Improved Marked Point Process Priors for Single Neurite Tracing Sreetama Basu∗ , Wei Tsang Ooi∗ and Daniel Racoceanu† ,‡ ∗ National

University of Singapore, Singapore Pierre and Marie Curie, France ‡ CNRS, France

† University

Abstract—Recent advances in neuroimaging has produced a spurt for automatic neuronal reconstruction algorithms for large scale data. A stochastic marked point process framework for unsupervised, automatic reconstruction of single neurons has been proposed. In this paper, we introduce improved priors modeling arborization patterns encountered in neurons for efficient detection of bifurcation junctions, terminal nodes, and intermediate points on neurite branches. These priors also enforce constraints for preserving the connectedness of the neuronal tree components in spite of imperfect labeling causing intensity inhomogeneity and discontinuities in branches. To demonstrate the effectiveness of the proposed priors, we performed neurite tracing on 3D light microscopy images of Olfactory Projection Fibre axons from the DIADEM data set and obtained good scores. We also analyzed the errors and their sources in the neurite tracing pipeline, in the hope of better integration of neuroimaging and automated tracing.

I.

I NTRODUCTION

Comprehending the structure and connections of the neurons is key to the study of brain development and functioning. The rapidly evolving field of neuroimaging has enables us to capture such neuronal morphology. The huge volume of rich and heterogeneous data generated, however, makes manual analysis of such data tedious, subjective and prohibitively expensive. Thus, fast, automatic, and computationally inexpensive algorithms are desired to analyze such data. Extraction and meaningful interpretation of neuronal morphology is a difficult task. These inherently 3D structures are difficult to capture faithfully as digital data. The resolution limit of microscopy technique and the slicing thickness of samples introduces a 2D projection effect occluding important nodal positions and connections. Imaging artifacts, such as structured noise, lighting gradation, cluttered backgrounds, and discontinuous beaded appearance of neurites due to uneven staining impose further challenges for automatic analysis. The scope of this work is limited to extraction of single neurons and its branching patterns. The case of multiple intertwined neurons presenting ambiguous crossover and bifurcations are resolved currently by applying varied arbitrary heuristics or resorting to manual post-editing, even by the most sophisticated automated reconstruction algorithms [1]. We believe that this issue is better tackled at the image acquisition step, by differently labeling separate neurons, as is successfully done by the Brainbow techniques. Existing neurite tracing algorithms can be broadly classified as global and local methods. Global methods such as skeletonization are resource intensive in both CPU and memory and are unattractive for big data sets. Local tracing

methods, on the other hand, are likely to result in disconnected components and large topological errors. Moreover, local explorative methods, such as active contours for connecting seed points, require careful guided initialization close to the optima for deterministic gradient descent. Hence, stochastic data exploration strategies are gaining popularity over these traditional deterministic image processing methods [2]. In this paper, we adopt the efficient stochastic framework for unsupervised neuronal network extraction proposed in [3] for analyzing neuron morphology from confocal microscopy data. The framework requires minimum parameterization, no user interaction or seed points, and adapts an object-based method embedded in a Marked Point Process (MPP). By using an energy function adapted for neuronal morphology, it can model the positional and connectivity information of neuronal trees. From the resulting optimal configuration of objects, we obtain information such as the centreline, local width, and orientation of fibres. A study of the sensitivity and robustness of the model parameters has lead us to propose improved priors, designed to reflect typical arborization patterns in single neuron data, in this paper. These priors model specific scenarios of single neurite branching, as opposed to a network of neurons with crossover of fibres from different neuronal sources. This specialization enables accurate identification of branch bifurcation and terminals and gives us uniformly spaced intermediate nodes anchored to lengths of neuronal fibres. We demonstrate the robustness of our method by generating a connected Minimum Spanning Tree (MST) representation of neuronal fibres. Such digital reconstruction of neuronal morphology from the microscopy data can be represented in the standard SWC format that is prevalent for storage, sharing, and analsys in the neuroimaging community. II.

P ROPOSED M ETHOD

The Point Process models were introduced in [4] to exploit random fields whose realizations are configurations of random points describing a spatial distribution of data. Under this view, images are considered as configurations of a Gibbs field, with the implicit assumption that, for a given problem, there exists a Gibbs field such that its ground states represent regularized solutions of the problem. A. Marked Point Process : Notations and Definitions We model our neuronal data as a set of spheres whose positions and parameters are realizations of a Marked Point Process (MPP) Y . Now, Y is also a random variable defined on K × M , where K is a bounded, connected subset of V3 , the image domain, and M is the mark space. We adopt

spheres as objects ωi = (xi , ri ), defined by their centres and radii sampled independently from image domain xi = [x, y, z]T , xi ∈ V3 and the radius range ri ∈ [rmin , rmax ]. The configuration space of the objects is given by Ω = ∪∞ n=0 Ωn where Ω0 is the empty set, each Ωn , n ∈ N is the set of unordered sets (configurations) containing n objects and γn ∈ Ωn , γn = {ω1 , . . . , ωn }. By the Gibbs energy formulation of process density, given a real, bounded below function U (γ) in Ω, the distribution µβ dµ in terms of the density p(γ) = dλβ (γ) w.r.t. Lebesgue-Poisson measure λ on Ω is defined as: z |γ| exp[−βU (γ)], where p(γ) = Zβ Z Zβ = z |γ| exp [−βU (γ)]dλ(γ),

(1)

and γ represents the configuration of objects, parameters z > 0, β > 0. U (γ) is the total global energy is defined as: X

X

Ud (ωi ) +

ωi γ

Ui (ωi , ωj ) +

ωi ,ωj ∈γ; ωi ∼ωj

X

Uc (ωi ),

(3)

ωi ∈γ

Ud represents the data energy, Ui and Uc are prior energy terms. In general, using Gibbs field approach we can control crucial information on extracting objects through the energy function. We seek to minimize the global energy U (γ) by finding γˆ with the number and positions of all n objects in the required configuration: γˆ = arg max p(γ) = arg min U (γ). γ

γ

Our aim is to abstract out the neuronal morphology from the microscopy data into a mathematical model to facilitate further analysis. We sample special configurations of objects and fit them to points of maximum medialness measure on the image volume. In the following section we describe the role each of the energy components for neurite tracing in detail. 1) Fit to data: Our data energy response is based on the tubularity filter proposed in [5]. The medialness measure M (ωi ) is obtained by taking an integral of the image gradient at a scale σG proportional to the object radii along the circumference of the cut of the spherical object on the normal plane by a rotating phasor Vθ = cos(θ)V1 + sin(θ)V2 Z

2π

∇I (σG ) (xi + ri Vθ )dθ|.

(5)

θ=0

An adaptive thresholding of the medialness response on the gradient response at the tube’s center Mc (ωi ) = |∇I (σH ) (xi )| enables to discriminate between “good” and “bad” objects. The data energy term is then defined as follows: Ud (ωi ) =

Ui (ωi , ωj ) =

( −(M (ωi ) − Mc (ωi )), if M (ωi ) > Mc (ωi ) 0, otherwise.

(6)

A good balance between the data energy Ud (γ) and the prior energy Up (γ) is necessary to get good results. Generally, this can be achieved by learning a weighting factor wd such that U (γ) = wd Ud (γ) + Up (γ). Learning wd , however, is challenging, as it depends on many factors (imaging modality, quality of data, properties of the neuron etc.) Further, wd should adapt to local sections of the images, rather than be a global parameter. Due to variations in density of labelled

 U+ , if d < dr U− , if dr ≤ d ≤ da  0, if d > d . a

(7)

Here, d is the Euclidean distance between the centers of the spheres; dr and da (dr < da ) are respectively the repulsive and attractive distances, dr , da are multiples of ri + rj . By varying dr and da , density of spheres along the neuronal branches can be controlled. 3) Spatial Configurations: The second prior is a multiobject interaction potential, incorporating constraints on the local sub-configurations depending on the number of immediate neighbors of an object, k(ωi ) = |ωj ∈ γ : dr < d(ωi , ωj ) < da |.  Uc (ωi ) =

(4)

B. Energy Model for 3D neuronal branches

π M (ωi ) = | 2

2) Connectedness: A pair-wise interaction potential for objects in each other’s zone of influence imposes continuity constraints on the configuration of objects modeling neuronal fibres. It favors objects in poorly stained, fragmented sections. U+ is a repulsive potential to penalize overlapping of objects, and U− is an attractive potential to favor objects in touching distances of each other. 

(2)

γ∈Ω

U (γ) =

neurons, varying depth of the image stack, and noise in the XY plane, even learning wd locally is difficult. This motivates us to improve our priors to obtain uniformly good results in all sections of the images.

 ∞, if k(ωi ) = 0 −E1 , if k(ωi ) = 1, 3  ∞, if k(ω ) > 3 i

(8)

The association of favorable negative energy potentials E1 with particular local sub-configurations (i) encourages survival of objects corresponding to critical nodes such as bifurcations and terminals, and (ii) discourages isolated objects, which likely corresponds to cell nuclei or other background structures. Our analysis of the gold standard reconstruction indicates there are no multi-furcations in the data considered. Thus, this method weeds out unusual local sub-configurations from the candidate global configurations. Using such local sub-configurations, we can generate further descriptors for the extracted neurons (average branch curvature, branching order, length etc.) besides identifying bifurcations, terminals, and points of high inflection along branches. C. Optimization The main idea of our proposed approach is to sample special configurations consisting of spherical objects and fit them to the microscopy data stacks to voxels of maximum neuriteness measures. These configurations are projected onto the image volume. The configurations are optimized by measuring the similarity between the projected model of the configuration and the neuronal data. A Gibbs energy is defined on the configuration space. The optimum global energy is defined over the space of union of all possible configurations, considering an unknown a-priori number of objects. As exhaustive search of the solution space is impractical, the approach uses Multiple Birth and Death (MBAD) dynamics [6] to find the Maximum A Posteriori (MAP) estimation (Eq.4), greatly reducing computational cost and speeding up convergence. The method optimizes the object configuration iteratively, where multiple random objects are proposed and removed independently and simultaneously in each iteration depending on the relative energy change due to their introduction. It samples from the probability distribution µβ using a Markov chain of the discrete-time MBD dynamics defined on Ω and apply a

Fig. 1.

OP5

Fig. 2.

OP8

Fig. 3.

OP1

The top row shows the MPP configuration fitted to the neuronal data on Maximum Intensity Projection of the stacks. The green nodes represent bifurcation, blue nodes terminals and the intermediate nodes on branches are red. Note that due to the projection effect some of the nodes appear mis-located. Generally, higher rates of mis-detection occurs in densely branches sections. The bottom row shows the reconstruction from MPP configuration in red. The Gold Standard is projected in cyan, with offset for ease of visualization. Fig. 4.

Simulated Annealing scheme. At every iteration,0 a transition 00 is0 considered 00from current configuration γ to γ ∪ γ where γ ⊂ γ and γ is any new configuration. The corresponding transition probability is given by: 0

00

P (γ → γ ∪ γ ) 00 Y ∼ (zδ)|γ | ωi ∈γγ

0

Y αβ (ωi , γ)δ 1 , 1 + αβ (ωi , γ)δ 1 + α β (ωi , γ)δ 0

(9)

ωi ∈γ

where αβ (ωi , γ) = exp(−β(U (γ \ ωi ) − U (γ))). The convergence properties of the Markov Chain to the global minimum under a decreasing scheme of parameters δ and β1 are proved in [6]. In this way, the iterative process finds a configuration γˆ minimizing the global energy Eq. 3. D. Parameters The parameters of the model can be categorized in three classes. Firstly, the marks of the objects, the radius range [rmin , rmax ], is set as [1,10] for our experiments (derived from domain information such as resolution of imaging and approximate dimension range of neurite fibres). Secondly, the parameters of the energy model U+ = 5, U− = −5, E1 = 2 are set so as to maintain a balance between the data term and prior terms. The third set are the parameters of the sampling dynamics — the birth intensity δ and the inverse temperature β. The model is sensitive to the initialization of the birth intensity, which is generally set as an over estimation of the expected number of objects. Let f (I) be the number of objects

expected from the radiometric term. This can be approximated from the number of voxels (P ) in the foreground intensity range by binary clustering algorithms. Using mean r from [rmin , rmax ] and assuming uniform distribution of objects — P f (I) = 4πr 3 . The advantage of the adopted MBD dynamics 3 is that the birth of objects is independent of temperature, allowing new objects to be added to the evolving configuration, even when the system is cool. Thus, β is not critical and is heuristically initialized such that the system has enough time to cool down slowly, allowing sufficient iterations for the evolution of the configuration. III.

E XPERIMENTS

We evaluated our proposed model by application to DIADEM Olfactory Projections Fibres acquired by 2-channel confocal microscopy [7]. While Markov Chain Monte Carlo methods are notorious for their slow convergence, the novel MBD sampling strategy enables our MATLAB implementation to converge under 5 mins on a machine with Intel Core i7 processor, 3.4 GHz with 8GB RAM. Note, we run our experiments with the data term pre-computed. It is observed with live computation of data term for the same initialization the time taken is roughly 2 hrs to converge for these datasets due to the way the 3D volume of images slices is handled in MATLAB. A. Reconstruction To further demonstrate the effectiveness of proposed model, we reconstruct the neuron trees as a Minimum Spanning Tree (MST). We extract the branches of the neurons by

The histogram shows euclidean distance of the extracted points set (P) using our proposed model from gold standard manually delineated centrelines (G). Fig. 5.

δ0 No. of objects in final configuration DIADEM Score

OP1 1200 266

OP4 1600 358

OP5 650 159

OP8 750 175

0.932

0.925

0.808

0.847

TABLE I. Summary of Reconstruction : The table shows the initialization of the birth intensity δ0 for each data set, the number of objects in the final configuration and the DIADEM metric score for the corresponding reconstruction.

fitting a configuration of spheres to the branch centrelines and simultaneously identifies the junctions and terminals. We build an adjacency matrix defining connections between nodes from the proposed priors according to neighboring objects criteria. A connected MST representation of the detected set of objects is derived from the adjacency matrix for visualization of the neuron trees. Fig. 4 illustrates the algorithmic steps in the reconstruction of the data-sets. Finally, a depth first traversal of the obtained MST, starting from the root, as provided with each data set, gives us the directed tree representation of the voxels representing the centreline of the neuronal structure. B. Evaluation and Discussion We score the performance of our reconstruction method with the DIADEM metric [8]. It gives a F-score for the positions and connections of the reconstructed neuron by comparing against a manual gold standard reconstruction. We obtained lower scores for the OP5 and OP8 data sets mainly because we trace the entire structure while the gold standard contains only a part of it. The DIADEM metric penalizes topological errors closer to the root more than errors further down the tree hierarchy. It ignores short branches of less than 6 pixel length in the scoring. Due to various inconsistencies, instead of using this metric, the community has proposed favoring visual inspection of results. We also compare the deviation of our extracted centreline against the Gold Standard in 5. We obtained sub-voxel deviation of reconstructed centreline from the Gold Standard for 50% of MPP nodes for OP1 and OP4. We observed better performance on data sets such as OP1 (1496 points) and OP4 (1383 points) in spite of them showing more complex structure due to denser reconstructed by the experts compared to other data sets OP5 (135 points) and OP8 (152 points). Generally, the Z-axis has much larger contribution to localization error compared to XY plane due to differential

resolution. Regions of the data with acute angles between branches and the main branch is more error prone. Due to a lack of image data consideration in neighbor identification, these regions prove to be tricky for our algorithm. The other error prone branches are ones oriented near perpendicular to the imaging plane (along our “Z” axis). There is a stark shortening effect of their length and they appear as blobs leading to misinterpretation. The primary source of such an error is the slicing thickness at the image acquisition stage. Such errors introduced during the image acquisition stage accumulates and culminates in large topological distortion in the final stages. Further, there is generally much noise in the top slices of confocal microscopy data. Such errors are better eliminated in preprocessing stages, ideally by imaging the volume from both end and compositing them together to get a relatively noise free data. IV.

C ONCLUSION

To conclude, we present an MPP model with specialized priors for accurate detection of bifurcation, terminals, and intermediate nodes and subsequent reconstruction of neuronal trees as a connected minimum spanning tree. The proposed framework combines the merits of local and global neurite tracing methods by optimizing a global energy function and simultaneously adapting the object configurations locally to better fit the neuronal data. The obtained results demonstrate its reliability and robustness for fully automated reconstruction of single neuron morphology. R EFERENCES [1]

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