We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the mixed finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure preserving: it reproduces curl-free fields precisely and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover’s distance computation, for efficient and robust computation.
Trajectories are usually collected with physical sensors, which are prone to errors and cause outliers in the data. We aim to identify such outliers via the physical properties of the tracked entity, that is, we consider its physical possibility to visit combinations of measurements. We describe optimal algorithms to compute maximum subsequences of measurements that are consistent with (simplified) physics models. Our results are output-sensitive with respect to the number k of outliers in a trajectory of n measurements. Specifically, we describe an O ( n log n log 2 k )-time algorithm for 2D trajectories using a model with unbounded acceleration but bounded velocity, and an O(nk) -time algorithm for any model where consistency is “concatenable”: a consistent subsequence that ends where another begins together form a consistent sequence. We also consider acceleration-bounded models that are not concatenable. We show how to compute the maximum subsequence for such models in O ( n k 2 log k ) time, under appropriate realism conditions. Finally, we experimentally explore the performance of our algorithms on several large real-world sets of trajectories. Our experiments show that we are generally able to retain larger fractions of noisy trajectories than previous work and simpler greedy approaches. We also observe that the speed-bounded model may in practice approximate the acceleration-bounded model quite well, though we observed some variation between datasets.
Purpose To derive a generic approach for accurate localization and characterization of susceptibility markers in MRI, compatible with many common types of pulse sequences, sampling trajectories, and acceleration methods. Theory and Methods A susceptibility marker’s dipolar phase evolution creates 3 saddles in the phase gradient of the spatial encoding, for each sampled data point in k‐space. The signal originating from these saddles can be focused at the location of the marker to create positive contrast. The required phase shift can be calculated from the scan parameters and the marker properties, providing a marker detection algorithm generic for different scan types. The method was validated numerically and experimentally for a broad range of spherical susceptibility markers (0.3 < radius < 1.6 mm, 10 < |∆χ| < 3300 ppm), under various conditions. Results For all numerical and experimental phantoms, the average localization error was below one third of the voxel size, whereas the average error in magnetic strength quantification was 7%. The experiments included different pulse sequences (gradient echo, spin echo [SE], and free induction decay scans), sampling strategies (Cartesian, radial), and acceleration methods (echo planar imaging EPI, turbo SE). Conclusion Spherical markers can be identified from their phase saddles, enabling clear visualization, precise localization, and accurate quantification of their magnetic strength, in a wide range of clinically relevant pulse sequences and sampling strategies.
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