Consistent segmentation using a Rician classifier
Graphical abstract
Highlights
► Noise in magnetic resonance images should be modeled as Rician distribution. ► Rician distribution fits the MR image histogram better than a Gaussian one. ► Cortical surfaces from the brain MR images can be better delineated using Rician models in a segmentation algorithm compared to a Gaussian one. ► Segmentations between same brain MR images acquired under different pulse sequences are more consistent using Rician modeling.
Introduction
Various automated segmentation techniques have been proposed to segment brain tissues—typically cerebrospinal fluid (CSF), gray matter (GM) and white matter (WM)—in magnetic resonance (MR) images. Accurate and reliable tissue segmentation is extremely important to the neuroscience community because it is a key step in nearly every image-based study of the brain in health and disease (Resnick et al., 2003, Querbes et al., 2009, Raz et al., 2003). Manual segmentation by experts is still considered to be the gold standard in brain quantification though automated or semi-automated segmentation is acceptable for large-scale studies in which the image acquisition parameters are identical and manual segmentation is impractical (Tu et al., 2007).
Fully automated brain tissue segmentation algorithms can be sensitive to noise, partial volume effects, acquisition protocols, scanner differences, and imaging artifacts such as intensity inhomogeneities, zippers, and ringing. Techniques have been proposed to address all of these limitations and have been very successful in large part. Most algorithms incorporate spatial smoothness to reduce isolated misclassification due to noise and local artifacts (cf. Li, 1995, Leemput et al., 1999). Intensity inhomogeneities are either estimated in preprocessing (e.g. Sled et al., 1998, Chang and Fitzpatrick, 1992, Vovk et al., 2004) or incorporated within the classification algorithm itself (e.g. Pham and Prince, 1999, Pham, 2001, Styner et al., 2000). Incorporation of statistical atlases (cf. Woolrich et al., 2009, Prastawa et al., 2004) and control of topology (Bazin and Pham, 2007) have been used to reduce misclassification error through incorporation of prior knowledge. The partial volume effect is typically addressed by producing a soft classification, i.e. one that provides membership functions or posterior densities associated with each tissue class (Leemput et al., 2003, Choi et al., 1991, Noe and Gee, 2002). The effect can also be addressed by super-resolution methods (Rousseau, 2008, Souza and Senn, 2008), probabilistic models, or topological methods (Bazin and Pham, 2007, Wua and Chung, 2009, Leemput et al., 2009).
Compensation for different acquisition protocols or scanner differences has been particularly problematic for tissue segmentation algorithms (Clark et al., 2006). Approaches to normalize histograms to a common scale have been proposed (Nyul and Udupa, 1999, Han and Fischl, 2007, He et al., 2008), and most recent algorithms use some kind of explicit or implicit intensity normalization preprocessing in practice. Achieving true pulse sequence independence, though, currently requires one to use special pulse sequences (Fischl et al., 2004) that permit computation of the underlying tissue parameters to which a segmentation algorithm can be applied (Prince et al., 1995). Though admirable in spirit and quite effective, common practice precludes routine use of special pulse sequences, and modern study designs have typically relied on the use of a multiple scanners or types of scanners or multiple structural acquisition protocols with fixed parameters (Shock et al., 1984, Mueller et al., 2005) in order to yield images whose segmentations can be quantitatively compared within a particular study (Wolz et al., 2010).
Two classes of tissue classification methods have emerged as leading algorithms for MR brain image segmentation: methods (Bezdek et al., 1993, Pham and Prince, 1999, Siyal and Yu, 2005) based on fuzzy c-means (FCM) (Bezdek, 1980) and methods based on a Bayesian framework using a finite Gaussian mixture model assumption (Leemput et al., 2003, Hong et al., 2007, Woolrich et al., 2009, Ashburner and Friston, 2005, Awate et al., 2006). Both approaches have been augmented to account for spatial smoothness (Pham, 2001, Held et al., 1997, Scherrer et al., 2008), most commonly using a Markov Random Field (MRF) (Li, 1995). At this time, the performances of these methods are very similar “across the board” and the algorithms are widely used in large-scale studies. Yet experience shows that algorithm parameters must be tuned in order to achieve satisfactory results when acquisition parameters change. We suggest in this paper that both classes of algorithms operate with a less accurate model of image intensity and that improving the model can provide improved segmentation and robustness to pulse sequence changes.
The FCM method is not based on an underlying intensity model, though one can tease apart the variational formulation in order to assert its basic assumptions. In its conventional formulation, FCM is a clustering method that associates voxels to all classes in proportion to the value of its computed membership functions. The clusters are uniformly spread around each center intensity, which is also estimated by the algorithm. The so-called “fuzziness parameter” in FCM, roughly speaking, determines how spread out the clusters are from their centroids (Yu et al., 2004, Roy et al., 2008). The basic formulation is not Bayesian, and there is no formula relating the underlying tissue intensities to the observed intensities and there is no explicit noise model. Accommodations have been made to account for clusters that might not have the same size (Cavalcanti and de Carvalho, 2005, Roy et al., 2008, Gustafson and Kessel, 1979), but the added parameters must generally be known in advance and tuned to any given pulse sequence.
The most common Bayesian formulations are based on a finite Gaussian mixture model, in which the conditional probability of the image intensity for a particular tissue type is Gaussian (Leemput et al., 1999). The parameters of the underlying Gaussian conditional probabilities (and often the mixture coefficients that proportionally weight these densities) are typically estimated using the expectation maximization (EM) algorithm (Dempster et al., 1977). If image smoothness is maintained through the use of an MRF, then the EM algorithm solves a maximum likelihood estimation problem and optimal estimates of both the mixture parameters and the posterior densities are found. The model choice together with the estimation procedure automatically accommodates for clusters that might be of different sizes and relative proportions (if the mixture coefficients are also estimated). It is logical to assume that the additional flexibility of this model together with the Bayesian optimality would lead to a better result than FCM. However, there are numerous papers that support the contrary opinion.
We are led to question the underlying assumption of a Gaussian model of the intensities in the current Bayesian methods. In conventional MR imaging, the acquired raw data is the underlying signal in “real” (in-phase) and “imaginary” (quadrature phase) channels, each of which is corrupted by additive zero-mean i.i.d. Gaussian noise. The complex image intensities are obtained using the Fourier transform, which preserves the Gaussian nature of the noise in the real and imaginary components of the image intensities (Bernstein et al., 1989). Since the observed image intensities are formed by taking the complex modulus of the real and imaginary parts of the complex image, each image voxel becomes a Rician random variable (Gudbjartsson and Patz, 1995, Henkelman, 1985). See Section 2 for more details.
The underlying signal values are generally different at each voxel because of biological variability. Therefore, the probability distribution that describes the collection of all voxels taken together is a Rician mixture model in which there is a different conditional Rician probability density function for each underlying signal value. By noting that within each tissue class the underlying signal intensities are close in value, this rich mixture model can be approximated by one that has only three conditional Rician probability densities, one for each tissue class. When the underlying signal values are large relative to the noise, it is known that a Rician distribution can be approximated by a Gaussian distribution (Sijbers et al., 1998). But since this approximation becomes less accurate with smaller underlying signal values, we can expect the greatest impact of using this Rician mixture model versus a Gaussian mixture model to be in the tissue classes having the smallest underlying signal values.
To illustrate this point, in Fig. 1a we show the smoothed histogram of intensities in an inhomogeneity corrected (Sled et al., 1998) Magnetization Prepared Rapid Gradient Echo (MPRAGE) image together with two fitted histograms, one using a mixture of Gaussians (blue1) and one using a mixture of Ricians (red). It is observed that the Rician fit is better, an observation that can be quantitatively verified by noting that the Kullback–Leibler (KL) distances (Kullback and Leibler, 1951) between the image histogram and the Gaussian fit is 0.0418 and between the image histogram and the Rician fit is 0.0097. In Fig. 1b, the fits of the individual class conditional probabilities derived from the Gaussian (blue) and Rician (red) fitting process. It is observed that the CSF densities show the most difference, which is to be expected since these intensities are the lowest. The WM densities are most similar, which makes sense since these tissues have the highest intensities in this T1-weighted pulse sequence, and are likely to be well approximated by a Gaussian as a result.
In this paper, we propose a brain image tissue segmentation algorithm based on an underlying finite Rician mixture model, which we call Rician Classifier using EM (RiCE). We primarily focus on the difference between Rician and Gaussian models of the tissue intensities. Consequently, we do not include any bias-field correction in our method, instead, we pre-process all the data using a non-parametric inhomogeneity correction method (N3) (Sled et al., 1998). Although the inhomogeneities in different MR sequences can depend on the sequence itself, N3 has been shown to work well on different sequences (Manjon et al., 2007, Mangin, 2000). In order to include smoothness on the resulting segmentation, the algorithm includes an MRF model. This fully automatic algorithm does not require parameter choices, relying instead on the assumption that cluster intensity distributions will be Rician regardless of the pulse sequence. The main contribution of this work is to improve segmentation consistency between different pulse sequences having T1-weighted (T1w) contrast. We compare our method with a Gaussian intensity model approach, SPM (spm_segment function) (Ashburner and Friston, 2000, Ashburner and Friston, 2005, Chard et al., 2002), a Gaussian model approach on log-transformed intensities, FAST (Woolrich et al., 2009) and two FCM based approaches, Freesurfer (Dale et al., 1999) (mri_ms_EM function) and FANTASM (Pham, 2001).
We outline our assumptions on noise models and EM are explained in Section 2 and the algorithm is described in Section 3. Validations on simulated and real data are presented in Sections 4.1 Brainweb phantom validation, 4.2 IBSR validation, respectively. Then we show the improvement in segmentation consistency of the Rician model over a comparable Gaussian model in Section 5 and the comparison of our method with other state of the art methods in Section 6.
Section snippets
Noise estimation
Magnitude images are most commonly used in MRI. They are acquired in two steps. Complex data is acquired in separate in-phase and quadrature phase channels. We assume that each channel is corrupted with uncorrelated additive Gaussian noise, having zero mean and the same variances (Gudbjartsson and Patz, 1995, Bernstein et al., 1989, Henkelman, 1985). Then real and imaginary images are reconstructed from the complex data by inverse Fourier transform. The inverse Fourier transform, being linear
A finite mixture model using Ricians
We now develop an EM classification algorithm for the Rician mixture model. The log-likelihood of Eq. (2) is extended to include random noise removal by introducing an MRF on the underlying segmentation zjk. The total log-likelihood after these modifications is given by,
The unknown prior probabilities πjk in Eq. (2) are replaced by a spatially varying function following the model described in (Nikou et al., 2007). In the following
Brainweb phantom validation
We first validate RiCE on the Brainweb phantom (Cocosco et al., 1997) and compare it with SPM (Ashburner and Friston, 2000), FAST (Zhang et al., 2001), FANTASM (Pham, 2001) and a FCM based segmentation from Freesurfer (Dale et al., 1999), (mri_ms_EM function). SPM uses a Gaussian intensity model and it tries to recover the non-Gaussianity of the intensity PDF by modeling it with multiple Gaussians. FAST uses a Gaussian model on the log transformed intensities. Freesurfer and FANTASM use
Segmentation consistency
We carry out a consistency performance experiment on a set of 3T data from the Baltimore Longitudinal Study of Aging (BLSA) (Shock et al., 1984, Resnick et al., 2003), comprised of T1w axial MPRAGE and SPGR acquisitions (256 × 256 × 124 volumes having the resolution of 0.9375 × 0.9375 × 1.5 mm) of 14 normal subjects, ages in the range of 69–92. The SPGR acquisitions are registered to their corresponding MPRAGE acquisition using a rigid registration (Jenkinson and Smith, 2001) and stripped using a hybrid
Comparison with other methods
In this section, we compare the overall performance of our method with other methods. Fig. 8 shows the comparison of the hard segmentations using the five algorithms. The Dice coefficients of the three classes and their volume weighted “average” Dice are shown in Table 5, which shows that both the CSF and GM segmentation are more similar in the case of RiCE. t-Tests comparing the overlap of CSF and GM show a significant improvement in consistency over the other four methods. This experiment
Summary and conclusion
This paper proposes a Rician PDF based brain MR segmentation technique. We have concentrated on consistent segmentation of three primary tissues, cerebrospinal fluid, gray matter and white matter, from T1-weighted MR images acquired with two different pulse sequences, MPRAGE and SPGR. The underlying acquisition parameters, like repetition time, inversion time or flip angle, are usually different from one sequence to another, which gives rise to the variability of the tissue contrast. With exact
Acknowledgments
This research was supported in part by the Intramural Research Program of the NIH, National Institute on Aging. We are grateful to all the participants of the Baltimore Longitudinal Study on Aging (BLSA), as well as the neuroimaging staff for their dedication to these studies. This work was also supported by the NIH/NINDS under Grant 5R01NS037747.
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