Improvements to the itk::KernelTransform and
subclasses
Release 1.10
Rupert Brooks and Tal Arbel
August 15, 2007
McGill Centre for Intelligent Machines
McGill University, Montreal, Canada
rupert.brooks@mcgill.ca, arbel@cim.mcgill.ca
Abstract
Kernel-based transforms such as the thin plate spline are frequently used to model deformations in med-
ical imaging. The existing implementation in ITK is capable of being used to warp images, but does not
work in the registration framework. The existing implementation is inefficient, requiring recomputation
of all cached values at every parameter change, and the Jacobian calculation is not implemented. By
reversing the roles of the fixed and moving parameters, the transform can be adapted for registration use.
We present modified classes which are more efficient, and calculate the Jacobian correctly.
Contents
1 Introduction 1
2 Kernel Transforms 2
3 Problems with the existing implementation 3
4 Proposed changes 4
5 Testing 6
6 Conclusion 6
1 Introduction
Deformable transformations based on a small set of matched points are extremely useful in medical imaging.
While the deformation at each point is explicitly defined by the correspondence, the deformation between

2points must be interpolated. There are a family of spline based approaches which interpolate a smooth de-
formation field, of which the most widely known is the thin plate spline [3, 1]. In the ITK, this family of
transformations is referred to as kernel transformations, because they can be expressed as linear combina-
tions of radially symmetric kernel functions centered on each point.
2 Kernel Transforms
We begin with two sets of n corresponding landmark points, in a space with dim dimensions. We will refer
to these as the source landmarks, PS = {pSi}, and the target landmarks, PT = {pTi}. A displacement vector,
di = pTi −pSi , can be defined at each point, and these vectors can be grouped into one long vector, D, as
follows:
D =
[
dT1 d
T
2 ... d
T
n
]
. (1)
The kernel function, g(x) maps dim-dimensional vectors onto dim×dim symmetric matrices.
In this model, a deformation field, F(x), is viewed as a set of dim independent functions of spatial position,x,
F(x) = [ f1(x), ..., fdim(x)]
T . Thus there is a separate, independent spline for the displacement in each coordi-
nate. Each such spline is considered to be a combination of a linear (affine) transformation, plus a weighted
combination of kernel functions centered at each point.
F(x) =CT ·G(x)+A ·x+b (2)
whereCT is a dim×n vector of weights, G is the dim×dim ·n matrix made by stacking the kernel functions
centered at each landmark point, evaluated at the point of interest, and A and b are a linear transformation
in the coordinates. Thus the complete spline is defined by n ·dim+dim2 +dim parameters.
For n landmark points the value of the displacement field at each point provides dim ·n constraints. There are
fewer equations than unknowns so this leaves the system underdetermined. A further constraint is applied
by requiring that the deformation should flatten out to an affine transformation far from all the landmarks.
This flatness constraint can be developed into the set of linear equations [3]
PTS ·C = 0, (3)
where PS is the source landmark coordinates arranged in a matrix, as follows:
PS =



pS11 I . . . pS1dim I I
... . . .
...
...
pSn1 I . . . pSndim I I


 (4)
This provides enough additional equations to make the system have full rank.