Point-set registration framework with Conditional ...

Point-set registration framework with Conditional Random Fields for automatic tracking of neurons in C. elegans whole-brain videos

Shivesh Chaudhary, Hang Lu Department of Chemical and Biomolecular Engineering Georgia Institute of Technology, Atlanta, GA 30332 [email protected]

Abstract Advanced microscopic techniques combined with microfluidics allow fast collection of whole brain functional recordings in C. elegans. However generating important neuroscience insights from whole-brain videos is still limited by the processing time of the videos. Significant head deformations during recordings make tracking of neurons throughout the video a challenging task, thus slowing down the process of extracting neuron activity traces from the videos. In this paper we present a framework for automatic tracking of neurons.

1

Introduction

C. elegans as a model organism offers two unique advantages that can be utlilized to investigate the relationship between complex behavior and whole-brain dynamics. First, quantitative analysis of complex behavior is possible e.g. [? ? ]. Second, functional activity of majority of neurons can be recorded with high spatiotemporal resolution [34, 22]. Compared to the fast collection rate of whole-brain recordings, processing videos to generate important neuroscience insights is slow. A major bottleneck in processing these videos is tracking neurons throughout the video length to accurately extract neuron activity traces. Large non-rigid head deformations during recording make tracking a challenging task. Errors in segmenting densely packed neurons and dim neurons generates outliers in some frames and missing neurons in others thus making tracking further difficult. In this work we address these issues and present a framework for automatic tracking of neurons.

2

Related Work

Object tracking in fluorescence microscopy images is achieved by either sequential correspondence estimation between objects in consecutive frames [1, 7, 11, 26] or global optimization [23][3, 51, 37, 17, 28, 29]. These methods fall into two categories. Methods that estimate only correspondence [1, 26] etc. and registration methods that estimate correspondence as well as optimal spatial transformation to register objects in consecutive frames e.g. [49]. Iterative estimation of correspondence in registration methods provide better results when large deviations are present in frames. Among general class of registration algorithms, soft-assign approaches [8] and gaussian mixture model (GMM) based methods [21, 31, 42] are well known. Further, several improvements have been proposed to registration algorithms both for improving correspondence estimation [10, 53, 36, 45, 52, 25] as well as improving spatial transformation[27, 35, 33, 13, 15, 6]. Sequential pairwise registration methods mentioned above are not suitable for tracking when the number of objects detected in each frame vary due to presence of outliers or missed detection. Joint registration methods which register multiple frames simultaneously [44, 43, 12] can handle such 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.

errors however their application for tracking has not been shown. Further, correspondence estimation in registration methods can be improved by using additional contextual features apart from spatial location such as neighborhood constraints [53, 14], shape context [52], color [9], texture [37] etc. Conditional Random Fields (CRF) based methods [32, 39, 48, 46, 50] that included higher order spatio-temporal features showed accuracy improvement in correspondence labelling and tracking tasks. However, their integration with registration methods has not been explored. The contribution of this work is two fold. First, we present a joint registration framework for simultaneously registering objects in all frames. This helps in tracking all possible objects detected throughout the video, thus keeping track of objects missing in few frames. Second, we integrate conditional random fields based labelling framework with joint registration to include more contextual features such as spatial relationships among objects and temporal consistency of correspondence. CRF framework of our algorithm and simultaneous tracking of objects in all frames have similarities to graphical model based global optimization methods [23, 3, 51, 37, 17, 28, 29]. However there are key differences. First, similar to MHT [4], these global methods optimize all possible tracks through all detected objects, which is more suitable for tracking randomly moving objects (e.g. pedestrians, cells). Our algorithm reduces this combinatorial complexity by utilizing coherent motion of nearby neurons in deforming C. elegans brain. This is done by specifying spatio-temporal constraints in CRF model on jointly registered point-sets. Second, tracking problem is formulated in Directed Acyclic Graph framework in these methods, which allows using combinatorial optimization techniques such as Integer Linear Programming [3, 23, 20], Min-Cost flow algorithms [37, 17, 51] and Viterbi algorithm [28, 37]. We formulate tracking problem in undirected cyclic graphs manner by including spatial neighborhood constraint and use Loopy Belief Propagation algorithm [30] for optimization. Third, these methods assume independence of tracks [37, 3, 51] whereas we account for spatial dependency of tracks in CRF framework.

3

Formulation

Let Xi = [xi1 , xi2 , . . . , xiNi ] ∈ RD×Ni be a matrix forming the the ith point-set that represents Ni neurons detected in the ith frame of the video. The set X = {Xi }N i=1 represents N frames of video in which neurons are to be tracked. We treat tracking problem as labelling problem where neurons detected in each frame are to be assigned a globally consistent label. Our algorithm consists of two steps. In the first step we build a global reference point-set and use this set to generate coarse correspondence between all point-sets. In the second step we assign a globally consistent label to each point using spatial and temporal contextual features. 3.1

Step - 1 : Joint registration

In joint registration framework [12] each point-set Xi is generated by applying a spatial transformation Φi (Gi , Wi ) : R3 → R3 to a realization of a GMM. The component centroids of the GMM are given by a global reference point-set Y = [y1 , y2 , . . . , yM ] ∈ RD×M . We assume isotropic covariances and equal membership probabilities for each GMM component. To account for outliers (misidentified neurons in a frame), we add an extra component with th th prior probabililty ω. Hence the probablity  of realization of j2  neuron in i frame is given by P kT (x ,W )−y k M 1 − i ij 2σ2i k P (xij ) = Nωi + 1−ω . Spatial transformation on xij is D exp k=1 M i (2πσi2 ) 2 PNi T parameterized as Ti (xij , Wi ) = xij + k=1 Gi (i, k)IWik . Here Gi ∈ RNi ×Ni is a gaussian kx −x k2

kernel given by Gi (p, q) = exp − ip2β 2iq and Wik is the k th row of transformation parameters matrix Wi ∈ RNi ×D for ith point-set. Gaussian kernel constrains the transformation to lie in a Reproducible Kernel Hilbert Space (RKHS) with gaussian reproducing kernel and ensures spatial smoothness of the transformation [31, 27]. We use modified Expectation Conditional Maximization algorithm [18] to learn parameters. The E-step calculates the expected value of complete data log likelihood equivalent to calculating the posterior probability that neuron j in ith frame is generated from k th spatial component. old αijk = P (k|xij )old = P M

k=1

exp − exp −

kTi (xij ,Wiold )−yk k2 2σi2

kTi (xij ,Wiold )−yk k2 2σi2

2

D

+ (2πσi2 ) 2

ω M 1−ω Ni

(1)

In the first M-step we obtain the spatial transformation parameters by maximizing the complete data log likelihood.  N,N N  i ,M 2 X X α λ old kTi (xij , Wi ) − yk k T T L=− αijk − tr(Wi Gi Wi ) + tr(Ti Mi Ti ) + c (2) 2σi2 2 2 i=1 i,j,k

where c =

PN,Ni ,M i,j,k

old αijk log

1−ω D (2πσi2 ) 2

+

PN,Ni

Ni

i,j

old αij,M +1 log

ω Ni .

The second term in equation

(2) minimizes the norm of Φi in RKHS [27], thus regularizes its smoothness. Further, the third term puts a constraint on the transfomation to preserve the local topology of the point-set [15]. Mi = (I − Li )T (I − Li ) where j th column of Li consists of weights of K nearest neighbors of xij that reconstruct xij [38]. These weights are obtained in least squares sense by minimizing PK kxij − k=1 xik Lik k2 . We solve for Wi and σi2 by setting the derivative of equation (2) to zero. (d(Pi 1)Gi + ασi2 I + λσi2 Mi Gi )Wi = PTi YT − (d(PTi 1) + λσi2 Mi )Xi T σi2 =

1 (tr(Ti d(PTi 1)TTi ) − 2tr(PTi YT Ti ) + tr(Yd(Pi 1)YT )) Npi D

(3) (4)

PN ,M old Here 1 denotes a vector of ones, d(a) is diagonal matrix formed by vector a, Npi = j,ki αijk old and Pi ∈ RM ×Ni is the matrix of posterior probabilities for ith point-set with Pi (k, j) = αijk . We repeat the E-step and the first M-step iteratively to obtain optimal transformation parameters Wi . After estimating all Wi , in the second M-step, we update the GMM centroids keeping other parameters fixed. Setting T = [T1 , T2 , . . . , TN ] and P = [P1 , P2 , . . . , PN ]. Y = TPT d(P1)−1

(5)

We intialize GMM centroids as neuron locations in a randomly selected frame. Since certain neurons may not have been detected in this frame, the number of GMM components may be fewer than the total number of neurons detected in all frames. Therefore, we update the number of components before the second M-step. We assign neurons in each frame to their maximum posterior probability component using the posterior probabilities Pi such that each component is assigned to only one neuron. Subsequently, we use kernel density estimation to estimate the distribution of unassigned neurons in all frames and initialize more components at local maxima of the estimated distribution. 3.2

Step - 2 : Labelling by Conditional Random Fields

In gaussian mixture models, labelling is achieved by assigning each neuron to its maximum posterior probability component. Such labelling uses only spatial location information. To achieve more spatially and temporally coherent labelling, we model the conditional distribution of labels of neurons given the registered point-sets as Conditional Random Field. Each neuron Ti (xij ) can take a label lij ∈ L{1, 2, . . . , M }. Therefore the joint distribution of labels is given as 1 P (l11 , . . . , liNi ) = exp Z

 N,N Xi ij

Φ(lij ) +

X s i,j,k∈Nij

X

Φ(lij , lik ) +

 Φ(lij , li+1,k )

(6)

t i,j,k∈Nij

Here Z is the partition function. We include three kinds of potentials in our model that specifiy different spatial and temporal contextual constraints. The unary potentials Φ(lij ) specifiy the likelihood that neuron j in frame i is assigned a label k ∈ L{1, . . . , M } on the basis of featural differences between j and yk . Various types of features can be used such as shape context [2], graph centrality measures [10] etc. We define Φ(lij ) on the basis of spatial affinity of registered neuron j to component k. Φ(lij = k) = exp(−λu kTi (xij ) − yk k2 )

(7)

We impose the local neighborhood structural constraint that labels assigned to neighboring neurons should preserve the distance between the labels [6, 41]. For this we include Potts like pairwise potentials Φ(lij , lik ) between each neuron j and its k spatial neighbors ∈ Nijs . Zheng et al. [53] proposed such constraint however CRF framework can generalize this to more complex constraints. Φ(lij = m, lik = n) = exp(−λs kd(Ti (xij ), Ti (xik )) − d(ym , yn )k2 ), k ∈ Nijs 3

(8)

To ensure temporal consistency of labelling, we build temporal neighbourhood graph by sequential pairwise registration of frame i to frame i + 1 and adding edges between neurons in frame i to their closest match in frame i + 1. Further we include Ising like pairwise potentials Φ(lij , li+1,k ) that favor that temporal neighbors are assigned same labels. 0

Φ(lij = m, li+1,k = n) = exp(−λt kTi (xij ) − xi+1,k k2 )1{m = n}, k ∈ Nijt (9) Finally marginal distribution of each neuron label is inferred by Loopy Belief Propagation algorithm [30] and neurons are assigned to the maximum marginal component.

4

Results

We set the following values for hyperparameters in our model - outlier ratio ω = 0.1, gaussian kernel scale β = 3, smoothness regularizer α = 3, local topology regularizer λ = 10. CRF parameters for unary and pairwise potentials were set as λu = 0.3, λs = 1, λt = 1. ω was set on the basis of average segmentation error rate in frames. λ was set high to preserve the local rigidity of transformation. This helps in maintaining the neighborhood constraint specified in CRF. λu , λs and λt were chosen on the basis of relative importance of respective constraint in labelling. Accuracy of the algorithm was assessed by comparing tracking results with a manually annotated video. Specifically, we analyzed how accurately each neuron was tracked throughout the video and number of incorrectly tracked neurons in each frame. We found that most of the neurons, 70%, were incorrectly tracked in very few frames 5% (Figure 1(a)). Also most of the frames in video, 88%, had less than 14% incorrectly tracked neurons (Figure 1(b)). Figure 1(c) shows qualitative results of labels assigned to neurons in a frame sequence. Here color encodes the label assigned to neuron.

Figure 1: a) Neuron tracking accuracy b) Frame tracking accuracy c) Qualitative tracking results

5

Conclusion

In this work we presented an algorithm for automatic tracking of neurons in C. elegans whole-brain videos. Validation on a manually annotated video showed that the algorithm provides promising accuracy. The algorithm is robust against very large deformations in consecutive frames but temporal constraint in CRF needs to be relaxed in this case. Also, large outlier ratio due to errors in segmenting densely packed neurons degrades accuracy. Further work is required for extensive validation on more videos and comparison with other methods. We speculate our algorithm to achieve better accuracy as compared to directed graphical models such as Kalman or Particle filtering [16, 19] that do sequential tracking. Further, CRF model does not assume independence of observations thus non-independent, higher order contextual features [24] can be specified. Hence our model is more general as compared to many cost optimization tracking methods e.g. nearest neighbor [40, 5], temporal association [47] methods. Our algorithm should improve the processing speed of whole-brain videos of C. elegans. Fast extraction of neuron activity traces from tracked videos will help in fully utilizing the large amount of whole-brain data being collected. 4

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