Input data: Without loss of generality, we assume that the input data is a piecewise linear (PL) scalar field defined on a PL -manifold with . Values are given at the vertices of and linearly interpolated on higher dimension simplices.
Critical points: Given an isovalue , the sub-level set of , noted , is the pre-image of the open interval onto through : . The sur-level set is symmetrically given by . These two objects serve as segmentation tools in many analysis tasks .
The points where the topology of differs from that of are the critical points of and their values are called critical values
. Critical points can be classified with theirindex , which is 0 for minima, 1 for 1-saddles, for -saddles and for maxima, with the dimension of .
Persistence diagrams: The distribution of critical points of can visually be represented by a topological abstraction called the persistence diagram [24, 15] (Fig. 1). By applying the Elder Rule , critical points can be arranged in a set of pairs, such that each critical point appears in only one pair with and . More precisely, the Elder Rule states that as the value increases, if two topological features of , for instance two connected components, meet at a given saddle of , the youngest of the two (the one with the highest minimal value, ) dies at the advantage of the oldest (the one with the lowest minimal value). Critical points and then form a persistence pair.
The persistence diagram embeds each pair in the plane such that its horizontal coordinate equals , and the vertical coordinate of both and is and , corresponding respectively to the birth and death of the pair. The height of the pair is called the persistence and denotes the life-span of the topological feature created at and destroyed at . In three dimensions, the persistence of the pairs linking critical points of index , and denotes the life-span of connected components, voids and non-collapsible cycles of .
In practice, persistence diagrams serve as an important visual representation of the distribution of critical points in a scalar data-set. Small oscillations in the data due to noise are typically represented by critical point pairs with low persistence, in the vicinity of the diagonal. In contrast, topological features that are the most prominent in the data are associated with large vertical bars (Fig. 1). In many applications, persistence diagrams help users as a visual guide to interactively tune simplification thresholds in multi-scale data segmentation tasks based on the Reeb graph [58, 12, 53, 85, 30, 83] or the Morse-Smale complex [32, 33, 65].
Wasserstein distance: Metrics have been defined to evaluate the distance between two scalar fields . The -norm , is a classical example. In the context of topological data analysis, multiple metrics [15, 14] have been introduced in order to compare persistence diagrams. In our context, such metrics are key to identifying zones in the data which are similar to one another. Persistence diagrams can be associated with a pointwise distance, noted inspired by the -norm. Given persistence pairs and , is given by Eq. 1:
where is the set of all possible bijections mapping each critical point of to a critical point of the same index in or to the diagonal, noted – which corresponds to the removal of the corresponding feature from the assignment, with a cost .
1.2 Assignment problem
The assignment problem is the problem of choosing an optimal assignment of workers to jobs , assuming numerical ratings are given for each worker’s performance on each job.
Given ratings are summed up in a cost matrix , finding an optimal assignment means choosing a set of independent entries of the matrix so that the sum of these elements is optimal. Independent means than no two such elements should belong to the same row or column (i.e. no two workers should be assigned to the same job and no worker should be given more than one job). In other words, one must find a map of workers and jobs for which the sum is optimal. There are possible assignments, of which several may be optimal, so that an exhaustive search is impractical as gets large.
Similarly, the unbalanced assignment problem is the problem of finding an optimal assignment of workers to jobs, where some jobs or workers might be left unassigned. This is the case of assignments between sets of persistence pairs; where costs are defined for leaving specific pairs unassigned.
If a number is added or subtracted from all the entries of any one row or column of a cost matrix, then an optimal assignment for the resulting cost matrix is also an optimal assignment for the original cost matrix.
This means that the cost matrix can be replaced with where (resp. ) is an arbitrary number which is fixed for the row (resp. the column).
If is a matrix and is the maximum number of independent zeros of (i.e. number of entries valued at 0), then there are lines (row or columns) which contain all the zeros of .
This allows to determine whether an optimal assignment has been found and thus constitutes the stop criterion.
The algorithm iteratively performs additions and subtractions on lines and columns of the cost matrix, in a way that globally decreases the matrix cost, until the optimal assignment has been found, that is, until the matrix contains a set of independent zeros.
In the remainder we consider the unbalanced Kuhn-Munkres algorithm [51, 7], an improvement over Kuhn’s original version which follows the same principles, with an enhanced strategy for finding independent elements. The goal is always to reduce the cost matrix and find a maximal set of independent zeros. These independent zeros are marked with a star: they are candidates for the optimal assignment. Zeros which are candidates for being swapped with a starred zero are marked with a prime. Throughout the algorithm, rows and columns of the matrix are marked as covered to restrict the search for candidate zeros.
1.3 Persistence assignment problem
The assignment problem for persistence pairs is very similar to the standard unbalanced assignment problem, except additional costs are defined for not assigning elements (i.e. matching persistence pairs with the diagonal). The assignment between diagrams and then involves the numerical rating associated with assigning with , along with (resp. ) the numerical rating associated with matching (resp. ) with the diagonal.
If and are sets of persistence pairs such that and , then it is possible to solve the persistence assignment problem using the standard Kuhn-Munkres algorithm with the cost matrix described by Eq. 3, as proposed in :
The first line corresponds to matching pairs from to pairs from ; the second one corresponds to the possibility of matching pairs from to the diagonal; the third one is for matching pairs of to the diagonal and the last one completes the cost matrix. The drawback of this approach is that it requires to solve the assignment problem on a cost matrix (that potentially contains two non-sparse blocks where persistence elements are located, see Fig. 4), though the number of distinct elements is at most . As seen in Sec. 2, our algorithm addresses this issue.
This section presents a quick overview of our tracking method, which is illustrated in Fig. 9. The input data is a time-varying PL scalar field defined on a PL -manifold with .
First, we compute the persistence diagram of the scalar field for every available timestep.
Next, for each pair of two consecutive timesteps and , we consider the two corresponding persistence diagrams and . For each couple of persistence pairs , we define a distance metric corresponding to the similarity of these pairs: (see Sec. 3).
For each pair of consecutive timesteps, we compute a matching function . Every persistence pair of is associated to to , which is either a persistence pair in or so as to minimize the total distance . Finding the optimal involves solving a variant of the classical Assignment Problem, as presented in Sec. 1.3. Only persistence pairs involving critical points of the same index are taken into account.
We compute tracking trajectories starting from the first timestep. If at timestep the matching associates with , then a segment is traced between and . If , the current trajectory ends. Trajectories are grown following this principle throughout all timesteps. Properties are associated to trajectories (time span, critical index), and to trajectory segments (matching cost, scalar value).
Finally, trajectories are post-processed to detect feature merging or splitting events with a user-defined geometric threshold.
2 Optimized persistence matching
This section presents our novel extension of the Kuhn-Munkres algorithm, which has been specifically designed to address the computation time bottleneck described in Sec. 1.2.
2.1 Reduced cost matrix
The classical persistence assignment algorithm based on Kuhn-Munkres considers , a cost matrix. We propose to work instead with , a reduced matrix defined in Eq. 4, where every zero appearing in the last row is considered independent. This amounts to considering that persistence pairs corresponding to rows are not assigned by default. Fig. 4 summarizes the matrices considered by each assignment method.
This last row, emulating the diagonal blocks of Fig. 4-b requires a specific handling in the optimization procedure. In particular, it requires the first step of the algorithm to subtract minimum elements from columns (and not rows) so as not to have negative elements in the matrix.
As a reminder, the original algorithm proceeds iteratively in two alternating phases: matrix reduction that makes new zeros appear, and augmenting path that finds a maximal set of independent zeros. At the iteration, the current maximal set of independent zeros is made of starred zeros. After a matrix reduction, new zeros appeared that can potentially belong to the new maximal set of independent zeros. Such candidates are primed. A single augmenting path (as in Fig. 3) replaces a set of starred zeros with primed zeros, forming a new set of independent zeros with one more element. Rows and columns of the matrix are marked as covered to restrict the search for candidates in the augmenting path phase. Blue blocks of Algorithm 1 indicate our extension of the Kuhn-Munkres algorithm.
In this novel extension, an augmenting path constructed in the corresponding phase can start from a starred zero in the last row (and then potentially find a primed zero in its column), but such a path can never access a starred zero in the last row at another step, for the corresponding column would have been covered prior to this (and thus cannot contain a primed zero, see Algorithm 1). A starred zero in the last row can then never be unstarred.
The Kuhn-Munkres approach has the property to only increase row values (and only decrease column values). When our algorithm working on the reduced matrix ends, it is therefore not possible that the elements on the top-right corner of the corresponding full matrix be negative. Furthermore, given Theorem 1, the resulting matrix corresponds to the same assignment problem.
Working with the reduced matrix , however, does not necessarily yield an optimal assignment. When assignments are found in the bottom row, if there has been additions to the matrix rows, then the corresponding matrix would contain a top-right block that is not zero, and a top-left block that is not zero either. Thus, the stop criterion stated by Theorem 2 may not be respected when lines are covered (as the real number of independent zeros in is ). Moreover, in our setup, a starred zero in the last column can never be unstarred; this is allowed in the approach on , owing to the bottom-right block, initially filled with zeros.
We therefore use the criterion stated in Eq. 5 to ensure that if, at any given iteration of the algorithm, a zero is starred in the last row of column , the cost of assigning the corresponding persistence pair to any other pair is higher than the cost of leaving both unassigned (0 for the -column pair and the residual value for -row pairs – see Algorithm 1). This specificity is illustrated in Fig. 5.
The Eq. 5 criterion is checked whenever a zero appears on the last row after a subtraction is performed on a column by the algorithm. If it is observed, the corresponding column is removed from the problem and the persistence pair is set unassigned.
If the criterion is not respected, we have to report back the reduced problem onto the full matrix (missing banned columns and with reported found residuals ). For this, we need to keep track of residuals, that is, values that have been added or subtracted from each row and column throughout the course of the algorithm. Once these residuals have been reported onto the full matrix, there can be no negative element, and all of the optimization work has been reported (so that we do not start all over again from the beginning, but we start from the optimized output of the first phase).
In practice for persistence diagrams, we always observed that the first phase is sufficient to find an optimal assignment. Using this approach prevents from working with two potentially large blocks of persistence elements, typically occurring with the complete matrix for and . This property is further motivated by the use of geometrical lifting (Sec. 3). The approach is detailed in Algorithm 1.
2.3 Sparse assignment
In practice, it is often observed that some assignments are not possible, and that reordering columns in the associated cost matrix would enable faster lookups and modifications , using sparse matrices. With persistence diagrams, the following simple criterion (Eq. 6) can be used to discard lookups for potential matchings.
Working with our version of the Kuhn-Munkres algorithm then becomes interesting for many assignments verify Eq. 6 (Fig. 6), hence greatly reducing the lookup time for zeros, minimal elements, and the access time for operations performed on rows or columns.
On the contrary, the original full-matrix version of Kuhn-Munkres deals with non-sparse blocks which have to be accessed and modified constantly throughout the course of the algorithm.
3 Lifted persistence Wasserstein metric
This section highlights the limitations of the natural Wasserstein metric applied to time-varying persistence diagrams and presents an extension that enhances its geometrical stability. Geometrical considerations are motivated, in terms of accuracy and performance.
Persistence diagrams can be embedded into the geometrical domain (Fig. 1). Doing so, one easily sees how different embeddings can correspond to similar persistence diagrams in the birth-death space. Working in this 2D space does yield irrelevant matchings: as can be seen in Fig. 7, when only the birth-death coordinates of persistence pairs are considered, a matching can be optimal even if it happens between geometrically distant zones. As a consequence, the distance metric between persistence pairs must be augmented with geometrical considerations.
where , and correspond to geometric distances between the extrema involved in the persistence pairs on a given axis. We process diagonal projections as follows (Eq. 8):
where the terms , and correspond to the geometric distance between the critical points of a given pair . Intuitively, it accounts for the distance between the critical points to cancel.
A lifted distance is considered by augmenting the geometric distance with coefficients . This aims at giving more or less importance to the birth, death or some of the coordinates during the matching, depending on applicative contexts. For instance, in practice it is desirable to give less importance to the birth coordinate when dealing with - persistence pairs (in other words, for tracking local maxima, see Fig. 8). For the remainder of the paper and the experiments, we used for maxima and for minima, for normalized geometrical extent and scalar values. We observed that using a lifted metric further increases the cost matrix sparsity, resulting in extra speedups.
4 Feature tracking
This section describes the four main stages of our tracking framework, relying on the discussed theoretical setup. Without loss of generality, we assume that the input data is a time-varying 2D or 3D scalar field defined on a PL-manifold. Topological features are extracted for all timesteps (Fig. 9, a-b), then matched (Fig. 9, c); trajectories are built from the successive matchings (Fig. 9, d) and post-processed to detect merging and splitting events.
4.1 Feature detection
First, we compute persistence diagrams for each timestep. We propose using the algorithm by Gueunet et al. , in which only - and - persistence pairs are considered.
When the data is noisy, it is possible to discard pairs of low persistence (typically induced by noise) by applying a simple threshold. In practice, this amounts to only considering the most prominent features. Using such a threshold accelerates the matching process, where for approaches based on overlaps, removing topological noise would require a topological simplification of the domain (for example using the approach in ), which is computationally expensive.
4.2 Feature matching
4.3 Trajectory extraction
Trajectories are constructed by simply attaching successively matched segments. For all timesteps , if the feature matching associates with , then a segment is traced between and , and is potentially connected back to the previous segment of ’s trajectory. If , the current trajectory ends. Properties are associated to trajectories (time span, critical point index) and to trajectory segments (matching cost, scalar value, persistence value, embedded volume).
4.4 Handling merging and splitting events
Given a user-defined geometrical threshold , we propose to detect events of merging or splitting along trajectories in the following manner. If are two trajectories spanning throughout and respectively, and if for some , , where is a lifted distance, then an event of merging (or splitting) is detected. We consider that a merging event occurs between and at time , when neither nor start at . We then consider that the oldest trajectory takes over the youngest. For example, and meet (according to the criterion) at the last timestep of , and started before , then we disconnect the remainder of from the trajectory before and we connect it so as to continue until ’s original end. Similarly, a splitting event occurs between and at time , when neither nor end at . The process is illustrated in Fig. 10. It is done separately for distinct critical point types: minima, maxima and saddles are not mixed.
This section presents experimental results obtained on a desktop computer with two Xeon CPUs (3.0 GHz, 4 cores each), with 64 GB of RAM. We report experiments on 2D and 3D time-varying datasets, that were either simulated with Gerris  (von Kármán Vortex street, Boussinesq flow, starting vortex), or acquired (Sea surface height, Isabel hurricane). Persistence diagrams are computed with the implementation of  available in the Topology ToolKit ; the tracking is restricted to - and - pairs. We implemented our matching (Sec. 2) and tracking approaches (Sec. 4) in C++ as a Topology ToolKit module.
5.1 Application to simulated and acquired datasets
We applied our tracking framework to both simulated and acquired time-varying datasets to outline specific phenomena.
In Fig. 11, we present the results of the tracking framework applied to an oceanographic dataset. The scalar field (sea surface height) is defined on 365 timesteps on a triangular mesh. We can see interesting trajectories corresponding to well-known oceanic currents. Drifting (a, b) and turbulent behaviors (c) of local extrema are highlighted. In Fig. 12, tracking is performed on the vorticity of highly unstable Boussinesq flow. Thanks to our analysis, trajectories can be filtered according to their temporal lifespan, revealing clearly different trajectory patterns among the turbulent features. This kind of analysis may be easily performed based on other trajectory attributes, depending on applicative contexts. In Fig. 13, we show our approach on a 3D hurricane dataset whose temporal resolution is such that a method based on overlaps of split-tree leaves (see Sec. 5.3) could not extract trajectories. In Fig. 14, our tracking framework correctly follows local extrema of the vorticity field in a simulated vortex street.
5.2 Tracking robustness
In the following two sections, we demonstrate the robustness and performance of our tracking framework.
We compare to the greedy approach based on the overlap of volumes [9, 8, 75, 66] of split-tree leaves, which amounts to tracking local maxima. In this approach, for every pair of consecutive timesteps , split-tree segmentations and are computed (these are a set of connected regions). Overlap scores are then computed for every pair of regions , as the number of common vertices between and . Scores are sorted and is considered matched to the highest positive scoring such that has not been matched before. Trajectories are extracted by repeating the process for all timesteps.
The robustness of our tracking framework is first assessed on a synthetic dataset consisting of whirling gaussians, on which we applied noise (Fig. 15). Identified trajectories are sensibly the same with a perturbation of 10% of the scalar range. The 75% most important features are still correctly tracked after a 25% random perturbation has been applied to the data.
In Fig. 14, our method is compared to the greedy approach, based on overlaps, while decreasing the temporal resolution. The overlap approach yields trajectories corresponding to noise (Fig. 14-e), which can be filtered by applying topological simplification beforehand (this would have a significant computational cost as it requires to modify the original function), or by associating the scalar value of the function to every point in the trajectory and then filtering the trajectory in a post-process step. In our setup, is is much simpler to discard this noise, by using a threshold for discarding small persistence pairs before the matching (implying a faster matching computation). When downsampling the temporal resolution to only 20% of the timesteps, our approach still gives the correct results (Fig. 14, d vs. e). With 15% of the timesteps, our approach (Fig. 14-f) still agrees with the tracking on the full-resolution data (Fig. 14-b), until preceding features begin to catch up, resulting in a zig-zag pattern. By comparison, the overlap method fails to correctly track meaningful regions from the beginning of the simulation to its end; it is indeed dependent on the geometry of overlaps, which is unstable. It can be argued that the locality captured by overlaps is emulated in our framework by embedding and lifting the Wasserstein metric, when the overlap method does not take persistence into account when matching regions. Also note that if the saddle component of persistence pairs associated to maxima is ignored (i.e. if in Eq. 7 and Eq. 8) during the matching, then the geometrical distance can be insufficient for correctly tracking these persistence pairs (c). Therefore, the problem of matching persistence pairs for tracking topological features cannot be reduced to the (unbalanced) problem of assigning critical points in 4 dimensions (3 for the geometrical extent, one for the scalar value).
Fig. 13 further illustrates the robustness of our approach when downsampling the data temporal resolution. In hurricane datasets, local maxima can be displaced to geometrically distant zones between timesteps if those are taken at multiple-day intervals. This unstable behavior and the absence of obvious overlaps makes it particularly difficult to track extrema; nevertheless, our framework managed to track them at a very low time-resolution.
5.3 Tracking performance
We then compare our framework with our implementation of the approach based on overlaps  on the ground of performance. Figures are given in Table 1. Note that our approach has the advantage of taking persistence diagrams as inputs, so it can be applied to unstructured or time-varying meshes, for which overlap computations are not trivial. Our approach is also relatively dimension-independent: though in 3D, computing overlaps is very time-consuming (Fig. 1-Isabel), the complexity of the Wasserstein matcher, which only takes embedded persistence diagrams as inputs, for a given number of persistence pairs is sensibly equivalent. For both Isabel and Sea surface height datasets, we applied a 4% persistence filtering on input persistence diagrams. As the experiments show, our approach is faster in practice than the overlap method with best-match search.
|Data-set||Pre-proc (s)||Matching (s)||Post-proc|
5.4 Matching performance
Next, we compare the performance of the matching method we introduced in Sec. 2 to two other state-of-the art algorithms.
|Data-set||Sizes of diagrams||Time (s)|
We compare it to the reference approach for the exact assignment problem  based on the Kuhn-Munkres algorithm, and to our implementation of the approximate approach based on the auction algorithm [4, 41], on the ground of performance.
Table 2 shows that our new assignment algorithm is up to two orders of magnitude faster than the classical exact approach . In particular, the best speedups occur for the larger datasets which indicates that our approach also benefits from an improved scaling.
It is often useful in practice to discard low-persistence pairs prior to any topological data analysis as these correspond to noise. Fig. 16 compares the running times of our approach,  and  as more and more low-persistence pairs are taken into account. When removing pairs whose persistence is below 5% of the scalar range, which is commonly accepted as a conservative threshold, our approach is faster than all competing alternatives. When considering more low-persistence features, below 4%, our approach is competitive with the approximated auction approach with 1% error. Below 2%, only noise is typically added in the process. The performance of our algorithm becomes comparable to that of the high-precision auction approximation although our approach guarantees exact results.
As we described, our framework enables the tracking of 0-1 and - persistence pairs. It would be interesting to extend it to support the tracking of saddle-saddle pairs (in 3D) and see its application to meaningful use cases.
Besides, the lifting coefficients proposed in our metric (Eq. 7) might be seen as supplementary parameters that have to be tuned according to the dataset and applicative domain. Nonetheless, we observed in our experiments that these parameters do not require fine-tuning to produce meaningful tracking trajectories. The extent to which these can be enhanced by fine-tuning is left to future work.
The lifted distance can be generalized to take other parameters, such as the geometrical volume, mass, feature speed, into account, and be fine-tuned to answer the specificity of various scientific domains. Merging and splitting might also be enhanced, or given more flexibility, for instance with additional criteria. We also believe that the performance of the post-processing phase can be improved.
Additionally, we believe that the approximate auction algorithm can also take the lifted persistence metric into account by performing Wasserstein matchings between persistence pairs in 5 dimensions, and possibly benefit from geometry-based lookup accelerations, as suggested in lower dimension in . It remains to be clarified how the quality of the matchings is affected in practice by using an approximate matching method, and how sparsity can enhance the research phase for the auction algorithm.
We note that the theoretical complexity of our matching method is, as the Munkres method, cubic; however, the two orders of magnitude speedups demonstrated in our experiments allow to study more challenging datasets. For very large case studies, the use of persistence thresholds could prove quite helpful for controlling the computing time of matchings. Among other non-trivial tracking methods, some graph matching methods are based on graph-edit distances [26, 1]. Their adaptation to the case of persistence diagrams or other topological structures (such as contour trees and Reeb graphs) may enable an additional structural regularization, this ought to be investigated in future work.
In this paper, we presented an original framework for tracking topological features in a robust and efficient way. It is the first approach combining topological data analysis and transport for feature tracking. As the kernel of our approach, we proposed a sparse-compliant extension of the seminal assignment algorithm for the exact matching of persistence diagrams, leveraging in practice important speedups. We introduced a new metric for persistence diagrams that enhances geometrical stability and further improves computation time. Overall, in comparison with overlap-based techniques, our approach displays improved performance and robustness to temporal downsampling, as experiments have shown.
We plan to release the implementation of our tracking framework open-source as a part of TTK  in the near future; we hope that it will be useful to the community with an interest for efficient tracking methods. We look forward to adapting it to tracking phenomena in in-situ contexts, where the large-scale time-varying data is accessed in a streaming fashion. As we are also interested in larger datasets, we are currently carrying out scaling tests on complex physical case studies available at Total S.A., for which one needs specifically adapted rendering techniques  to apprehend the resulting graphical complexity of the topology evolution.
We also believe that the application potential of our matching framework can be studied for tasks other than time-tracking, for instance, self-pattern matching and symmetry detection, or feature comparisons in ensemble data.
Acknowledgements.This work is partially supported by the Bpifrance grant “AVIDO” (Programme d’Investissements d’Avenir, reference P112017-2661376/DOS0021427) and by the French National Association for Research and Technology (ANRT), in the framework of the LIP6 - Total SA CIFRE partnership reference 2016/0010. The authors would like to thank the anonymous reviewers for their thoughtful remarks and suggestions.
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