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Spectral Algorithms for Segmenting Neurons in their Three-dimensional Space
Ioannis
a
Koutis ,
Richard
a
a
Garcia-Lebron ,
Jose
a
Farrington-Zapata ,
Jose L.
b
Serrano-Velez ,
Eduardo
b
Rosa-Molinar
Computer Science Department, Biology Department – Biological Imaging Group,
University of Puerto Rico-Rio Piedras
b
Abstract
The adapted Random Walker method
Results
Automating segmentation of individual neurons in electron microscopic (EM) images
is a crucial step in the acquisition and analysis of connectomes. It is commonly
thought that approaches which use contextual information from distant parts of the
image to make local decisions, should be computationally infeasible. Combined with
the topological complexity of three-dimensional (3D) space, this belief has been
deterring the development of algorithms that work genuinely in 3D.
The Random Walker method introduced by Leo Grady [7] solves the following
problem. First a simple affinity graph is constructed. For the affinity construction we
use a rather simple function. A number of pixels/nodes are taken as seeds. Each
seed is assigned label. We can have an arbitrary number of labels, but for our
purpose let’s assume that there are two kinds: inside seeds lying in the interior of the
neuron in focus, and outside seeds that are pixels in the exterior of the neuron. The
rest of the pixels are unlabeled. One can imagine a particle starting at an undecided
pixel and performing a random walk on the affinity graph. If the particle at time t is in
a vertex i, the particle selects one of the neighboring vertices j, with probability
proportional to wij. One can ask the question:
The method handles easily very noisy frames as the ones shown in Figure 1
However, recent breakthrough results in spectral graph theory show that this intuition
is wrong. It is in fact possible to solve linear systems of matrices associated with the
affinity graphs derived from the images in time that essentially scales with their size.
This renders feasible a multitude of previously proposed algorithms for image
segmentation, and in particular algorithms based on the computation of fundamental
spectral properties of the graphs, which encode information valuable for
segmentation.
In this work we adapt the Random Walker method to expand a rough shape of the
neuron into a significantly more precise segmentation. We apply the methods to
customized registered 3D EM images of retrograde-filled spinal motor neurons with
in bloc heavy atom staining. Supplemented with the recently discovered linear
system solvers our algorithms make efficient use of 3D contextual information to
generate noise-insensitive neuron segmentation that delivers the surface of the
neuron as whole, rather than as a stack of 2D boundaries.
The synergy of advanced customized EM imaging techniques and recent
breakthroughs in spectral graph theory enables the development of powerful and
efficient segmentation algorithms that operate genuinely on 3D images to deliver
accurate segmentations.
Three-Dimensional (3D) Algorithms
Assuming cross-sectional images are registered into a volume, the direct
segmentation of the 3D volume has several advantages over linking individual 2D
segmentations, or `propagating’ an initial 2D segmentation through the volume.
Dendrites of the same neuron that appear disconnected or ambiguously connected in
individual 2D frames, are ultimately connected through the soma. In addition, the
expected noise artefacts and the natural fluctuations in pixel brightness have strong
adversary effect on connectivity properties in 2D but they are insignificant in a 3D
context.
What is the probability that the random walker will reach first an inside seed?
This probability clearly depends in a very involved and global way on all paths from
the pixel to all the inside and outside seeds. It is however possible to compute these
probabilities very fast using our algorithmic primitive:
The probabilities can be calculated by solving two SDD linear systems
that inherit the connections from the graph
and are of dimension equal to the number of undecided pixels.
1.
While there are unlabeled pixels
2.
Compute the above probabilities for all such pixels
3.
Assign the inside (outside) label to all pixels with probability >0.75 (<0.25)
Computing Seeds
We find seeds by performing a simple threshold cut of edges in the affinity graph,
accompanied with the computation of a connected component inside the neuron.
Outside seeds are produced by using a combination of the L1 distance of pixels
from the connected component and their intensity values. Final inside seeds are
calculated by sparsifying the connected component and keeping a percentage of the
nodes in it based on brightness. Care is taken so that dendrite pixels are not entirely
cut out but still maintain a 50% representation
Figure 4. Cross-sectional view of 3D segmentation.
Future Work
The method involves a small number of parameters and a possible way of
computing the best values for any given instance can be achieved by training a
neural network. We also plan to explore further the use of spectral primitives for the
construction of the affinity graphs, using the effective resistance between two
neighboring pixels as a more robust measure for affinity.
Fast Solvers: An Algorithmic Primitive
Despite these advantages several works insist in taking the 2D/linking approach.
Typically the reason is a combination of two factors. (i) The segmentation algorithm at
hand does not scale with the input size, and so computation is prohibitively expensive
on larger inputs. (ii) The algorithms and their implementation rely on concepts that
depend on planar geometry and don’t generalize readily to the much more complex
3D space. This is especially true for algorithms that rely on topology [1].
The method delivers in less than 10 minutes accurate segmentations that deliver the
surface directly in 3D. .
We adapt the random walker method to perform the following loop:
Fig2.The labeled and unlabeled areas. Red marks the computed boundary.
Fig1. Charging artefacts and chatter noise. Typically, 2D segmentation algorithms
can’t `break’ through the walls of noise. Such scans are However, neighbouring lessnoisy frames, or frames with non-aligning noise provide `routes of escape’.
Figure 3. The algorithm is insensitive to charging artifacts and chatter noise.
Recent breakthroughs allow now the solution of systems of linear equations Ax=b,
where A is a Symmtric Diagonally Dominant matrix (SDD), in time O(m logm),
where m is the number of the edges in the graph, or equivalently the number of
edges in the corresponding graph [3]. The condition defining an SDD matrix A is:
Fast implementations are already available [4]. The implications in algorithm design
are very wide [6]. In image processing in particular, algorithms such as ncuts,
considered widely to be “at least quadratic” [2], can now be implemented to run in
near-linear time. Also, the intuition that graph diameter is a bottleneck in the
propagation of distant information --implicitly expressed in [1] ” Iteration of the
dynamics propagates information over long distances.” -- is not correct. The solution
of a linear system depends fundamentally on the whole matrix, yet it can be found in
time that essentially scales with the input size.
References
[1] Machines that learn to segment images: a crucial technology for connectomics.
Jain, Seung, Turaga
[2] Convolutional Networks Can Learn to Generate Affinity Graphs for Image
Segmentation. Turaga, Murray, Jain, Roth, Helmstaedter, Briggman, Denk, Seung
[3] A near-mlogn solver for SDD linear systems, Koutis, Miller, Peng.
[4] Combinatorial preconditioners and multilevel solvers for problems in computer vision
and image processing, Koutis, Miller, Tolliver
[5] Spectral Graph Theory, Chung
[6] A breakthrough in algorithm design, Kroeker
[7] Random Walks for Image Segmentation, Grady
[8] Rayburst sampling, an algorithm for automated three-dimensional shape analysis
from laser scanning microscopy images, Rodriguez, Ehlenberger, Hof, Wearne
[9] Automatic 3D neuron tracing using all-path pruning, Peng, Long, Myers
Acknowledgments
I. Koutis is partially supported by NSF CCF-1018463. and UPRRP seed funds.
II. R. Garcia is supported by an NSF Bridge to the Doctorate Fellowships.
III. The biological imaging group is supported by MH-086994, NSF-1039620, and NSF0964114..