DEMONS REGISTRATION WITH LOCAL AFFINE ADAPTIVE REGULARIZA TION:APPLICATION TO REGISTRATION OF ABDOMINAL STRUCTURES

Moti Freiman1, Stephan D. Voss2, Simon K. Warfield1

1 Computational Radiology Laboratory, Children’s Hospital, Harvard Medical School, Boston, MA, USA2 Department of Radiology, Children’s Hospital, Harvard Medical School, Boston, MA, USA

E-mail: moti.freiman@childrens.harvard.edu

ABSTRACT

We present a local affine based adaptive regularization ap-proach as an alternative to the homogeneous regularizationused in Thirion’s demons non-rigid registration algorithm[1].The original homogeneous regularization does not preservethe deformation field discontinuities that are related to theindependent motion of different organs, as commonly oc-curs during CT and MR imaging of the abdomen and pelvis.Instead, our method fits local affine transformation to eachvoxel, identifies the discontinuities between the locally fittedaffine transformations and adaptively smooths the deforma-tion field, while preserving its local affine discontinuities.Experimental results on both synthetic and real CT imagesdemonstrate that our method yields a smoother deformationfield while preserving the registration accuracy.

Index Terms— Locally affine registration, Demons non-rigid registration, Adaptive regularization

1. INTRODUCTION

Abdominal image registration is a prerequisite task for at-las based abdominal image segmentation, which is utilizedby many clinical applications, such as radiotherapy planning,surgical simulation, and for the derivation of volumetric basedbio-markers.

Abdominal image non-rigid registration is a particularlychallenging task due to the presence of multiple organs, manyof which move independently or are influenced by the move-ment of adjacent structures (i.e. the diaphragm), contributingto independent deformations.

Among many existing registration methods, Thirion’sdemons algorithm [1] is the most widely used for the compu-tation of dense deformation between images [2] because of itslinear complexity and simple implementation. This algorithmiteratively computes the best dense deformation field thatminimizes the intensity dissimilarity between the images,andregularizes the deformation field to ensure spatial coherencyby homogeneous isotropic smoothing with a Gaussian kernel.

This investigation was supported in part by NIH grants R01 RR021885,R01 EB008015, R03 EB008680 and R01 LM010033.

However, the homogeneous smoothing step does not pre-serve possible discontinuities in the deformation field, whichmay be related to the independent movement of different or-gans. Thus, the resultant deformation field may be inaccu-rate. Several adaptive regularization approaches have beenrecently developed. They can be roughly classified into twocategories: 1) Image-driven and 2) deformation field driven.

Image-driven adaptive regularization methods use aweighted regularization approach, where the regularization isinversely proportional to discontinuities derived from the im-age intensity domain. In [3], a tumor region is automaticallysegmented from the intensity image and used to control thesmoothing operator. In [4], a stiffness parameter derived fromthe moving image intensities is used to control the smoothingprocess. In [5], a reliability-measure is used to control thesmoothing amount with respect to the estimated image noise.In [6], the amount of smoothing is inversely proportional tothe curvature of the image. In [7], a Hessian based structuredetector is used to control the smoothing amount. In [8] aphysical model generated from prior manual segmentationwas used to control the smoothness amount and direction.

Deformation field driven methods use deformation field-based information to reduce the smoothing effect along thedeformation field discontinuities. In [9] a specific filter isusedto prevent over-smoothing along deformation-field disconti-nuities that are related to resections or retractions. However,this approach cannot be used to ensure coherent and inde-pendent deformation of different organs. In [10], a machine-learning approach is used to encode a task-specific deforma-tion prior, that can be used later for the regularization. Thisapproach requires accurate construction of the deformationprior for each registration task.

In this paper we present a local-affine adaptive smoothingapproach for the regularization in the demons algorithm. Ourapproach models the dense deformation as a set of local affinetransformations, and adaptively smooths the dense deforma-tion field while preserving the discontinuities along the localaffine components using the anisotropic smoothing approach[11]. We assume that voxels that are close, both spatially andin their intensity value, represent the same structure, andthus

(a) Refernce image (b) Moving image

(c) Demons registration result (d) Local affine adaptiveregularization result

Fig. 1. Synthetic example: (a)-(b) Axial slices from the ref-erence (a) and moving (b) images; (c)-(d) Registration resultsof the demons algorithm (c) compared to our result (d), withthe fixed image overlaid in red.

have similar affine motion. Thus, the local-affine modelingprocess assigns an affine transform to each voxel, by consid-ering its local neighborhood voxels, weighted by their inten-sity similarity.

Fig. 1 illustrates the effect of our method compared tothe original demons algorithm on a synthetic example. Thecoupling of efficient dense deformation estimation with lo-cal affine adaptive smoothing yields a better registration algo-rithm which is more suitable for the registration of abdominalimages.

2. METHOD

Given a patient imageIp, the goal is to find a dense deforma-tion field Dr

p that minimizes its dissimilarity to the referenceimageIr. Thirion’s demons algorithm [1] computes the de-formation field that minimizes the energy:

Drp = argmin

Drp

E(Ip, Ir, Drp) + S(Dr

p) (1)

whereE(Ip, Ir, Drp) is the dissimilarity measure between the

reference and deformed images, andS(Drp) is a regularization

term that controls the smoothness of the resultant deformationfield. The solution is found by applying the following twosuccessive steps iteratively:

1. Compute an unconstrained dense deformation field thatminimizes the dissimilarity between the patient and thereference images.

2. Regularize the deformation field by homogeneous isotropicGaussian smoothing to ensure its spatial coherence.

Since the smoothing step may over-smooth the deforma-tion field discontinuities that are related to independent move-ments of different organs, we replace the smoothing step (2)with a new anisotropic smoothing filter that is inversely pro-portional to the differences between the local affine transfor-mations. Our smoothing step consists of the following com-ponents:

2.a) Fit an affine transformationA(~x) to each voxel~x basedon its neighborhoodΩ~x.

2.b) Compute the gradient of the affine transformations do-main.

2.c) Apply adaptive smoothing [11] to the deformation field,based on the affine transformations gradient.

We will describe these components in detail next.

2.1. Local affine transformation fitting

The fitting of a local affine transformation to each voxel is de-fined as follows: For each voxel~x, find an affine transformA(~x) that best approximates the deformation of the struc-ture in this region with respect to the neighboring voxels~y ∈Ω~x and the current estimation of their deformation vectorsDr

p(~y). Since considering voxels in the neighborhoodΩ~x of~x equally may introduce motions of different organs, we usea weighting function that prefers voxels with similar intensityvalues over voxels with large intensity differences. Conse-quently, the local affine fitting is formulated as a weightedleast-squares problem:

A(~x) = argminA

∑

~y∈Ω~x

w~y · ‖ A(~y) − Drp(~y) ‖2 (2)

The weightsw~y are defined as:

wy = exp

(−

(I(~x) − I(~y))2

2σ2

)(3)

whereσ is a predefined scaling parameter, representing theexpected signal intensity homogeneity within same structure.The solution can be found efficiently using Horn’s method[12], or using the SVD method [13].

2.2. Local affine domain gradient computation

We use the Frobenius metric between matrices:

‖ A1 − A2 ‖=

√√√√N∑

i

M∑

j

(A1(i, j) − A2(i, j))2 (4)

(a) Before (b) After our registration

(c) After demons registration (d) After diffeomorphic demonsregistration

Fig. 2. Clinical registration example. Axial slice from an ab-dominal CT scan with the manual annotation from the mov-ing image overlaid in red: (a) before registration; (b) afterour registration; (c) after demons registration, and; (d) afterdiffeomorphic demons registration.

with appropriate scaling for the translational componentsofthe affine transformation, to define the gradient of the localaffine domain. The gradient is then computed using the finitedifferences approach.

2.3. Local affine adaptive smoothing

Given the local affine domain gradient∇A(~x), we use theanisotropic diffusion operator to adaptively smooth the defor-mation vectors associated with each voxel, while preservingthe discontinuities between local affine motions. Followingthe anisotropic diffusion approach presented in [11], our dif-fusion operator is defined as follows:

∂Drp(~x)

∂t= ∇c(~x) · ∇Dr

p + c(~x)∆Drp( ~X) (5)

wherec(~x) is the diffusion coefficient defined as:

c(~x) = exp

(−‖ ∇A(~x) ‖2

k2

)(6)

andk is a predefined constant scaling parameter that differen-tiates between local affine differences that relate to indepen-dent organ movements and noise.

Dice Harmonic(%) energy

Kidney 1Our 0.8 0.28

Demons 0.8 0.31Diff demons 0.59 4.53

Kidney 2Our 0.87 0.44

Demons 0.87 0.58Diff demons 0.86 2.73

PelvisOur 0.53 0.27

Demons 0.53 0.3Diff demons 0.55 0.29

Table 1. Registration results. The first column is the struc-ture that was registered. The second column is the methodthat was used for the registration. The third column is theDice similarity measure between the manual segmentation ofthe structure and the deformed segmentation from the movingimage. The fourth column is the harmonic energy of the resul-tant deformation field. For each dataset, the first row presentsthe results obtained using our method. The additional tworows present the results of the other algorithms.

3. RESULTS

3.1. Synthetic example

We used our algorithm to register a 3D image with twocrosses to a similar image, where each of the crosses hasa translational movement of 1 voxel in each axis, in oppo-site directions. Fig. 1 presents the reference and movingimages, and the registration results of the original demonsalgorithm [1] compared to our algorithm results. Our algo-rithm preserves the object boundaries much better comparedto the demons result. The final Sum of Squared intensity Dif-ferences (SSD) between the fixed and the registered imageusing our method was 7.3, while the demons algorithm resultyielded a SSD of 45.04.

3.2. Abdominal CT registration

To evaluate our method on real images, we randomly col-lected three CT scans of patients with various pathologiesfrom the hospital archive, and applied inter-patient registra-tion. Initially, all the images were globally aligned usingaffine registration with the elastix toolbox [14]. Next, thekidney and pelvis regions were extracted from the entire ab-dominal scan. The properties of the images that were used forthe non-rigid registration experiment were as follows: Thekidney image size was 96x96x150 voxels, and the pelvis im-age size was 234x162x128. Both had physical spacing of1.17x1.17x1.2mm3.

Finally, we applied non-rigid registration to the pelvis andthe two kidneys using our method, and compared its results to

that of the original demon’s registration [1] and to the diffeo-morphic demons algorithm [15]. To accelerate the registrationprocess, we used our anisotropic smoothing approach only asa post-processing step on the demons registration result. Theparameters that were used for our anisotropic smoothing wereas follows:σ = 15, k = 1, the neighborhood radius used foraffine tranformation fitting was 1 and the number of smooth-ing iterations was 30. Fig. 2 presents a kidney registrationexample.

To quantitatively measure our method’s performance,the kidney and the pelvis were segmented manually on bothimages, and aligned using the resultant deformation fields.The Dice similarity measure was used to evaluate the accu-racy of the registration, and the Harmonic energy measure[15], which is inversely proportional to the deformation fieldsmoothness, was used to quantify the smoothness of the re-sultant deformation field. Table 1 presents the results for boththe two kidneys and the pelvis experiments. For all exper-iments, our method yielded a smoother deformation field,while preserving the registration accuracy. This is in contrastto the existing regularization methods that suffer from a com-promise between the accuracy of the resulting deformationfield and its smoothness [16].

4. CONCLUSIONS

We have presented a local-affine adaptive smoothing ap-proach for the regularization of dense deformation field dur-ing demons registration. Our approach locally fits the affinetransformation that best represents the local motion of dif-ferent objects. Then, it smooths the original deformationfield while preserving its local affine discontinuities. Rep-resentative results on both a synthetic example and clinicalabdominal CT images demonstrate that our method producessmoother deformation-fields while preserving their accuracy.In the future we plan to use this method for multiple organatlas-based abdominal image segmentation.

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