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					Spatial Preprocessing II


     Jesper L.R. Andersson
 Karolinska MR-Centre, Stockholm,
             Sweden
                      Outline
• Spatial Normalisation
   – Why?
   – How?
• Distortion Correction
   – Why?
   – How?
• Movement-by-susceptibility interaction
   – What is?
   – How?
         Movement Correction: How?
  How to make a new image when we know the movement

     Start with a       Write (inferred) value into blank sheet
     blank sheet



                                Transform
                                co-ordinate
                                         x
For each voxel-centre               
                                                       Find co-ordinate
     co-ordinate                                       in original image
                            x=10mm
                            y=0
                            =10º
 Movement Correction: How?
             How do we know the movement?
                           f1            f2
Let’s say we                                        And we want to find the
have two                                            difference in position
images: f1 and f2                                   between them

                           f1                       f1                   f1
  f2-f1                    x                        y                    


              x                    +y                       +


                    “The way the              “The way the           “The way the
Observed            difference image          difference image       difference image
difference          would have looked         would have looked      would have looked
                    had there been a          had there been a       had there been a
                    1mm x-translation”        1mm y-translation”     1 degree rotation”
    Spatial Normalisation: Why?
• We want to pool results across subjects
• We want to report results in a concise format
      Me         Someone else
                                • Should we pool
                                  “activations” in
                                  the yellow voxel?
                                • Is it meaningful
                                  to say I activated
                                  voxel [25 60 20]?
         Spatial Normalisation: How?
                    What is a “displacement-map”?
                                                       Go to original image
   For each voxel-centre                                and find intensity at
      in blank sheet.                                  “warped” co-ordinate


                                    Get position in
                                   original space by
                                   adding pertinent
                                    displacement.



 x '   x   d x ( x, y ) 
 y '    y    d ( x, y ) 
     y                    
      Spatial Normalisation: How?
               Example: Rectangle->Ellipse
                                             Go to original image
For every voxel
                                             and find intensity at
position in blank                            warped co-ordinate
      sheet

                       Get position in
                      original space by
                      adding pertinent
                        displacement



  Template
    Spatial Normalisation: How?
   Right, but how do we get the displacement field?

• First we must find a good way to represent the field.

                                            Original “image”
 Warped image

                   Silly displacement-map



   Template
Spatial Normalisation: How?
       Component displacement-maps.

                             y-displacement,
                             black: downward
                             translation
                             white: upward
                             translation
                             grey: no translation




                             x-displacement,
                             black: leftward
                             translation
                             white: rightward
                             translation
            y-displacement   gray: no translation

x-displacement
     Spatial Normalisation: How?
        Representing the field with basis-warps.


• To prevent                                 ...
  impossible
  deformations we                            ...
  restrict it to be a
  linear combination                         ...
  of permitted basis-
  warps.                   ...
                                 ...
                                       ...


                                                   ...
• One can for
  example use the                            ...
  discrete cosine set
 Spatial Normalisation: How?
       Remember the square->ellipse map?
                                                    square->ellipse map

Each basis-warp multiplied by a weight

   +   +    +    +     +   +    +    +

   +   +    +    +     +   +    +    +

   +   +    +    +     +   +    +    +
                                               x-component of
   +   +    +    +     +   +    +    +       square->ellipse map

   +   +    +    +     +   +    +    +

   +   +    +    +     +   +    +    +

   +   +    +    +     +   +    +    +

   +   +    +    +     +   +    +        =
   Spatial Normalisation: How?
    But how do we find the displacement-map?

• Remember realignment? We assumed that
  the observed difference was a linear
  combination of different “causes”.
                      f1         f1         f1
        f2-f1         x          y          

                x         +y         +




• But what are the “causes” in this case??
      Spatial Normalisation: How?
       But how do we find the displacement-map?
• The “causes” in this case are differences in shape. These
  are represented by the basis-warps
     Me    Not me
                        Lets try and explain the difference
                        between “me” and “someone else”
                        by this y-component basis-warp.


                         y            f  y                f  

                                                    
     Not me - Me    y-displacement   Intensity change per   Intensity change
                         per           y-displacement            per 
          Spatial Normalisation: How?
                 And with more basis functions

     f2-f1         y  1 f  y            y   2 f  y
              1                        +2                     +...

                                   f                        f
Unravel                             1                      2

                          Ditto
                                      1 
                  f1   f1        
    f2  f1                      2   e  Aβ  e          Ring a bell?
                  1    2        
                                      
        Spatial Normalisation: How?
        Remember, that was only the y-components.

      f2-f1         y  1 f  y             y   2 f  y
                1                       +2                      +...

                                   f                          f
                                    1                        2
And the x-
components
              x   65 f  x            x   66 f  x
                                                                      But that
    ...+65                      +66                       +...    doesn’t really
                                                                      change the
                                                                      maths.
                           f                          f
                           65                        66
Distortion Correction: Why?
         Distortion Correction: How?
            We can measure the field at each point.
    GE or GE-EPI                                  Post-processing

Short TE     Long TE


 Re Im        Re Im

                                         3D watershed           Inversion into
    s           l                      based                    undistorted
                                         phase-                    space (EPI
                         Phase wrap      unwrapping                     only)
  1                 (because -<<)
                                                    Weighted least-
                                                     square fit of
                                                     spatial basis-
                                                       functions
Phase evolution during TE
EPI images are
  distorted

Distorted image   Field-map tells us how each
                    voxel should be moved




Corrected image

                         Correction
Movement Correction: Revisited
• Sensitivity: Large error variance may prevent us
  from finding activations.
• Specificity: Task correlated motion may pose as
  activations.
                                              “Large” Activation




                        Intensity in voxel




                                             Scan #
                   BUT!
• This is known as “residual movement-related
  variance”.

                                            “Large” Activation



                      Intensity in voxel




                                           Scan #
   More BUT!
x-, y- and z-   x-, y- and z-
 translation      rotation
Now, why on earth is that?
• Movement-by-susceptibility-distortion
  interaction
• Movement-by-susceptibility-dropout
  interaction
• Spin-history effects
• Interpolation errors
• Movement during acquisition of volume
    What can we do about it?
I: The sledgehammer (regression)
• Include movement parameters as confounds
  in statistical model
  + Will remove all variance that correlates with
    movements. Protects against false positives.
  - Will remove all variance that correlates with
    movements. May remove activations.
     What can we do about it?
     II: Modelling the effects
• Using physics based models to assess and
  correct for all adverse effects of motion.
  + Correct thing to do
  - Cannot yet model all effects
     - Movement-by-susceptibility-distortion interaction.
       modelled by SPM (Unwarp).
     - Movement-by-susceptibility-dropout interaction. ?
     - Spin-history effects. Prospective motion correction?
     - Interpolation errors. Not a problem!?
     - Movement during acquisition of volume. Modelled
       by Peter Bannister and Mark Jenkinson (FSL?)
 Movement-by-susceptibility-distortion interaction
• The subject will disrupt the B0 field, rendering it
  inhomogeneous.
• This will cause spatial distortions in EPI images.
• The distortions             Original           Realigned
  vary with subject
  orientation.
• Hence, the shape
  of the subject will
  appear to vary
  when imaged at
  different positions.
            Describing the field
• In principle there is a unique B0 field for each
  subject position. How can we describe the problem to
  make it mathematically tractable?
             Displacement
                                            Slope goes to derivative field



                            Pitch () Intercepts
                                        goes to
                                        “constant”
                                        field
             Displacement




                                                     “Taylor expansion”
                            Pitch ()
So, how does it affect the data?
f1         f2           B0        f1 y

     -           2•            
                                              2= 4.7°
     ...




                             ...
f1         fi           B0        f1 y

     -           i•                        i= -4.1°
     But we don’t know the field, right?
f1         f2          B0 1   f1 y        B0 2   f1 y             B0 5   f1 y

     -           12•                    +22•                   +...+52•                    +...
                                    f 2                      f 2                           f 2
                                     1                       2                            5




                                                     ...




                                                                                    ...
                           ...
     ...




f1         fi          B0 1   f1 y        B0 2   f1 y             B0 5   f1 y

     -           1i•                    +2i•                   +...+5i•                    +...
                                     f i                      f i                           f i
                                      1                       3                            5

                                                                          1  
                                                                                A   
           f1  f2  f2 1 f 2  2         f2 5                 2 
                                                                 2    
                                                                      
Or                  
            f1  fi   fi 1 fi  2        fi  5
                                                                           
                                                                         
                                                                                       β  Aβ
                     
                                                                                A   
                                                                         5   i 
                                                              
                                                                             
                                                                                 
 Validation against measured field maps.
• Dual echo-time EPI data collected at each time-point.
• Phase-maps estimated using “standard” techniques.
                                                                       Time series
 TE=30ms      TE=40ms                  TE=30ms      TE=40ms


  Re Im        Re Im                   Re Im         Re Im
                            1st   TR                          2nd TR     ...

     s           l                      s             l

                                                     2
 1

                                       Weighted least-        Inversion
                                       square fit of          into
       3D watershed based              spatial basis-         undistorted
       phase-unwrapping                functions              space
  Are the field-maps properly explained
    by a 1st order Taylor expansion?
             1st
Measured
field maps
                                                                   40th
Taylor
says
                               +            +            + error

             B0{i}       B0         B0          B0 
Thus
predicting
these

With this
error
Comparing estimated and measured
        derivative fields.
• Derivative field with respect to
  pitch estimated directly from
  time series and from field-maps.
       Estimated                     Measured
 Derivative-fields estimated from
      different time-series
• Derivative field with respect to
  pitch estimated from two
  different time series on the
  same subject
   Small movements 2º        Large movements 6º
 Derivative-fields from different subjects
               Subject #1      Subject #2


Derivative
with respect
to pitch




Derivative
with respect
to roll
            More Movies
Realigned           Realigned & Unwarped
  A (slightly contrived) example:
Movement and task uncorrelated




       tmax=9.80
Movement and task uncorrelated
   Regression        Unwarp




   tmax=8.03        tmax=9.76
 A (slightly contrived) example:
Movement and task correlated




     tmax=13.38
Movement and task correlated
  Regression                    Unwarp




  tmax=5.06                    tmax=9.57
               Spin-history or
               “must remember to move”?
                Conclusion
• Subjects move! There will be movement-
  related variance in the data.
• If movement uncorrelated with task.
  – Slight loss of sensitivity. Regression, Unwarp
    or “ignoring it” all fine.
• If movement correlated with task.
  – Loss of specificity. Must be remedied.
  – Regression always restores specificity, but may
    cause large loss of sensitivity.
  – Unwarp partially restores specificity. Causes no
    loss of sensitivity.

				
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posted:3/6/2010
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