Image  registra-on  is  considered  as  to  maximize  the   amount  of  shared  informa-on  in  two  images   –  reducing  the  amount  of  informa-on  in  the  combined   image     •  Algorithms  used   –  Joint  entropy   •  Joint  entropy  measures  the  amount  of  informa-on  in   the  two  images  combined     –  Mutual  informa,on   •  A  measure  of  how  well  one  image  explains  the  other,   and  is  maximized  at  the  op,mal  alignment   –  Normalized  Mutual  Informa-on
7. 7. Measures  of  Informa-on   •  Hartley  deﬁned  the  ﬁrst  informa-on  measure:   –  H  =  n  log  s   –  n  is  the  length  of  the  message  and  s  is  the  number  of   possible  values  for  each  symbol  in  the  message   –  Assumes  all  symbols  equally  likely  to  occur   •  Shannon  proposed  variant  (Shannon’s  Entropy)   •  weighs  the  informa-on  based  on  the  probability  that  an  outcome   will  occur   •  second  term  shows  the  amount  of  informa-on  an  event  provides   is  inversely  propor-onal  to  its  probability  of  occurring
8. 8. Three  Interpreta-ons  of  Entropy   •  The  amount  of  informa-on  an  event  provides   –  An  infrequently  occurring  event  provides  more   informa-on  than  a  frequently  occurring  event   •  The  uncertainty  in  the  outcome  of  an  event   –  Systems  with  one  very  common  event  have  less   entropy  than  systems  with  many  equally  probable   events   •  The  dispersion  in  the  probability  distribu-on   –  An  image  of  a  single  amplitude  has  a  less  disperse   histogram  than  an  image  of  many  greyscales   •  the  lower  dispersion  implies  lower  entropy
9. 9. Joint  Entropy  for  Image  Registra-on   •  Deﬁne  a  joint  probability  distribu-on:   –  Generate  a  2-­‐D  histogram  where  each  axis  is  the   number  of  possible  greyscale  values  in  each  image   –  each  histogram  cell  is  incremented  each  -me  a  pair           (I_1(x,y),  I_2(x,y))  occurs  in  the  pair  of  images   •  If  the  images  are  perfectly  aligned  then  the  histogram  is  highly   focused.    As  the  images  mis-­‐align  the  dispersion  grows   •  recall  Entropy  is  a  measure  of  histogram  dispersion
10. 10. Entropy  for  Image  Registra-on   •  Using  joint  entropy  for  registra-on   –  Deﬁne  joint  entropy  to  be:   –  Images  are  registered  when  one  is  transformed  rela-ve   to  the  other  to  minimize  the  joint  entropy   –  The  dispersion  in  the  joint  histogram  is  thus  minimized
11. 11. Joint  entropy:  overlap  problem   aligned   MR/MR   MR/CT   MR/PET   2mm   5mm   •  Joint  entropy  very   sensi-ve  to   mapping  of   posi-on  and   intensity   •  ‘blur’  with   increasing   misregistra-on   •  May  lead  to   incorrect  solu-on   Figure  from  Hill  et.al.,  Voxel   Similarity  measures  for  automated   image  registra3on,  1994,  Proc.  SPIE,   2359
12. 12. Solu-on:  Mutual  Informa-on    A  solu-on  to  the  overlap  problem  from  which   joint  entropy  suﬀers  is  to  consider  the   informa-on  contributed  to  the  overlapping   volume  by  each  image  being  registered   together  with  the  joint  informa-on.  The   informa-on  contributed  by  the  individual   images  is  simply  the  entropy  of  the  por-on  of   the  image  that  overlaps  with  the  other  image   volume
13. 13. Deﬁni-ons  of  Mutual  Informa-on   •  Three  commonly  used  deﬁni-ons:   –  1)  MI(A,B)  =  H(B)  -­‐  H(B|A)  =  H(A)  -­‐  H(A|B)   •  Mutual  informa-on  is  the  amount  that  the  uncertainty  in  B  (or   A)  is  reduced  when  A  (or  B)  is  known.   –  2)  MI(A,B)  =  H(A)  +  H(B)  -­‐  H(A,B)   •  Maximizing  the  mutual  info  is  equivalent  to  minimizing  the   joint  entropy  (last  term)   •  Advantage  in  using  mutual  info  over  joint  entropy  is  it  includes   the  individual  input’s  entropy   •  Works  beZer  than  simply  joint  entropy  in  regions  of  image   background  (low  contrast)  where  there  will  be  low  joint   entropy  but  this  is  oﬀset  by  low  individual  entropies  as  well  so   the  overall  mutual  informa-on  will  be  low
14. 14. Deﬁni-ons  of  Mutual  Informa-on  II   –  3)   € ⎛ p(a,b) ⎞ I(A,B) = ∑ p(a,b) ⋅ log⎜ ⎟ p(a) p(b) ⎠ ⎝ a,b •  This  deﬁni-on  is  related  to  the  Kullback-­‐Leibler  distance   between  two  distribu-ons   •  Measures  the  dependence  of  the  two  distribu-ons   •  In  image  registra-on  I(A,B)    will  be  maximized  when  the   images  are  aligned   •  In  feature  selec-on  choose  the  features  that  minimize  I(A,B)   to  ensure  they  are  not  related.
15. 15. MI  algorithm
16. 16. Registra-on  using  MI  Maximiza-on   Rigid  Registra,on   Aﬃne  Registra-on   Non-­‐rigid  Registra-on