![]() ![]() For small velocities and acceleration (simulation 1), 95% of all simulated trajectories could be tracked (no. Each row in the three-dimensional plot corresponds to one simulation (for dynamic parameter settings, see Table 1). ![]() ( B) Accuracy of the object tracking module in dependence of different dynamic models. In contrast, the mean squared distances for unidirectional vesicles show an upwardly curving parabolic curve, indicating a motor protein driven motility model. The diffusion constants (D = 9.260 × 10 −11 cm 2/sec for stationary vesicles and D = 8.223 × 10 −11 cm 2/sec for vesicles in nocodazole treated cells) show a time-independent correspondence to the slope of the mean squared distance plots, indicating that those vesicles undergo Brownian motion. Note that the mean squared distances for stationary vesicles are very much similar to those for vesicles in nocodazole treated cells ( ![]() According to the classification of trajectories, a clear separation between plots for stationary (♦), bi-directional (▴), and unidirectional (■) moving vesicles is observed. The mean squared change in distances was computed as 〈Δ d 2〉 = 〈 2〉, where d( t) denotes the trajectory length at time step t, and Δ t denotes the time interval. 1 and 2 and a nocodazole treated cell with disrupted microtubules (not shown). DYNAMIC PROCESSES MOVIEA movie showing the multidimensional visualization of trajectories is published as supplemental data on the PNAS web site ( Figure 4 ( A) Time-dependent mean squared distances for the cell shown in Figs. ( D) Within the scene viewer, the user may view the time–space from different directions and may zoom and assign different colors and textures (not shown) to selected trajectories. For display reason, only 20% of the stationary trajectories are shown. Fast moving trajectories are color-encoded whereas stationary trajectories are visualized in gray. ( C) Visualization of selected trajectories within the Open Inventor Scene Viewer. ( B) After time–space interpolation, trajectories were categorized into three classes: stationary ( Top), unidirectional ( Middle), and bi-directional ( Bottom). The highlighted rings on trajectories correspond to intersections of the image sequence with interpolated trajectories. The sequence of original images is embedded into the continuous time–space where time evolves along the vertical axis. ( A) Tracking and interpolation between consecutive time steps is demonstrated for 4 of 40 sections at indicated time steps. 2 B).įigure 2 Time–space tracking of vesicles for the cell shown in Fig. The reconstruction procedure was embedded in a powerful multidimensional viewer (Open Inventor Scene Viewer, Silicon Graphics, Mountain View, CA available for almost all hardware platforms), providing a differentiated visualization of a large number of trajectories in time–space (Fig. Note that the first and second derivative correspond to velocity and acceleration, respectively, which are crucial for quantification of dynamics (see below). Furthermore, continuous derivatives exist up to the order of two. Cubic b-splines were chosen for interpolation between corresponding points because they are stable in a geometric sense: That is, they do not tend to oscillations even for a large number of sample points. By cubic b-spline interpolation between corresponding gravity centers, they are subsequently transformed into continuous trajectories (Fig. According to the correspondence map provided by the particle, tracking module discrete trajectories are formed in time–space. The dynamics of objects with constant shape over time are well described by the dynamic repositioning of their gravity centers. We developed two different approaches for continuous time–space reconstruction. Regions of interest are finally detected as regions with locally maximal intensity (Fig. Each node of the graph is associated with one region and is assigned morphological parameters such as mean intensity, shape, and size of the respective region. Based on the image partition, a region neighborhood graph is built. As a result, closed borderlines enclosing homogeneous regions are obtained. Because borderlines induce a partition of the image, the second condition implies that edge pixels need to belong to adjacent partitions with equal probability. This induces a local coordinate system such that the orthogonal axis aligns with the isophote (line of constant intensity) parallel to the borderline. For small scale structures, local orientation can be approximated by the direction of gradient. Edges were traced based on two parameters, namely local orientation and equal probability of belonging to adjacent regions. A region was considered homogenous if its intensity values could be modeled by a Gaussian distribution. To obtain closed borderlines, edges were assumed to separate two neighboring homogeneous regions. ![]()
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