Volume 13 Supplement 2
Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine (BIBM 2013): Biomedical Engineering
Inverse consistent non-rigid image registration based on robust point set matching
- Xuan Yang^{1}Email author,
- Jihong Pei^{2} and
- Jingli Shi^{1}
https://doi.org/10.1186/1475-925X-13-S2-S2
© Yang et al.; licensee BioMed Central Ltd. 2014
Published: 11 December 2014
Abstract
Background
Robust point matching (RPM) has been extensively used in non-rigid registration of images to robustly register two sets of image points. However, except for the location at control points, RPM cannot estimate the consistent correspondence between two images because RPM is a unidirectional image matching approach. Therefore, it is an important issue to make an improvement in image registration based on RPM.
Methods
In our work, a consistent image registration approach based on the point sets matching is proposed to incorporate the property of inverse consistency and improve registration accuracy. Instead of only estimating the forward transformation between the source point sets and the target point sets in state-of-the-art RPM algorithms, the forward and backward transformations between two point sets are estimated concurrently in our algorithm. The inverse consistency constraints are introduced to the cost function of RPM and the fuzzy correspondences between two point sets are estimated based on both the forward and backward transformations simultaneously. A modified consistent landmark thin-plate spline registration is discussed in detail to find the forward and backward transformations during the optimization of RPM. The similarity of image content is also incorporated into point matching in order to improve image matching.
Results
Synthetic data sets, medical images are employed to demonstrate and validate the performance of our approach. The inverse consistent errors of our algorithm are smaller than RPM. Especially, the topology of transformations is preserved well for our algorithm for the large deformation between point sets. Moreover, the distance errors of our algorithm are similar to that of RPM, and they maintain a downward trend as whole, which demonstrates the convergence of our algorithm. The registration errors for image registrations are evaluated also. Again, our algorithm achieves the lower registration errors in same iteration number. The determinant of the Jacobian matrix of the deformation field is used to analyse the smoothness of the forward and backward transformations. The forward and backward transformations estimated by our algorithm are smooth for small deformation. For registration of lung slices and individual brain slices, large or small determinant of the Jacobian matrix of the deformation fields are observed.
Conclusions
Results indicate the improvement of the proposed algorithm in bi-directional image registration and the decrease of the inverse consistent errors of the forward and the reverse transformations between two images.
Keywords
Consistent image registration Robust point matching Correspondence Forward transformation Backward transformationIntroduction
Point set matching is a kind of image registration method used widely in the areas of shape matching, motion correction, object recognition and other computer vision applications. The aim of point set matching is to find spatial transformations between two point sets extracted from two images, where the correspondence relationship of points is unknown. Many approaches attempted to solve for point set matching under the affine or projective transformation [1–3]. Recently, there has been considerable interest in point set matching for non-rigid objects [4–10].
The robust point matching (RPM) has become a popular point matching method due to its robustness to disturbances such as noise and outliers. There are two issues needed to be settled for RPM: the correspondence and the transformation. RPM handled these issues generally based on an iterative estimation framework. It utilizes similarity constraints to compute a set of putative correspondences, which include inlier points that there are true correspondence relationship with points in other point set and exclude outlier points without corresponding ones in other point set [5, 6, 8, 9]. And then, under the current estimat of the correspondence, the transformation may be estimated and used to update the correspondence.
Transformations used in RPM can be classified into two categories: non-parametric and parametric. The non-parametric transformation is the one where the geometric deformation is not any parametric mapping functions, such as elastic, fluid and diffusive deformation field. Generally, geometric constraints are needed to estimate the non-parameter transformations. Ma et al.[8] used a non-parametric geometrical mapping to formulate the point matching problem as robust vector field interpolation, which took the advantage of regularization the vector field when nonparametric geometric constraint is required. Although point set matching algorithms with non-parametric transformation lead to a globally smooth dense deformation field, they cannot preserve topology of the deformed field.
The parametric transformation is the one where the geometric deformation is represented as parametric mapping functions, such as thin-plate splines (TPS), radial basis function based and affine transformations. Chui et al. [4] proposed the TPS-RPM algorithm using TPS to map the source point set to the target point set. Wang et al. [7] chose the TPS as the non-rigid deformation function to achieve group-wise registration of a set of shapes represented by unlabelled point-sets. Jian et al. [9] employed the TPS and the Gaussian radial basis functions respectively to implement three different cost functions used in RPM. Lian et al. [10] applied linear transformation in RPM and reduced the energy function of RPM to a concave function with very few non-rigid terms.
However, whether non-parametric or parametric RPM, the majority of the existing RPM algorithms are asymmetric, that is, the changes measured from transformations are dependent of the order in which the images are registered. When interchanging the order of register images, the RPM algorithm cannot estimate the inverse transformation. Asymmetry problem of registration algorithms lead to biased results when statistical analysis is performed after registration [11]. In order to tackle the asymmetric problem in image registration, symmetric algorithms and inverse consistent algorithms are proposed. Symmetric algorithms optimize cost functions without explicitly penalizing asymmetry. They construct symmetric cost functions by estimating one transformation from one image to another, or construct ordinary cost function by estimating bidirectional transformations to map two images to a common domain using iterative method [12–22]. Bondar et al. [12] imposed a symmetry constraint to TPS-RPM by evaluating the correspondence matrix based on the forward and the backward transformations, but the transformation used in TPS-RPM still is unidirectional. Bhagalia et al. [16] introduced a bi-directionality term to the RPM objective function, their aim is to reduce the mapping errors in both forward and backward directions for points only, instead of enforcing the forward and the backward transformation to be inverse to each other.
Alternatively, inverse consistent algorithms introduce consistency constraints to the cost function and estimate the forward and backward transformations at the same time [23–32]. Consistency of the forward and backward transformations constrains the forward and backward transformations to be inverses to each other, which ensures that the correspondence produced by the forward transformation is consistent with the correspondence produced by the backward transformation. The idea of inverse consistent image registration is first proposed by Christensen et al. [23], in which inverse consistency constraint introduced and added to the matching criteria of images. Johnson et al. [24] developed the idea of Christensen and other authors. They proposed the Consistent Landmark Thin-Plate Spline (CLTPS) registration algorithm to estimate the forward and backward transformations between two images based on the correspondence of landmarks. However, the correspondence of control points cannot be ensured during the iterative procedure of the CLTPS algorithm. Furthermore, Christensen et al. [25] employed Johnson et al.'s algorithm to track lung motion using CT images of multiple breathing periods. He and others [26] concatenated a sequence of small deformation transformations using Johnson et al.'s algorithm to estimate the forward and backward large deformation transformations concurrently. Gholipour et al. [27] introduced the inverse consistency to a cost function based on a parametric free-form deformation model with a regular grid of control points. Algorithms in [23–27] are based on a parameterized function model. On the other hand, the consistency constraints are also introduced into the registration algorithms based on the dense non-parametric model. Zhang et al. [28] employed consistency constraints in a variational framework for multi-modal images registration. Leow et al. [29] only solved the forward transformation by directly modelling the backward transformation using the inverse of the forward transformation in unbiased MRI registration. They employed the symmetrizing Kullback-Leibler(KL) distance between the identity map and the transformation, and showed that symmetrizing KL distance is equivalent to considering both the forward and backward transformations in image registration. Tao et al. [30] implemented a symmetric and inverse consistent diffeomorphic registration algorithm by avoiding explicit calculation of the inverse deformation. The inverse consistent registration algorithms produce the kind of deformation results that maintain the neighbourhood relationship and present more biological meaning. They produce better correspondence between medical images and smoother displacement fields compared with unidirectional registration algorithms.
The main focus of the paper is to estimate the inverse consistent parametric transformations in RPM. The TPS is the most commonly used parametric transformation in RPM. Although TPS produces a smooth transformation from one image to another, it does not define a consistent correspondence between the two images except at the location of control points [24]. Correspondingly, the transformation solved by the TPS-RPM is unidirectional, that is, the forward and the backward transformations cannot be ensured to be inverted to each other, and the correspondence defined by the forward transformation is different from the correspondence defined by the backward transformation in TPS-RPM.
Presently, to the best of our knowledge, there is no an inverse consistent registration method that can find the forward and backward transformation between two images by matching the sets of points of two images. In this paper, we present an inverse consistent registration algorithm based on robust point matching. The main contributions of this paper as follow. Firstly, we introduce inverse consistency constraint in the RPM cost function, and estimate the forward and backward transformations for two sets of point simultaneously using modified CLTPS. We modify the CLTPS algorithm to improve accuracy of point-to-point mapping in consistent transformations. Secondly, the fuzzy correspondence relationships between points are estimated based on both the forward and backward transformations. Image similarity is also incorporated to the corresponding relationship between points in order to reduce the mismatch of points.
An earlier version of this article was published in the IEEE International Conference on Bioinformatics and Biomedicine (BIBM) hold on 18-21 December 2013 [31] and the sections about consistent robust point matching are from that article. In this paper, we introduce the regularized TPS to preserve the topology of the deformation fields, and estimate the forward and backward transformations during the complete iterative process of point matching, instead of at the end of the iterative process. We further introduce the modified consistent landmark thin-plate spline registration to the complete iterative process of robust point matching. The convergence of our algorithm is demonstrated by experiments. Additionally, we correct the experiment results of RPM in [31] and conduct some new experiments to further compare the performance of inverse consistent RPM using CLTPS in results.
Methods
TPS-RPM review
where a and W are affine transform matrix and warp coefficient matrix respectively, w _{ j } is an element of matrix W, r _{ ij } = ||x _{ i } − x _{ j }|| is the distance norm between point x _{ i } and x _{ j } , ϕ(r _{ ij }) is the basis function of TPS.
where T is the temperature in the anneal procedure of TPS-RPM. The fuzzy correspondence matrix is subject to ${\sum}_{i=1}^{K+1}{m}_{ij}=1$ for j ∈ {1, 2, . . . , N}, ${\sum}_{j=1}^{N+1}{m}_{ij}=1$ for i ∈ {1, 2, . . . , K}, and m _{ ij } ∈ [0, 1]. The nearer the distance between the mapped x _{ i } and y _{ j } is, the more likely a corresponding relationship exists between x _{ i } and y _{ j } .
The cost function is derived from a statistical physics model. The term ${\sum}_{i=1}^{K}{\sum}_{j=1}^{N}{m}_{ij}\text{log}{m}_{ij}$ is a barrier function, which is used to push the minimum of the cost function away from the discrete points. The temperature T contorls the degree of convexity of the cost function [3]. When T is sufficiently small, the cost function is ensured to be convex. λ and ζ are regularization parameters. In the TPS-RPM algorithm, Expectation-maximization (EM) algorithm is adopted to solve M and h iteratively, the detailed process can be found in literature [4].
When TPS-RPM is used to register the source image I and target image J , the source point set X and the target point set Y are extracted from I and J respectively. Next, TPS-RPM is employed to estimate the forward transformation h : X → Y , which is the transformation to map the source image I to the target image J so that I(h(x)) = J . When image J is registered to image I, the backward transformation g : Y → X maps the image J to image I so that J (g(x)) = I. As previously mentioned, it is required that the forward transformation and the backward transformation are inversely consistent, i.e. g ○ h = id and h ○ g = id, where id is the identity map, to ensure the correspondence between the two images to be consistent. However, the forward transformation h and the backward transformation g is not dependent to each other for TPS-RPM, since TPS is an unidirectional function which results in a non-consistent correspondence between the two images except at the control points, that is, g ○ h ≠ id, h ○ g ≠ id and g ○ h ≠ h ○ g. Furthermore, the value of the fuzzy correspondence matrix M is computed based on the mapping errors in the forward transformation only, the mapping errors from Y to X will not be penalized, which leads to a bias matching result.
Inverse consistent robust point matching
In (5), the mapping errors between two point sets are extended as the combination of distance between the target point and the mapped position of the source point using the forward transformation, and the distance between the source point and the mapped position of the target point using the backward transformation, instead of only using the forward mapping errors. Both the smoothness of the forward and backward transformations ||Lh||^{2} + ||Lg||^{2} are included in the cost function. Χ is the weighting parameters to make a trade-off between the inverse consistent error and other terms.
where, I(x _{ i }) and J (y _{ j } ) are two local regions centred at x _{ i } in image I and y _{ j } in image J . I(h(x)) and J (g(x)) are deformed images of I and J using the forward transformation and the backward transformation respectively. corr is the correlation coefficient used to measure the similarity between two local regions. T _{ s } is the temperature parameters of image similarity. By introducing image information to the fuzzy correspondence matrix, improvement of image matching is achieved for the inverse consistent RPM.
where ${v}_{i}={\sum}_{j=1}^{N}{m}_{ij}{y}_{j}$ , i = 1, 2, . . . , K and ${z}_{j}={\sum}_{i=1}^{K}{m}_{ij}{x}_{i}$, i = 1, 2, . . . , N are the virtual points computed in the forward and backward directions respectively. Moreover, v _{ i } is expected to be corresponding to x _{ i }, and z _{ j } is expected to be corresponding to y _{ j } also. v _{ i } and z _{ j } are held fixed during the procedure of the M step. Then, the optimization problem is to find the optimal forward and backward transformations h and g given four point sets {x _{ i }}, {y _{ j }}, {v _{ i }} and {z _{ j }}, where {x _{ i }} are corresponding to {v _{ i }}, and {y _{ j }} is corresponding to {z _{ j }}. The iterative process continuously alternates the E step with the M step until it converges. Next, we will discuss how to calculate transformations h and g at the same time by optimizing Ec(h, g).
Modified consistent landmark thin-plate spline registration
Given two point sets with known correspondence relationship, the Consistent Landmark Thin-Plate Spline (CLTPS) registration algorithm [24] was originally proposed to solve the inversely consistent transformations between these two point sets. During the procedure of CLTPS, only two point sets are used to estimate the forward and backward transformations simultaneously. However, there are four point sets {x _{ i }}, {y _{ j }}, {v _{ i }} and {z _{ j }} in RPM. Based on the correspondence between {x _{ i }} and {v _{ i }}, and the correspondence between {y _{ j }} and {z _{ j }}, an intuitive approach to estimate the forward transformation h is to let {x _{ i }} be the source point set and {v _{ i }} be the target point set. Conversely, let {y _{ j }} be the source point set and {z _{ j }} be the target point set to estimate the backward transformation g. Details of CLTPS can be referred in [24].
However, there several existed problems in CLTPS: (1) the mapped positions of control points are oscillated near their target positions, instead of mapping exactly to the target positions [31]; (2) topology of the forward and backward transformations cannot be ensured to be preserved.
Firstly, there is a minor oscillation problem in CLTPS algorithm. In CLTPS, the forward and backward displacements are updated iteratively using the temporary forward and temporary backward transformations f _{1}(x) and f _{2}(x), where f _{1}(x) is estimated by considering the current mapped position of {x _{ i }} and {v _{ i }} as the source and target control point sets respectively, and f _{2}(x) is estimated by considering the current mapped position of {y _{ j }} and {z _{ j }} as the source and target control point sets respectively. However, in CLPTS, x _{ i } can be mapped to a location near to v _{ i }, but cannot be mapped to v _{ i } exactly. The same goes for y _{ j } also. To tackle the oscillation problem of CLTPS, we propose a new approach to update the forward and backward displacements iteratively. Denote r _{ i } and s _{ j } as the temporary mapped positions of x _{ i } and y _{ j } respectively. After the k th iteration, x _{ i } is mapped to r _{ i } using the current forward displacement u _{ k }(x), and y _{ j } is mapped to s _{ j } using the current backward displacement w _{ k }(x). We update the forward and backward displacements iteratively as follows:
$\begin{array}{c}{u}_{k+1}\left(x\right)={u}_{k}\left(x\right)+\alpha {u}_{k}^{*}\left(x\right),\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{u}_{k}^{*}\left(x\right)={u}_{t}\left({u}_{k}\left(x\right)+x\right),\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{u}_{t}\left(x\right)={f}_{1}\left(x\right)-x,\\ {w}_{k+1}\left(x\right)={w}_{k}\left(x\right)+\alpha {w}_{k}^{*}\left(x\right),\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{w}_{k}^{*}\left(x\right)={w}_{t}\left({w}_{k}\left(x\right)+x\right),\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{w}_{t}\left(x\right)={f}_{2}\left(x\right)-x.\end{array}$(9)
It implies that x _{ i } is mapped to v _{ i } exactly using our approach. Similarly, we can prove that y _{ j } is mapped exactly to z _{ j } using the backward displacement.
Finally, r _{ i } and s _{ j } are required to be updated as the newest mapped positions of x _{ i } and y _{ j } for each iteration. So, after the update of the forward and backward transformations, r _{ i } and s _{ j } are updated correspondingly using the latest transformations respectively. More importantly, r _{ i } is closer and closer to v _{ i } with the increase in the number of iterations, rather than swinging nearby v _{ i } as CLTPS. Similarly, s _{ j } is closer and closer to z _{ j } in the iteration process. All these ensures that x _{ i } is mapped exactly to its target position v _{ i }, and y _{ j } is mapped exactly to its target position z _{ j } using the modified consistent landmark thin-plate spline registration algorithm.
Details of the modified consistent landmark thin-plate spline registration are described in algorithm 1.
Algorithm 1 Modified Consistent Landmark Thin-Plate Spline (CLTPS) registration algorithm using four points sets.
1: Let r _{ i } = x _{ i } , s _{ j } = y _{ j } ; u(x) = 0, w(x) = 0, the steps α and β, the mapping error threshold ξ of control point, and the maximum number of iteration m _{ iter } , k = 1.
2: Regularized TPS is performed to estimate the temporary forward transformation f _{1}(x) based on the correspondence between r _{ i } and v _{ i }, and the temporary backward transformation f _{2}(x) based on the correspondence between s _{ j } and z _{ j }.
3: Update the displacements,
u(x) = u(x) + αu^{ ∗ }(x), u^{ ∗ }(x) = u _{ t }(u(x) + x), u _{ t }(x) = f _{1}(x) − x,
w(x) = w(x) + αw^{ ∗ }(x), w^{ ∗ }(x) = w _{ t }(w(x) + x), w _{ t }(x) = f _{2}(x) − x.
4: Compute the Jacobian values of the forward and backward transformations, if the minimum Jacobian values of the forward or backward transformations are negative, the iteration is terminated.
5: Get h^{ − 1}(x), the inverse function of forward transformation and g^{ − 1}(x), the inverse function of backward transformation.
6: Update displacement field of forward and backward transformation. u(x) = u(x) − β[u(x) − g^{ − 1}(x) + x], meanwhile, w(x) = w(x) − β[w(x) − h^{ − 1}(x) + x].
7: r _{ i } and s _{ j } are updated as r _{ i } = x _{ i } + u(x _{ i }), s _{ j } = y _{ j } + w(y _{ j }).
8: Check whether the termination condition is met. If k > m _{ iter } or |u(x _{ i }) − (v _{ i } − x _{ i })| < ξ or |w(y _{ j }) − (z _{ j } − y _{ j })| < ξ, the iteration is terminated; otherwise, k = k + 1, go to step 2.
Results
In this section, we will evaluate the performance of inverse consistent RPM algorithm with simulated data and medical images, and also illustrate the efficacy of the image information in estimating the correspondence of points for image registration.
Synthetic data
To determine the behaviours of the forward and backward transformations, a uniform grid in size of 100 × 100 is employed to be the deformation field of transformations. The inverse consistent error (ICE) of the forward and backward transformations is evaluated by summing the forward consistency error and the backward consistency error, ICE = ||h − g^{ −1 }|| + ||g − h^{ −1 }||. Considering that the transformation used in RPM is uni-directional, we perform RPM in the forward and backward directions simultaneously to estimate h and g respectively. The weighted mapping errors between the target points and mapped source points using the forward transformation, and the source points and mapped target points using the backward transformation are used to define the distance error (DE), $DE={\sum}_{i}{\sum}_{j}{m}_{ij}\left(||{y}_{j}-h\left({x}_{i}\right)||+||{x}_{i}-g\left({y}_{j}\right)||\right)$.
In order to compare the performance of MCRPM, CRPM and RPM, same iteration number is used for three algorithms, the results are shown in Figure 2 (there are some errors in the experiment results of RPM in [31], we correct these errors here). It can be seen that the forward and backward registration results using MCRPM are similar to those using RPM, which implies that the forward registration accuracy of MCRPM is equivalent to that of RPM. Furthermore, both the forward and backward registration accuracy using MCRPM are satisfied. Especially, it is noted that the backward registration error using RPM is not better than that using MCRPM for data 1 and data 2, which demonstrate the advantages of the MCRPM in the bidirectional registration. The bidirectional registration error using CRPM is large for the first and the third point sets, since there is a significant deformation between these two point sets, and the oscillation problem leads the mapped positions of points are not corresponding to their target positions obviously in these cases.
Inverse consistent errors of MCRPM, CRPM and RPM ('-' denotes the topology of transformation is not preserved).
Data | MCRPM | CRPM | RPM |
---|---|---|---|
1 | 0.0150 | 0.2430 | - |
2 | 0.0157 | 0.0841 | 0.0423 |
3 | 0.0395 | 0.1538 | 0.0706 |
4 | 0.0137 | 0.0349 | 0.0423 |
Small deformation registration of brain Images
The mean square deviation and mean inverse consistent error of registration results of Figure 5 and Figure 6.
MSD | ICE | |||||
---|---|---|---|---|---|---|
MCRPM | CRPM | RPM | MCRPM | CRPM | RPM | |
Figure 5 | 22.98 | 23.76 | 29.63 | 0.0092 | 0.0107 | 0.0242 |
Figure 6 | 26.23 | 31.01 | 54.13 | 0.0054 | 0.0074 | 0.0233 |
The Jacobian values of the forward and backward transformations of Figure5 and Figure 6.
|Det(h) − 1| | |Det(g) − 1| | |||||
---|---|---|---|---|---|---|
MCRPM | CRPM | RPM | MCRPM | CRPM | RPM | |
Figure 5 | 0.0098 | 0.0150 | 0.0342 | 0.0201 | 0.0167 | 0.0365 |
Figure 6 | 0.0163 | 0.0236 | 0.0920 | 0.0198 | 0.0264 | 0.0871 |
Lung Slices
The Jacobian values of the forward and backward transformations.
|Det(h) − 1| | |Det(g) − 1| | |||||
---|---|---|---|---|---|---|
Case | MCRPM | CRPM | RPM | MCRPM | CRPM | RPM |
1 | 0.1159 | 0.0986 | 0.1275 | 0.1324 | 0.0991 | 0.1357 |
2 | 0.1104 | 0.0903 | 0.1026 | 0.1186 | 0.0906 | 0.0896 |
3 | 0.1336 | 0.1435 | 0.1553 | 0.1544 | 0.1639 | 0.1540 |
4 | 0.2086 | 0.2157 | 0.2593 | 0.2351 | 0.2349 | 0.2028 |
5 | 0.1640 | 0.1338 | 0.2396 | 0.1713 | 0.1316 | 0.2863 |
6 | 0.1492 | 0.1716 | 0.1947 | 0.1709 | 0.1771 | 0.1494 |
7 | 0.1034 | 0.1185 | 0.1467 | 0.1054 | 0.1137 | 0.0711 |
8 | 0.2659 | 0.2083 | 0.2259 | 0.3125 | 0.2356 | 0.2913 |
9 | 0.0997 | 0.1125 | 0.0791 | 0.1096 | 0.1261 | 0.0808 |
10 | 0.1445 | 0.1332 | 0.2197 | 0.1290 | 0.1065 | 0.1433 |
Individual brain images
The fourth experiment contains the same slices extracted from 10 subjects of Brain Web. This experiment is used to demonstrate the performance of our approach for inter-subject image registration when the deformations of images are large. One subject serves as the target image and another image is aligned to the target image. Registration described as subject 1-2 means that subject 1 and subject 2 are used for evaluation.
Conclusions
We proposed a consistent image registration approach by combining the RPM algorithm and modified consistent landmark thin-plate spline algorithm together. It introduced the forward and the backward transformations to the cost function of points matching, and estimated the correspondence matrix based not only on bi-directional transformations but also on the correlation of image content. The forward and backward transformations were estimated during the complete iterative process of point matching. The regularized TPS was introudced to our algorithm to produce topology-preserving transformations for image registration with large deformation, and produce smooth transformations for image registration with small deformation. The modified consistent landmark thin-plate spline algorithm improved the correspondence between points, and significantly reduced the inverse consistent error between the forward and backward transformations. Experiment results demonstrated the convergence of our algorithm, and medical images registration results showed that our algorithm was superior to RPM in aspect of intensity matching between images. A desired improvement in our approach would be to reduce computational time to estimate the inversely consistent transformations.
Declarations
Acknowledgements
This paper is supported by National Natural Science Foundation of China (U1301251) and Shenzhen Science and Technology Projection (JCYJ20130326112132687).
Declarations
Publication of this article has been funded by National Natural Science Foundation of China (U1301251) and Shenzhen Science and Technology Projection (JCYJ20130326112132687).
This article has been published as part of BioMedical Engineering OnLine Volume 13 Supplement 2, 2014: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine (BIBM 2013): BioMedical Engineering OnLine. The full contents of the supplement are available online at http://www.biomedical-engineering-online.com/supplements/13/S2.
Authors’ Affiliations
References
- Besl PJ, McKay ND: A method for registration of 3-d shapes. IEEE Trans Pattern Anal Mach Intell 1992,14(2):239–256. 10.1109/34.121791View ArticleGoogle Scholar
- Xue Z, Shen D, Teoh EK: An efficient fuzzy algorithm for aligning shapes under affine transformations. Pattern Recognition 2001,34(6):1171–1180. 10.1016/S0031-3203(00)00067-4View ArticleGoogle Scholar
- Gold S, Rangarajan A, Lu Cp, Mjolsness E: New algorithms for 2d and 3d point matching: Pose estimation and correspondence. Pattern Recognition 1997, 31: 957–964.Google Scholar
- Chui H, Rangarajan A: A new point matching algorithm for non-rigid registration. Comput Vis Image Underst 2003,89(2–3):114–141. 10.1016/S1077-3142(03)00009-2View ArticleGoogle Scholar
- Yang J: The thin plate spline robust point matching (tps-rpm) algorithm: A revisit. Pattern Recogn Lett 2011,32(7):910–918. 10.1016/j.patrec.2011.01.015View ArticleGoogle Scholar
- Zheng Y, Doermann D: Robust point matching for nonrigid shapes by preserving local neighborhood structures. IEEE Trans Pattern Anal Mach Intell 2006,28(4):643–649.View ArticleGoogle Scholar
- Wang F, Vemuri BC, Rangarajan A, Eisenschenk SJ: Simultaneous nonrigid registration of multiple point sets and atlas construction. IEEE Trans Pattern Anal Mach Intell 2008,30(11):2011–2022.View ArticleGoogle Scholar
- Ma J, Zhao J, Tian J, Yuille AL, Tu Z: Robust point matching via vector field consensus. IEEE Transactions on Image Processing 2014,23(4):1706–1721.View ArticleGoogle Scholar
- Jian B, Vemuri BC: Robust point set registration using gaussian mixture models. IEEE Trans Pattern Anal Mach Intell 2011,33(8):1633–1645.View ArticleGoogle Scholar
- Lian Wei, Zhang Lei: Robust point matching revisited: A concave optimization approach. In Proceedings of the 12th European Conference on Computer Vision - Volume Part II: 2012; Florence, Italy. Edited by: Fitzgibbon, A.W., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. Springer, Berlin, Heidelberg; 2012:259–272.Google Scholar
- Sotiras A, Davatzikos C, Paragios N: Deformable medical image registration: A survey. IEEE Trans Med Imaging 2013,32(7):1153–1190.View ArticleGoogle Scholar
- Bondar L, Hoogeman MS, Osorio EMV, Heijmen BJ: A symmetric nonrigid registration method to handle large organ deformations in cervical cancer patients. Medical physics 2010,37(7):3760–3772. 10.1118/1.3443436View ArticleGoogle Scholar
- Guetter Christoph, Xue Hui, Chefd'Hotel Christophe, Guehring Jens: Efficient symmetric and inverse-consistent deformable registration through interleaved optimization. Proceedings of the 8th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, ISBI 2011 2011, 590–593. March 30 - April 2, 2011, Chicago, Illinois, USA 10.1109/ISBI.2011.5872476View ArticleGoogle Scholar
- Reuter M, Rosas HD, Fischl B: Highly accurate inverse consistent registration: A robust approach. NeuroImage 2010,53(4):1181–1196. 10.1016/j.neuroimage.2010.07.020View ArticleGoogle Scholar
- Yang D, Li H, Low DA, Deasy JO, El Naqa I: A fast inverse consistent deformable image registration method based on symmetric optical flow computation. Physics in medicine and biology 2008,53(21):6143. 10.1088/0031-9155/53/21/017View ArticleGoogle Scholar
- Bhagalia Roshni, Miller James, Roy Arunabha: Bi-directional labeled point matching. Proceedings of the 2010 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, Rotterdam, The Netherlands 2010, 380–383. 14–17 April, 2010 10.1109/ISBI.2010.5490331View ArticleGoogle Scholar
- Beg MF, Khan AR: Symmetric data attachment terms for large deformation image registration. IEEE Trans Med Imaging 2007,26(9):1179–1189.View ArticleGoogle Scholar
- Rogelj P, Kovacic S: Symmetric image registration. Medical Image Analysis 2006,10(3):484–493. 10.1016/j.media.2005.03.003View ArticleGoogle Scholar
- Ashburner J, Ridgway GR: Symmetric diffeomorphic modeling of longitudinal structural mri. Frontiers in neuroscience 2012., 6: Google Scholar
- Bender ET, Hardcastle N, Tome WA: On the dosimetric effect and reduction of inverse consistency and transitivity errors in deformable image registration for dose accumulation. Medical physics 2012,39(1):272–280. 10.1118/1.3666948View ArticleGoogle Scholar
- Xue Hui, Ding Yu, Guetter Christoph, Jolly Marie-Pierre, Guehring Jens, Zühlsdorff Sven, Simonetti Orlando: Motion compensated magnetic resonance reconstruction using inverse-consistent deformable registration: Application to real-time cine imaging. Medical Image Computing and Computer-Assisted Intervention - MICCAI 2011 - 14th International Conference 2011, 564–572. Toronto, Canada, September 18–22, 2011, Proceedings, Part IView ArticleGoogle Scholar
- Avants BB, Epstein CL, Grossman M, Gee JC: Symmetric diffeomorphic image registration with cross-correlation: Evaluating automated labeling of elderly and neurodegenerative brain. Medical Image Analysis 2008,12(1):26–41. 10.1016/j.media.2007.06.004View ArticleGoogle Scholar
- Christensen GE, Johnson HJ: Consistent image registration. IEEE Trans Med Imaging 2001,20(7):568–582. 10.1109/42.932742View ArticleGoogle Scholar
- Johnson HJ, Christensen GE: Consistent landmark and intensity-based image registration. IEEE Trans Med Imaging 2002,21(5):450–461. 10.1109/TMI.2002.1009381View ArticleGoogle Scholar
- Christensen GE, Song JH, Lu W, El Naqa I, Low DA: Tracking lung tissue motion and expansion/compression with inverse consistent image registration and spirometry. Medical physics 34(6):2155–2163.Google Scholar
- He Jianchun, Christensen Gary: Large deformation inverse consistent elastic image registration. Proceedings of the Information Processing in Medical Imaging: IPMI 2003 2003, 438–449. Ambleside, UK, July 20–25, 2003, ProceedingsView ArticleGoogle Scholar
- Gholipour A, Kehtarnavaz N, Yousefi S, Gopinath KS, Briggs RW: Symmetric deformable image registration via optimization of information theoretic measures. Image Vision Comput 2010,28(6):965–975. 10.1016/j.imavis.2009.11.012View ArticleGoogle Scholar
- Zhang Z, Jiang Y, Tsui H: Consistent multi-modal non-rigid registration based on a variational approach. Pattern Recognition Letters 2006,27(7):715–725. 10.1016/j.patrec.2005.10.018View ArticleGoogle Scholar
- Leow AD, Yanovsky I, Chiang M, Lee AD, Klunder AD, Lu A, Becker JT, Davis SW, Toga AW, Thompson PM: Statistical properties of jacobian maps and the realization of unbiased large-deformation nonlinear image registration. IEEE Trans Med Imaging 2007,26(6):822–832.View ArticleGoogle Scholar
- Tao G, He R, Datta S, Narayana PA: Symmetric inverse consistent nonlinear registration driven by mutual information. Computer Methods and Programs in Biomedicine 2009,95(2):105–115. 10.1016/j.cmpb.2009.01.011View ArticleGoogle Scholar
- Yang Xuan, Pei Jihong, Shi Jingli: Image registration using consistent robust point matching. Proceedings of the 2013 {IEEE} International Conference on Bioinformatics and Biomedicine 2013, 273–278. Shanghai, China, December 18–21, 2013 10.1109/BIBM.2013.6732502View ArticleGoogle Scholar
- Bender ET, Tomé WA: The utilization of consistency metrics for error analysis in deformable image registration. Physics in medicine and biology 2009,54(18):5561. 10.1088/0031-9155/54/18/014View ArticleGoogle Scholar
- Musse O, Heitz F, Armspach J: Topology preserving deformable image matching using constrained hierarchical parametric models. IEEE Transactions on Image Processing 2001,10(7):1081–1093. 10.1109/83.931102View ArticleGoogle Scholar
- Joshi SC, Miller MI: Landmark matching via large deformation diffeomorphisms. IEEE Transactions on Image Processing 2000,9(8):1357–1370. 10.1109/83.855431View ArticleGoogle Scholar
- Christensen GE, Rabbitt RD, Miller MI: Deformable templates using large deformation kinematics. IEEE Transactions on Image Processing 1996,5(10):1435–1447. 10.1109/83.536892View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.