ELASTIC IMAGE WARPING USING A NEW RADIAL BASIC

FUNCTION WITH COMPACT SUPPORT

Zhixiong Zhang and Xuan Yang

College of Information Engineering, Shenzhen University, GuangDong province, 518060 China

Keywords: Image registration, Compact support, Elastic registration, Bending energy, Biharmonic equation.

Abstract: Thin plate spline (TPS) and compact support radial basis functions (CSRBF) are well-known and successful

tools in medical image elastic registration base on landmark. TPS minimizes the bending energy of the

whole image. However, in real application, such scheme would deform the image globally when

deformation is local. Although CSRBF can limit the effect of the deformation locally, it cost more bending

energy which means more information was lost. A new radial basic function named ‘Compact Support Thin

Plate Spline Radial Basic Function’ (CSTPF) has been proposed in this paper. It costs less bending energy

than CSRBF in deforming image locally and its global deformation effect is similar to TPS. Numerous

experimental results show that CSTPF performs outstanding in both global and local image deformation.

1 INTRODUCTION

Elastic image registration is a significant content in

medical image registration. And image deformation

plays an important part in elastic image registration.

The use of TPS for point-based elastic registration

was first proposed by Bookstein (Bookstein, 1989).

TPS forces the corresponding landmark to match

each other exactly and minimizes the utilization of

the bending energy of the whole image, therefore, it

is widely used in various fields (Brown and

Rusinkiewicz, 2004). However, the deformation of

TPS is global, it would be problematic when only

local difference exists (Ruprecht and Müller, 1993).

N. Arad, D. Reisfeld(Arad and Reisfeld, 1995) has

investigated Gaussian radial basis function(RBF)

which reduces the global influence. And M.

Fornefett, H.S.Stiehl (Fornefett et

al.,1999),(Fornefett et al., 2001) used compact

support radial basis function(CSRBF) in medical

image registration.

Although TPS can minimize the bending energy

of the whole image, it can not deform the image

locally. CSRBF can deform the image locally;

however, it costs much bending energy which means

that the warping loses lots of information form

original image. This weakness is especially distinct

while the deformation is globally.

In this paper, we proposed a new compact

support radial basic function to deform the elastic

image. This function not only limits the image’s

deformation in a local domain, but also is a

fundamental solution to the biharmonic equation . Its

bending costs less energy consequently.

Simultaneously, when the support set is wide, its

warp effect is similar to TPS. Therefore, this new

compact support radial basic function performs well

in the local and global registration experiments.

2 THE LIMITATION OF ELASTIC

IMAGE REGISTRATION

PRESENTLY

TPS models the deformations by interpolating

displacements between source and target points. The

basic function of TPS is

() log

ii i

Ur r r= ,

r is

the distance form the cartesian origin.

TPS’s basic

function is a so-called fundamental solution to the

biharmonic equation (Bookstein, 1989), which can

minimize the bending energy(1).

222

()

TPS

fff

Ef dxdy

xxyy

∂∂∂

=++

∂∂∂∂

∫∫

(1)

Local elastic image registration which bases on

RBFs has the same interpolation function as TPS. It

can not eliminate the global effect of the

deformation. M. Fornefett and K. Rohr(Fornefett et

216

Zhang Z. and Yang X. (2008).

ELASTIC IMAGE WARPING USING A NEW RADIAL BASIC FUNCTION WITH COMPACT SUPPORT.

In Proceedings of the First International Conference on Bio-inspired Systems and Signal Processing, pages 216-219

DOI: 10.5220/0001058402160219

 SciTePress

al.,1999), (Fornefett et al., 2001) applied

function of Wendland as RBFs for elastic

registration of medical image. These radial basic

functions have compact support.

These compact support radial basic functions can

limit the deformation in a local domain. However,

they are not the fundamental solution to the

biharmonic equation. So they cost more bending

energy in deforming the image, especially when

their support radius are enormous, the loss of

information of the source image is considerable.

3 COMPACT SUPPORT THIN

PLATE SPLINE RADIAL BASIC

FUNCTION

3.1 Compact Support Thin Plate Spline

Radial Basic Function

In this paper, we aim to find out a function that not

only is the solution to the biharmonic equation

which can limit the bending energy, but have the

characteristic of the functions which can deform

image locally as well. Therefore, we use TPS’s basic

function:

() log

ii i

Ur r r=

to construct a new

compact support radial basic function:

(a) (b) (c)

Figure 1: Compact support thin plate spline radial basic

function ： (a)

() log

ii i

Ur r r=

(b) fig.(a) displace 1/e

upward (c) CSTPF.

As proven in figure.1 (a), TPS’s original basic

function

() log

ii i

Ur r r=

decreases at

e/10 −

after

e/1

, the function increases rapidly

afterwards. It can be noticed that the decreasing part

(

e/10 −

part) of this function is similar to compact

support radial basic functions which can deform the

image locally. So we add a constant (constant value

is the min value of this function:

e/1

) to this

function (View at fig.1 (b)), and let its increasing

part become zero. Then the function becomes:

⎩

⎨

⎧

≤+

,/1log

)(

rrerr

(2)

Figure.1 (c) shows that presently this function (2)

has the characteristic of CSRBF. It has the max

value at

, it is a decreasing function which is

compact supported meanwhile. Therefore, this

function can be used for local deform interpolation.

Furthermore, this function is the solution to the

biharmonic equation and is able to decrease the

bending energy. Because this function comes from

TPS’s original interpolate function, we name it as

CSTPF (compact support thin plate spline radial

basic function).

3.2 Local Deformation using CSTPF

For the purpose of further investigating elastic image

deformation, we hypothesized that the source images

have already been rigidly registered with the target

image, ignoring the affine part of the interpolation

function. In case of affine free deformation, the

interpolation function is illustrated as follow: (This

interpolation function was utilized in the following 2

chapters.)

|)),((|),(

yxPUwxyxf

xix

−+=

∑

(3)

First of all, we compared the performance of local

deformation of CSTPF and CSRBF（

2,3,a

）at the

same support set. Figure.2 explains the result of the

deformation using the elastic registration approach

base on CSTPF and CSRBF with four pairs of

manual landmarks. It can be noticed that CSTPF is

the solution to the biharmonic equation,

correspondingly, less bending energy is required

than using CSRBF, which means source image’s

information is better saved and the deformation has

been greatly improved.

(a) CSTPF (b) CSRBF

Figure 2: Local deformation results (support radius

100

) (a)Deformation using CSTPF, the cost of

bending energy is3.342; (b) Deformation using CSRBF,

the cost of bending energy is 6.323.

ELASTIC IMAGE WARPING USING A NEW RADIAL BASIC FUNCTION WITH COMPACT SUPPORT

217

3.3 Global Deformation using CSTPF

It has been proved that CSFPF preformed well in

local deformation in last paragraph, now let’s

discuss how it perform in global deformation.

Figure.3 is a contrast of the global deformations

using the elastic registration approach base on TPS

、CSTPF、CSRBF with manual landmarks. In this

Figure shows we can see that the deformations using

the elastic registration approach base on CSTPF

(Fig.3 (b)) and TPS (Fig.3 (c)) are almost the same.

Figure.3(e) takes one line out of the deformation’s

results and makes a comparison. It is shown that the

deformation’s lines of CSTPF and TPS are almost

superposed. This result illustrated that image

deformation using CSTPF can keep the advantage of

TPS in global deformation, which can not be

achieved by using CSRBF.

(a) Landmarks (b) TPS (c) CSTPF

(d) CSRBF (e)

Figure 3: Global deformation contrast (support radius

1000

) ： (a) Landmarks’ position; (b) global

deformations using TPS; (c) global deformations using

CSTPF; (d) global deformations using CSRBF; Contrast

of the third line of Figure 3 (b)(c)(d), Notice Figure 3 (b)

and (c) are almost superposed.

Experimental results have proved that image

deformation using the elastic registration approach

base on CSTPF is better than CSRBF.

To better illuminate the problem and aiming to

compare the bending energy cost at different support

radii, we experimented on 6 groups of deformations

with random landmarks using the elastic registration

approach based on CSTPF and CSRBF. Figure.4

shows the deformations’ bending energy cost at

different radii. In this graph, it is evident that image

deformation using CSTPF costs less bending energy

than CSRBF.

Consequently, the analysis and experiments in

this chapter indicate that image global deformation

using the elastic registration approach bases on

CSTPF is similar to those on TPS. Furthermore, it is

capable to localize the image deformation domain

while TPS can not. In local image deformation,

utilization of the elastic registration approach bases

on CSTPF costs less bending energy than CSRBF

with the same support radius.

(a) Deformations with 6 random landmarks

(b) Deformations with 10 random landmarks

Figure 4: Contrast of deformations’ bending energy cost

with random landmarks usizng CSTPF and CSRBF (X-

axis: support radius, Y-axis is deformation’s bending

energy cost. real line in figure: CSRBF’s energy cost，

broken line in figure: CSTPF’s energy cost): (a)

Deformations with 6 random landmarks; (b) Deformations

with 10 random landmarks.

4 EXPERIMENTAL RESULTS OF

MEDICAL IMAGES

In this chapter, we have prepared two experiments in

practical situations. With manual landmark, we

compared the registration results for medical images

using the elastic registration approach base on

CSTPF and CSRBF.

In Figure 5, we can compare the results of

deformation using CSTPF and CSRBF. They look

similar but definitely not the same. Observing their

edge comparison (Figure 5 (d) and (e)), it is revealed

that after deformation, figure 5(d) has more edge

information than figure 5(e) (as shown by the

arrowhead), which means more information was

saved by using CSTPF than CSRBF.

Finally, we employed another experiment to

demonstrate that global deformation using CSTPF is

better than CSRBF. In this experiment, we used an

image of deferent mode, figure.6 (a) is MRI image

and figure 6 (b) is CT image. It can be easily noticed

that the source image and target image are just the

same as they have no deformation. However,

because we get landmarks manually, it is liable to

have some artificial errors which are, however,

considered as allowable errors. Given that these

BIOSIGNALS 2008 - International Conference on Bio-inspired Systems and Signal Processing

218

allowable errors are unavoidable, the source image

deform globally.

(a) (b) (c)

(d) (e)

Figure 5: Comparison of local deformation using CSTPF

and CSRBF(

2,3,a

) (

40=r

): (a) Source Image(with

landmarks); (b) Target Image (with landmarks); (c)

Comparison of edge of (a) and (b); (d)Deformation using

CSTPF and the edge comparison of deformed image and

original image(top left corner); (e)Deformation using

CSTPF and the edge comparison of deformed image and

original image(top left corner).

(a) (b) (c)

(d) (e)

Figure 6: Comparison of global deformation using CSTPF

and CSRBF(

2,3,a

) (

1000=r

):(a) Source Image(with

landmarks); (b) Target Image (with landmarks); (c)

Comparison of edge of (a) and (b); (d)Deformation using

CSTPF and the edge comparison of deformed image and

original image(global), the deformation cost 0.051327

bending-energy; (e)Deformation using CSTPF and the

edge comparison of deformed image and original

image(global), the deformation cost 0.19555 bending-

energy.

It can be seen in figure.6 (d) that the image after

global deformation using CSRBF has changed its

shape significantly, while CSTPF kept the shape of

source image satisfied. This demonstrated that

CSTPF has stronger capability in global deformation

than CSRBF.

5 CONCLUSIONS

In conclusion, TPS performs better in image global

deformation, but it is not suitable for the local elastic

registration. CSRBF can be used in the local

registration, but it cost more bending energy which

means that it will lose more information during the

deformation. Moreover, its global deformation is not

as well as TPS. In this paper, we proposed a new

radial basic function ‘CSTPF’ which is a solution to

the biharmonic equation. In local deformation, using

CSTPF cost less bending energy. In global

deformation, using CSTPF can keep the topology of

source image. Additionally, it can save the

information more integrated and it is approaches to

TPS. Hence CSTPF is considered superior in image

deformation.

ACKNOWLEDGEMENTS

This paper is sponsored by Nation Natural Science

Funds (NO 60572101) of China.

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