Symmetric deformable image registration via optimization of information theoretic measures

https://doi.org/10.1016/j.imavis.2009.11.012Get rights and content

Abstract

The use of information theoretic measures (ITMs) has been steadily growing in image processing, bioinformatics, and pattern classification. Although the ITMs have been extensively used in rigid and affine registration of multi-modal images, their computation and accuracy are critical issues in deformable image registration. Three important aspects of using ITMs in multi-modal deformable image registration are considered in this paper: computation, inverse consistency, and accuracy; a symmetric formulation of the deformable image registration problem through the computation of derivatives and resampling on both source and target images, and sufficient criteria for inverse consistency are presented for the purpose of achieving more accurate registration. The techniques of estimating ITMs are examined and analytical derivatives are derived for carrying out the optimization in a computationally efficient manner. ITMs based on Shannon’s and Renyi’s definitions are considered and compared. The obtained evaluation results via registration functions, and controlled deformable registration of multi-modal digital brain phantom and in vivo magnetic resonance brain images show the improved accuracy and efficiency of the developed formulation. The results also indicate that despite the recent favorable studies towards the use of ITMs based on Renyi’s definitions, these measures are seen not to provide improvements in this type of deformable registration as compared to ITMs based on Shannon’s definitions.

Introduction

Information theoretic measures (ITMs) have been used to measure nonlinear statistical dependency between two or more variables, as compared to linear correlation for measuring linear statistical dependencies [1]. These measures have also been successfully used in multi-modality medical image registration [2], [3], as well as in information theoretic learning (ITL) [1], [4], in such applications as nonlinear pattern classification and bioinformatics [5], [6]. A challenging application addressed here is deformable image registration, in which local spatial correspondences are quantified by similarity measures established between multi-modal images or images obtained under different scanning conditions [2], [7], [8]. The use of deformable registration in medical image processing includes the registration of deformable tissue and organs, such as breast, chest, heart, etc.; brain shift due to neurosurgery; longitudinal changes due to aging, development, and disease; as well as spatial normalization [9] and inter-subject registration [2].

The idea of using ITMs in image registration originates from the observation that the joint histogram of two images is generally more dispersed when the images are not registered [7], [11]. The correspondences between multi-modal image intensity values are normally nonlinear and complex, therefore correlation-based measures are not generally accurate for multi-modal image registration [2], [7], [8], [11]. The ITMs, i.e. entropy and joint entropy of images [11], quantify the amount of information shared between images. The lower the joint entropy of two images, the lower the dispersion of their information. The minimization of joint entropy is thus used to achieve registration. ITMs based on both joint and marginal entropies provide alternative metrics for representing the mutual information content of images and may provide better performance metrics for registration or classification.

There has been a considerable amount of research on the use of mutual information (MI) in medical image registration [7], [12]. The literature supports an extensive use of MI for rigid and affine registration [2], [13], [14], [15], and particularly indicates that maximization of MI generates quite accurate results in rigid registration of multi-modal brain images [13]. Different optimization techniques [16], efficient multi-resolution estimation and optimization [16], [17], and extensions of MI such as normalized MI (NMI) [18], have been devised to achieve sub-voxel accuracy in rigid registration. This is rather straightforward as the large number of samples typically used for MI estimation provides a reliable basis for optimizing the six parameters of a rigid or the 12 parameters of an affine transformation. The situation becomes much more challenging in deformable registration, when a high-dimensional deformation field needs to be optimized. In order to compute a high-dimensional deformation field based on registration, the resolution of image correspondences, i.e. the existing image features, should be higher than the resolution of local deformations. When these conditions hold, with appropriate estimation and computation, ITMs can be used for local deformable image registration.

The use of ITMs in deformable image registration has been considered in a few studies; in [19] MI and NMI were used with a B-Spline free-form deformation model for deformable registration of breast MR images. The performance of MI was compared to that of cross correlation and correlation ratio in [8]. Skerl et al. [21], have recently developed protocols for evaluation of similarity measures for rigid and non-rigid multi-modality registration. Their study on several sets of multi-modality images indicates the superiority of MI and NMI as compared to correlation ratio and entropy in deformable image registration. An efficient estimation and optimization framework was developed for MI-based deformable registration in [22] for PET-CT chest image registration. The use of ITMs has been recently discussed in a tutorial on nonlinear information processing [1]. Literature suggests that the higher order ITMs based on Renyi’s definitions or distance measures may provide better performance in signal and image processing applications [1], [23]. An evaluation of higher order ITMs has been previously done only for rigid registration in [3] which concludes that some higher order ITMs may potentially perform better in rigid registration as compared to Shannon’s mutual information. Nevertheless, there has not been such an evaluation and comparison for deformable image registration. These aspects are addressed in this paper, noting that the computation and optimization of ITMs for deformable image registration and their performance analysis is much more challenging as compared to rigid and affine registration.

Initially, inspired by the work of Erdogmus and Principe [1], Shannon’s and Renyi’s definitions of joint entropy and mutual information, as well as the NMI [18] and Cauchy-Schwartz Quadratic Mutual Information (CS-QMI) [23] are examined here. Various approaches exist for efficient estimation of ITMs, among which a fuzzy histogram binning estimation approach [6], [24] is used in this paper. The fuzzy histogram binning estimation approach can be considered as a generalization of the B-Spline method that Thevenaz and Unser developed for MI-based rigid registration [17], and Mattes et al. utilized for deformable registration [22]. The fuzzy histogram binning estimation approach share similarities with the partial volume interpolation (PVI) [16] and the generalized partial volume estimation (GPVE) [25] approaches. The comparison in [26] shows that the B-Spline method of Thevenaz and Unser [17] outperforms the GPVE method, while GPVE is more robust.

On the basis of the B-Spline fuzzy histogram binning estimation, the development in this paper involves analytical formulations for an efficient computation of the derivatives of various ITMs. Inverse consistency is obtained for ITMs through the computation of symmetric cost functions similar to the symmetric sum of square differences formulation in [27], [28]. This formulation requires the transformation to be invertible. This condition is guaranteed here by using sufficient bound parameter constraints within the optimization framework. The devised symmetric registration formulation provides inverse consistency and accurate results as it combines the estimation, interpolation, and analytical derivatives computation on both the source and target images.

The development and formulation of the registration framework based on the optimization of ITMs, estimation and derivatives computation are presented in Section 2. An evaluation based on digital brain phantom and real magnetic resonance (MR) brain images is presented in Section 3. MR brain images typically provide high-resolution features that make the evaluation relatively accurate. In addition to high-resolution MR images, we have intentionally utilized down-sampled digital phantom images and low-resolution echo-planar images (EPI) to compare the performance of the similarity measures in more challenging cases. Digital phantom images provide ground truth for quantitative evaluation; and in vivo experiments involve the registration of functional to anatomic MRI. The paper is concluded in Section 4.

Section snippets

Image registration

Image registration is normally formulated as an optimization problem to find the best global or local spatial transformation or deformation model that matches a source image to a target image. The source and target images can be represented by Is:R3RandIt:R3R, which associate scalar intensity values to points described by x = (x, y, z) in the three-dimensional vector space R3. The target image is fixed and the source image undergoes a spatial vector transformation of the form T(x̲):R3R3 with

Registration functions

In order to evaluate and compare the performance of the ITMs considered in this paper, rigid, affine, and deformable registration functions were computed for the digital brain phantom images of the Brainweb database [36]. T1-weighted (T1w), T2-weighted (T2w) and Proton Density (PD) MR brain images with 1 mm resolution, 3% noise, and 20% intensity non-uniformity (default values) were obtained from the Brainweb database (http://www.bic.mni.mcgill.ca/brainweb/). In order to consider the effect of

Conclusion

The development and analysis carried out in this paper indicates that: (1) the fuzzy histogram binning estimation approach along with the analytical computation of derivatives yield an efficient framework for optimization of ITMs, (2) among various ITMs, MI and NMI based on Shannon’s definitions lead to the most accurate results, and (3) the introduced symmetric formulation provides inverse consistency and more accurate results for deformable image registration by symmetric computation of the

Acknowledgements

This study was jointly supported by the UTD Erik Jonsson School of Engineering and Computer Science, and a subcontract from UT Southwestern Medical Center at Dallas, funded by the Department of Veterans Affairs through VA IDIQ Contract No. VA549-P-0027 awarded and administered by the VA Medical Center, Dallas, TX. The content of this paper does not necessarily reflect the position or the policy of the Veterans Administration or the Federal government, and no official endorsement should be

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