Medical Image Registration Based on Mutual ... - IEEE Xplore

Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference Shanghai, China, September 1-4, 2005

Medical Image Registration Based on Mutual Information Using Kriging Probability Density Estimation Chuanxin Niu

Abstract — This paper proposes a new approach of Kriging Estimation (KE) to the problem of probability density estimation, which is critical during the registration of medical images using Mutual Information (MI). The author proposed that the linear coefficients of Kriging Interpolation have “probability character” which makes them appropriate to be utilized in probability estimation. KE is a complement of existent probability estimation methods, the experiments provided have proved its efficacy.

I. INTRODUCTION

T

HE purpose of image registration is to spatially align two or more single modality images taken at different times or several images acquired by multiple imaging modalities [2]. Image registration requires maximization of a measure of similarity, Mutual Information (MI) for two images is among the measures focused these years and has been proved efficient by many researches. The estimation of probability density plays an important role in the MI-based registration. W. M. Wells III et al. [1] employed Parzen Window estimation which was a widely used non-parameterized method, whereas Maes et al.[3] employed histogram estimation without evaluate the MI’s derivative. In this paper, we proposed an improvement of the latter one by introducing the Kriging Estimation (KE) which is derived from Kriging Interpolation (KI). The paper is organized as follows. The basic concept and method of MI-based registration is described in Section II, especially the requirement for interpolation method during the probability density estimation. The introduction of Kriging Estimation is in Section III, this part focuses on how to embed the Kriging technique into the probability histogram estimation (See Maes [3]) whereas the detailed description of Kringing isn’t mentioned much. Section IV gives the report of experiments on the registration among CT, MR, PET images, section V draws some conclusions from the comparison experiment above, which generally proved the efficacy of our approach.

II. METHODOLOGY A. Registration The idea is that two images are optimally matched when their mutual information is maximum. In the following derivation, two volumes of image data that are to be registered will be mentioned, which are: reference volume and the floating volume, each of which was formed by images for reference and for geometric alignment, respectively. A voxel’s gray level of the reference volume is denoted r (t ) , where t are the coordinates of the voxel. A voxel’s gray level of the floating volume is denoted similarly as f ( s ) . Given that T is a transformation from the coordinate frame of the reference volume to the floating volume, r (T ( s ) ) is the reference volume voxel associated with floating volume f (s ) . In order to distinguish the transformation and its parameterization, we will use T to denote the transformation and α for its parameterization. That is, the counterpart of f (s ) is r (Tα ( s) ) . (See Maes [3]) The goal of registration is to find a geometrical transformation T , as well as its parameter α , that maximize the registration measure M , which is:

α * = arg max M ( f ( s ), r (Tα s )) α

B. Mutual Information Mutual Information (MI) is fundamentally based on the concept of entropy. The entropy of a image can be interpreted as a measure of uncertainty, variability, or complexity. The MI for Image A and B f (s ) and r (Tα (s )) is defined in the following way:

I ( A, B ) ≡ h ( A) + h ( B ) − h ( A, B ) (1)

with h( A) , h ( B ) being the entropy of A and B, respectively, h( A, B ) their joint entropy.

h( A) , h ( B ) and h( A, B ) are defined as follows: Manuscript received April 15, 2005. This work was supported in part by the National Color Science and Engineering Laboratory in Beijing Institute of Technology. Chuanxin Niu is an undergraduate with the Electronic Engineering Department of Beijing Institute of Technology, Beijing 100081 P.R.C. (phone: 86-010-68942308; e-mail: minos.niu@ gmail.com).

0-7803-8740-6/05/$20.00 ©2005 IEEE.

h( A) = − ³ p (a ) ln p (a )da (2)

h( B) = − ³ p(b) ln p(b)db (3) h( A, B) = − ³ p(a, b) ln p(a, b)da db (4) According to the statements above, the registration can be

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re-denoted as

α = arg max I ( f ( s), r (Tα s ) ) (5) *

I ( f (s ), r (Tα (s)) )

α

≡ h ( f ( s ) ) + h ( r ( Tα ( s ) ) ) − h ( f ( s ), r ( Tα ( s) ) ) (6)

³

Where hF ( f ) = − pF ( f ) ln pF ( f )df (7)

hR (r ) = − ³ pF (r ) ln pF (r )dr (8) hFR ( f , r ) = − ³ pFR ( f , r ) ln pFR ( f , r )df dr (9) C. Estimation of Probability Density Obviously, to evaluate the joint probability pFR ( f , r ) and the marginal probability pF ( f ) and pR ( r ) , is a critical step of the registration. To this problem, W. M. Wells III et al. [1] employed Parzen Window estimation which was a widely used non-parameterized method, whereas Maes et al.[3] employed histogram estimation. Here in this paper, we focus on the latter one. The fundamental idea of the histogram estimation is to determine the discrete probability distribution based on frequency of every existent gray level value. It’s especially to estimate the joint probability pFR ( f , r ) because the marginal probability can be easily get by a simple summation at the margin. Suppose H α ( f , r ) denotes the image gray level histogram, it can be computed by binning the gray level pairs ( f ( s), r (Tα ( s)) ) for all s in the floating image. D. Histogram Modifying with Interpolation In general, Tα ( s) will not coincide with a known point of the reference image and thus interpolation is need to obtain the gray level r ( Tα ( s ) ) . Nevertheless, the histogram should not be simply added a certain value to Hα ( f ( s ), r (Tα s ) ) because this may lead to unpredictable changes in the marginal distribution [3]. Instead, the histogram should only be modified at existent values with specified weight, namely, only Hα ( f ( s), r (ti ) ) , i = 1, 2,! , n should be updated where r ( ti ) denotes each of the pre-known (say, sampled) voxels’

gray levels of the reference image.

this paper will not mention much of the principle and derivation of the technique, what we concerned more is how to use its interpolation results to modify the histogram estimation, which is so-called Kriging Estimation. A. Expression Suppose there are n sampled points ti (i = 1, 2,! , n) of the reference image, Kriging Interpolation shows that the gray level of an unknown point Tα ( s) could be expressed as a linear summation of r ( ti ) , the final expression of the interpolation is: n

r ( Tα s ) = ¦ λi r (ti ) , with

= 1 (10)

i

i =1

i =1

B. Probabilistic Character Consider V as a random variable over {r (ti )} , i = 1, 2,!, n , then the mathematical expectation of V could be denoted as: n

E (V ) = ¦ p [ r (ti ) ] ⋅ r (ti ) , with

n

¦ p [ r (t )] = 1 (11). i

i =1

i =1

Compare the equations (10) and (11), if we regard r (Tα s ) as the combine-acted effect of all the possible samples (i.e. r (ti ), i = 1, 2,! , n ), λi and p [ r (ti )] could be easily associated together, namely λi represents the probability of p [ r (ti )] well, which is proposed as the probabilistic character of linear sum coefficient λi . As a result, it can be inferred that

λi will be adaptable to modify the histogram. C. Modifying of Histogram Taking λi as a practicable weight for the histogram, we propose that for r ( Tα s ) =

n

¦ λ r (t ) , i

i

i =1

∀i : H α ( f ( s ), r (ti )) + = λi ,

n

¦λ

i

= 1 (12)

i =1

Then the estimation for the joint and marginal image gray level distributions pFR ,α ( f , r ) , pF ,α ( f , r ) , and pr ,α ( f , r ) are obtained by normalization of Hα ( f , r ) :

Hα ( f , r ) (13) ¦ f ,r H α ( f , r )

pFR ,α ( f , r ) =

III. KRIGING ESTIMATION Kriging is called the best linear unbiased interpolator because it is theoretically capable of minimizing the estimation error variance while being a completely unbiased estimation procedure [4]. This technique was originally developed for estimating ore grade thicknesses or accumulations in mining deposits, but is applicable to any spatially dependent data. Historically, Kriging technique was first introduced to the medical image process by Rob W. Parrott et al. in 1993 [4],

n

¦λ

pF ,α ( f ) = ¦ pFR ,α ( f , r ) (14) r

pr ,α ( r ) = ¦ pFR ,α ( f , r ) (15) f

According to the definition of MI, it can be evaluated that

I (α ) = ¦ pFR ,α ( f , r ) log 2 f ,r

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pFR ,α ( f , r ) pF ,α ( f ) pR ,α ( r )

(16)

Then equation (5) becomes α * = arg max I (α ) α

IV. EXPERIMENTS We’ve practiced registrations on MR-PET images as well as CT-MR images using the Powell optimize approach. The experiments were performed on Intel Pentium IV 2.8GHz, 512MB Ram, and visualized results are presented in Fig 1 and Fig 2. Up to the submission, the result provided by “Evaluation of Retrospective Image Registration” of Vanderbuilt Univ. hasn’t been published, which means our approach remains undergoing the check of “Gold Criterion”.

REFERENCES [1]

[2]

[3]

[4]

(a) (b) (c) (d) Fig. 1. The visual result of registration between of MR and PET images. (a) Original MR image (b) Original PET image (c) Transformed MR image (d) Superposition of transformed MR&PET images

(a) (b) (c) (d) Fig. 2. The visual result of registration between of CT and MR images. (a) Original CT image (b) Original MR image (c) Transformed CT image (d) Superposition of transformed CT&MR images

V. CONCLUSION AND FUTURE WORK Kriging Estimation is a complement of existent probability estimation methods, it provides high accuracy when images for registration are provided with scattered sampled points. With an interpolated point is related with all pre-known points, however, the cost of this method is usually higher than Nearest Neighbor, Trilinear, etc. Future work includes the comparison between ordinary Kriging and universal Kriging, try to get draw some conclusion on how to chose the most appropriate one at different occasion. Furthermore, the optimization and reduction of computation scale remains lots of work to do.

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W. M. Wells III, P. Viola, H. Atsumi, S. Nakajima, and R. Kikinis, “Multi-modal volume registration by maximization of mutual information,” Med. Imag. Anal., vol. 1, pp. 35–51, 1996. Josien P. W. Pluim, J. B. Antoine Maintz, and Max A. Viergever, “Interpolation Artefacts in Mutual Information-Based Image Registration,” Computer Vision and Image Understanding, 77, pp. 211–232, 2000. F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Trans. Medical Imaging, 16(2), pp. 187–198, 1997. Rob W. Parrott, Martin R. Stytz, Phillip Amburn, David Robinson, “Towards Statistically Optimal interpolation for 3-D Medical Imaging,” IEEE Engineering in Medicine and Biology, 93, pp. 49-59, 1993.