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Figure 2. Computational methods.Top and middle: Illustration of two neighboring sections (view from top) and main processing steps. Bottom: algorithmic details and references. Endpoints are illustrated in blue and green. The initial alignment computes a coarse, linear alignment from a subset of endpoints. The fine alignment moves endpoints closer in two steps: a linear alignment followed by an elastic alignment. The matching determines pairs of corresponding endpoints (indicated in different colors), which are finally connected across sections (not illustrated).
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Figure 3. Model for section boundary.Top: View from side at traced microtubule centerlines for two small regions from two neighboring sections of a C. elegans sample and a X. laevis sample. For the purpose of illustration, the sections have been aligned. Many microtubules are regularly oriented in parallel bundles in the X. laevis sample. Microtubules are less regularly oriented in the C. elegans sample. Bottom left: Part of a centerline together with the corresponding tomogram slice. Bottom right: Model for boundary between two neighboring sections. Microtubule centerlines are modeled as straight lines with a single orientation (illustrated as dashed arrows). Endpoints close to a boundary (gray area) are described by two-dimensional coordinates as if they were located in a single plane. The transformation acts on one section boundary (the 's in section 2). A matching consists of pairs of corresponding endpoints (one pair indicated as dotted line). The goal is to find a transformation and a matching such that corresponding centerlines can be connected across several sections. See main text for detailed notation.
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Figure 4. Illustration of the expectation maximization algorithm for a Gaussian mixture model.Dots represent the centroids of the Gaussians which are located at . Crosses represent the points in X. E-step: Arrows indicate values of the posterior for each point in Y. M-step: Steps towards convergence move the points closer and reduce the width of the Gaussians. Upon convergence, corresponding points should be close to each other and the Gaussian should be sharply peaked.
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Figure 5. Illustration of the expectation maximization algorithm for periodic variables.The dashed circles illustrate the unit sphere. Arrows indicate the orientation unit vectors projected onto the plane (blue: , black: ). The green circles depict values of the Fisher-Mises distribution with centers located at the arrow tips. In the M-step, a rotation around the center is computed that moves the arrow tips closer. Simultaneously, the concentration parameter is updated and the width of the Fisher-Mises distribution is reduced. Left: before M-step. Right: after M-step.
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Figure 6. Examples of matching configurations.Centerlines from two consecutive sections are depicted in blue and green. The random variables are illustrated on the left of each panel. Connections to and indicate assignments to endpoints in the neighboring section. Connections to indicate assignments to the placeholder. A and C: consistent assignments. C should be preferred because the corresponding points are closer. B and D: assignments that should be penalized.
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Figure 7. Endpoint distances. is the direct distance, which is computed as the horizontal distance between and . is the projected distance, which is computed by projecting along the line orientation at onto a plane through that is perpendicular to the orientation at .
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Figure 8. Illustration of a factor graph with message passing.Circles depict singleton factors. Squares depict pair factors. Red arrows and numbers indicate a possible message passing schedule.
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Figure 9. Endpoint positions and orientations of the evaluation samples.Endpoint positions and orientations on the facing boundaries of two consecutive unaligned sections. The different sections are indicated by color. Left: endpoint positions (view from top). Right: line orientations plotted in polar coordinates. Top: C. elegans. Middle: X. laevis. Bottom: T. brucei. Scale bars 2 m.
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Figure 10. Alignment of T. brucei.The view is from the top. Unless stated otherwise, colors indicate 6 different sections. The stacking order is best seen in the final result (bottom right) at the right side: green, cyan, purple, yellow, blue, green. The length of the microtubule array is approximately 8m. Top left: initial position of unaligned sections. Only endpoints are displayed. Top right: The result after applying the initial alignment algorithm. Endpoints in all sections are depicted in black; lines in gray. The subset of endpoints that were used for computing the transformation are highlighted in red. Some pairs of sections are reasonably aligned. Other pairs are unaligned. Bottom left: endpoints after applying our algorithm for linear alignment from orientation (Figure S1). All sections are reasonably rotated. Bottom right: result after applying our algorithm for linear alignment from position and orientation (Figure S2). Endpoints and lines are displayed, both colored by section. All sections are reasonably aligned.
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Figure 11. Stability of alignment.We rotated one section of a pair of correctly aligned sections for several samples over a range of 360° at 5° steps, as depicted by the circles, and tested whether the algorithms recovered the originally correct orientation, which is indicated by a vertical bar at the bottom of each circle. Dark gray indicates failure of the algorithm to recover the original rotation angle. Light gray indicates that the algorithm discovered the correct rotation. All algorithms found correct alignments for small rotations (light gray area in bottom half of each circle). The algorithm for linear alignment from orientation was most successful (largest light gray areas in middle row). Top: Rigid point set registration algorithm by Myronenko and Song (Figure 2 in [28]). Middle: Algorithm for linear alignment from orientation (Figure S1). Bottom: Algorithm for linear alignment from position and orientation (Figure S2).
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Figure 12. Distances before and after alignment.Histogram of the distances between corresponding endpoints for the X. laevis ground truth before and after applying the non-rigid point set registration algorithm by Myronenko and Song (Figure 4 in [28]) and our algorithm for elastic alignment.
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Figure 13. Matching performance over parameter variations.Displayed are false negatives (FN), false positives (FP), number of manual assignments, number of iterations, and number of disagreements when varying matching parameters around their maximum-likelihood estimates (indicated by solid vertical line) for the X. laevis sample: top left: by direct position distance parameter ; top right: by shift difference parameter (the dashed vertical line indicates the value that we chose instead of the maximum-likelihood estimate); bottom left: by angle difference parameter ; bottom right: by projected distance parameter .
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Figure 1. Starting point and output of our computational methods.Results are displayed for a stack of 3 00 nm thick and 5m wide sections of microtubule centerlines in the mitotic spindle of a C. elegans early embryo. Each section contains approximately 1500 microtubules, which were traced in electron tomograms. Top: Side view of unaligned centerlines before stitching. Colors indicate different sections. Bottom: Perspective view of the stitched microtubules after applying our algorithms. Color indicates length of the stitched microtubules: greenish lines are longer; bluish lines are shorter. The orange bounding boxes help getting an impression of the alignment transformation. The boxes were oriented parallel to the main coordinate axes before alignment. Some lines do not have a continuation in the next section (for example bluish lines at top right), because the area covered by the tomograms varied from section to section.
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