# Three papers on Nonrigid registration using free-form deformations

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```					Informatics and Mathematical Modelling / Image Analysis

Three papers on Nonrigid
registration using free-form
deformations
Presented by Hildur Ólafsdóttir
Informatics and Mathematical Modelling / Image Analysis

Papers presented
 Rueckert et al. , TMI 1999: Nonrigid registration using free-
form deformations: Application to breast MR images

 Schnabel et al. , MICCAI 2001: A Generic Framework for Non-
rigid Registration Based on Non-uniform Multi-level Free-form
Deformations

 Rueckert et al., TMI 2003: Automatic Construction of 3-D
Statistical Deformation Models of the Brain Using Nonrigid
Registration
Informatics and Mathematical Modelling / Image Analysis

Overview
 Introduction – image registration
 Tasks within medical image analysis
 TMI paper 1999:
– Nonrigid registration using free-form deformations
   Global model
   Local model
   Regularisation
   Similarity measures
   Optimisation
– Results on Breast MRI
 MICCAI paper 2001:
– Multi-resolution registration
– Non-uniform control point spacing
– Applications
 TMI paper 2003:
– Statistical deformation models
– Statistical deformation models of the brain
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Introduction – image registration
 Image registration - general definition:
– Image registration is the process of determining a mapping
between the coordinates in one image and those in another,
to achieve biological, anatomical or functional
correspondence
 Purpose of image registration in medical image
analysis
–   Monitoring of changes in an individual
–   Fusion of information from multiple sources
–   Comparison of one subject to another
–   Comparison of one group to another
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Introcution – image registration
 Registration of one image to the coordinate system of another
image by a transformation, T: (x,y,z)  (x0, y 0, z 0)
(x’,y’,z’)

T(x,y,z)
(x,y,z)

Image A               Image B

 Types of transformation
– Rigid – rotation, translation
– Affine – rotation, translation, scaling, shearing
– Nonrigid – All sorts of nonlinear deformations
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Tasks within medical image analysis I
 Rigid registration
– Bones
 Affine registration
– If scale changes are expected
   Growth
   Inter-subject registration
 Nonrigid registration
– Correction for tissue deformation
   Breast MRI
   Liver MRI
   Brain shift modelling
– Modelling of tissue motion
   Cardiac motion
   Respiratory motion
– Modelling of growth and atrophy
   Brain development
   Dementia or schizophrenia
– Fusion of different modalities
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Tasks within medical image analysis II

intra-modality            inter-modality

Intra-subject                  MR1 ! MR2,                MR1 ! CT1,
CT1 ! CT2,                PET1 ! CT1, etc
PET1 ! PET2, etc

Inter-subject                  S1MR ! S2MR, S1CT
! S2CT, etc
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I. Nonrigid registration using
free-form deformations:
Application to breast MR images
By D. Rueckert, L. I. Sonoda, C. Hayes, D.L. G. Hill,
M. O. Leach and D.J. Hawkes

IEEE Transactions on Medical Imaging 1999
Informatics and Mathematical Modelling / Image Analysis

Nonrigid registration using free-form deformations

 A combined transformation consisting of both a local and a
global transformation
 T(x,y,z) = Tglobal(x,y,z) + Tlocal(x,y,z)
 Global: Accounts for the overall motion of the object
 Local: Accounts for local deformations of the object
 Cost function: C = Csimilarity+¢ Csmooth
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Global motion model
 Simplest choice: Rigid transformation
– Rotation, translation ) 6 degrees of freedom (d.o.f)
 More general: Affine transformation
– Rotation, translation, scaling and shearing ) 12 d.o.f
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Local motion model I
 Free form deformations (FFDs) based on cubic B-splines
 Basic idea: To deform an object by manipulating an underlying
nx£ ny£ nz mesh of control points , with spacing .
– Control points can be displaced from their original location
– Control points provide a compact parameterisation of the
transformation
Informatics and Mathematical Modelling / Image Analysis

Local motion model II
 Bi represents the ith basis function of the B-spline
B0(u) = (1-u)3/6
B1(u) = (3u3-6u2+4)/6
B2(u) = (-3u3+3u2+3u+1)/6
B3(u) = u3/6
 B-splines are locally controlled – computationally
efficient
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Local motion model III
 Hierarchical approach
 A hierarchy of control point meshes 1,…L at increasing
resolutions
 At each resolution we have a transformation Tllocal

 Represented by a single B-spline FFD
– Control point mesh progressively refined
– New control points inserted at each level
– Spacing is halved in every step
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Regularisation of the local transformation
 Constrain to a smooth transformation
– Penalty term:
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Similarity measures I
 How do we know when we have a good fit between
two images??
 Depends on the type of images you are registering
 Similarity assumptions
– Identity
 Single-modality, only differ by gaussian noise
– Linear
 Single-modality, differ by constant intensity
– Information theoretic/probabilistic
 multi-modality, intensity changing, related by some statistical
or functional relationship
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Similarity measures II – identity or linearity
assumption
 Sums of squared differences

 Normalised cross correlation
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Similarity measures III – Information theoretic

 Entropy

 Joint entropy

 Mutual information (MI)
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Similarity measures IV – Information theoretic
 Mutual information is still sensitive to overlapping
 Normalised mutual information

– Is robust to the amount of overlap between images
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Final Cost function
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Optimisation
calculate the optimal affine transformation parameters  by
maximising Csimilarity

initialise the control points 
repeat
calculate the gradient vector of the cost function, C(,) with respect
to the nonrigid transformation parameters, : rC = C(,l)/l

while ||rC||> do
recalculate the control points  = +rC/||rC||
recalculate the gradient vector rC
increase the control point resolution by calculating new control points
l+1 from l
increase the image resolution
until finest level of resolution is reached
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Application: Breast MRI
 9.5% of women in the UK develop breast cancer
 Examination
–   Currently: X-ray mammography
–   Two 3D MR scans
–   Pre- and post-contrast
–   Rate of uptake is determined by the difference between the two
different scans
 Problems: Motion of the patient, respiratory and cardiac motion
 Registration of the pre- and post contrast images is required
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Breast MRI without contrast agent
No registration

Before motion                After motion          Difference
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Breast MRI without contrast agent
Registration

Trans-
formed
image

Difference

Rigid                 Affine                Nonrigid
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Breast MRI with contrast agent – tumour detection
No registration

Pre-contrast                  Post-contrast     Difference
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Breast MRI – with contrast – tumour detection
Maximum intensity projection

No registration                   Rigid

Affine                        Nonrigid
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Further validation
   Correlation coefficient
   Squared sum of differences
   Ranking of radiologists
   See tables I and II pg. 719
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II. A Generic Framework for
Non-rigid Registration Based on
Non-uniform Multi-level Free-
form Deformations
By Julia A. Schnabel, Daniel Rueckert, Marcel Quist, Jane M.
Blackall, Andy D. Castellano-Smith, Thomas Hartkens,
Graeme P. Penney, Walter A. Hall, Haiying Liu, Charles L.
Truiwit, Frans A. Gerritsen, Derek L. G. Hill, and David J.
Hawkes

MICCAI 2001
Informatics and Mathematical Modelling / Image Analysis

Introduction
 An extension and generalisation of the first article
 Multi-resolution registration
– Smoothing term is no longer needed
 Non-uniform control point spacing
– Improved computation time
 Tested on many different applications
Informatics and Mathematical Modelling / Image Analysis

Multi-resolution registration
2D slices through 3D single- and multi-resolution FFDs for the

Single-
resolution

Multi-
resolution
Informatics and Mathematical Modelling / Image Analysis

Non-uniform control spacing
Passive control point
Active control point

Reduces computation time considerably
Informatics and Mathematical Modelling / Image Analysis

Non-uniform control point spacing II
 Determine the status of each control point at each level
: Threshold
M: Statistical
measure
 M : Reference image measure
– Calculated prior to registration
– Local characterisation of the reference image
– Entropy, variance,…
 M : Joint image pair measure
– Recomputed at each step
– Degree of local image alignment
– Here: Local gradient of the cost function  C/ i,j,k
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Intra-subject registration of brain MRI
 Intra-subject registration of pre- to post-operative brain
MRI
Difference images

Rigid                Non-rigid 15mm          Non-rigid 7.5mm
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Inter-subject registration of brain MRI

Reference           Affine        Uniform, non-rigid, 20mm

Uniform,          Non-uniform,   Non-uniform,
non-rigid         non-rigid, 5mm non-rigid, 5mm
5mm               M = entropy    M = variance
Informatics and Mathematical Modelling / Image Analysis

III. Automatic Construction of
3-D Statistical Deformation
Models of the Brain Using
Nonrigid Registration
By Daniel Rueckert, Alejandro F. Frangi and Julia
Schnabel

IEEE Transactions on Medical Imaging 2003
Informatics and Mathematical Modelling / Image Analysis

Statistical Deformation Models (SDMs)
 Similar to Statistical Shape Models
 Not based on landmarks but on deformation fields
obtained from the nonrigid registration
– ) Fully automatic
– ) Describe not only a single structure but the full
surrounding anatomy
Informatics and Mathematical Modelling / Image Analysis

Building a SDM from FFDs
 Control points represent the deformation
 Apply a PCA to the control points
– 1,2,..,n where n is the number of subjects
– i is the concatenation of nx£ ny£ nz control points
P: Eigenvectors
 Result:                                           b: model parameter

 By varying b we can generate different instances of
FFD which describe the respective class of anatomies
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SDM of the brain – average intensity atlas

Reference subject                          Average intensity after
nonrigid registration of all
subjects to the reference
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SDM of the brain – Modes of variation, corpus
callosum

Mode 1                        Mode 2            Mode 3
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Software
 The source code for FFDs can be obtained from
Daniel Rueckerts webpage
– http://www.doc.ic.ac.uk/~dr/software

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