Coupled Bayesian Framework for Dual Energy Image Registration

Transcription

1 in Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), Vol. 2, pp , 2006 Coupled Bayesian Framework for Dual Energy Image Registration Hao Wu University of Maryland College Park, MD wh2003 AT cfar.umd.edu Yunqiang Chen and Tong Fang Siemens Corporate Research 755 College Rd E, Princeton, NJ {yunqiang.chen, tong.fang} AT siemens.com Abstract Image registration for X-ray dual energy imaging is challenging due to the overlaid transparent layers (i.e., the bone and soft tissue) and the different appearances between the dual images acquired with X-rays at different energy spectra. Moreover, subpixel accuracy is necessary for good reconstruction of the bone and soft-tissue layers. This paper addresses these problems with a novel coupled Bayesian framework, in which the registration and reconstruction can effectively reinforce each other. With the reconstruction results, we can design accurate matching criteria for aligning the dual images, instead of treating them as multi-modality registration. Furthermore, prior knowledge of the bone and soft tissue can be exploited to detect poor reconstruction due to inaccurate registration; and hence correct registration errors in the coupled framework. A multiscale freeform registration algorithm is implemented to achieve subpixel registration accuracy. Promising results are obtained in the experiments. 1. Introduction Image registration finds various applications in computer vision and medical imaging. Different registration methods have been developed for rigid or non-rigid deformations [7, 8, 4, 1]. Some recent research exts image registration to fuse images from different modalities [14, 6, 10, 16]. However, it remains a challenging task to achieve robust and accurate registration in X-ray dual energy imaging. Dual images obtained with X-rays at different energy spectra have different appearances, which complicates the designing of appropriate similarity measurements for image alignment. Mutual information has been proposed for multimodality image registration [7, 8, 4, 1], but it is difficult to meet the subpixel-accuracy requirement for good reconstruction of the bone and soft-tissue layers the final goal of the dual energy imaging. In this paper we propose a coupled Bayesian method based on the image formation model of dual energy imaging to solve image registration and reconstruction jointly. Dual energy imaging is designed to improve conventional chest radiography, which suffers from low sensitivity for detecting lung nodules or other subtle details due to the overlap of bone structures and soft tissue. Dual energy imaging is designed to separate the bone and soft tissue from two X-ray images acquired at different energy spectra. Since the attenuation coefficients of bone and soft tissue follow different functions of the energy, the dual images can be weighted and subtracted to acquire the separated soft tissue specific and bone specific images, thereby allowing better evaluations of the lung nodules or pleural calcification [15]. Image registration is a key component in the dualexposure method, which performs the image acquisition procedure at two different kv(p) levels in two exposures and provides better image quality than the one-shot method [12]. The time gap between the two exposures can be about ms, during which patient or anatomical motions may result in motion artifacts in the weighted subtraction. It is necessary to align the two images before subtraction. The dual images are overlays of two layers (i.e., the bone B and soft tissue S). The image formation model is: I 1 = a B + b S I 2 = c T (B) + d T (S) where a, b, c, and d are known constants reflecting the attenuation coefficients of the bone and soft tissue to the X-ray at different spectra. Based on the two observed images I 1 and I 2, we need to reconstruct the bone B and the soft tissue S as well as the non-rigid motion T. For the simplest case, where there is no motion between the dual images, the bone and soft tissue can be easily obtained through weighted subtraction of I 1 and I 2 : B = (d I 1 b I 2 )/(a d b c) S = (a I 2 c I 1 )/(a d b c) In traditional dual energy imaging, image registration is a preprocessing step to align I 1 and I 2, followed by the weighted subtraction in Eq. (2). However, there are several problems in that scheme: (1) (2)

2 The images I 1 and I 2 have different appearances due to different attenuation coefficients. No simple similarity measurement can be designed to guide the registration process. Cross-correlation or mutual information [5] might be used, but they assume only very general depencies between the two images and neglect the image formation model in Eq. (1). It is more difficult to achieve robust registration with high accuracy. Reconstruction of bone B and soft tissue S is highly depent on the accuracy of the non-rigid registration. From Eq. (2), we can see that any registration error will be amplified by 1/(a d b c), which could be very large when the ratio of the coefficients (a/c and b/d) are similar. Our experiments show that subpixel accuracy is necessary for good reconstruction. The subtraction procedure in Eq. (2) assumes perfectly aligned dual images. No registration error can be corrected in this step. Also, the subtraction does not utilize any prior knowledge of the bone and soft tissue (e.g., smoothness constraint and edge modeling), which have been very helpful in image restoration. In this paper, we propose a coupled Bayesian framework to register the dual images and reconstruct the bone and soft-tissue layers jointly. In the coupled framework, the two processes reinforce each other and can achieve more robust and accurate results. First, with explicit modeling of the bone and soft tissue, we can design accurate similarity measurements for the registration process instead of treating the dual images as if from different modalities and rely on more difficult multi-modality registration techniques. Second, prior knowledge and constraints of the bone and soft tissue can be easily integrated to check the validity of the reconstruction results. Most important, registration error that causes invalid reconstruction (e.g. highly correlated bone and soft-tissue layers) can be detected and minimized in the coupled framework. To achieve subpixel accuracy in nonrigid registration, we design a hierarchical free-form registration algorithm with successive accuracy adjustment. The organization of this paper is as follows. In section 2 we first formulate the dual energy imaging problem with a coupled Bayesian framework. In section 3, hierarchical free-form non-rigid registration algorithm is presented to achieve sub-pixel accuracy. Promising experimental results and conclusions are given in sections 4 and 5, respectively. 2. Coupled Bayesian Framework for Registration and Reconstruction 2.1. Traditional Bayesian Framework Image registration is an ill-posed problem, in the sense that it is usually under-determined and there may be many possible solutions. The problem becomes even more complicated for non-rigid registration. Prior knowledge/constraints of the imaging process and possible deformation are necessary for robust and accurate registration methods. A Bayesian framework provides a natural way to incorporate various constraints based on prior knowledge [2, 16]. The objective of image registration is to find the most probable deformation T which maximizes the posterior probability P (T I 1, I 2 ). From Bayesian rule, we have: P (T I 1, I 2 ) = P (I 1, I 2 T )P (T ) P (I 1, I 2 ) where P (I 1, I 2 ) is a constant term with respect to T. Therefore, the maximum a posterior (MAP) solution to the registration problem can be obtained as follows: T = arg max T log P (I 1, I 2 T ) + log P (T ) where P (I 1, I 2 T ) defines the similarity measurement to judge how well the deformation T aligns the two images. For example, sum of squared difference (SSD) is used for single modality and mutual information term is widely used for multimodality images. P (T ) represents the prior knowledge of the deformation field T, e.g. smoothness. Dual energy imaging is usually done in two steps. First, the dual images are registered by a multimodality registration method. Then, a weighted subtraction in Eq. (2) is used to reconstruct the underlying bone and soft-tissue layers. We can see that this is not an optimal solution. The final goal of dual energy imaging is to reconstruct the bone and soft-tissue layers. The subtraction procedure assumes no error in the registration. It tries to reconstruct B and S to literally match every pixel of I 1 and I 2 based on T. A unique solution of B and S can always be found, no matter how poorly T is estimated, and there is no scheme to correct the registration error when the reconstruction is not good. Combining registration and reconstruction is necessary to form a closed loop to adjust the registration result if the reconstructed bone and soft-tissue layers are poor (e.g., highly depent). Furthermore, without knowing the bone and soft-tissue layers, it is very difficult to define a similarity measurement P (I 1, I 2 T ), because the images are generated with X-rays at different energy spectra and have different intensity values even without deformation. It is possible to treat the dual images as if they are from different modalities and to design the similarity measurement based on mutual information or cross correlation. Those measurements neglect the image formation model in Eq. (1) and assume only very general depencies in the dual images. Hence, they are more susceptible to false matching when dealing with spatial variant bone and soft-tissue layers.

3 by: Figure 1. Dual Energy Image Formation Model It is obvious that registration and reconstruction can be better addressed jointly. A related idea is proposed in [13], where registration and segmentation are solved jointly by coupled partial differential equations. Segmentation is represented by a level-set function to separate the image into exclusive parts. In our case, the bone and soft-tissue layers are transparent and overlaid together to generate the acquired images. Hence, we need to model the underlying bone and soft-tissue layers with two extra appearance templates and handle the layer reconstruction and the registration jointly Coupled Registration and Reconstruction As we have pointed out, the composition characteristics of dual energy imaging make it very challenging for traditional methods. We propose a new method based on the image formation model of the dual imaging process, which leads naturally to a coupled framework integrating registration and reconstruction together. The dual images are generated according to Eq. (1) and can be illustrated by Fig. 1. It becomes clear that all the difficulties arise from the fact that the bone B and soft tissue S cannot be observed directly. Instead they are overlaid with different coefficients to form images with different appearances. Considering the hidden variables B and S jointly during the registration process, we can formulate the dual energy image registration as follows: P (T, B, S I 1, I 2 ) = P (I 1, I 2 T, B, S)P (T, B, S) P (I 1, I 2 ) With the assumption that T, B and S are mutually indepent, the MAP solution to this problem can be obtained (3) T = arg max log P (I 1, I 2 T, B, S)+log[P (T )P (B)P (S)] T,B,S (4) This MAP solution leads naturally to the conclusion that the registration and reconstruction are to be solved jointly. The deformation T, the bone B and the soft tissue S should be updated together to match the acquired dual images I 1 and I 2. The prior knowledge about the deformation and the appearance of the layers (i.e., P (T ), P (B) and P (S)) can be easily integrated to refine the reconstruction. For example, P (B) and P (S) can model the smoothness constraint with edge modeling and P (T ) can regularize the smoothness of the deformation field. Unlike the traditional subtraction method, if the reconstructed B and S do not satisfy the prior knowledge due to registration error, T will also be updated and corrected to achieve the optimal solution. Furthermore, by introducing bone and soft-tissue layers explicitly, the similarity measurement P (I 1, I 2 T, B, S) can be easily derived. We do not have to rely on the mutual information or cross correlation to judge the matching of the dual images even though they have different appearances. Assuming the imaging noise is zero mean Gaussian, a good estimation of B, S and T should allow us to synthesize the dual images and therefore minimize the following cost function: log P (I 1, I 2 T, B, S) I 1 a B b S 2 + I 2 c T (B) d T (S) 2 This is more accurate matching criteria than mutual information or cross correlation and less susceptible to false matching, and hence allows more robust registration with sub-pixel accuracy. It can be seen that, in the proposed coupled Bayesian framework, the registration and reconstruction reinforce each other and can provide better results than traditional schemes. In the following subsections, we explain the prior knowledge and constraints we enforce and a detailed implementation of the proposed registration method Enforcing Prior Knowledge and Constraints Based on the proposed coupled Bayesian framework, we can easily enforce various prior knowledge or constraints into this optimization scheme and can guarantee a more stable and physically meaningful registration and reconstruction result. In our experiments, we make the following assumptions and constraints which are generally true in dual energy imaging: The deformation between the dual images (e.g., patient aspiration) can be modeled by a non-rigid dense deformation field T. We assume that the deformation field

4 is smooth across the image. With this assumption, we can choose the P (T ) in Eq. (4) as: log P (T ) T T 2 where T is the average deformation within the neighborhood (i.e., the low-pass filtered version of T ). The bone and soft-tissue layers should satisfy the smoothness constraint with edge modeling. Let e be the edge map, where e(p) = 1 means the corresponding pixel p is an edge point and the smoothness constraint should be suppressed. To prevent all the pixels from being classified as edge points, λ e is used to penalize the edge points. Then we have: log P (B) ((1 e) B B 2 + λ e e) log P (S) ((1 e ) S S 2 + λ ee ) where B and S are the average bone and soft tissue within the neighborhood. Registration error is a major factor in poor reconstruction. In the coupled framework, it is possible to correct the misalignment that causes significant reconstruction error. Usually, misaligned dual images cannot cancel out the bone or soft-tissue and cause highly correlated artifacts in reconstructed bone and soft-tissue layers. Hence, indepent analysis between the bone and soft tissue can help detect and correct the registration error. As pointed out in [3], Bayesian frameworks alone cannot guarantee the indepence constraint even though the indepence is used to factorize the joint probabilities. We should explicitly enforce the indepence by minimizing mutual information MI(B, S). Using λ i to control the weighting of each constraint, the complete objective function, C, to be minimized can be derived as follows: C = I 1 a B b S 2 + I 2 c T (B) d T (S) 2 + λ 1 ((1 e) B B 2 + λ e e + (1 e ) S S 2 + λ ee ) + λ 2 T T 2 + λ 3 MI(B, S) (5) 2.4. Optimization The previous cost function is optimized using a variational approach in an iterative manner. To initialize the algorithm, we do a rough registration based on Harris corner detection on the dual images. Correspondence is found based on the cross correlation in the neighborhood around the detected corners. Then the initial deformation of each pixel is approximated by the weighted average of the nearest four corners. Based on this deformation field, we can also generate the initial reconstruction of the bone and soft tissue, using the weighted subtraction in Eq. (2). We first fix the T and try to optimize the B and S. For each pixel, we search within a search range B k+1 = B k + δ 1 and S k+1 = S k + δ 2 to find the B k+1 and S k+1 that minimize the cost function in Eq. (5). We then fix the B and S and try to optimize the deformation field T. We adopt a free-form deformation model that is controlled by regularly distributed control points (i.e., rectangular grids). Hierarchical searching strategy is used to optimize the deformation model to achieve sub-pixel accuracy successively as explained in next section in detail. This optimization strategy is described as following algorithm: Algorithm 1: Optimization Scheme Data: Given Dual Energy images I 1, I 2. Result: Reconstructed Deformation Field T, bone layer B and soft-tissue layer S. Use some traditional geometric based registration algorithm followed by simple subtraction process to get T 0, B 0, S 0. while stop do Fix T k, find the B k+1, S k+1 for each pixel p do In a search range m δ 1 m, n δ 2 n, for each pixel p, each term in Eq. (5) is computed corresponding to Bp k + δ 1, Sp k + δ 2 to find the optimal δ opt 1 and δ opt 2. Update: Bp k+1 = Bp k + δ opt 1 ; = Sp k + δ opt S k+1 p 2 ; If B k+1 B k 2 ε B and S k+1 S k 2 ε S stop. Otherwise, continue. Fix B k+1, S k+1, find T k+1 to optimize the cost function in Eq. (5) using the hierarchical searching strategy described in section 3 and Algorithm 2. If T k+1 T k 2 ε T, stop. Otherwise, continue. In the above optimization scheme, the mutual information is computationally very expensive. We adopt the approximation described in [3], which approximate the entropy term, e.g. H(X) = p(x) log p(x) by Taylor expansion: x log x = x 0 + (1 + log x 0 )x + O(x x 0 ) 2 where the entropy and joint entropy can be approximated

5 Figure 2. Rectangular free-form deformation model and Gaussian pyramid structures for fast and stable registration by a summation within a neighborhood and hence can be calculated efficiently. For more details, refer to [3]. 3. Sub-pixel Free-form Non-rigid Registration In the previous section, we defined the objective function in Eq. (5) and the iterative optimization scheme in Algorithm 1. For each iteration, based on the refined bone B k and soft tissue S k from the previous iteration, we can further refine the registration result T k+1. The algorithm is detailed in this section. A non-rigid registration algorithm for Digital Subtraction Angiography based on geometric features matching is proposed in [9]. Geometric feature points are extracted and matched, based on which triangle meshes are built to define the deformation model. However, the geometric features are depent on the image content and may not be dense enough and accurate enough for dual energy image registration. We take a similar approach to [11], where a region-based free-form registration algorithm is used to register breast MR images. Regularly distributed control points are used to control the deformation of images for the alignment. It can utilize all the information in the image, rather than being based on the sparse feature points. Instead of using cubic B-splines to interpolate the control points, we use rectangularly spaced control points to control the free-form deformation (FFD), which has better localization and less computation. The performance is almost the same when the control points are dense enough. A Gaussian Pyramid structure is used for the hierarchical searching strategy in our algorithm to improve the speed and stability of the optimization procedure. Figure 2 illustrates our deformation model and the hierarchical optimization strategy. The registration is refined in a coarse-to-fine manner. When the optimal solution for the coarse level is reached, we double the density of the control points and map it to the next finer level and further refine the control points based on the finer resolution image. This scheme converges faster and is less susceptible to local minima. For each layer, we deform the bone B and soft tissue S (i.e. c B + d S) to match the observed image I 2 by the control points. For each control point, we search within a local search region to map it to a new coordinate in image I 2 and update the matching cost between c B + d S and I 2 within the affected regions (each control point affects the four rectangular regions around it). For subpixel accuracy, the search step is set to be smaller than one pixel. In our experiments, we use 0.25 pixel as the search step when we reach the finest level. Suppose the control points on c B +d S are at (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ), (x 4, y 4 ) and correspond to (x 1, y 1), (x 2, y 2), (x 3, y 3) and (x 4, y 4) in I 2 respectively. The pixel (x p, y p ) inside the grid on c B+d S will be map to (x p, y p) in I 2 based on bilinear interpolation as follows: where [x p, y p] T = (1 b) V 1 T + b V 2 T a = (y p y 1 )/(y 2 y 1 ), b = (x p x 1 )/(x 3 x 1 ) [ ] [ ] x V 1 = [1 a, a] 1 y 1 x 2 y 2, V x 2 = [1 a, a] 3 y 3 x 4 y 4 There are other ways to find the coordinate mapping, such as Homography matrix transformation. From our experiments, we find bilinear geometric interpolation is stable and accurate compared to other transformation methods and more computationally efficient. Given pixel correspondence, we can easily compute the cost function I 2 c T (B) d T (S) 2 + λ 2 T T 2 and find the control mesh that minimizes the cost function. The searching strategy for finding the optimal deformation model T is given in Algorithm 2: 4. Experiments The proposed algorithm is applied to the X-ray dual energy chest imaging. Experiments and comparisons on both real images and synthesized images show the improvement of the proposed coupled registration method on registration accuracy and the reconstruction results. For comparison, we also implemented a separated method, which registers the dual images first and then uses weighted subtraction to reconstruct the bone and soft-tissue layers. To register the dual images, maximization of mutual information is used to guide the registration [5]. The non-parametric density estimation technique for calculating the joint entropy between the dual images can, to some extent, handle the non-stationary mapping function between the dual images. We replace our cost function in Eq. (5)

6 Algorithm 2: Hierarchical free-form non-rigid registration In the Gaussian Pyramid, from coarse level 0 to fine level N, do the following: while k N do while stop do Update deformation M i+1 k (x, y) for kth level for each control point Mk i (x, y) do Searching in its neighborhood to find the optimal control mesh that minimizes the matching cost. Update M i+1 k (x, y) to the optimal position; If M i+1 k Mk i 2 ε k, stop. increase the density of control points to generate the initial Mk+1 0 i+1 (x, y), map Mk (x, y) to (x, y); M 0 k+1 increase k to k + 1 and reset i to 0; Figure 3. X-ray dual energy chest imaging by maximizing the mutual information between dual images and apply the same hierarchical free-form registration method to align the dual images. Based on the registration results, weighted subtraction is performed to separate the bone and soft-tissue layers. Fig. 3 shows the dual images for chest imaging. There are both bone and soft-tissue structures overlaid in the images, which makes detecting lung nodules or other subtle details more difficult. Reconstruction of the bone and soft tissue specific images can greatly increase its diagnostic value. Accurate registration is necessary for good reconstruction of the bone and soft tissue. Otherwise, the image difference caused by registration error can easily become much more significant than the different characteristics of the bone and soft tissue. Now we compare the separated method with our coupled method. In the separated method, maximization of the mutual information handles the different appearances between the dual images reasonably well. But when the bone and soft tissue both have complex structures overlaid together, it is very difficult to estimate the mapping function robustly and accurately. Also, there is no scheme to refine the registration, even if the reconstruction results do not satisfy our prior knowledge in the separated scheme. The reconstruction results of the separated method are shown in Fig. 4 (a) and (b). There are some noticeable artifacts when bone edges and soft-tissue structures are overlaid together. The coupled method provides much better results in Fig. 4 (c) and (d). It can be seen that the reconstructed results are much smoother and cleaner. To compare the results quantitatively, we also do some tests with synthesized motion so that we have ground truth for accurate error analysis. We select the previous reconstructed bone and soft tissue as ground truth to generate a pair of synthesized images. A transformation field that expands the lung region is applied to simulate an aspiration motion. Quantitative results and comparisons between the separated method and our coupled method are summarized in the following tables. First we compare the registration accuracy. The estimated deformation field T is compared with the ground truth (the synthesized motion). The average and maximum absolute registration error (in pixels) is listed as follows: Error in T Average Max Variance Separated Method Coupled Method It can be seen that the separated registration method achieves reasonably good results with maximum registration error of only pixels. However, the proposed coupled framework further improves the registration accuracy and provides consistently better results throughout the image. The mean and the variance of the registration error are much smaller in the coupled method. We also compare the error in the reconstructed bone and soft-tissue layers. The absolute difference between the reconstructed results and the ground truth is normalized by the maximal intensity value of the bone and soft-tissue images. The errors in different methods are listed in the following table: Error in B Average Max Variance Separated Method e-4 Coupled Method e-4 Error in S Average Max Variance Separated Method e-4 Coupled Method e-5 It is clearly shown that our coupled method generates consistently better reconstruction results. In Fig. 5, comparison on another pair of dual images is shown. The coupled method provides better results on dif-

7 Figure 4. Comparison between the traditional separated registration method and our coupled method: (a) and (b) are the reconstructed bone and soft-tissue layers based on the traditional separated registration method. (c) and (d) are the results from our coupled method. ferent data set under different imaging conditions. 5. Conclusions In this paper, we propose a coupled Bayesian framework for registering dual energy images and reconstruction of the overlaid bone and soft-tissue layers jointly. It is a considerable improvement over the traditional separated scheme where multi-modality image registration is first applied and followed by a simple weighted subtraction to reconstruct the bone and soft tissue. More prior knowledge is included in the proposed framework and results in more stable and physically meaningful results. The proposed coupled algorithm could be more helpful for low-dose X-ray imaging to reduce radiation to the patients. In low-dose X-ray imaging, the signal/noise ratio drops significantly and causes severe challenges. We plan to look in that direction in the future. References [1] L. G. Brown. A survey of image registration techniques. ACM Computing Surveys (CSUR), ACM Press, 24(4): , [2] P. Cachier and N. Ayache. Regularization in image non-rigid registration: I. trade-off between smoothness and intensity similarity. technical report, INRIA, [3] Y. Chen, H. Wang, T. Fang, and J. Tyan. Mutual information regularized bayesian framework for multiple image restoration. In Proc. IEEE Int l Conf. on Computer Vision, volume 1, pages , [4] W. Crum, T. Hartkens, and D. Hill. Non-rigid image registration: theory and practice. British Journal of Radiology, 77: , [5] J. Kim, V. Kolmogorov, and R. Zabih. Visual correspondence using energy minimization and mutual information. In Proc. IEEE Int l Conf. on Computer Vision, 2003.

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