 Methodology article
 Open Access
Spatial registration of neuron morphologies based on maximization of volume overlap
 Ajayrama Kumaraswamy^{1}Email authorView ORCID ID profile,
 Kazuki Kai^{2},
 Hiroyuki Ai^{2},
 Hidetoshi Ikeno^{3} and
 Thomas Wachtler^{1}
https://doi.org/10.1186/s128590182136z
© Kumaraswamy et al. 2018
 Received: 13 April 2017
 Accepted: 26 March 2018
 Published: 18 April 2018
Abstract
Background
Morphological features are widely used in the study of neuronal function and pathology. Invertebrate neurons are often structurally stereotypical, showing little variance in gross spatial features but larger variance in their fine features. Such variability can be quantified using detailed spatial analysis, which however requires the morphologies to be registered to a common frame of reference.
Results
We outline here new algorithms — RegMaxS and RegMaxSN — for coregistering pairs and groups of morphologies, respectively. RegMaxS applies a sequence of translation, rotation and scaling transformations, estimating at each step the transformation parameters that maximize spatial overlap between the volumes occupied by the morphologies. We test this algorithm with synthetic morphologies, showing that it can account for a wide range of transformation differences and is robust to noise. RegMaxSN coregisters groups of more than two morphologies by iteratively calculating an average volume and registering all morphologies to this average using RegMaxS. We test RegMaxSN using five groups of morphologies from the Droshophila melanogaster brain and identify the cases for which it outperforms existing algorithms and produce morphologies very similar to those obtained from registration to a standard brain atlas.
Conclusions
We have described and tested algorithms for coregistering pairs and groups of neuron morphologies. We have demonstrated their application to spatial comparison of stereotypic morphologies and calculation of dendritic density profiles, showing how our algorithms for registering neuron morphologies can enable new approaches in comparative morphological analyses and visualization.
Keywords
 Spatial registration
 Neuron morphology
Background
Since Ramon y Cajal’s ‘Neuron Theory’ [1], neuronal morphology has been a prominent field of study in Neuroscience. With early handdrawn illustrations, later camera lucida tracings and more recent digital reconstructions [2], scientists have investigated the structure of individual nerve cells to better understand its role in neuronal function and pathology. Using modern imaging techniques and reconstruction algorithms, labs from around the world are producing huge numbers of detailed 3D morphologies [3, 4], and databases have been developed to collect and host such data [5].
A prominent application of neuron morphology is in comparative studies aiming to quantify the intergroup and intragroup variability of neurons. Neuronal shape and structure have been known to vary widely, even across specimens of a single species, making their characterization and classification a very difficult task [6]. Although long investigated [7, 8], the general principles underlying such diverse structures have largely been elusive, with a few widely applicable ones being uncovered only in the last decade [9–12]. Many different approaches with increasingly complex methods have therefore been used in the investigation of neuronal shape and structure.
A common approach has been to statistically test the variance of whole cell scalar measures ([13, 14]) of neuronal morphologies within and between groups. Although these methods have been successful in some cases [15–17], they have proven unsuitable for quantifying finer changes in topology and morphology [15, 18].
The next finer level of quantification involves dividing each morphology into concentric disks or shells about preidentified centering points, grouping topologically or morphologically equidistant regions from different individuals and computing statistical variability of morphological and topological measures like the number of dendrites [19–21] within and across groups to characterize morphologies. For each such set of corresponding regions, statistical variability of morphological and topological measures like the number of dendrites [19–21] are used to characterize morphologies. Although this approach has been successfully used to quantify intergroup and intragroup variability in several studies of specific cell types [22–25], it has been found to be inadequate for morphologies that have similarly complex structures but differ in fine spatial distributions of morphological and topological features [15, 18]. For such cases, Mizrahi et al. [18] illustrated the use of Hausdorff Distance based features by quantifying the overall spatial dissimilarity between morphologies at different spatial scales. More recently, Kanari et al. [26] proposed a novel feature based on radial distance and topological “persistence” of dendrites and showed that a distance measure based on it could distinguish groups of complex morphologies with fine differences. A shortcoming of these approaches is that regions that are morphologically or topologically equidistant are lumped together for analysis, which can lead to dilution or cancellation of differences. Another drawback of this approach is the requirement for identification of corresponding centering points across different specimens, especially for invertebrates for which the somas are “segregated” [27] and variably located (for example, see Additional file 1 that visualizes classified groups of morphologies from Drosophila melanogaster).
For localization of intergroup and intragroup differences in morphological features, a spatial correspondence needs to be established between regions, in other words, the morphologies need to be coaligned or coregistered. Several recent studies have proposed methods for such coregistration of morphologies and used them to compare morphologies.
Fiduciary markers can be used to register the original image data to a standard brain before extracting morphologies [28, 29]. Although this approach is very effective for brain regions with an existing standard brain [30–32], construction of a new standard brain is beyond the means of individual researchers as it requires a huge concerted effort. Furthermore, even for the cases where brain atlases are available, registration of individual morphologies can be ineffective due to lack of sufficient fiduciary markers in the brain region of interest. Hence methods that coregister morphologies without requiring external information are needed.
Methods
We describe here algorithms for coregistration of morphologies based on maximizing spatial overlap and such an approach requires defining a measure of spatial dissimilarity between morphologies and describing a strategy for finding a transformation that minimizes this dissimilarity. We discuss these aspects in the following subsections.
Measures of spatial dissimilarity
Our algorithms approach spatial dissimilarity based on the overlap between volumes occupied by morphologies at different spatial scales. The following definition for volume occupied by morphologies is used.
Representing the volume of a morphology
A common way of representing a neuron’s three dimensional structure is by using the SWC format [14, 39], which represents a binary tree embedded in three dimensional space. Each point or node has, apart from its three spatial coordinates, a radius associated with it. With these features, every parentchild pair of nodes can be used to construct a frustrum, and consequently a set of connected frustra can be constructed from a tree structure which then represents the neuronal morphology. In our algorithms, to extract a volume representation of a morphology described in the SWC format, the three dimensional space containing the morphology is discretized into a set of equally sized cubic voxels (Fig. 1 top row). The voxels are positioned so that there is a voxel with its center at the origin of the space and the edge length of a voxel, which we term “voxel size”, is the most important parameter of this volume discretization. Among these voxels, those that contain at least one point of the morphology are identified and the resulting set of voxels is used to represent its volume.
Measures of spatial dissimilarity for two morphologies
where n() represents the number of elements in a set, and ∪ and ∩ represent the set union and set intersection operators, respectively. This measure essentially quantifies the spatial overlap between two morphologies normalized by their total volume.
Our algorithms use two measures of spatial dissimilarity, which we call “centric” and “noncentric” measures. The noncentric measure calculates the spatial dissimilarity between morphologies based on the values given, without applying any transformations. This measure is used when estimating translation and rotation differences between morphologies. The centric measure first translates one of the morphologies so that its centroid coincides with that of the other and calculates spatial dissimilarity using the volumes of the resulting morphologies. This measure is used when estimating scaling differences.
Measures of spatial dissimilarity for a group of morphologies
We define a measure for more than two morphologies based on voxel occupancy in the following paragraphs.
Given a group of morphologies, occupancy of a voxel is defined as the total number of morphologies of the group that have at least one point belonging to the voxel. A histogram of voxel occupancy values is calculated using all voxels with nonzero occupancy. A weighted histogram is created by multiplying each count of the histogram by its voxel occupancy. A normalized histogram is created by normalizing the weighted histogram by its sum.
It is desirable that a perfectly coregistered group of morphologies, i.e., a group with each morphology occupying the same set of voxels, has a spatial dissimilarity of zero. The normalized histogram of such a group would have a value of one at voxel occupancy equal to the size of the group and zero for all other values of voxel occupancy. Larger deviation from such a normalized histogram indicates larger spatial dissimilarity. Therefore, we define spatial dissimilarity of a group of morphologies as the distance between its normalized histogram and the normalized histogram corresponding to perfect spatial overlap, quantified by EarthMoverDistance [40].
Estimating best transformations
In our approach, morphologies are coregistered by repeatedly removing rotation, scaling or translation differences. These differences are estimated using a multiscale method based on exhaustive searches, which are described in the following paragraphs. Since the measures defined above show multiple local minima over the space of transformations, especially when working at low voxel sizes (Fig. 1), gradient based optimization techniques are not suitable.
Exhaustive search
Exhaustive search is a basic search algorithm where all candidates from the search space are sequentially generated and tested to find the solution which optimizes a certain criteria. To illustrate this with the example of estimating the rotational difference between two morphologies, exhaustive search can be formulated as sequentially generating all possible rotations, applying them to one of the morphologies, calculating spatial dissimilarity for each of them with the reference and choosing that rotation which leads to the least dissimilarity. However, the number of possible rotations is infinite. Therefore, an approximate estimate is obtained by generating a discrete set of equally spaced rotations from a plausible region of the rotation space and exhaustively searching among these rotations for the optimal rotation. This can be implemented by parametrizing rotation, sampling the plausible range of each parameter uniformly with a certain intersampleinterval, and exhaustively searching all combinations of the resulting parameters (for implementation details see Additional file 2).
Multiscale estimation
Using exhaustive search on a discretized search space imposes a tradeoff between accuracy of the resulting estimate and the computational cost associated with its calculation. To reduce this computational cost, our algorithms use the strategy of hierarchical or multiresolution matching [41, 42] which has been successfully used to speed up and reduce errors of 3D image registration methods. Starting at the largest voxel size, it runs an exhaustive search over an equally spaced discrete set of plausible parameters to find an estimate. The exhaustive search at the next lower voxel size is run over a smaller region around this estimate determined by its uncertainty (see Additional file 2 for more details). Thus estimates are progressively refined by running exhaustive searches over a sequence of discretized volumes generated using decreasing values of voxel size.
RegMaxS
Using this multiscale estimation method to determine transformation differences between morphologies, RegMaxS iteratively applies transformations to remove determined differences until no transformation reduces the spatial dissimilarity between the morphologies any further. It first translates one of the morphologies so that its center coincides with the other. It then applies a sequence of translation, rotation and scaling transforms to minimize the spatial dissimilarity between morphologies. The order in which the different transformations are applied is determined based on how the application of one transformation influences the subsequent estimation of another transformation difference.
Rotation and translation do not affect each other, i.e., if there are only rotation and translation differences between morphologies, it does not matter whether the rotation difference is removed first and then the translation difference or vice versa. However, scaling and rotation/translation affect each other, i.e., applying a scaling affects a subsequent estimation of a translation/rotation difference and vice versa. Hence, RegMaxS applies a sequence of translation/rotation transforms until no translation or rotation can reduce spatial dissimilarity further. Then it applies a scaling transform. This is followed again by a set of translation/rotation transforms which is then followed again by a scaling. This iteration of alternatively applying a set of translation/rotation and scaling is continued until none of the transforms can decrease the spatial dissimilarity between the morphologies any further. Finally, the iteration at which spatial dissimilarity was minimized is chosen as the final solution. (see Additional file 2 for actual algorithm). Note that RegMaxS does not handle reflections. Any reflections must be removed before the algorithm is applied.
RegMaxSN
RegMaxSN is an algorithm for coregistering multiple morphologies. It uses RegMaxS for coregistering pairs of morphologies and is based on “iterative averaging” [43] which has been successfully used to generate several standard brain atlases [43–45]. It is an iterative algorithm, which in each iteration uses a reference volume and registers all morphologies to it. From the resulting registered morphologies, it generates an “average volume”, which is then used as the reference in the following iteration. For the first iteration, volume occupied of one of the morphologies to be registered is chosen as the initial reference. The iteration stops when all pairwise registrations of an iteration are rejected (see “Accepting a pairwise registration” section) Finally, the iteration at which the occupancy based measure of the morphologies was minimized is chosen as the final solution (see Additional file 2 for actual algorithm).
Computing the average volume
There are several ways of generating an average volume from a group of registered morphologies. In image stack registration paradigms, where voxel values are multivalued and numerical (E.g.: for grayscale image stacks), an average of a set of images is generated by averaging the value for each voxel across the set of images. In other problems where voxel values are nonnumerical (string labels for example, as in [43]), a democratic policy is used, where the most frequently occurring value is chosen for each voxel. However, in our formulation each voxel takes one of two values, ’1’ or ’0’, indicating whether it contains at least one point of the morphology or not. Using a democratic policy would mean that the average retains only those voxels for which more morphologies have ’1’s than ’0’s. For those cases where some parts of the morphologies have not yet overlapped at the end of the first iteration, this policy would remove those parts from the average. Since the morphologies are registered to this average in the following iteration, those parts would no longer be taken into account for registration. Instead, we use a more conservative approach and assign a voxel in the average volume to be ’1’ if at least one of the morphologies being averaged has a value ’1’. In other words, the average volume of a given set of morphologies is calculated as the union of the voxel sets of all the morphologies. This ensures that each morphology is completely represented in the average and thereby contributes equally in determining the final registration.
Initial approximate registration
For the first iteration, an initial approximate registration is performed by matching centroids. For all subsequent iterations, no initial registration is applied.
Restricting total scaling
In every iteration, RegMaxSN uses RegMaxS for registering morphologies to an average volume. A parameter of RegMaxS is the range of values of scales over which RegMaxS searches to find the scale that, when applied to the test morphology, minimizes its spatial dissimilarity with the reference. However, if this range of possible scales is constant, and RegMaxSN repeatedly aligns the morphologies to the average volume of the previous iteration, it would scale the morphologies larger and larger to stretch the dimensions which show high spatial dissimilarity. If such scaling is not constrained, the morphologies would become disproportionately and unrealistically large to achieve a high similarity value. Hence, RegMaxSN constrains the total scaling that is applied to a morphology. It keeps track of the total scaling that has been already applied to a morphology at the end of each iteration and reduces the amount of scaling that can be applied to it in the next iteration. This prevents the total scaling from becoming unrealistic.
Normalizing final morphologies
As explained above, since RegMaxSN repeatedly registers morphologies to the average of the previous iteration, the final morphologies would have translation, rotation and scaling differences with the initial reference morphology, i.e., the reference morphology of the first iteration. For further analysis on these final registered morphologies, it is convenient to transform them such that they are comparable to the original reference morphology. Thus, RegMaxSN calculates the sum total of all translation, rotation and scaling transforms applied to the original reference morphology over all iterations and applies the inverse of this total transformation to all the final registered morphologies. This makes all of them comparable with the original reference morphology.
Accepting a pairwise registration
At each step, RegMaxS uses the multiscale method for determining transformation differences. In the multiscale method, the final estimate is determined at the lowest voxel size of the algorithm. Thus, RegMaxS tries to minimize spatial dissimilarity between two morphologies at this lowest voxel size. Doing so could lead to an increase in spatial dissimilarity at a higher voxel size. This is acceptable, since we want an exact or a very large overlap between the volumes of the morphologies. However, when working iteratively with a group of morphologies, the reference corresponds to an actual morphology only for the first iteration. For all other iterations, it is a conservative “average” representing the union of the volumes of several morphologies, which does not represent any single morphology. Sacrificing spatial overlap at a higher voxel size for spatial overlap at a lower voxel size can cause overfitting, in the sense that parts which do not necessarily correspond to each other would end up being randomly matched. Hence, a morphology registered to an average is accepted only if spatial dissimilarity at the highest voxel size has decreased. If the spatial dissimilarity at the highest voxel size has remained the same, then the spatial dissimilarity at the next highest voxel size is considered, and so on. When a registration is not accepted, the test morphology is itself designated as the registered morphology.
Testing the methods
To validate RegMaxS and RegMaxSN, we tested them on several groups of morphologies. We defined measures for quantifying performance and calculated them for each of the test cases. Comparing these measures, we identified the cases where the algorithms performed poorly and investigated the reason behind them. In this section, we describe the morphologies and performance measures used for testing the algorithms.
Morphologies used for testing
Synthetic Morphologies used to test RegMaxS
We first created a set of 10 noisy morphologies by adding independent zeromean Gaussian noise of standard deviations (std) 1, 3, 5,...,17, 19 μm to the points of the morphology. Next, 100 different random transformations were constructed by drawing translations from a uniform distribution over [20, 20] μm, rotations from a uniform distribution over [30, 30] degrees and scaling from a uniform distribution over [0.5, 1/0.5]. Each transformation was applied to the set of ten noisy morphologies to generate one hundred such sets. In addition, 1000 noiseless morphologies were generated by applying 1000 different random transformations constructed as above to the original noiseless morphology. To summarize, we used 2000 transformed morphologies: (1000 without noise) + (100 with noise of std 1 μm) + (100 with noise of std 3 μm) +.... + (100 with noise of std 19 μm).
Morphologies used to test RegMaxS and RegMaxSN
Neurons from Drosophila melanogaster used for testing RegMaxS and RegMaxSN
Group name  No. of morphologies  Description  NBLAST Cluster [46] 

LCInt  8  Interneuron of the fly Lobula complex  246 
ALPN  14  Neuron projecting from the antennal lobe to the mushroom body  458 
OPInt  23  Interneuron of the fly Optic lobe  209 
AA1  12  Interneuron of fly ventrolateral protocerebum  921 
AA2  9  Neuron of the fly antennal mechanosensory and motor center  803 
All the morphologies were generated from image stacks of the FlyCircuit Database [31]. The morphologies reconstructed without registering to any standard brain atlas (“nonstandard” morphologies) were obtained from NeuroMorpho.org [2]. Morphologies which were reconstructed after registering to a Drosophila standard brain [30, 46] (“standardized” morphologies) were obtained from Dr. Gregory Jefferis.
Measures for quantifying performance of RegMaxS
RegMaxS was evaluated by applying it to register a test morphology to a reference and calculating residual errors based on the Euclidean distances of corresponding point pairs between result and reference morphologies. When synthetic morphologies were used, the test morphologies were randomly transformed versions of the reference and hence a pointwise correspondence was readily available. When real morphologies were used, test and reference morphologies were from the group ‘LCInt’ and no such correspondence was available. In this case, correspondences were defined by choosing the nearest neighbor among the test SWC points for every SWC point of the reference morphology.
Measures of performance:
 1.
Performance for every test across SWC points, using
\(\left \{\left \{d_{1}^{Q_{1}}, d_{2}^{Q_{1}}, \cdots, d_{m}^{Q_{1}}\right \}, \left \{d_{1}^{Q_{2}}, d_{2}^{Q_{2}}, \cdots, d_{m}^{Q_{2}}\right \}, \cdots, \left \{d_{1}^{Q_{n}}, d_{2}^{Q_{n}}, \cdots, d_{m}^{Q_{n}}\right \}\right \}\)
 2.
Performance for every SWC point of the reference morphology across tests, using,
\(\left \{\left \{d_{1}^{Q_{1}}, d_{1}^{Q_{2}}, \cdots, d_{1}^{Q_{n}}\right \},\left \{d_{2}^{Q_{1}}, d_{2}^{Q_{2}}, \cdots, d_{2}^{Q_{n}}\right \}, \cdots, \left \{d_{m}^{Q_{1}}, d_{m}^{Q_{2}}, \cdots, d_{m}^{Q_{n}}\right \}\right \}\)
These performance measures were calculated as the percentage of tests or SWC points for which distances were significantly smaller than the smallest voxel size used. Since only distance values smaller than the smallest voxel size were relevant, we used the onetailed Wilcoxon test, also known as the Signs test with a significance level cutoff of one percent.
Measure of anisotropic scaling:
Comparing RegMaxSN with other methods

PCA: A method using Principal Component Analysis based on a similar method for image stacks [47].

PCA + RobartsICP: The PCA method above followed by AnisotropicScaled Iterative Closed Point [36].

BlastNeuron: The affine transformation step of BlastNeuron [33].

Standardized: A method using a standard brain [30].
Code for BlastNeuron and RobartsICP was obtained from the respective authors. Morphologies registered to a standard brain were provided by Dr. Gregory Jefferis. The PCA method was implemented as follows. Given a test and a reference morphologies, we assumed that they have similar dendritic density profiles and were oriented similarly in space. Based on this, the method assumes a correspondence between the first principal axes (principal axes corresponding to the largest principal factors), second principal axes and the third principal axes of the two morphologies. This method translates the test morphology so that its center coincides with that of the reference and rotates it so that their corresponding principal axes align. Scaling differences are determined based on the variances of the morphologies along the corresponding principal axes and the test morphology is appropriately scaled.
Each registration method was applied to each of the five groups of morphologies with the standardized version of one of the morphologies as the initial reference. Performance was quantified using the occupancybased measure defined above. The results of PCA, PCA + RobartsICP, RegMaxS and RegMaxSN were in the same frame of reference as the standardized morphologies allowing direct comparison. The results of BlastNeuron however were in a different frame of reference.
In addition, the above registration tests were repeated three times for each method and each group using different morphologies as initial references and performances were quantified in each case.
Computing density profiles from sets of registered morphologies for visualization
We visualized the results of PCA, BlastNeuron and RegMaxSN along with the standardized morphologies by constructing density profiles from each of them and by maximal projections of these density profiles along two orthogonal planes. These density profiles were generated using the method described in [30]. For each set of morphologies that were coregistered, a density profile was constructed discretized with a voxel size of 0.25μm×0.25μm×0.25μm. Each morphology was resampled so that the distance between any pair of connected points was at most 0.1μm. Each voxel that contained at least one point of the morphology was assigned a value of 1 and all others were assigned 0. This binary density profile was smoothed using a unity sum 3D discrete Gaussian Kernel. The standard deviation of this kernel was chosen individually for each group of morphologies. Density profiles so calculated for each morphology were averaged across morphologies to obtain a density profile for the set of morphologies.
Results
Testing RegMaxS with synthetic morphologies
Testing RegMaxS with noiseless morphologies
We first used the synthetically generated noiseless morphologies for testing RegMaxS. In each of these test registrations, the respective original morphology was always used as the reference while a transformed version of the original morphology was used as the test. The smallest voxel size used was 10 μm for all the tests. When pointwise distance statistics were calculated for each test registration across SWC points, 675 of 1000 tests (67.5%) had final distances that were significantly smaller than the smallest voxel size (n =1290, Signs Test, 1% significance level). When pointwise distance statistics were calculated for each SWC point across test registrations, 1287 of 1290 SWC points (99.76%) had final distances that were significantly smaller than the smallest voxel size (n =1000, Signs Test, 1% significance level). Thus, although RegMaxS fails to register a significant number of SWC points in a third of the test registrations, the number of points for which it consistently fails across tests is small.
Three example tests are illustrated in Fig. 2. RegMaxS failed for the test morphology “Example3”, especially in removing scaling differences. This was caused by the heavy anisotropic scaling in this morphology (scaling differences: 1.12 along X, 0.61 along Y and 1.27 along Z, MAS =0.37). We analyzed this further by separating morphologies based on their level of anisotropic scaling (see “Effect of anisotropic scaling” section below).
In these tests the morphologies used had nearly planar densities. However, RegMaxS also performed well on morphologies with 3D extent. This is demonstrated in the “Testing RegMaxS with real reconstructions” section using LCInt morphologies which have a nonplanar dendritic density profile.
Effect of anisotropic scaling
To investigate the effect of the level of anisotropic scaling on the performance of RegMaxS, we calculated statistics only for the tests with low levels of anisotropic scaling, i.e., for cases where Measure of Anisotropic Scaling (MAS) was less than 0.2. Across SWC points, 166 of 193 tests (86%) had significant numbers of final distances smaller than the smallest voxel size (n =1290, Signs Test, 1% significance level). Across test registrations, 1290 of 1290 SWC points (100%) had final distances less than smallest voxel size (n =193, Signs Tests, 1% significance level). This shows that RegMaxS performs better for cases with low levels of anisotropic scaling, i.e, for cases where the MAS is less than 0.2.
Testing RegMaxS with noisy morphologies
RegMaxS was designed to coregister morphologies so that their spatial characteristics can be compared, assuming that the morphologies have very similar structure and belong to the same stereotypic neuron group but are obtained from different specimens. Even stereotypical neurons exhibit natural biological variability in the exact location of their dendrites from individual to individual, especially for higher order dendrites. Thus, in order to properly register such morphologies, RegMaxS must be able to tolerate such variability in dendritic position. We tested this by applying RegMaxS to morphologies where noise was added to each point of the morphology.
Testing RegMaxS with real reconstructions
RegMaxS applies affine transforms for reducing spatial dissimilarity between morphologies. However, multiple morphologies of the same stereotypical neuron obtained from different specimens could show nonaffine differences as well, if the brains of the specimens show nonaffine differences. This is taken into account while constructing brain atlases that use both affine and nonaffine transforms (e.g., [43]). To test if the limitation to affine transforms is a major drawback for RegMaxS, we registered nonstandard versions of LCInt morphologies (see Additional file 1 for its 3D structure) to their corresponding standardized versions. Since a pointwise correspondence between the morphologies was not available in this case, we used distance statistics of nearest point pairs of the reference morphology and the registered morphology for quantifying algorithm performance. The algorithm performed well on all neurons, with significant number of nearest point pairs closer than the smallest voxel size (117≤n≤276, Signs test, 1% significance level). However, these tests showed slightly larger final distances (5.51 ± 4.49 μm) compared to tests using noiseless synthetic morphologies with only affine transformation differences (3.08 ± 3.35 μm). The distributions of nearest point distances also showed more outliers compared to noiseless synthetic tests because of nonrigid differences between the nonstandard and standardized morphologies.
Testing RegMaxSN with groups of morphologies
Discussion
We have presented RegMaxS and RegMaxSN, algorithms for coregistering pairs and groups of neuron morphologies, respectively, by maximizing spatial overlap. We have quantified the performance of RegMaxS using synthetic and real morphologies. We have tested RegMaxSN on different groups of morphologies with different initial references and quantified its performance for each case.
Initialization
Spatial registration is a global optimization problem usually consisting of multiple local minima. Most registration algorithms therefore initialize using an approximate solution before minimizing dissimilarity. Several different strategies have been developed for initialization of registration algorithms [48]. However, initialization is required only when the objects being registered are expected to have large transformation differences. Neuron morphologies of the same type obtained from different individuals do no usually have large transformation differences other than translations caused by arbitrary choice of origin. Hence RegMaxS uses centroid alignment for initialization. Nonetheless, RegMaxS can be easily modified to include an appropriate initialization if an application demands it.
RegMaxS vs RegMaxSN
Compared to RegMaxS, RegMaxSN has mainly two additional components in its procedure — iterative registration and final normalization. While RegMaxS registers all morphologies once to the initial reference, RegMaxSN applies multiple iterations of such registrations, calculating a new reference in each iteration. This iterative strategy reduces the effect of the choice of initial reference on algorithm performance. In our tests, RegMaxSN performed better than RegMaxS for most cases, and showed less variability across different initial references compared to RegMaxS (see Figs. A31 and A35 of Additional file 3), indicating better suitability for these cases.
For ALPN, OPInt and AA1 morphologies, the performance of RegMaxSN was nearly the same as that of RegMaxS. In these cases, RegMaxSN chose the morphologies at the end of its first iteration as the solution, i.e., the same solution as RegMaxS. However, the solution morphologies for RegMaxSN were additionally normalized so that they were comparable to the initial reference and this caused the observed reduction in performance of RegMaxSN compared to RegMaxS in some of these cases. The normalization was applied mainly for the purpose of visualization and comparison with other methods, and can therefore be excluded when analyzing single groups of morphologies.
Computational cost
RegMaxS applies a sequence of transformations for maximizing spatial overlap between two morphologies. It estimates transformation differences at each step using a measure of spatial overlap based on the set of voxels occupied by each morphology. However, the set of voxels occupied by a morphology can change with every rotation or scaling. This makes it hard to predict the computational cost of estimating transformation differences at each step and thus to estimate the total computational cost of RegMaxS. Furthermore, RegMaxS and RegMaxSN are both iterative algorithms which stop only when spatial overlap between morphologies cannot be improved further. This further complicates the prediction of total number of iterations and total computational cost.
Comparison of average runtimes per morphology for different registration algorithms
Average runtimes per morphology (s)  

Method  LCInt  ALPN  OPInt  AA1  AA2 
PCA  0.07  0.08  0.45  0.59  0.25 
PCA + RobartsICP  189.63  318.18  287.02  68.07  39.03 
BlastNeuron  19.48  319.32  N.A.  115.95  185.78 
RegMaxS  98.44  239.20  474.88  90.45  46.89 
RegMaxSN  1141.81  605.82  2796.96  883.670  633.44 
Choice of voxel sizes
The most important parameters of RegMaxS and RegMaxSN are the set of voxel sizes over which transformation difference estimates are refined during coregistration of morphologies. The largest and the smallest voxel sizes define the coarsest and the finest spatial scales, respectively, at which the algorithms register morphologies. The algorithms consider a voxel to be occupied by a morphology if it contains one or more of its SWC points and align morphologies by applying transformations to match the sets of occupied voxels. Thus morphological features at scales finer than that defined by the smallest voxel size are ignored by the algorithms. Therefore, a good choice for the smallest voxel size is the spatial scale below which morphological features are not expected to match.
In our preliminary tests involving morphologies of different sizes and dendritic densities, we found a smallest voxel size of 10μm to be a good compromise and therefore used it for evaluating algorithm performances. To investigate the effect of reducing the value of the smallest voxel size, we repeated the tests by setting the value of smallest voxel size to 5μm. The results are summarized in Additional file 4. For pairwise coregistration of test morphologies that were larger in size and that had fewer features at scales smaller than 10μm than other test morphologies, the performance of RegMaxS reduced from 67.5% at 10μm to 32.2% at 5μm. On the other hand, for pairwise coregistration of test morphologies that were smaller in size and had more features at scales smaller than 10μm, performance of RegMaxS showed only a minor improvement. Furthermore, performance of RegMaxS and RegMaxSN in coregistration of groups of morphologies did not show any substantial changes when smallest voxel size was changed from 10μm to 5μm (Additional file 4, Fig. A41). Thus, the value of smallest voxel size can influence the performance of our algorithms depending on the size and the sparsity of structural features of morphologies being registered, and should be chosen accordingly.
Applicability
RegMaxS repeatedly applies a set of rotation/translations followed by a scaling to maximize spatial overlap between morphologies. Scales are estimated after aligning centroids of morphologies. In other words, RegMaxS seeks a solution of close centroid alignment. Therefore RegMaxS and consequently RegMaxSN are best applicable to morphologies that are complete and have similarly situated centroids. Their application to partial morphologies or largely incomplete reconstructions is not straightforward and requires caution and consideration. For more efficient handling of such cases, the algorithms could be modified so that they do not depend heavily on centroid alignment.
RegMaxSN was outperformed by PCA + RobartsICP and BlastNeuron for one out of five of our test groups of morphologies, AA2. Importantly, this was not due to poor performance of RegMaxSN, but due to untypically good performance of BlastNeuron and PCA+RobartsICP. A reason for this could lie in the unusually high structural stereotypy of AA2 morphologies, which is also reflected by lower values of occupancybased dissimilarity compared to other groups (Fig. 5, also see Fig. A35 of Additional file 3). This high structural stereotypy indicates the existence of a solution with very close pointtopoint alignment, and hence BlastNeuron and PCA + RobartsICP, which are based on pointwise distance statistics, performed better. Under most realistic conditions, however, neuron morphologies will have a nonnegligible biological variability in their fine spatial features, and therefore we would expect RegMaxSN to perform better than the other methods considered here, as was the case for the other four test groups. However, since our sample sizes were small (n =4) we could not establish statistical significance for the differences in performance.
Calculating dendritic density profiles using RegMaxSN
Applying RegMaxSN to three groups of stereotypic neuron morphologies from the Droshophila melanogaster brain, we have shown that RegMaxSN can coregister groups of neuron morphologies. Without the need for an external reference like a standard brain atlas, the registration results were very similar to morphologies registered conventionally, using such a reference. Dendritic density profiles can be calculated from groups of registered morphologies by spatial averaging (see “Methods” section). Thus RegMaxSN can be used to calculate dendritic density from profiles of stereotypic neurons (Fig. 6). Such density profiles are useful in analyzing spatial variances in different subregions of neurons and can provide insights about the brain regions surrounding neurons [11]. Furthermore, density profiles so calculated could be used in generative models of neuron morphology [10, 49, 50]. Such models usually assume simple density profiles like a uniform density over the region of arborization. The availability of better spatial density profile estimates can improve such existing models and also enable the development of new models.
Possible improvements
RegMaxS applies a sequence of translation, rotation and scaling transformations to maximize the spatial overlap between morphologies. We tested RegMaxS with synthetic morphologies that had random translation, rotation and scaling differences and demonstrated its ability to revert these transformations. Other affine differences like shear would be expected to be compensated approximately by combinations of rotation and anisotropic scaling transformations. However, specifically including shear in the sequence of transformations applied could speed up the registration process and possibly result in better performance.
Topological features play an important role in determining neuronal function [51, 52] and hence are indispensable in the study of neuron morphology. Some recent studies [26, 33] have illustrated the effectiveness of the combined use of spatial and topological features for characterization and classification of morphology. Since RegMaxSN can provide better spatial registration of morphologies than existing methods, it could be used as preprocessing to remove spatial differences for algorithms that subsequently estimate topological differences. Further, incorporating topological features into its formulation could lead to even more powerful methods for analyzing neuron morphologies.
Conclusion
We have addressed the problem of coregistering neuron morphologies, which is a crucial requirement for visualization and spatial analysis of stereotypical neurons, by formulating algorithms based on maximizing spatial overlap. Our tests using synthetic and real groups of morphologies have indicated that our algorithms can be used for registering stereotypic neuron morphologies that show considerable spatial variability in their fine structures as long as they are similarly scaled along different axes. The dendritic densities of stereotypic neurons calculated using our algorithms were very similar to those produced using a standard brain, demonstrating the potential of our algorithms in detailed spatial comparison of neuron morphologies.
Declarations
Acknowledgements
The authors would like to thank Gregory Jefferis for providing standardized neuron morphologies extracted from the Flycircuit database before they were published and also for insightful discussions. The authors would like to thank Philipp Rautenberg, Christian Kellner, Christian Garbers, Andreas Herz and Dinu Patirniche for constructive comments and fruitful discussions.
The authors would like to also thank the NeuroMorpho.org team for hosting neuronal morphologies and the FlyCircuit team for making the large collection of single neuron image stacks and their reconstructions available.
Funding
This study was supported by German Federal Ministry of Education and Research (BMBF) (grants 01GQ1116 and 01GQ1302) and the Japan Science and Technology Agency (JST) through the "German  Japanese Collaborations in Computational Neuroscience.
Availability of data materials
The morphologies reconstructed without registering to any standard brain atlas (“nonstandard” morphologies) were obtained from NeuroMorpho.org (http://dx.doi.org/10.1038/nrn1885). The corresponding registered (“standardized”) morphologies were part of NBLAST (http://dx.doi.org/10.1016/j.neuron.2016.06.012) and were obtained from Dr. Gregory Jefferis (jefferis@mrclmb.cam.ac.uk). The names of all the neurons used in this study are provided below.
Visual Neuron From Blow Fly: HSNfluoro01;
LCInt: Gad1F000062, ChaF000012, ChaF300331, Gad1F600000, ChaF000018, ChaF300051, ChaF400051, ChaF200000;
ALPN: VGlutF700500, VGlutF700567, VGlutF500471, ChaF000353, VGlutF600253, VGlutF400434, VGlutF600379, VGlutF700558, VGlutF500183, VGlutF300628, VGlutF500085, VGlutF500031, VGlutF500852, VGlutF600366;
OPInt: TrhF000047, TrhM000143, TrhF000092, TrhF700009, TrhM000013, TrhM000146, TrhM100009, TrhF000019, TrhM000081, TrhM900003, TrhF200035, TrhF200015, TrhM000040, TrhM600023, TrhM100048, TrhM700019, TrhF100009, TrhM400000, TrhM000067, TrhM000114, TrhM100018, TrhM000141, TrhM900019, TrhM800002;
AA1: VGlutF300181.CNG, VGlutF400545.CNG, VGlutF500778.CNG, VGlutF300196.CNG, VGlutF300288.CNG, VGlutF600290.CNG, VGlutF600499.CNG, VGlutF400665.CNG, VGlutF300142.CNG, VGlutF500147.CNG, VGlutF600181.CNG, VGlutF700190.CNG
AA2: TrhF700063.CNG, TrhF500050.CNG, TrhF500106.CNG, TrhM500051.CNG, TrhF600071.CNG, TrhF500093.CNG, TrhF500148.CNG, TrhF500154.CNG, TrhF700018.CNG
Programs for applying RegMaxS and RegMaxSN to SWC files are available at https://github.com/wachtlerlab/RegMaxS. doi: https://doi.org/10.12751/gnode.feee47.
Authors’ contributions
All the authors contributed to the conceptualization of the methods. AK implemented the method, carried out the analysis. TW guided the study. HI and TW contributed to improvement of the methods. AK and TW prepared the original draft. HA, HI and TW reviewed and edited the manuscript. All authors approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Images used in Additional file 1 have been reproduced from http://flybrain.mrclmb.cam.ac.uk with written permission from Dr. Gregory Jefferis.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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