Automated Neuron Reconstruction from 3D Fluorescence Microscopy Images Using Sequential Monte Carlo Estimation
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Abstract
Microscopic images of neuronal cells provide essential structural information about the key constituents of the brain and form the basis of many neuroscientific studies. Computational analyses of the morphological properties of the captured neurons require first converting the structural information into digital tree-like reconstructions. Many dedicated computational methods and corresponding software tools have been and are continuously being developed with the aim to automate this step while achieving human-comparable reconstruction accuracy. This pursuit is hampered by the immense diversity and intricacy of neuronal morphologies as well as the often low quality and ambiguity of the images. Here we present a novel method we developed in an effort to improve the robustness of digital reconstruction against these complicating factors. The method is based on probabilistic filtering by sequential Monte Carlo estimation and uses prediction and update models designed specifically for tracing neuronal branches in microscopic image stacks. Moreover, it uses multiple probabilistic traces to arrive at a more robust, ensemble reconstruction. The proposed method was evaluated on fluorescence microscopy image stacks of single neurons and dense neuronal networks with expert manual annotations serving as the gold standard, as well as on synthetic images with known ground truth. The results indicate that our method performs well under varying experimental conditions and compares favorably to state-of-the-art alternative methods.
Keywords
Neuron reconstruction Bayesian filtering Sequential Monte Carlo estimation Particle filtering Fluorescence microscopyIntroduction
The brain is regarded as one of the most complex and enigmatic biological structures. Composed of an intricate network of tree-shaped neuronal cells (Ascoli 2015), together forming a powerful information processing unit, it performs a myriad of functions that are essential to living organisms (Kandel et al. 2012). Obtaining a blue print of the architecture of this network, including the morphologies and interconnectivities of the neurons in various subunits, helps to understand how the brain works (Ascoli 2002; Donohue and Ascoli 2008; Cuntz et al. 2010), including how neurodegenerative disease processes alter its function. A key instrument in this endeavor is microscopic imaging, as it allows detailed visualization of neuronal cells in isolation and in tissue, thus providing the means to study their structural properties quantitatively (Senft 2011).
Quantitative measurement and statistical analysis of neuronal cell and network properties from microscopic data rely on the ability to obtain accurate digital reconstructions of the branching structures (Halavi et al. 2012) in the form of a directional tree of connected nodes (Ascoli et al. 2007). The ever increasing amount of available image data calls for automated computational methods and software tools for this purpose, as manual delineation of neurons is extremely cumbersome even in single image stacks, and is downright infeasible in processing large numbers of images (Svoboda 2011; Senft 2011). Automating neuron reconstruction requires solving fundamental computer vision problems such as detecting and segmenting tree-like image structures (Meijering 2010; Donohue and Ascoli 2011; Acciai et al. 2016). This is complicated by the large diversity of neuron types, imperfections in cell staining, optical distortions, inevitable image noise, and other causes of ambiguity in the image data. Consequently, with the current state-of-the-art, manual proof-editing of automatically obtained digital reconstructions is often necessary (Peng et al. 2011b). Recent international initiatives such as the DIADEM challenge (Gillette et al. 2011) and the BigNeuron project (Peng et al. 2015a, 2015b) have catalyzed research in automated neuron reconstruction but have also clearly revealed that further improvement is still very much needed before computers can fully replace manual labor in performing this task.
Related Work
Early methods and tools for digital neuron reconstruction were semi-automatic and required extensive manual intervention for their initialization and operation or the curation of faulty results (Glaser and Van der Loos 1965; Capowski and Sedivec 1981; Glaser and Glaser 1990; Masseroli et al. 1993). With the increasing capabilities of computers it became possible to store and process 3D images of neurons (Cohen et al. 1994; Belichenko and Dahlström 1995). More recently, the state-of-the-art in the field has moved towards full automation of neuron reconstruction, and various freely available software tools are now available for this purpose (Peng et al. 2010; Longair et al. 2011; Peng et al.2014a, 2014a), though the need for flexible editing tools has remained unabated (Luisi et al. 2011; Dercksen et al. 2014).
Neuron reconstruction methods typically have a modular design where each module or stage of the processing pipeline deals with different structural objects. Depending on the subproblems being solved, modules can operate independently, or work together for example to combine local and global processing, possibly requiring multiple iterations. Several subproblems that can be identified in the literature include image prefiltering and segmentation (Zhou et al. 2015; Türetken et al. 2011; Sironi et al. 2016; Mukherjee and Acton 2013), soma (cell body) detection and segmentation (Quan et al. 2013), landmark points extraction (Al-Kofahi et al. 2008; Wang et al. 2011; Choromanska et al. 2012; Baboiu and Hamarneh 2012; Su et al. 2012; Radojević et al. 2016), neuron arbor tracing (Zhao et al. 2011; Liu et al. 2016; Leandro et al. 2009; Radojević and Meijering 2017a; Xiao and Peng 2013), and assembling the final tree-like graph structure (Zhou et al. 2016; Türetken et al. 2011; Yuan et al. 2009). In the remainder of this section we briefly review techniques for solving each of these subproblems. Since our primary goal in this paper is to present a new method, the review is not meant to be exhaustive, but to put our method into context.
The pool of neuron reconstruction methods is very diverse (Meijering 2010; Donohue and Ascoli 2011; Acciai et al. 2016; Peng et al. 2015a) but there are also many commonalities. For example, image prefiltering to enhance tubular structures is typically carried out using Hessian or Jacobian based processing (Xiong et al. 2006; Al-Kofahi et al. 2008; Yuan et al. 2009; Wang et al. 2011). And to cope with uneven staining, adaptive thresholding (Zhou et al. 2015), perceptual grouping (Narayanaswamy et al. 2011), and vector field convolution (Mukherjee et al. 2015) have been used. For image segmentation (separating foreground from background), a wide variety of methods has been proposed, including the use of feature-based classifiers (Türetken et al. 2011; Chen et al. 2015; Jiménez et al. 2015), tubularity based supervised regression (Sironi et al. 2016), and even deep learning (Li et al. 2017). The general difficulty of supervised methods, however, is their need for extensive manual annotation for training to arrive at usable segmentation models. In our proposed method we have chosen to avoid this by using carefully designed explicit models.
For the detection and segmentation of the neuronal somas, which typically have a much larger diameter than the dendritic and axonal branches, a simple and efficient solution is to apply morphological closing and adaptive thresholding (Yan et al. 2013). An alternative is to use shape fitting approaches (Quan et al. 2013). Next, to initialize and/or guide the segmentation of the arbor, landmark points are often extracted using image filters that specifically enhance tubular structures (Wang et al. 2011; Türetken et al. 2011; Choromanska et al. 2012; Su et al. 2012; Radojević et al. 2016), a popular one being the so-called “vesselness filter” (Frangi et al. 1998). In our proposed method we have adopted classical approaches for soma and seed point detection as detailed in the next section.
Segmentation or tracing of all branches of the dendritic and axonal trees is the main challenge of the reconstruction problem. A widely used approach to overcome the difficulties caused by imperfect staining and image noise is to use techniques that find globally optimal paths between seed points by minimizing a predefined cost function (Meijering et al. 2004; Peng et al. 2011a; Longair et al. 2011; Quan et al. 2016). But many other concepts have been proposed as well, including model fitting (Schmitt et al. 2004; Zhao et al. 2011), contour extraction (Leandro et al. 2009), active contour segmentation (Wang et al. 2011; Luo et al. 2015), level-set or fast-marching approaches (Xiao and Peng 2013; Basu and Racoceanu 2014), path-pruning from oversegmentation (Peng et al. 2011a), distance field tracing (Yang et al. 2013), marching rayburst sampling (Ming et al. 2013), marked point processing (Basu et al. 2016), iterative back-tracking (Liu et al. 2016), and learning based approaches (Chen et al. 2015; Gala et al. 2014; Santamaría-Pang et al. 2015). In recent works we have shown the great potential of probabilistic approaches to neuron tracing (Radojević et al. 2015; Radojević and Meijering 2017a, 2017a) which formed the basis for the new fully automated neuron reconstruction method presented and evaluated in the next sections.
The final aspect of neuron reconstruction is the assembling of the complete neuronal tree structure from possibly many partial or overlapping traces and putting it into a format that is both representative and suitable for further automated analysis. This is typically solved by graph optimization strategies such as the minimum spanning tree (MST), the alternative K-MST (Türetken et al. 2011; González et al. 2010), or integer programming (Türetken et al. 2013). To deal with very large data sets it has also been proposed to assemble the 3D graph representation through tracing in 2D projections and applying reverse mapping (Zhou et al. 2016). However, with the advent of sophisticated assemblers such as UltraTracer (Peng et al. 2017), it is possible to extend any base tracing algorithm to deal with arbitrarily large volumes of neuronal image data (Peng et al. 2017). Therefore, in our proposed method, we do not use projections but perform the tracing in the original image (sub)volumes. And to obtain the graph representation we propose a new approach to refining and grouping the individual traces.
Proposed Method
The pipeline of our proposed method consists of six steps (Fig. 1) described in detail in the following subsections. We assume that image stacks contain a single neuron (one soma) or just an arbor (no soma) as in the DIADEM (Brown et al. 2011) and BigNeuron data (Peng et al. 2015a). In short, we first extract the soma and a set of seeds, which serve to initialize our probabilistic branch tracing scheme. The resulting traces are iteratively refined and their corresponding nodes spatially grouped into a representative node set that is traversed to form the final reconstruction.
Soma Extraction
Seed Extraction
From the resulting tubularity map, initial seed points s_{i} = [p_{i},v_{i},σ_{i}] are selected whose tubularity value is the highest in a cylindrical neighborhood with radius 3σ_{i} and length σ_{i}, centered at p_{i}, and oriented along v_{i}. For this purpose we use a find-maxima function ported from ImageJ, which applies a noise tolerance τ to prune insignificant local maxima (Ferreira and Rasband 2012). The final set of seeds is subsequently obtained by excluding the maxima where the correlation of the image with a cylindrical template model is too low, using exactly the same criterion as for termination of branch tracing, described next.
Branch Tracing
Trace Refinement
Node Grouping
Parameters of our method and their default values and grid search values used for each data set in the experiments
Parameter | Value | Description | ||||
---|---|---|---|---|---|---|
Default | OPF | NCL1A | BGN | Synthetic | ||
r _{ s} | 6 | 0 | 0 | 0, 4, 8, 12 | 0, 4 | Erosion radius [voxels] |
σ | {2, 4, 6} | {2},{2, 4},{2, 4, 6} | {2},{2, 4},{2, 4, 6} | {2},{2, 4},{2, 4, 6} | {2, 4, 6} | Scale combinations [voxels] |
τ | 10 | 4, 6, 8, 10, 12 | 6, 8, 10 | 6, 8, 10 | 6, 8, 10 | Local maxima tolerance [8-bit scale] |
N | 20 | 20 | 20 | 20 | 20 | Number of samples |
κ | 3 | 3 | 3 | 3 | 3 | Circular variance [voxels] |
d | 3 | 3 | 3 | 3 | 3 | Tracing step size [voxels] |
ζ | 1 | 1 | 1 | 1 | 1 | Scale variance [voxels] |
K | 20 | 20, 50 | 20, 50 | 20 | 20, 50 | Likelihood sensitivity |
\(c_{\min }\) | 0.5 | 0.4, 0.5 | 0.3, 0.4, 0.5 | 0.3, 0.4, 0.5 | 0.3, 0.4, 0.5 | Correlation threshold |
L | 200 | 200 | 200 | 200 | 200 | Iteration limit |
δ _{ n} | δ_{9} = 4 | δ_{9} = 4 | δ_{9} = 4 | δ_{1} = 3, δ_{9} = 4 | δ_{1} = 3, δ_{9} = 4 | Node density limit [count/volume] |
r _{ g} | 2 | 2 | 2 | 2 | 2 | Grouping radius [voxels] |
Tree Construction
The final step of our method is the construction of a graph representing the complete neuronal arbor. This is facilitated by the bidirectional connectivity of the group nodes in \(\hat {\mathcal {N}}\). However, similar to a real neuron, the final graph must be a tree, in which the nodes are unidirectionally linked (Figs. 1f and 4d), as also required by the SWC file format for storing digital neuron reconstructions (Stockley et al. 1993; Cannon et al. 1998). Starting from the soma node, or from the group node with the highest cross-correlation value if no soma was found in the image, the nodes in \(\hat {\mathcal {N}}\) are iteratively traversed using a breadth-first search (BFS) algorithm. In this process it is possible to discard any isolated branches and single-node terminal branches (false positives).
Implementation Details
Our method, which we call Probabilistic Neuron Reconstructor (PNR), was implemented in C++ as a plugin for the freely available and extendable bioimage visualization and analysis tool Vaa3D (Peng et al. 2010; 2014a).^{2} The source code of PNR is freely available for non-commercial use.^{3} As mentioned in the preceding sections, the method has a number of free parameters, which are summarized in Table 1, where we also list default values.
Experimental Results
The performance of our PNR method was evaluated using both synthetic and real fluorescence microscopy image stacks of single neurons and was compared to several alternative 3D neuron reconstruction methods that yielded favorable performance in the BigNeuron project (Peng et al. 2015a). These included the second all-path pruning method (APP2) (Xiao and Peng 2013), NeuroGPS-Tree (GPS) (Quan et al. 2016), BigNeuron’s minimum spanning tree (MST) method, and we also added our recently published alternative probabilistic method based on probability hypothesis density filtering (PHD) (Radojević and Meijering 2017a).
To quantify performance we adopted the commonly used measures of distance and overlap of neuron reconstructions with respect to the ground truth (in the case of synthetic images) or the gold-standard reconstructions obtained by manual annotation (in the case of real images). The distance measures were the average minimal reciprocal spatial distance (SD) between nodes in the reconstructions being compared, the substantial spatial distance (SSD) using only the nodes with a spatial distance larger than a threshold S, and the percentage of these substantially distant nodes (%SSD), all computed after densely resampling each reconstruction to reduce the distance between its adjacent nodes to one voxel (see Peng et al. 2010 for details). The overlap measures were precision (P), recall (R), and the F score (Powers 2011), computed from the numbers of true-positive (TP), false-positive (FP) and false-negative (FN) nodes according to the spatial distance threshold S.
All experiments were performed on a MacBook Pro with 2.2 GHz Intel Core i7 processor and 16 GB RAM memory to test the practicality of the methods on a typical computer system. For each method we optimized the score for each performance measure by exploring a grid of possible parameter values around the default ones (see Table 1 for our method and the cited papers for the other methods). To keep the experiments feasible, we set the maximum allowed processing time per stack and method to 2 hours. In the sequel, to save space, we show only the F scores (higher is better) and SSD scores (lower is better), while the P, R, SD, and %SSD scores are given in the supplement. Our conclusions are based on the complete body of results.
Experiments on Synthetic Neuron Images
We developed a plugin for ImageJ (Schneider et al. 2012) called SWC2IMG,^{4} which takes any SWC file as input and simulates fluorescence microscopy imaging of all neuronal branches in the file at a specified SNR and COR level, producing an image stack whose true digital reconstruction is the very input. It assumes that in practice, because of the relatively large spatial extent of even a single neuron with its complete arbor, the combination of optical magnification factor and digital image matrix size in real neuron images is typically such that the voxel size is larger than the point spread function (PSF), implying that the partial-volume effect of digitization is more prominent than the optical blurring by the microscope. Based on this, the plugin simulates the imaging simply by estimating for each voxel which fraction of its volume is occupied by the neuron. Next, it simulates noise by using the Poisson noise model representative of optical imaging, which defines SNR as the image intensity inside the neuron above the background, divided by the standard deviation of the noise inside (Sheppard et al. 2006). And finally, to allow for correlated signal and noise, which we found to improve the visual realism of the simulated images, the plugin also offers the possibility to apply Gaussian smoothing at a specified scale, being the COR parameter, while preserving the SNR level. Generally, the lower the SNR and/or the higher the COR level, the more challenging the data and the reconstruction problem.
Experiments on Real Neuron Images
In addition to synthetic data we also used three real neuron image data sets to evaluate the absolute and relative performance of our method. The first two are the olfactory projection fibers (OPF) data set (9 image stacks) and neocortical layer-1 axons (NCL1A) data set (16 image stacks) from the DIADEM challenge (Brown et al. 2011), and the third is part of the BigNeuron (BGN) training data set (76 image stacks) (Peng et al. 2015a), all imaged with fluorescence microscopy (confocal or two-photon) and manually annotated as described in detail in the cited works and corresponding resources. Being the smallest of the three, in terms of both neuronal volume and complexity, OPF is probably the most often used data set in the field. NCL1A is often used as it contains neuronal network-like structures and no clear somas. And BGN is the largest, most diverse, and thus most challenging data set for evaluating neuron reconstruction methods. Together, the 100+ image stacks in these data sets have a wide variety of image qualities and volumes (10 MB to 2 GB per stack) and portray a wide range of neuronal shapes and complexities (Fig. 13), representative of many studies. For some stacks in the BGN data set, the voxel size was unknown, and in these cases we used a default x:y:z voxel aspect ratio of 1:1:2, reflecting the typically lower resolution in the depth dimension. Also, because of the mentioned processing time constraint, 3 of the 76 image stacks could not be reconstructed by all methods (see Supplementary Fig. ?? for these and other hard cases), so the presented results are based on the remaining 73.
Finally we investigated the sensitivity of PNR with respect to two of its parameters that one might suspect to be rather critical. The first is the noise tolerance parameter τ used to prune insignificant local maxima in the seed extraction (Seed Extraction). To obtain the best possible results while keeping the computation times manageable, we considered different sets of values for this parameter, depending on the data set (Table 1). For example, in the case of the relatively small-sized OPF data set we considered values τ = 4,6,8,10,12, while for the larger NCL1A and BGN data sets and the synthetic images we examined τ = 6,8,10. The results (Supplementary Figs. ??-??) show that τ is in fact not a very sensitive parameter of the method and that the suggested default value is a suitable choice. The second parameter is the node density limit δ_{n} used to terminate the tracing (Branch Tracing). Here, too, we considered different sets of values depending on the data set, for n = 5 and n = 9. The results (Supplementary Fig. ??) show that the method is also not very sensitive to this parameter and its default value is suitable. Notice that due to the probabilistic nature of the method there is inherently some randomness in the results. But altogether we believe the results justify the conclusion that PNR is a robust method and a valuable addition to the neuron reconstruction toolbox.
Conclusions
We have presented a new fully automated probabilistic neuron reconstruction method (PNR) based on sequential Monte Carlo filtering. It traces individual neuron branches from automatically detected seed points repeatedly but statistically independently to acquire more evidence and to be more robust to noise and other artifacts. The traces are subsequently refined, merged, and put into a tree representation for further analysis. We evaluated the method on both synthetic and real neuron images and compared it to various other state-of-the-art neuron reconstruction methods (APP2, GPS, MST, PHD) using commonly used quantitative performance measures (we presented F and SSD scores). To obtain realistic synthetic data we developed a novel simulator (SWC2IMG) that can turn any given SWC file into an image stack of specified quality whose ground truth reconstruction is the input. For the evaluation on real data we used about 100 single-neuron fluorescence microscopy image stacks of widely varying quality and complexity, with corresponding manual reconstructions serving as the gold standard, from three different data sets used in the DIADEM and BigNeuron studies. The results show conclusively that the proposed method is generally favorable and also outperforms our own alternative neuron reconstruction method based on probability hypothesis density (PHD) filtering we presented recently. Nevertheless, there still remains much room for further improvement, as none of the quantitative scores were near perfect for any of the considered methods even for high SNR levels and very lenient distance thresholds. Possible directions for future work within the presented probabilistic framework would be to explore other state transition and measurement models. Alternatively, since no single method always performs best on all images of a given data set, and the results of different methods are likely complementary, another possible direction could be to combine multiple methods either during tracing or in a post-processing step. The latter approach is already being explored in the BigNeuron project. But regardless of the outcome of this effort we conclude that the method proposed in this paper may already prove to be of great use in many cases. Our software implementation of the method will be made freely available for non-commercial use upon publication.
Information Sharing Statement
The source code of the presented method is freely available for non-commercial use from https://bitbucket.org/miroslavradojevic/pnr.
Footnotes
Notes
Acknowledgements
This work was funded by the Netherlands Organization for Scientific Research (NWO grant 612.001.018 awarded to Erik Meijering).
Supplementary material
References
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