A UNIFIED SEGMENTATION METHOD FOR DETECTING SUBCELLULAR COMPARTMENTS IN IMMUNOFLUORESCENTLY LABELED TISSUE IMAGES Ali Can, Musodiq Bello, Harvey E. Cline, Xiaodong Tao, Paulo Mendonca and Michael Gerdes GE Global Research Center, Niskayuna, NY, 12309 ABSTRACT We present a unified segmentation framework for detecting both membrane and nuclei structures in microscopy images of immunofluorescently labeled histological tissue sections. The non-parametric method presented can handle arbitrary mixtures of blob- and ridge-like structures, which is commonly found in tissue images. The algorithm iteratively estimates the empirical likelihood functions of curvature and intensity based features of nuclei and membrane structures. The method was compared to manual segmentation, binary thresholding and watershed, and achieved up to 97.1% sensitivity and 94.4% specificity compared to manual segmentation. Scores measuring target protein expressions in each of the segmented subcellular compartments were also computed. For estrogen receptor (ER), the automatically obtained expression scores achieved 96% sensitivity and 90% specificity compared to manual assessment. 1. INTRODUCTION Morphological features of tissues and cells can be described as blobs (i.e. nuclei of cells) and ridges (such as cellular membranes). Segmentation of ridge-like and blob-like structures is a common task in medical and life sciences imaging applications. These applications require detecting vessels, bronchial tree, bones, nodules in medical applications [1-6], and detecting neurons, nuclei and membrane [7-10] structures in microscopy applications. Partitioning a multiple channel digital image into multiple regions or compartments is one of the most critical steps for quantifying one or more biomarkers in molecular cell biology, molecular pathology, and pharmaceutical research. Identification of protein pathways at the molecular level has become crucial for understanding diseases. A good example of this is in the diagnosis of breast cancer where tests are done to help determine the course of treatment. Breast tumorigenesis does not follow a single line of defined changes, but is heterogeneous in terms of cell types and protein expression. For example, a short course of trastuzumab administered with docetaxel is effective in breast cancer patients who are Her2/neu positive, while counter-indicated for Her2/neu negative patients, which comprise 75% of the cases [11, 12]. In addition to Her2/neu, other markers such as ER, PR, and p53, have been used to define breast tumors and determine therapy options [13]. Protein expression is commonly assessed qualitatively by pathologists. Accurate segmentation of membrane and nuclei compartments forms the basis for higher level protein quantification and statistical analysis. For example, the distribution of a target protein on each of the segmented compartments can be quantified to reveal activation of specific protein signaling pathways. The pathway can then be related to clinical outcomes and therapeutic planning. Our segmentation algorithm accurately detects the compartments, hence enabling fully automated quantitative analysis. The quantitation of biomarkers can be accomplished without giving definite decisions for each pixel, but rather computing the likelihood of a pixel belonging to a region. Such probability maps can be computed using the intensity and geometry information provided by each channel. In this work, we present a likelihood function estimator that calculates the probability maps of membranes and nuclei structures in images. Starting from known initial geometric constraints, the algorithm iteratively estimates empirical likelihood functions of curvature and intensity based features. The distribution functions are learned from the data. This is different from existing parametric approaches, since it can handle arbitrary mixtures of blob and ridge-like structures. In tissue imaging, a nuclei image in an epithelial tissue comprises both ridge-like and blob-like structures. The network of membrane structures in tissue images is another example where the intersection of ridges can form partly blob structures.
Corresponding Author: Ali Can,
[email protected]
2. METHODS Encoding the curvature information with eigenvalues of the Hessian matrix ( λ1 ( x, y ) ≤ λ2 ( x, y ) ) is one of the most commonly used methods for detecting ridge-like and blob-like structures [2, 14-16], due to their invariance to rigid transformations. However the eigenvalues are dependent on image brightness. Given an image I ( x, y ) , we define two curvature-based features independent of image brightness: λ1 ( x, y ) , λ2 ( x, y )
θ ( x, y ) = tan −1
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Table 1: Steps for estimation of likelihood functions Define f1 ( x, y ) = I ( x, y ), f 2 ( x, y ) = φ ( x, y ), f 3 ( x , y ) = θ ( x, y ) Compute initial log-likelihood functions L( f 2 ( x, y )) , and
(2)
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do for k=1:3 A. Estimate the foreground and background sets
and refer to them as shape index, and normalized-curvature index respectively. This is essentially the same as defining the S F = {( x, y ) : L( fi ( x, y )) ≥ ε i , L( f j ( x, y )) ≥ ε j } eigenvalues in a polar coordinate system. This transformation also results in bounded features, − 3π / 4 ≤ θ ( x, y ) ≤ π / 4 , and S B = {( x, y ) : L( f i ( x, y )) ≤ −ε i , L( f j ( x, y )) ≤ −ε j } which is critical in interpreting the results 0 ≤ φ ( x, y ) ≤ π / 2 where (i, j ) ∈ {1, 2,3} , i ≠ j and formulating a unified approach to segment both membrane and nuclei structures. These curvature features are used in B. Estimate the decision boundaries Tˆk addition to image intensity. Note that due to illumination variations, tissue autofluorescence, scattering, and out-ofC. Estimate the log-likelihood function focus light, the use of intensity alone results in poor P (( x, y ) ∈ S F / f k ( x, y )) L( f k ( x, y )) = log segmentation. A slight change in the selected intensity P(( x, y ) ∈ S B / f k ( x, y )) threshold results in dramatic differences in the segmented P ( f k ( x , y ) / ( x, y ) ∈ S F ) images. ≈ log B P( f k ( x, y ) / ( x, y ) ∈ S ) From the images, a probability map that represents the probability of a pixel being a foreground is calculated, starting Enforce monotonic increasing constraint for the from known geometric cues. An initial segmentation based on intensity and the normalized-curvature index the shape index and the normalized-curvature index separates end for the image pixels into three subsets: background, foreground, until stopping criteria met and indeterminate. The indeterminate subset comprises all the pixels that are not included in the background or foreground subsets. From these subsets, the background and foreground intensity distributions and the intensity log-likelihood functions are estimated. The algorithm keeps iterating by using two out of the three features at a time to estimate the distribution of the feature that is left out. Finally, these log-likelihood functions are combined to determine the overall likelihood function. 2.1. Estimation of the log-likelihood functions The log-likelihood functions are estimated based on the assumption that the intensity and the feature vectors defined in Equations 1 and 2 are independent. Notice that these equations are normalized such that they measure a ratio rather than absolute values. If the overall image brightness is increased or decreased, these metrics stay unchanged. Starting with initial log-likelihoods determined based on the known geometry of the ridge-like or blob-like structures, the algorithm uses two out of these three feature sets to estimate the class membership of each pixel (foreground, background, or indeterminate), and use the pixel classes to estimate the class conditional probability and the log-likelihood of the third feature. This procedure is repeated either for a certain number of iterations or until convergence in log-likelihood functions is achieved. In our experiments, the algorithm typically converges in three iterations. Table 1 shows details of the algorithm. In Step-A the class memberships are determined based on two of the three features. Note that the union of the foreground pixels, S F , and the background pixels, S B , is a subset of all the pixels. Therefore, we subsample only from the dataset where we have more confidence about the class membership and use only these points to estimate the log-likelihood function of the other feature. In Step-B we estimate the decision boundary along the direction of the feature that is not used in Step-A. Although not necessary for the estimation of the log-likelihood
functions, the decision boundaries can be used to enforce monotonicity constraints for some of the log-likelihood functions. Step-C estimates the log-likelihood functions as a function of the class conditional probabilities. For the intensity and normalized-curvature index, the monotonicity constraints are then enforced. This implies that, in the case of the intensity feature, the brighter a pixel is, the more likely it is to be on the foreground. The initial log-likelihood functions are defined as (3) L( f 2 ( x, y )) = 2ε 2 (U (φ ( x, y ) − φM ) − 0.5) . L( f 3 ( x, y )) = ε 3 (U (θ ( x, y ) − θ L ) − U (θ ( x, y ) − θU ) − U (θ ( x, y )) ) ,
(4)
where U is the unit step function, and ε i are the likelihood thresholds for each feature. Now using these initial loglikelihoods, the sets in Step-A would be equivalent to the following sets, S F = {( x , y ) : θ L ≤ θ ( x, y ) ≤ θU , φ ( x , y ) > φM }
(5)
S B = {( x, y ) : θ ( x , y ) ≥ 0, φ ( x, y ) ≤ φM },
(6)
where θ L = −3π / 2, θU = −π / 2 for blobs, and θ L = −π / 2 − ∆1 , θU = −π / 2 + ∆ 2 for ridges. These parameters can be easily derived for different geometric structures. For example, for bright blobs on a dark background, both eigenvalues are negative, hence the angle between them is less than − π / 2 . Since the angle is relative to the larger eigenvalue, it is bounded by − 3π / 2 . The ridge margins are at small angles, ∆1 and ∆ 2 , for straight ridges they are equal. For the initial sets, we subsample from θ ≥ 0 to observe background pixels. Note that due to noise, the background pixels can have any curvature index. However only a subset with positive polar curvature is sufficient to estimate the intensity distribution for the background pixels. An initial threshold for normalized-curvature index, φ M , is set to the median value of all the normalized-curvature index values. Figure 1b shows the initial background (black), foreground (white), and indeterminate (gray) subsets computed using the shape index and the normalized-curvature index for the image shown in Figure 1a. These initial subsets are far from being complete (has many false negatives), but they have very few false positives, therefore provide enough information to estimate the distribution of the feature (intensity) that is left out. From these subsets, class conditional distribution functions, and the log-likelihood functions of the intensity for the background and the foreground are estimated and shown in Figure 2a (dashed plot) and (dotted plot), respectively. Mathematically, given the estimated initial sets, S F , and S B , the class conditional intensity distribution of the foreground, P ( I ( x, y ) / ( x, y ) ∈ S F ) , and the background, P ( I ( x, y ) / ( x , y ) ∈ S B ) are estimated. Given the initial log-likelihood function of the shape index, and the estimated log-likelihood function of the intensity, we recompute the background/foreground subsets (Figure 1c). Then using these subsets we estimate the class conditional distribution functions (Figure 2b), and the log-likelihood function for the normalized-curvature index (Figure 2e). The monotonicity constraint is imposed for the log-likelihood function of the normalized-curvature index, implying that the foreground has a higher curvature for a given intensity value than the background. Figure 1c show the subsets derived from intensity and shape index. The class conditional density functions are shown in Figure 2c; and the log-likelihood functions are shown in Figure 2d. The same procedure is repeated for the shape index; the estimated log-likelihood functions for the intensity and the normalized-curvature index are used to form the background/foreground subsets (Figure 1d). Then, based on these subsets, the class conditional functions, and log-likelihood functions are estimated (Figure 2f). Although the proportion of the mixtures between blobs and ridges is different, the algorithm learns the likelihood densities from the data. The monotonicity constraint, which is crucial to stabilize the convergence of the algorithm, is imposed by first estimating the decision boundaries. An optimal intensity threshold for the intensity and the normalized-curvature index are estimated by maximizing the a Posteriori Probabilities (MAP), (7) Tˆk = arg max P ( I ( x, y ) ≥ T /( x, y ) ∈ S F ) + P ( I ( x, y ) < T /( x, y ) ∈ S B ) for k = 1, 2 . T
Note that this is equivalent to minimizing the overall error criteria when the a priori distributions for the background and the foreground are equal. Since we have an estimate for the class conditional distributions, the value of the decision threshold is determined by a one-dimensional exhaustive search, rather than any parametric approximations. While there is only one decision boundary along the intensity, and normalized-curvature index dimensions, there can be multiple boundaries along the shape index feature. Therefore we do not impose any monotonicity constraint on the shape index log-likelihood function.
In Step-C, we estimate the log-likelihood functions. However, for small values of numerator and denominator this expression can become undefined or unstable. Therefore, we defined a modified empirical log-likelihood function by imposing the non-decreasing constraint as follows, sup( L( f k ( x, y )), L* ( f k ( x, y ) − ∆ )) f k ( x, y ) > Tˆk (8) L* ( f k ( x, y )) = L( f k ( x, y )) f k ( x, y ) = Tˆk , for k = 1, 2 inf( L( f ( x, y )), L* ( f ( x, y ) + ∆ )) f ( x, y ) < Tˆ k k k k where ∆ is the bin size of the histogram used to estimate the intensity distributions. Equation 8 is calculated recursively starting from Tˆk estimated by Equation 7. This guarantees that the estimated empirical log-likelihood function does not change the decision boundary when the log-likelihood function ( L* ( f k ( x, y )) = 0 ) is used for decision. In the above equation note that the index, k, is defined for the first two features, not for all of them, therefore excluding the shape index. Examples of empirical non-decreasing intensity log-likelihood functions are shown in Figures 2d and 2e for nuclei structures. The algorithm repeats Steps A-C for all features until a stopping criterion is met. Different stopping criteria can be defined, such as the rate of change in the estimated decision boundaries. We tested the algorithm in more than 600 membrane and nuclei images, and in all case the algorithm converged in three iterations.
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(e) (d) (f) Figure 1. (a) Image of a nuclei marker (DAPI). Estimated foreground subsets (white color), background subsets (black color), and indeterminate pixels not included in either foreground or background subsets (gray color) based on two of the features – (b) shape index and Normalized-curvature index; (c) Shape index and Intensity; and (d) Intensity and Normalized-curvature index. Estimated Probability maps from (e) empirical log-likelihood function and (f) parametric log-likelihood function.
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(f) Figure 2. Estimated class conditional distribution and log-likelihood functions of the (a) intensity, (b) normalized-curvature index, and (c) shape index for the nuclei image shown in Figure 1. The distribution of foreground, background and all pixels are plotted with dotted, dashed, and solid lines, respectively. The estimated log-likelihood functions based on the (d) intensity, and (e) normalized-curvature index. In (f), the empirical and the model based Log-likelihood functions of the shape index are represented with solid and dashed lines, respectively.
2.2. Detection of cytoplasm and epithelial nuclei A number of clinically relevant biomarkers are expressed in the epithelial nuclei. When quantitatively analyzing nuclear content, it is important to distinguish epithelial nuclei from surrounding stromal nuclei. Current practice in molecular imaging is to use biomarkers such as keratin to differentiate the epithelial tissue from the stromal tissue. An alternative is to use computational techniques that utilize the fact that epithelial nuclei are typically surrounded by membrane structures. Since we already have a marker and an algorithm for detecting epithelial membranes, this information can be leveraged to specify the nuclei specific to epithelial cells. The nuclei in the epithelial tissue are typically larger and more densely populated compared to stromal tissue where fibroblast nuclei are dispersed throughout the extracellular matrix. The morphological differences between epithelial and stromal nuclei can be explored by defining a superset of the nuclei, cytoplasm, and membrane set. Let S ( x, y ) denote this superset, defined as the union of the detected compartments, S ( x, y ) = C ( x , y ) ∪ M ( x , y ) ∪ N ( x, y ) ,
(9)
where C ( x , y ) , M ( x, y ) , and N ( x , y ) denote cytoplasm, membrane, and nuclei pixels respectively. Cytoplasm is defined as the union of the sets of small regions between membrane and nuclei pixels. Since the stromal nuclei are not connected through membrane structures and are sparsely distributed, they can be detected by a connected component analysis of S ( x, y ) . An epithelial mask, E ( x, y ) , is generated as a union of large connected components of S ( x, y ) . The nuclei set is then separated into epithelial nuclei and stromal nuclei by masking with E ( x, y ) .
3. RESULTS AND VALIDATION One hundred and twenty three Tissue Microarray (TMA) images from 55 patients (some patients were represented as multiple tissue cores) are stained with DAPI (nuclei), pan-cadherin (Membrane), cytokeratin (Tumor/Epithelial Mask), and estrogen receptor (ER) markers. DAPI (blue) and pan-cadherin (red) shown in Figure 3a are used to segment the subcellular compartments, and cytokeratin to segment the epithelial mask. Red, blue and green colors in Figure 3b show the segmented membrane, nuclei and cytoplasm compartments, respectively. The dark shades of the colors in Figures 1b and 1d indicate the non-epithelial regions. Figures 3c and d illustrate detailed enlarged regions. The distribution of the ER protein (green in Figure 3a) are calculated on each of the epithelial subcellular regions. ER is a nuclear marker and is expected to partially or fully express in the nuclear region in ER-positive patients. Therefore, the nuclear ER distribution is defined as the output distribution comprising a mixture of real expression, non-specific expression and autofluorescence (AF). Membrane ER distribution is defined as the input distribution comprising non-specific binding and AF. Quantitation metrics for all the images are automatically obtained based on a signed Kolmogorov Smirnov (sKS) Statistics [17]. KS distance is a well-known metric to test the statistical significance of the difference between two distributions. We developed a signed KS distance in which the sign indicates which compartment is expressed more.
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(d) Figure 3. (a) Original image of membrane (red) and nuclei (blue) markers. Green represents a target protein (ER). (b) Segmentation results membrane (red), nuclei (blue), cytoplasm (green). (c, d) Magnified views of the highlighted area in (a, b respectively. We validated the algorithm by comparing the automated segmentation to manual segmentation by human observers. Three biologists were segmented cropped regions of nuclei and membrane images manually using an in-house graphical tool. Another set of three people used LiveWire [18] semi-automated tool and sensitivity and specificity performance was assessed using Simultaneous Truth and Performance Level Estimation (STAPLE) [19], a widely used tool for comparing image segmentations. The specificity of the automated segmentation ranged from 93.7% to 97.1% while the specificity ranged from 90.3% to 94.4% when compared to available manual segmentations. Figures 4a and b show a typical cropped region used for verification and the ground truth segmentation obtained by STAPLE from all available segmentation masks. The curvaturebased algorithm was also validated by comparing to binary thresholding and watershed segmentation methods as implemented in Insight Toolkit (ITK). Figure 4c shows the Receiver Operating Characteristic (ROC) curves for the three algorithms using the ground truth from the STAPLE algorithm. Furthermore, 19 observers were requested to score the images as ER positive if they visually identified more than 10% of nuclei ER expression (following the standard scoring method by pathologists). Images scored positive by more than 50% of the observers were taken as positive, and images with sKS statistic greater than -3% were automatically classified as positive (the negative sign implies a nuclear expression relative to the membrane). The automated score correlated well with the manual score, yielding only 8 false positives, and 2 false negatives out of the 123 samples i.e., 96% sensitivity and 90% specificity for the quantitation based on the segmentation. Two serial sections (for the same patients) were used to assess the robustness of the image based subcellular quantitation to staining and tissue variations. Figure 4d shows the scores from the two serial sections. The slope of the orthogonal regression is close to 1 (0.926), and the intercept is close to 0. The mean orthogonal distance (0.078) to the fitted line is an indication of the robustness of the segmentation and quantitation methods over time.
C om parison of scor es fro m S e rial S ect ions
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0. 4 0. 3 0. 2 0. 1 0 -0. 1 -0. 2 -0. 3 -0. 4 -0. 5 S lope = 0.926 460, Interce pt = - 0.013 004 M eanO rt hogonalEr ror = 0. 078027 - 0.5 0 0. 5 M S K 3 ER K S E nuc Em em score
(d) Figure 4. (a) Typical cropped region used for verification. (b) Ground truth segmentation generated by STAPLE from multiple segmentations. (c) ROC curve comparing the results of the curvature-based algorithm (CBS) with binary thresholding (TH) and two versions of the watershed algorithm (WS) and manual segmentations by individuals labeled with initials. (d) Comparison of the automated scores between two serial tissue sections.
4. CONCLUSION We have presented a unified segmentation algorithm to detect subcellular compartments in fluorescently-labeled tissue images. The expression levels of target proteins are also quantified in each of the compartments. The segmentation was verified by comparing to several manual segmentations and also he results of other segmentation algorithms. In addition, nuclear expression of ER was accurately calculated based on pathologist assessment standards. The overall staining, segmentation, and quantitation system was also validated using adjacent serial tissue sections.
5. ACKNOWLEDGMENTS This work is supported by General Electric Global Research Center. Thanks to M. Seel, F. Ginty, M. Montalto, Z. Pang, N. Barnhardt, A. Bhaduri, A. Sood, T. Treynor, S. Dinn, F. Pavan-Woolfe, S. Abbot, B. Sarachan, J. Rittscher, Q. Li, C. Bilgin, K. Whittemore and J. Klimash for discussions and participation in the observer study. Dr. William Gerald, Memorial Sloan-Kettering Cancer Center, provided the tissue sections.
REFERENCES [1] A. Can, H. Shen, J. N. Turner, H. L. Tanenbaum, and B. Roysam, "Rapid automated tracing and feature extraction from retinal fundus images using direct exploratory algorithms," IEEE TITB, vol. 3, pp. 125-138, 1999. [2] C. Kirbas and F. Quek, "A review of vessel extraction techniques and algorithms," ACM Computing Surveys, vol. 36, pp. 81-121, 2004. [3] K. Krissian, G. Malandain, N. Ayache, R. Vaillant, and Y. Trousset, "Model-based detection of tubular structures in 3 D images," CVIU, vol. 80, pp. 130-171, 2000. [4] D. L. Pham, C. Xu, and J. L. Prince, "A survey of current methods in medical image segmentation," Annual Review of Biomedical Engineering, vol. 2, pp. 315-337, 2000. [5] F. Zana and J. C. Klein, "Segmentation of Vessel-Like Patterns Using Mathematical Morphology and Curvature Evaluation," IEEE Trans. on Image Processing, vol. 10, 2001. [6] P. R. Mendonca, R. Bhotika, S. A. Sirohey, W. D. Turner, J. V. Miller, and R. S. Avila, "Model-based analysis of local shape for lesion detection in CT scans," Int. Conf. on Medical Image Computing and Computer-Assisted Intervention, CA, USA, 2005. [7] K. A. Al-Kofahi, A. Can, S. Lasek, D. H. Szarowski, N. Dowell-Mesfin, W. Shain, J. N. Turner, and B. Roysam, "Median-based robust algorithms for tracing neurons from noisy confocal microscope images," IEEE TITB, vol. 7, pp. 302-317, 2003. [8] U. Adiga, R. Malladi, R. Fernandez-Gonzalez, and C. Ortiz de Solorzano, "High-throughput analysis of multispectral images of breast cancer tissue," IEEE Trans. Image Process, vol. 15, pp. 2259-68, 2006. [9] D. Comaniciu and P. Meer, "Cell image segmentation for diagnostic pathology," Advanced Algorithmic Approaches to Medical
Image Segmentation:, Springer, pp. 541—558, 2001. [10] T. Mouroutis, S. J. Roberts, and A. A. Bharath, "Robust cell nuclei segmentation using statistical modelling," Bioimaging, vol. 6, pp. 79-91, 1998.
[11] R. L. Camp, M. Dolled-Filhart, B. L. King, and D. L. Rimm, "Quantitative analysis of breast cancer tissue microarrays shows that both high and normal levels of HER2 expression are associated with poor outcome," Cancer Res, vol. 63, 1445-8, 2003. [12] H. Joensuu, et. al. "Adjuvant docetaxel or vinorelbine with or without trastuzumab for breast cancer," N Engl J Med, vol. 354, pp. 809-20, 2006. [13] G. G. Chung, et. al. "Beta-Catenin and p53 analyses of a breast carcinoma tissue microarray," Cancer, vol. 100, pp. 2084-92, 2004. [14] A. Frangi, W. Niessen, R. Hoogeveen, T. van Walsum, and M. Viergever, "Model-Based Quantitation of 3-D Magnetic Resonance Angiographic Images," IEEE Trans. Medical Imaging, vol. 18, 1999. [15] A. F. Frangi, et. al. "Quantitative analysis of vascular morphology from 3 D MR angiograms: In vitro and in vivo results," Magnetic Resonance in Med., vol. 45, pp. 311-322, 2001. [16] A. F. Frangi, W. J. Niessen, K. L. Vincken, and M. A. Viergever, "Multiscale vessel enhancement filtering," Lecture Notes in Computer Science, vol. 1496, pp. 1, 1998. [17] R. D'Agostino and M. Stephens. Goodness-of-fit Techniques, CRC Press, 1986. [18] W. Barret and E. Mortensen. “Interactive live-wire boundary extraction”, Medical Image Analysis, vol. 1, pp. 331-341, 1997. [19] S. Warfield, K. Zou, W. Wells. "Simultaneous truth and performance level estimation (STAPLE): an algorithm for the validation of image segmentation," IEEE Trans. Medical Imaging, vol. 23, no. 7, pp. 903-921, 2004.
