Log-Demons with Driving Force for Large Deformation Image Registration

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1 Log-Demons with Driving Force for Large Deformation Image Registration Le Zhang and Ying Wen* Shanghai Key Laboratory of Multidimensional Information Processing, Department of Computer Science and Technology, East China Normal University, Shanghai, China, * Abstract Capturing large diffeomorphic deformations is d- ifficult for many non-rigid registration methods. In this paper, we propose Log-Demons with driving force for large deformation image registration. The driving force obtained by boundary points correspondence exerts influence on continuous optimization of Log-Demons to improve the motion direction of points. We utilize MROGH descriptor matching to obtain points correspondence as driving force, then the driving force is added to the optimization of Log-Demons. We integrate the driving force in an exponentially decreasing form with velocity field of Log-Demons to drive the points moving globally and to speed up the convergence. Experiments performed on synthetic images, real scene images and brain images demonstrate that the proposed method can not only capture large deformations but also preserve details and register images at a higher accuracy. I. INTRODUCTION Non-rigid image registration technique is a significant field of image processing research, which is widely used in military, remote sensing and medicine. Among them, large deformation image registration is a big challenge. Up to now, much work has been done and a large family of algorithms have been put forward[1]-[4]. A predominant way for non-rigid registration is based on Demons algorithm as introduced in the seminal work by Thirion, where he considered image registration as a diffusion process. Thirion considered one image as a deformable model grid, diffusing through the boundaries of the other image which are considered as semi-permeable membranes[2]. The original Demons algorithm is based on gradient information, when the gradient information is low or missing, Demons may lead to wrong registration results. So several kinds of improved Demons algorithms have been proposed[3][4][9]. Vercauteren[16] demonstrated that the above different Demons variants are related to the use of different optimizers. And to improve the search direction of Gauss-Newton-like method, Vercauteren introduced Efficient Second-order Minimization(ESM)[12] into Demons, showing that symmetric forces can speed up convergence rate. However, a widely acknowledged issue with Demons algorithm is that the estimated transformation is not diffeomorphic and may destroy the topological structure which is vital in some practical applications such as computational anatomy. An approach to solve this issue is to compose deformations as a sequence of diffeomorphic free-form deformations modelled by B-splines[6], but the spline transformations become difficult to obtain diffeomorphisms under very large deformation conditions. The large deformation diffeomorphic metric mapping(lddmm)[7][8] was proposed by Beg et al., which seeks a time-dependent velocity field representation of a diffeomorphism and can achieve a high registration accuracy. Since gradient optimization is over the entire space of diffeomorphisms, LDDMM is time consumption and high memory intensive. LDDMM s limitation can be tackled using stationary velocity fields, so Log-Demons[14][15] was proposed by Vercauteren, which is an efficient non-parametric diffeomorphic image registration framework based on Thirion s Demons algorithm. Log-Demons algorithm is appealing as it not only ensures diffeomorphic mappings but also its computation is in linear complexity. Log-Demons achieves diffeomorphism by performing optimization procedure on a Lie group. But for some algorithms based on Log-Demons[5][31][30], it is still difficult to acquire very large and complex deformations. Conventional methods to deal with large deformations are to adopt coarse-to-fine warping framework, but it may still lead gradient-based methods to a local optimal solution, even in coarse resolutions[29]. To solve this problem, many methods based on regional descriptor matching are proposed. Brox[10] and Liu[11] proposed to integrate descriptors such as HOG[22] and dense SIFT[19] into the variational optical flow to preserve details for large displacement. But due to the mismatching of corresponding points, many motion details are still missed and the transformations are not diffeomorphic. Cifor[26] proposed hybrid feature-based Log-Demons registration which combined block-matching scheme with Log-Demons for tumour tracking. The blocks matching affects the motion direction of points in blocks, but it is hard to process images with complex scenario. Lombaert[28][29] also introduced spectral representation of shapes into Log-Demons, using the properties of spectral global matching to capture very large deformations between images, but the computation of spectral coordinates is time-consuming. The above methods, either do not preserve details, or are not diffeomorphic, or time-consuming, so it is not satisfactory for these methods to deal with very large deformation image registration. In order to overcome above mentioned problems, we propose Log-Demons with driving force for large deformation image registration. We first extract images boundaries using Canny algorithm as driving points, and then employ MROGH descriptor to describe each driving point. MROGH descriptor[21] is intrinsic rotation invariant and performs well in points matching and object recognition etc. Hence, the /16/$31.00 c 2016 IEEE 3052

2 characteristics of boundary can be well described by the MROGH descriptor. By descriptor matching, we can obtain corresponding relations which are defined as driving force in this paper. After that, we cast driving force into Log- Demons. Rather than directly replace the displacement vectors with matching results, we add driving force to the velocity field in each iteration to promote the registration. The main contribution of our work is that we employ boundary points correspondence as driving force to affect the update of Log- Demons in a global scope. And we integrate the driving force in an exponentially decreasing form with velocity field of Log- Demons to drive the movements of points and to speed up the convergence. Extensive experiments performed on synthetic images, real scene images and brain images show that our method has an excellent performance, especially for large deformation registration. The rest of the paper is organized as follows: Section II gives a detailed introduction of our method. The experimental results are presented in Section III. Finally, we draw a conclusion in Section IV. II. METHODOLOGY Before we present our method, we give a brief introduction of Log-Demons, since our method derives from Log-Demons. A. Log-Demons Given a fixed image F and a moving image M, Log- Demons estimates a dense diffeomorphic spatial transformation s that aligns M to F best: E(s, c) = 1 λ 2 Sim(F, M c)+ 1 i λ 2 dist(s, c) x λ 2 Reg(s) (1) T where λ i stands for the noise, λ T controls the degree of regularization and λ x stands for a spatial uncertainty on the correspondences. The variable c is exact spatial transformation that allows error for s[3], leading Demons algorithm to a well-posed problem. The similarity term Sim measures the resemblance between fixed image and warped image, and dist describes the distance between two transformations and ensures that s is close to c, and Reg term is used as a priori knowledge and keeps spatial transformation smooth. The transformation s is parameterized by the stationary velocity field v through exponential map s = exp(v). The velocities v is in the space of log-domain. As v is stationary, exp(v) is efficiently computed using the scaling-and-squaring algorithm[14]. The energy of the Log-Demons can be written in log domain: E(u) = 1 λ 2 isim(f, M s exp(u)) + 1 λ 2 dist(s, s exp(u)) x λ 2 Reg(s) T The optimization of energy equation can be decoupled into two steps. The first step is optimizing 1 Sim(F, M λ 2 i s exp(u)) + 1 λ dist(s, s exp(u)) 2 to obtain update field u 2 x with s being given. The update field u is calculated in the Lie algebra and then is mapped in the space of diffeomorphisms via exponential mapping. Thus transformation s is updated in (2) Fig. 1. The sketch of driving force of two corresponding boundary points. (a) Fixed image. (b) Moving image. (c) The correspondence of p and p. (d) The registration result of Log-Demons. the form of s = s exp(u). It should be noted that different optimization strategies lead to different expressions of Demons force[14]. By adapting Gauss-Newton-like approaches, we can get update field in each iteration as: F (x) M s(x) u(x) = J(x) (3) J(x) 2 + λ2 i (x) λ 2 x where J(x) is the gradient, defined as J(x) = M s(x). The second step is usually smooth the update u, using Gaussian kernels K fluid and K diffusion with standard deviations λ fluid and λ diffusion respectively. B. The Proposed Log-Demons with Driving Force 1) Motivation: Capturing very large deformations is a difficult task for many non-rigid image registration since the gradient computation may fall into local optimum. A popular solution is to establish accurate pointwise correspondence by feature matching technique[29] to push points moving in global scope forcibly. Log-Demons is unable to deal with images with very large deformations such as severe image distortion, whose boundaries are hard to match precisely. Considering that the image boundaries are rich in details and critical to diffusing models of Demons, we establish the correspondence between boundary points of two images and take the correspondence as driving force. In our method, We utilize Canny detector to extract boundary points as driving points. The correspondence between driving points of two images can be viewed as the driving force, which is integrated into the Demons force to push the points moving together. To the end, integration of Log-Demons and driving force can lead to a more precise registration. As shown in Fig.1, Fig.1(a) and Fig.1(b) are a fixed image and a moving image, respectively. Fig.1(c) shows the sketch of driving force of two corresponding boundary points. If the registration method can make the point p move to the position of the point p, the result is satisfactory. However, Log-Demons is unable to obtain the exact result shown in Fig.1(d), so we hope the driving force can be added into Demons force to guide the point rightly moving as indicated by the red arrows. In our method, there are two techniques need to be considered: how to integrate the driving force into Log-Demons algorithm and how to get the driving force. We will describe the two techniques in detail in the following sections. 2) Function of Log-Demons with Driving Force: Given a moving image M and a fixed image F, we can obtain the driving force u c of driving points. The u c can be calculated independent of Log-Demons and then integrated into the 2016 International Joint Conference on Neural Networks (IJCNN) 3053

3 continuous optimization of Log-Demons as the driving force. The objective function of our method is as follows: E(u) = Sim(F, M s exp(u) exp( 2 λ i λ uc )) k + 1 λ 2 dist(s, s exp(u) exp( 1 2 x λ uc )) (4) k λ 2 Reg(s) T It is noted that driving force u c is added to the Demons force. u c can be viewed as a constant variable. λ k controls the influence of driving force on update field u and is set as λ k =2 (k 1)/2, where k is the iteration number in Log- Demons. This leads to a high influence at the beginning and decreases rapidly along with iteration, thus the ill effects of inaccurate descriptor matching can be avoided especially when warped image is close to fixed image. Assuming that descriptor matching in global scope has high accuracy, the driving force u c enables global optimization of Log-Demons and pushes points moving towards right direction rapidly and accurately. According to the Baker-Campbell-Hausdorff(BCH) formula[15], the update should be exp(u(u, u c )) exp(u) exp(( 1 λ 2 k u c ), in order to compute simply, we replace it as U(u, u c ) u + 1 λ 2 k u c. Through first order expansion of intensity difference on the first step of Log- Demons[14]: F (x) M s exp(u)(x) F (x) M s(x)+j(x).u(x) (5) and approximation of distance between two diffeomorphisms: dist(s, s exp(u)) u (6) The energy function of first step of Log-Demons can be rewritten as: Algorithm 1 Log-Demons with Driving Force for Large Deformation Image Registration Input: Fixed image F, moving image M and initial velocity field v Output: Transformation s = exp(v) from M to F 1: Use Canny detector on F and M to obtain two boundary point sets as driving points 2: Compute MROGH descriptor M(x p ) for each driving point 3: Obtain correspondence u c between driving points through descriptor matching using Euclidean distance 4: repeat 5: Add u c to the update u mapping M exp(v) to F using Eq.8 6: Smooth the update u K fluid u for fluid-like regularization 7: Update velocity field:v log(exp(v) exp(u)) 8: Smooth velocity field v K diffusion v for diffusionlike regularization 9: until convergence (MROGH)[21] due to its excellent performance in finding corresponding points which have large orientation change. MROGH is intrinsic rotation invariant and do not need to assign a reference orientation for each driving point which may be a major error source for most of the existing methods[21], such as SIFT[19], HOG[22] and DAISY[20], etc. E(u) 1 [ ] F (x) M s(x) 2 Ω x + 0 x Ω x [ J(x) λ i(x) λ x I ] U(u, u c ) (7) where Ω x is the overlap between F and M s. By minimizing the equation Eq.7, at each pixel x, we can get 2 F (x) M s(x) u(x) = J(x) 1 J(x) 2 + λ2 i (x) λ 2 u c (8) λ 2 k x thus, u(x) is our update rule in each iteration. The proposed Log-Demons with driving force for large deformation image registration is summarized in Algorithm 1. 3) Driving Force: In this section, we present the method how to obtain the driving force. We take boundary points obtained by Canny detector as driving points. The information of driving points can be described by the feature descriptor. And then, the points correspondence can be obtained by descriptor matching. In our method, the driving force u c is defined as the correspondence of boundary points. The driving force can adjust points direction of gradient descent in the process of continuously calculating update field u. When describing a point information and performing descriptors matching in large deformation images, we select the Multisupport Region Order Based Gradient Histogram Fig. 2. Description of MROGH descriptor. (a) Construction of local coordinate system. (b) Multisupport Regions. For each driving point, we use a circular region as the support region with n points denoted by R = {x 1,x 2,..., x n }, I(x i ) is the intensity of sample point x i. Intensities of sample points are sorted in nondescending order and then these points are approximate equally divided into g partitions. Then we construct a rotation invariant coordinate system for each sample point when computing descriptors. As shown in Fig.2(a), suppose x p is a driving point and x i is one of the sample points in its support region. Then, we construct local coordinate system by setting x p x i as the positive y-axis and the corresponding vertical line as x-axis for the sample point x i. Local features are calculated in this local coordinate system to obtain rotation invariant. For each sample point x i, we compute gradient in its local International Joint Conference on Neural Networks (IJCNN)

4 Fig. 3. The synthetic images registration results of different methods. From top to bottom, the names and size of images are Heart( ), Lena(85 85), Hand(95 140), Tennis( ), Shoes( ) and Marble( ). And from left to right, images of each row are (a) fixed image, (b) moving image, registration results of (c) Log-Demons, (d) LDDMM, (e) Spectral Log-Demons and (f) Our method respectively. coordinate system: D x (x i )=I(x 1 i ) I(x 5 i ) (9) D y (x i )=I(x 3 i ) I(x 7 i ) (10) where x j i,j =1, 3, 5, 7, are x i s neighboring points along the x-axis and y-axis in the local x-coordinate system. The gradient magnitude m(x i ) and orientation θ(x i ) can be computed as m(x i )= D x (x i ) 2 + D y (x i ) 2 (11) θ(x i )=tan 1 (D y (x i )/D x (x i )) (12) Then the gradient is transformed to a d-dimensional vector, denoted by F G (x i )=(f1 G,f2 G,..., fd G), fj G = m(x i ) (2π/d α(θ(x i), dir j )) (13) 2π/d where dir j =(2π/d) (i 1),i=1, 2,...d, and α(θ(x i ), dir j ) is the angle between θ(x i ) and dir j. In order to reduce the number of incorrect matches, we choose support regions as the N nested regions centered at the driving point with an equal increment of radius as Fig.2(b) shows. In each support region, the local features of all the sample points are then pooled by their intensity orders to form a vector, then we accumulate vectors of different partitions of each support region to represent this support region, defined as D(R) = (F (R 1 ),F(R 2 ),..., F (R g )). F (R i ) is the accumulated vector of partition R i, i.e., F (R i )= F G (x), (14) x R i Finally, all support regions are pooled together to form final descriptor: M(x p )={D 1 D 2...D N }. We use the default parameter as d =8, g =6, N =4, thus the dimensionality of MROGH descriptor is International Joint Conference on Neural Networks (IJCNN) 3055

5 After constructing descriptor for every driving point, for point x i in moving image, we try to find its corresponding point x j in fixed image by MROGH descriptor matching. The point in the fixed image has a corresponding point in the moving image followed by that x i is the optimal correspondence for x j and x j is also the optimal correspondence to the other. If x i and x j are mutually optimal corresponding points, the displacement vector of point x i is obtained by u c i = x j x i, otherwise u c i =0. III. EXPERIMENTS In this section, we perform experiments on three types of images, i.e., synthetic images, real scene images and brain images to investigate the performance of our proposed method. We compare the proposed method with some popular registration algorithms, such as Log-Demons, Spectral Log-Demons and LDDMM. In the following, we use the default parameters of LDDMM, and all experiments are carried on the following empirical parameters: λ i =1, λ fluid =1, λ diffusion =1. We utilize Mean Square Error(MSE) to evaluate the intensity difference between the fixed image and the warped image. A. Experiments on Synthetic Images In this experiment, we focus on image registration with very large deformations. The moving images are all synthetic images. The max step λ x is set as λ x =2. After performing 100 iterations, the methods have been reached convergence, so we compare them within the same level of resolution. We select six fixed images and warp them with randomly large deformations, and the final registration results are shown in Fig.3. The first image of each row is a fixed image and the second is a deformed moving image, and the followings are registration results of Log-Demons, LDDMM, Spectral Log- Demons, and ours respectively. From Fig.3, it can be seen that in example of Heart, the moving image has some severely distorted parts marked by red arrows. Log-Demons, LDDMM and Spectral Log-Demons register unsuccessfully in these regions, while our method outperforms the other three methods and pushes all boundaries aligned. For the images of Hand, ring finger in the moving image and middle finger in the fixed image are overlapped. LDDMM and Spectral Log-Demons are not capable of moving fingers back to original position. Log-Demons wrongly aligns ring finger to middle finger, resulting in overlap and discontinuous of fingers, since the optimization may be limited in local scope. Due to the correspondence of boundary points, driving force of our method can push the finger back to the right position. The final MSE result of LDDMM is , Log- Demons is , Spectral Log-Demons is and ours is The intensity difference between fixed and warped images of our method is reduced by 59.66% compared with Log-Demons, 57.93% compared with Spectral Log-Demons and 86% compared with LDDMM. As for Lena, Log-Demons, Spectral Log-Demons and LDDMM can not align face or hair, while our method registers all correctly. For Tennis and Marble, twisted lines are hard task for registration methods. For example, the railings in Tennis are moved to wrong places by the other three methods since the lack of external force pulls them aligned with corresponding distant lines. While our method restores the lines successfully, demonstrating that Fig. 5. The colormaps of intensity difference for Hand. Blue means no difference and red means that intensity difference is biggest. From top to bottom, colormaps are obtained by Log-Demons, LDDMM, Spectral Log- Demons and ours respectively when iteration numbers are 1, 5, 10, 20, 100. driving force obtained by boundary points correspondence can restrain the movement of points and is crucial to keep complete object outline and details in the process of registration. In order to show more details in the process of registration, we take Hand as example to show intensity differences in refine iterations. For the sake of visual observation, we transform the intensity differences of Hand into colormap. Fig.5 shows the colormap of warped images when iteration numbers are 1, 5, 10, 20, 100. In the first 10 iterations, the fingers are nearly correctly restored by our method while others fail. It is noted that the driving force should have a great impact on Log-Demons at the beginning, and declines with the decrease of the intensity difference. The MSE curves of registration of Fig.3 are shown in Fig.4, in which the blue, black, green and red curves stand for Spectral Log-Demons, Log-Demons, LDDMM and our method respectively. For six experiments, our MSE curve drops fastest to approach convergence and is nearly always bellow the other methods, indicating that our method registers effectively and accurately. B. Experiments on Real Scene Images In this experiment, we investigate the performance of our proposed method on real scene images. We randomly choose 30 images from MIT[35] database and 20 images from Brox datasets[10] to evaluate. This experiment is conducted in four levels scheme. Due to the limitation of paper length, we present some sample images and registration results in Fig.6. The first image of each row is a fixed image and the second image is a moving image, and the followings are registration results of Log-Demons, LDDMM, Spectral Log-Demons, and ours respectively. As shown in Fig.6, all final registration results are very close in vision, but MSE of ours is superior to that of Log- Demons and Spectral Log-Demons, and LDDMM performs unstably. Table I presents the mean and standard deviation of MSE. It can be seen that the performance of our method on International Joint Conference on Neural Networks (IJCNN)

6 Fig. 4. MSE curves of different registration methods, the blue, black, green and red curves stand for Spectral Log-Demons, Log-Demons, LDDMM and our method, respectively. real scene images still has advantages, while LDDMM has large fluctuations. We can conclude that the performance of our method is superior to Log-Demons based methods on real scene images, and the driving force integrated into Log- Demons can lead to a more precise result. Mean MSE TABLE I. EXPERIMENTS ON REAL SCENE IMAGES Log-Demons LDDMM Spectral Log-Demons Ours Hand 33.95(±16.80) 21.65(±6.08) 36.20(±20.84) 32.27(±14.54) Table (±8.66) 84.35(±7.82) (±6.88) (±6.08) Toys (±41.97) (±80.11) (±40.02) (±38.28) Tennis (±9.22) (±14.65) (±8.54) (±10.61) Shoes (±12.66) (±85.53) (±11.17) (±15.49) C. Experiments on MR Images Demons algorithm is widely used in brain image registration, so we experimentally evaluate the performance of proposed scheme on MR brain images. The brains present complex structures across individuals, so we focus on the warping accuracy of the tissue structure in this experiments. We apply the proposed method to MR brain images registration and 35 T1 images[33] are randomly selected for evaluation. These brains vary significantly in morphology. We also conduct the experiment in four levels scheme and the maximal step is adjusted to 1.5. Fig.7 shows the registration results, the first image of each row is a fixed image and second is a moving image, and the followings are final results of Log-Demons, LDDMM, Spectral Log-Demons and ours respectively. The registration results of all methods have similar observations. In order to evaluate the registration accuracy, the brain image is segmented into three tissues: white matter(wm), gray matter(gm), and cerebrospinal fluid(csf) by the method of FSL[36]. Then, we measured MSE and Dice coefficient[34] to estimate the accuracy, and the Dice coefficient is defined as: ρ =2 O 1 O 2 /( O 1 + O 2 ) (15) where O1 and O2 denote region of interest (ROI) in the fixed and warped images. The MSE and dice coefficient of the four methods are listed in Table II. Our method has a slightly advantage over Log-Demons and Spectral Log- Demons in GM and CSF, while LDDMM performs the worst. The experimental results show that our method is comparable to other state of the art registration methods. IV. CONCLUSION In this paper, we propose Log-Demons with driving force for large deformation image registration. The correspondence of boundary points obtained by MROGH descriptor matching is defined as the driving force which is integrated into Log- Demons to improve the update way of Log-Demons. Thus the proposed method can not only capture large deformations but also preserve details and register images at a higher accuracy. Experiments on synthetic images, real scene images and brain images demonstrate that our method has a good performance for image registration. In particular, our method is superior to other methods for large deformation image registration. ACKNOWLEDGMENT The authors would like to thank Herve Lombaert, Laurent Risser and Bin Fan for sharing their code. This work was supported by The National Natural Science Foundation of China (No ), Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (ZF1213), Open Project of the Key Laboratory of Embedded System and Service Computing Ministry of Education (Tongji University) 2016 International Joint Conference on Neural Networks (IJCNN) 3057

7 Fig. 6. Experiments on real scene images. The first image of each row is a fixed image, and the second is a moving image. The followings are registration results of Log-Demons, LDDMM, Spectral Log-Demons, and Ours respectively. MSE is below images. Fig. 7. Experiments on MR brain images. The first image of each row is a fixed image, and the second is a moving image. The followings are registration results of Log-Demons, LDDMM, Spectral Log-Demons, and Ours respectively International Joint Conference on Neural Networks (IJCNN)

8 TABLE II. EXPERIMENTS ON BRAIN IMAGE REGISTRATION Dice Log-Demons LDDMM Spectral Log-Demons Ours coefficient WM (±0.0656) (±0.0778) (±0.0662) (±0.0655) GM (±0.0153) (±0.0170) (±0.0159) (±0.0155) CSF (±0.0242) (±0.0286) (±0.0235) (±0.0235) MSE (± ) (± ) (± ) (± ) and the Science and Technology Commission of Shanghai Municipality under research grant no.14dz REFERENCES [1] G. E. Christensen, P. Yin, M. W. Vannier, et al, Large-deformation image registration using fluid landmarks, Image Analysis and Interpretation, Proceedings. 4th IEEE Southwest Symposium. pp , [2] J. P. Thirion, Image matching as a diffusion process: an analogy with Maxwell s demons, Medical Image Analysis, vol. 2, pp , [3] P. Cachier, E. Bardinet, D. Dormont, et al, Iconic Feature Based Nonrigid Registration: The PASHA Algorithm, Computer Vision and Image Understanding, vol. 89, pp , [4] X. Pennec, P. Cachier, N. 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