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Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Curvature-Based Non-Rigid Image Registration
Gabriel Mañana and Eduardo Romero
BioIngenium Group, National University, Colombia
October 2, 2008
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
1
Mathematical Setting
2
Parametric Image Registration
3
Non-Parametric Image Registration
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
1.1 Some definitions
Definition
Definition 1.1 Let d ∈ N. A function b : Rd → R is called a
d -dimensional image, if
1. b is compactly supporteda ,
2. ´0 ≤ b(x) < ∞ for all x ∈ Rd , and
3. Rd b(x)k dx is finite, for k > 0.
a
b(x) = image intensity at spatial position x ∈ Rd
The set of all images is denoted by
!
"
Img (d ) := b : Rd → R | b is a d-dimensional image .
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Let d ∈ N, Ω := ]0, 1[d , and n1 , . . . , nd ∈ N be some given
numbers.
Definition
Definition 1.2 The points
xj1 ,...,jd = (xj1 , . . . , xjd )T ∈ Ω ∪ ∂Ω,
where 1 ≤ jg ≤ ng and 1 ≤ g ≤ d , are called grid points.
Definition
Definition 1.3 The array
X : = (xj1 , . . . , xjd ) 1≤ j! ≤ n! , ∈ Rn1 ×···×nd
!=1,...,d
is called the grid matrix.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Definition
Definition 1.4 Let N := n1 , . . . , nd and let the numbers
d
j ∈ N (1 ≤ j ≤ N) and (j1 , . . . , jd ) ∈ N (1 ≤ j! ≤ n! , 1 ≤ " ≤ d )
be related by the one-to-one lexicographical ordering,
j=
d−1
#
(jν+1 − 1)
ν=1
ν
$
nµ + j1 .
µ=1
−
→
The vector X := (xj )j=1,...,N ∈ RN , where xj = xj1 ,...,jd is called the
grid vector.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Definition
Definition 1.5 Let d ∈ N, Ω := ]0, 1[d , and n1 , . . . , nd ∈ N, be
some given numbers. For 1 ≤ j! ≤ n! , 1 ≤ " ≤ d , let
xjD
xjN
xjp
:=
%
%
j1
jd
,...,
n1 + 1
nd + 1
&T
,
&
2j1 − 1
2jd − 1 T
:=
,...,
, and
2n1
2nd
%
&
j1 − 1
jd − 1 T
:=
,...,
n1
nd
be the Dirichlet (ΩD ), Neumann (ΩN ), and Periodic (Ωp ) grids,
respectively.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Example
Let d = 2, Ω := ]0, 1[d , and n1 = n2 = 8. For
1 ≤ j! ≤ n! , 1 ≤ " ≤ 2,
xjD =
xjN
%
j1 j2
,
9 9
&T
&
2j1 − 1 2j2 − 1 T
,
=
16
16
%
&
j1 − 1 j2 − 1 T
p
,
xj =
8
8
%
Gabriel Mañana and Eduardo Romero
=
0.111 0.222 . . . 0.888
0.111 0.222 . . . 0.888
=
0.063 0.188 . . . 0.938
0.063 0.188 . . . 0.938
=
0.000 0.125 . . . 0.875
0.000 0.125 . . . 0.875
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Figure: Dirichlet (red), Neumann (gray), and Periodic (blue) grids.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
1.2 From continuous to discrete images
Two common models for the discretization of a continuous image b
on a discrete grid Ωd :
Meshpoint Model
bj• = bj•1 ,...,jd := b(xj ) for all xj = Ωd
Midpoint Model
bj× = bj×1 ,...,jd :=
´
cj
b(x)dx for all j = 1, . . . , N
where cj denotes a small region with center xj .
For both models we define b(x) = 0 for all x ∈
/ Ω.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Figure: Meshpoint and midpoint digitized images.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
1.3 From discrete to continuous images
Interpolation scheme I,
I : Rn1 ×···×nd × Rd → R.
For a discrete image B, the scheme assigns an intensity value
I(B, x) to any position x ∈ Rd .
There are local methods (Next-neighbor, d-Linear, etc.), and
global methods based on a set of basis functions (Lagrange
polynomials, sinc functions, B-splines, wavelets, etc.).
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
1.4 Lagrange and Euler frames
For image interpolation there are two reference frames, the
Lagrange coordinates, and the Euler coordinates. Assuming the
transformation ϕ is invertible, we can write '
x := ϕ(x) or
−1
equivalently x := ϕ('
' x ) := ϕ ('
x ).
From a computational point of view
B Lagrange (ϕ (i, j )) := B(i, j )
and
B Euler (i, j ) := B(ϕ−1 (i, j )).
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Figure: Left: Original image. Middle: Next-neighbor interpolation using
Lagrange coordinates. Right: Next-neighbor interpolation using Euler
coordinates.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Figure: Right-Top: Next-neighbor interpolation. Bottom-Left: Bi-linear
interpolation. Bottom-Right: Interpolation based on cosine basis
functions.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
1.5 The registration problem
Definition
Definition 1.6 Given a distance measure D : Img (d )2 → R, and two
images R, T ∈ Img (d ), find a mapping ϕ : Rd → Rd and a
mapping g : R → R such that D (R, g ◦ T ◦ ϕ) = min.a
a
The problem is ill-posed, thus, a direct approach is impossible.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
2. Parametric Image Registration
Parametric image registration techniques are all based on the
minimization of a certain distance measure, and the distance
measure, in turn, is based on a finite set of parameters (e.g. image
intensities) and/or a finite set of so-called image features (e.g.
landmarks, principal axes).
Typical examples of intensity-based distance measures are the sum
of squared differences (SSD), correlation ratio (CR), and mutual
information (MI).
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
2.1 Landmark-based parametric registration
Supply features F(R, j ) = x R,j and F(T , j ) = x T ,j ,
j = 1, . . . , m. Choose a set of basis functions ψ. Find
T
n
parameters
(n α = (α1 , . . . , αn ) ∈ R such that for
ϕ = j=1 αj ψj ,
D LM [ϕ] =
m
#
)F(R, j ) − ϕ (F(T , j )))f = min.
j=1
Needs landmarks.
Simple.
Only needs the numerical solution of a linear system of
equations.
Results might be arbitrarily awful.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
2.2 Landmark-based smooth registration
Supply features F(R, j ) = x R,j and F(T , j ) = x T ,j ,
j = 1, . . . , m. Choose a regularizer (e.g.
S TPS [ψ] := 12 *ψ, ψ+2q ) and a regularizing parameter α ≤ 0.
For α = 0 find ϕsuch that
S TPS [ϕ] = min
subject to ϕ (F(T , j )) = F(R, j ), j = 1, . . . , m.
Alternatively, for α > 0, find ϕsuch that
αS TPS [ϕ] + D LM [ϕ] = min.
Needs landmarks.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Only needs the numerical solution of a linear system of
equations, essentialy m unknowns and m equations; the
system is always non-singular.
Physically meaningful transformation, minimizes curvature.
Results may be bad.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
2.3 Principal axes-based registration
) * +
Compute ϕ ∈ d1 Rd , such that F(T ◦ ϕ) = FR where FB
is a feature vector containing the center of gravity, the
standard deviations, and the principal axis based on an
appropriate density class, e.g., Gauss or Cauchy density. A
solution can be deduced from an eigenvalue decomposition of
the moment matrices of R and T .
Simple, fast, easy to understand and to interpret.
Needs moment matrix and eigenvalues decomposition of two
d -by-d matrices.
Not suitable for multimodal densities/images.
Very few registration parameters.
Ambiguous results.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
2.4 Optimal parametric registration
Choose an appropriate distance measure D. Choose a set of
T
n
basis functions. Find
(nparameters α = (α1 , . . . , αn ) ∈ R
such that for ϕ = j=1 αj ψj ,
D [ϕ] = min.
A numerical solution can be found using optimization
methods, e.g., a Gauss-Jordan, Levenberg-Marquardt or
evolutionary method.
General, flexible.
No physical, meaningful transformation.
Optimization can be very slow, specially for high-dimensional
spline spaces.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3. Non-Parametric Image Registration
The idea behind non-parametric registration is to device an
appropriate measure, both for the similarity of images as well as for
the likelihood of a non-parametric transformation. This approach is
based on a variational formulation of the registration problem,
and the numerical schemes used are based on the Euler-Lagrange
equations that characterize a minimizer.
The basic components are a distance measure D and a regularizer
S. The distance measure is the driving force of the registration,
and the regularizer is used to restrict the transformation to an
appropriate behavior (i.e., no foldings or breaks).
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
The required displacement u is a solution of
D [R, T ; u] + αS [u] = min,
with some additional boundary conditions. The distance measure
used is given by
D [R, T ; u] :=
1
)Tu − R)L2 (Ω) .
2
By changing the regularizer S, different registration schemes can be
implemented. Particularly, we will review Elastic, Fluid, Diffusion
and Curvature registration schemes.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
The numerical approach used is based on the corresponding
Euler-Lagrange equations, which lead to a system of non-linear
partial differential equations, A [u] = f ( · , u). A fixed-point or time
marching algorithm can be used to circumvent the non-linearity of
f with respect to u.
3.1 General Framework
Given two images, R and T , find a transformation ϕ, such that Tϕ ,
whereTϕ (x) := T (ϕ(x)), is similar to R. For this, it is convenient
to split the transformation ϕ into the identity part and the
displacement u, u : Rd → Rd ,
ϕ(x) = x − u(x),
(1)
where an Eulerian viewpoint can be exploited.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Let x̂, û, and ϕ̂ denote the point, displacement, and transformation
with respect to the Lagrange coordinates and let x, u, and ϕ
denote the same with respect to the Euler coordinates. Thus,
x = ϕ̂(x̂) = x̂ + û(x̂), and
x̂ = ϕ(x) = x − u(x),
where ϕ = ϕ̂−1 , û(x̂) = u(x), and ϕ is assumed to be
diffeomorphic.
Definition
Definition 3.1 Given two manifolds M and N, a bijective map f
from M to N is called a diffeomorphism if both f : M → N and its
inverse f −1 : N → M are differentiable.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
The most intuitive way of approaching the registration problem is
then to design a suitable distance measure D and to minimize the
distance between R and Tu with respect to u,
u
D [R, T ; u] := D [R, Tu ] −→ min,
(2)
Tu (x) = T (x − u(x)).
(3)
where
A direct minimization of D presents some drawbacks: the problem
is ill-posed since small changes in the input data may lead to large
changes of the output data, the solution is not unique since the
problem is not convex, and the deformation may not even be
continuous.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
The solution is to add an additional regularizing term or smoother
S. With an appropriate smoother it becomes possible to distinguish
particular transformations that are more likely than others:
Fact
Problem 3.1
Given two images R, T , and a positive regularizing parameter α ∈
R>0 , find a deformation u, such that
J [u] := D [R, T ; u] + αS [u] = min.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Typical choices of smoothers S are based on bi-linear forms a, i.e.,
1
S [u] = a [u, u] ,
2
where
* dthe
+ bi-linear forms can be traced back to the inner product in
L2 R .
A necessary condition for a minimizer u is that the G âteaux
derivative d J [u, v ] of J vanishes for all suitable perturbations v .
This derivative is also known as the first variation of J in the
direction of v .
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
For the Gâteaux derivative of J we have
1
(a [u + hv , u + hv ] − a [u, u])
h→0 2h
= a [u, v ]
ˆ
*A [u] (x), v (x)+ Rd dx,
=
d S [u, v ] =
lim
(4)
Rd
where A is a partial differential operator.
For the Gâteaux derivative of D we have
ˆ
*f (x, u(x), v (x)+ Rd dx,
d D [u, v ] =
Rd
where the force f depends on the particular distance measure.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Fact
The minimization problem has been formulated without explicit
boundary conditions on the minimizer u, however, in order to
device efficient numerical schemes, explicit boundary conditions
have to be used.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3.2 General solution schemes
Different numerical schemes can be found in the literature for the
computation of a numerical solution of Problem 3.1 (e.g,
gradient-based steepest methods, Henn, Thirion). The approach to
be discussed here is based on the characterization of the minimizer
by the Euler-Lagrange equations, that for Problem 1 are of the form
A [u] (x) − f (x, u(x)) = 0, for all x ∈ Ω,
(5)
where Ω =]0, 1[d is the region under consideration, and d denotes
the spatial dimension of the images.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Fact
The partial differential operator A is related to the smoother S,
and the force f is related to the distance measure D.
A convenient way of solving this semi-linear partial differential
equation and to by-pass this non-linearity is to exploit a fixed-point
iteration scheme. Starting with an initial u (0) (e.g., u (0) ≡ 0), we
define u (k+1) implicitly by
,
.
/
(6)
A u (k+1) (x) = f x, u (k) (x) , x ∈ Ω, k ∈ N0 .
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
A slight modification of this iteration can be used to stabilize the
scheme,
,
.
/
u (k+1) (x) + τ A u (k+1) (x) = u (k) (x) + τ f x, u (k) (x) , (7)
x ∈ Ω, k ∈ N0 ,
that may also be rewritten as
.
/
,
u (k+1) (x) − u (k) (x)
+ A u (k+1) (x) = f x, u (k) (x) , (8)
τ
x ∈ Ω, k ∈ N0 .
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Introducing an artificial time t, making the displacement u time
dependent, u = u(x, t), and setting u (k) (x) = u(x, kτ ), where τ
denotes a fixed time-step, equation (7) may also be viewed as a
semi-implicit scheme for the time-dependent partial differential
equation
∂t u(x, t) + A [u] (x, t) = f (x, u(x, t)), x ∈ Ω.
(9)
Fact
A steady-state solution of the previous equation also fulfills the
neccesary condition for a minimizer of Problem 3.1.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Fact
For a numerical treatment of either the fixed-point-type equation
(6) or the time dependent partial differential equation (9), two
problems (which are related to discretization) have to be solved:
the computation of the force and the numerical solution of a partial
differential equation.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3.3 Computing the force
With the purpose of evaluating different smoothers, we will focus
here on the distance measure D = D SSD . Different distance
measures will be evaluated a posteriori. The force can be then
deduced from its Gâteaux derivative.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Theorem
Let d ∈ N and R, T ∈ Img (d ), T ∈ C 2 (Rd ), u, v : Rd → Rd ,
Ω :=]0, 1[d . The Gâteaux derivative of dD [R, T ; u],
d D [R, T ; u] :=
1
)Tu − R)L2 (Ω)
2
(10)
with respect to v is given by
ˆ
*(R(x) − Tu (x))∇Tu (x), v (x)+ Rd dx,
d D [R, T ; u; v ] =
Ω
where f : Rd × Rd → Rd ,
f (x, u(x)) := (R(x) − Tu (x))∇Tu (x)
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Since R and T are digital images, interpolation has to be used to
compute Tu (x) for non-integer values of u(x).
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3.4 Solving the PDE numerically
Fact
Since the underlying domain is very simple, Ω =]0, 1[d , a finite
difference scheme a can be used to solve the PDE numerically. To
this end, we will use the grids previously described, i.e., Dirichlet,
Neumann, and Periodic grids.
a
A finite difference is an expression of the form f (x + b) − f (x + a).
A finite difference divided by (b − a), is a difference quotient. The
approximation of derivatives by finite differences plays a central role in finite
difference methods for the numerical solution of differential equations,
especially boundary value problems.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3.5 Notation
For a function g : Rd → R, we use
' ) := (g (xj ))N ∈ RN ,
'g := g (X
j=1
and for a vector field v : Rd → Rd , we use
.
/T
'v := 'v1T , . . . , 'vdT
∈ RdN .
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
From the Taylor expansion of g : Rd → R, we get the standard
finite difference approximation for second order derivatives,
∂xj xj g (x) =
g (x + hj ej ) − 2g (x) + g (x − hj ej )
+ O(hj2 ),
hj2
∂xj xk g (x) =
1
g (x + hj ej + hk ek )
4hj hk
−g (x − hj ej + hk ek )
(11)
(12)
−g (x + hj ej − hk ek )
+g (x − hj ej − hk ek ) + O(hj2 + hk2 ),
where O is a function belonging to a class of certain decay,
f : R → R is O(hp ) if limh→0 f (h)h−p−1 = 0.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Fact
We assume that a finite difference approximation of A[u] at any
grid point xj can be obtained using a convolution filter S A ,
A[g ](xj ) ≈
#
SkA g (xk ) =: (S A ∗ g )(xj )
k∈ N (j)∪j
where N (j ) is a neighborhood of the grid point j and S A denotes a
filter connected with the partial differential operator A via its finite
difference approximation.
Now, the boundary conditions attached to the Problem 1 have to
be incorporated into the discrete formulation. In practice, this
can be done by a zero padding (Dirichlet), mirroring
(Neumann), or adequate copying (Periodic).
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
With the lexicographical ordering defined in (1.4), we can also
deduce a matrix representation of the convolution with S A ,
*
+
' ).
A · 'g := S A ∗ g (X
Using the fixed-point iteration scheme defined in (6) we end up
with the following overall algorithm:
General registration algorithm
' (k) , and U
' (k) = 0.
Initialize X
For k = 0, 1, 2, . . .
' (k) = f (X
' ,U
' (k) );
compute force F
' (k+1) = F
' (k) ;
solve partial differential equation AU
if converged, stop, end;
end.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3.6 Elastic Registration (ER)
The essential difference between a rigid and an elastic body is that
the relation of particles in the latter is no longer fixed but can vary
according to the elastic properties of the body.
Applying an external force to an elastic body results in a
deformation or strain of the body. The strain is related to tension
or stress of the body, and the shape of the body results from an
equilibrium of outer forces and inner stress.
The connection between strain and stress is provided by the
Hook’s law, that claims that strain is a reaction to stress.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Broit (1981) was the first to study elastic registration and his
approach is based on
D[R, T ; u] = D SSD [R, T ; u] =
1
)Tu − R)L2 (Ω)
2
and S[u] := P[u] where P denotes the linearized elastic potential
of the displacement u,
P[u] =
ˆ
Ω
d
µ #
λ
(∂xj uk + ∂xk uj )2 + (div u)2 dx,
4
2
(13)
j,k=1
where λ and µ are the so-called Lamé constants.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
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Non-Parametric Image Registration
The numerical treatment is based on the Euler-Lagrange equations,
which coincide with the Navier-Lamé equations for this particular
regularizer
f = µ0u + (λ + µ)∇div u.
(14)
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
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Non-Parametric Image Registration
Elastic Registration Summary
Regularization is based on the elastic potential
S elas [u] = P[u],
Aelas [u] = µ0u + (λ + µ)∇div u.
Two parameters, the Lamé constants µ and λ describe the
material properties, tipically λ = 0 is used to obtain maximal
expansion of the elastic body.
Physically meaningful transformation.
Transformation is restricted to small, local deformations.
Fast O(N log N) can be achieved using FFT-type factorization
for periodic boundary conditions.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3.7 Fluid Registration (FR)
The main difference between elastic and fluid registration is that for
elastic registration the regularization is based on S elas [u] = P[u],
where in the fluid approach it is based on S fluid [u] = P[∂t u].
In contrast to the elastic model, the strain in a fluid model depends
on the rate of change. Elastic models are characterized by a spatial
smoothing of the displacement field. In contrast, fluid models are
characterized by a spatial smoothing of the velocity field.
Thus, in principle, any displacement can be obtained given enough
time.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
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Fluid Registration Summary
Regularization is based on the elastic potential of the time
derivative of the displacement S fluid [u] = P[∂t u],
Afluid [v ] = f ( · , u), v =
d
u = ∂t u + ∇u v
dt
Two parameters, the Lamé constants µ and λ describe the
material properties, tipically λ = 0 is used to obtain maximal
expansion of the elastic body.
Requieres an additional Euler step.
Physically meaningful transformation.
Extremely flexible.
Fast O(N log N) can be achieved using FFT-type factorization
for periodic boundary conditions.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3.8 Diffusion Registration (DR)
This approach uses a gradient-based regularization term and a
finite difference approximation of the underlying partial differential
equation. Since this PDE may be viewed as a generalized diffusion
equation, this scheme is called diffusion registration.
In contrast to the physically motivated elastic and fluid
registrations, the idea behind the regularizer used in DR is to
privilege smooth deformations while minimizing oscillations of the
components of the displacement.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
One important advantage of this regularization is that the
Euler-Lagrange equations decouple with respect to the spatial
directions. As a consequence, the matrix representation A of the
finite difference-based discretized derivative of this smoother is a
d -by-d block diagonal matrix.
Another important advantage is that a registration step can be
performed in O(N), where N denotes the number of unknowns.
The tool used here is the so-called additive operator splitting
(AOS) scheme.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Diffusion Registration Summary
Regularization is based on first order spatial derivatives of the
displacement
d
S
diff
[u] =
1#
2
!=1
ˆ
Ω
*∇u! , ∇u! +Rd dx;
Adiff [u] = 0u.
Two parameters, the regularizating parameter α and the
time-step τ .
Transformation is restricted to small deformations.
Can be combined with the fluid idea to allow for larger
deformations.
O(N) implementation based on the AOS scheme.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
3.9 Curvature-based registration
Curvature registration is based on the distance measure D
introduced in Equation (10)
d D [R, T ; u] :=
1
)Tu − R)L2 (Ω)
2
and the regularizer
1
a[u, u]
2
where the bi-linear form a is defined by
S curv [u] :=
a[u, v ] =
d ˆ
#
!=1
Gabriel Mañana and Eduardo Romero
(15)
0u! 0v! dx
Ω
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
and Neumann boundary conditions
∇u! = ∇0u! = 0 for x ∈ ∂Ω, " = 1, . . . , d
are imposed.
The integrand (0u! )2 of S curv can be viewed as an approximation
to the curvature.
Fact
Thus, the idea of this regularizer is to minimize the curvature of the
components of the displacement.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Theorem
The Euler-Lagrange equations for J curv = D SSD + αS curv , where
D SSD is defined by Equation (10) and S curv is defined by Equation
(15) are
f (x, u(x)) + α02 u(x) = 0, x ∈ Ω,
∇u! = ∇0u! = 0 for x ∈ ∂Ω, " = 1, . . . , d .
The Euler-Lagrange equations for the curvature registration
functional are also known as the bi-harmonic equation.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Matrix stencils S curv ,d for the bi-harmonic
d = 2, 3

0 0
1
 0 2 −8

S curv ,2 = 
 1 −8 20
 0 2 −8
0 0
1
Gabriel Mañana and Eduardo Romero
operator and dimensions
0
2
−8
2
0
0
0
1
0
0



,


Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Matrix stencils S curv ,d for the bi-harmonic operator and dimensions
d = 2, 3
/
curv ,3
Sj,k,!
j,k,!=1,...,5


42,
j = k = " = 2,





−12, |j − 3| + |k − 3| + |" − 3| = 1,





|j − 3| + |k − 3| + |" − 3| = 2
2,
=
∧ max {|j − 3| + |k − 3| + |" − 3|} = 1



|j − 3| + |k − 3| + |" − 3| = 2
1,




∧ max {|j − 3| + |k − 3| + |" − 3|} = 2




0,
otherwise
S curv ,3 =
.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Curvature Registration Summary
Regularization based on second order spatial derivatives of the
displacement
d
S
curv
A
[u] =
curv
1#
2
!=1
2
ˆ
(0u! )2 dx;
Ω
= 0 u.
Two parameters, the regularizing parameter α and the
time-step τ .
Kernel contains affine linear transformations.
Transformation is restricted small deformations.
Fast O(N log N) implementation based on DCT-type
techniques.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
Fact
The regularization of elastic, fluid, diffusion, and curvature
registration can be applied to the displacement as well as to the
update of the displacement.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration
Mathematical Setting
Parametric Image Registration
Non-Parametric Image Registration
References
1
Fischer, B. and Modersitzki, J. (1999). Fast inversion of
matrices arising in image processing. Numerical Algorithms 22,
1-11.
2
Fischer, B. and Modersitzki, J. (2003). Curvature based image
registration. Journal of Mathematical Imaging and Vision
18(1), 81-85.
3
Modersitzki, J. (2004). Numerical methods for image
registration. Numerical Mathematics and Scientific
Computation, Oxford Science Publications.
Gabriel Mañana and Eduardo Romero
Curvature-Based Non-Rigid Image Registration