DebrisTracer: Reliable Tracking in Hypervelocity Impact Fast Imaging

DebrisTracer: Reliable Tracking in Hypervelocity Impact Fast Imaging Théophane Loloum, Fabien Vivodtzev, David Hébert, Baptiste Reynier, Michel Arrigoni, and Julien Tierny

arXiv:2607.15986v1 [cs.LG] 17 Jul 2026

Fig. 1: In experimental hypervelocity impacts, a projectile is launched at 5 km/s to the material under study (left, black rectangle), e.g., to emulate satellite collisions. This produces many debris fragments, of distinct masses and speeds (a)-(c). The reliable estimation of these mass and speed distributions is of major importance in aerospace design. DebrisTracer extends an off-the-shelf topology tracking approach [72] to incorporate domain knowledge and physical assumptions. It automatically produces an accurate and reliable debris tracking (d)-(g), supporting interpretable visual analyses, facilitating the understanding of this complex space-time phenomenon. In this experiment, it enables distinguishing and exploring three debris types (h): fast (yellow), medium (purple) and slow (blue) ejections. Abstract—This application paper presents DebrisTracer, a framework for the reliable tracking of debris in hypervelocity impact fast imaging. These noisy and highly specific datasets capture the ejection of a large number of debris fragments after the impact of a projectile launched at hypervelocity into a target material. The reliable estimation of debris mass and speed distributions is of major importance in aerospace applications. We document how to extend an off-the-shelf topology tracking framework [72] based on critical point extraction and matching, in order to incorporate domain knowledge and physical assumptions. Our approach automatically produces an accurate and reliable debris tracking, enabling an interpretable visual analysis of this complex space-time phenomenon. Extensive experiments demonstrate the accuracy improvements provided by our approach over established tools used by domain experts [23, 76] in terms of physical validation, specifically via the prediction of the experimental ejected mass and crater depth profiles. We illustrate the utility of our approach across several use cases (with varying impact angles and physics). We show that our statistical summaries enable the visual identification of distinct regimes within the debris population, corroborating and refining prior expectations of domain experts. Our database and our C++ implementation are available at this address: https://github.com/tloloum/DebrisTracer. Index Terms—Hypervelocity impact, feature tracking, topological data analysis.

1

I NTRODUCTION

The study of HyperVelocity Impacts (HVI) is of major importance in material science, with applications in aerospace design (e.g., to emulate satellite collisions [11, 30]), planetary defense (to study asteroid deflection [10,31]), or high energy density physics (for shielding in confinement fusion [34, 45]). Upon such impacts, the material under study undergoes a process called shock-wave compression, leading to the rapid formation of a crater, as well as the high-speed ejection of debris. In particular, the mass and velocity distributions of these debris are critical for modeling secondary damage, debris cloud evolution, shielding degradation or trajectory perturbations. Therefore, it is crucial for aerospace engineers to obtain accurate estimates of these distributions. For that, several diagnostics can be used, among which high-frame-rate videos of HVI experiments performed in laboratory conditions [29] (Fig. 2). However, the resulting datasets are particularly challenging to analyze, and existing frameworks used by domain experts [23, 76] often provide sub-optimal results, due to the specifics of HVI. Specifically, given the technical complexity and costs of HVI controlled experiments, only few datasets are available, preventing the

• Théophane Loloum, Fabien Vivodtzev, Baptiste Reynier, and David Hébert are with the CEA. E-mail: {firstname.lastname}@cea.fr • Michel Arrigoni is with ENSTA. E-mail: {firstname.lastname}@ensta.fr • Julien Tierny is with the CNRS and Sorbonne University. E-mail: {firstname.lastname}@sorbonne-universite.fr Manuscript received xx xxx. 201x; accepted xx xxx. 201x. Date of Publication xx xxx. 201x; date of current version xx xxx. 201x. For information on obtaining reprints of this article, please send e-mail to: [email protected]. Digital Object Identifier: xx.xxxx/TVCG.201x.xxxxxxx

reliable training of machine learning models. Moreover, for these very specific datasets, general-purpose segmentation models [58] (trained on generic images) exhibit identification rates that are both too low and inconsistent across consecutive frames (Fig. 3). These failures can be partly explained by the challenges of HVI imaging (Sec. 2.3). For instance, due to the high-frame-rate camera setups, these acquisitions are typically affected by strong vignetting artifacts, resulting in large contrast and brightness variations across the image. Specifically, these variations make it difficult to segment debris based on pixel intensity thresholding (Fig. 3). This challenges (i) debris characterization. In addition, for fast debris, the acquisition frequency of the employed cameras may be insufficient, leading to large displacements for a single debris fragment between consecutive frames. Moreover, due to occlusion, several debris may overlap in the image while they are distinct objects in 3D. These two difficulties confound tracking approaches based on object overlap detection, challenging (ii) debris tracking. Overall, the above two issues challenge the existing tracking approaches which use intensity thresholding for object characterization and overlap estimation for object tracking. In topological data analysis [17], several methods leveraged (i) persistent homology and (ii) transportation metrics to address these respective issues, as documented in recent surveys and benchmarks [37, 49, 85]. While a subset of these methods focus on tracking of complex structural patterns (e.g., based on merge trees), the binary characterization of objects in our application (debris or background) allows for a more straightforward and less constrained strategy. Specifically, we focus in this paper on a simple pre-existing approach [72] based on critical point extraction and tracking, and discuss its limitations for our application. We also document the process of extending this off-the-shelf topology tracking framework

Fig. 2: Experimental setup for HVI acquisitions: a projectile (a) is launched to the target (cross) at a typical speed of 5 km/s. This results in a cone-shaped ejection of many debris fragments (b). The impact leaves a crater on the target, captured by a laser scan (c), enabling the modeling of crater depth profiles (d) along the target segment.

to incorporate the necessary domain knowledge (Sec. 4) and physical assumptions (Sec. 5) to improve its accuracy and usability, while supporting trustworthiness and interpretability for domain experts. Extensive experiments (Sec. 6) demonstrate the accuracy improvements, both in terms of debris characterization and physical validation (via the estimated ejected mass and reconstructed crater profiles), over established tools used by domain experts [23, 76] and over our selected off-the-shelf topology tracking framework [72]. Our improved tracking offers new capabilities for analyzing the main trends in debris types, yielding new visual insights for experts, as documented in several use cases (Sec. 6.2), studying several impact angles and physics (projectile versus laser). We provide in additional material our C++ implementation, along with our database of hypervelocity impact acquisitions, which may constitute a benchmark for future work in feature tracking. 1.1

Related work

This section reviews the literature related to our work, which can be classified into the following two categories. (i) High velocity impact analysis: With the development of the aerospace industry since the twentieth century, hypervelocity impacts (HVI) have received significant attention in material science and reference textbooks are available [33]. Experimental protocols have been proposed to assess the size and/or velocity distribution of ejecta, based on devices in charge of physically collecting the debris (e.g., with gel or aerogels soft-recovery collectors [38]). In some cases, gravity can be used to separate the ejecta and collect them into bins [44]. Other authors analyze the secondary craters observed on witness plates [51]. However, these methods sometimes require dedicated experiments to ensure a good knowledge of the collector material behavior under impact [61,83]. Because of the difficulty to extract both size and velocity distributions from a single diagnostic, recent experiments have shown the value of combining different diagnostics [46, 60, 84]. Another line of experiments aim at providing “in-flight” diagnostics. For instance, Photonic Doppler Velocimetry (PDV) allows measurement of multiple fragments velocities with a laser probe [57]. Acquisitions based on high-speed cameras are becoming widely used in HVI experiments. However, the quantitative extraction of debris distributions from such videos remains challenging, as it requires a reliable frame-to-frame tracking method, for capturing the trajectories of the individual debris fragments. A few authors proposed custom tracking approaches [42, 78], while several studies rely on established tracking frameworks [21, 24]. In particular, the Fiji/ImageJ [65] plugin TrackMate [23, 76], provides a generic framework for particle tracking in 2D videos. It is considered as a go-to key tool by HVI experts, used frequently in the literature, e.g., [21, 24]. It implements a variety of algorithms for particle characterization (intensity thresholding, derivative analysis, etc.) and tracking (by overlaps or optimal assignment, etc.). However, to our knowledge, TrackMate does not support topological methods for characterization and tracking. Sec. 6.1 compares TrackMate and our framework, demonstrating substantial gains for our approach, in terms of physical validation. (ii) Topology tracking: Topological methods [17] have considerably developed over the last two decades in data visualization [28], for the robust and multi-scale representations of structural patterns in complex data. They rely on a family of concise topological descriptors, such as the persistence diagram [18], the merge and contour trees [9], Reeb graphs [54] or Morse-Smale complexes [69]. These descriptors have been successfully applied in a variety of applications, including com-

Fig. 3: Challenges of HVI acquisitions: given the specifics of this imaging modality (noise, vignetting artifacts, occlusion, debris shape variability), general-purpose segmentation models [58] provide correct classification rates that are low (a) and inconsistent across time steps (top versus bottom). Vignetting artifacts challenge automatic thresholding (blue) for debris characterization (b), as a given threshold may both under-segment (i.e., capture multiple debris within a single connected component), and miss certain debris, irrespective of the selected isovalue. Insufficient frame-rates result in debris which do not overlap in consecutive time steps (c), challenging overlap-based tracking approaches.

bustion [8], material science [71], fluid dynamics [47], bioimaging [7], astrophysics [70] or chemistry [14]. Since topological methods have a strong focus on structure rather than geometry, they naturally appear as promising candidates for characterizing and tracking debris, which are disconnected pieces exhibiting a wide variety of shapes. Several approaches have been proposed for tracking particles in simulated vector field data [27, 59, 74, 80]. However, since the input data is provided in our work in the form of gray-scale 2D videos, we will focus in the remainder on tracking approaches for scalar data. In scalar data analysis, several methods have been designed based on an object segmentation (e.g., using merge or contour trees), and relying on overlap detection between consecutive time steps to perform tracking [8, 41, 52, 63, 64, 66, 82]. However, HVI images challenge debris characterization based on intensity thresholding due to important vignetting artifacts (Fig. 3, Sec. 2.3). Moreover, insufficient frame-rates may prevent debris overlap over consecutive time steps, challenging overlap-based approaches. Alternative strategies for object characterization rely on topological persistence [18], which can be interpreted as a robust importance measure, enabling the identification of salient features, irrespective of vignetting artifacts (see Sec. 4.1). In this framework, feature tracking is no longer estimated via overlap detection, but by optimizing an assignment between the topological descriptors of consecutive time steps. This overall strategy has been employed in a variety of application domains, including fluid dynamics [71], cloud tracking [16,39] or cyclone tracing [22, 48, 77]. We refer the reader to recent surveys for further details on this overall strategy [37, 49, 85]. Nilsson et al. introduce an approach for matching extrema in timevarying scalar fields [50], based on gradient path integration. However, the local nature of gradient and the cumulative error of its integration may be challenged by fast moving objects, in particular in the presence of noise. Soler et al. introduce an assignment optimization approach between persistent extrema [72], combining criteria based on persistence and distances in the original domain. Several assignment approaches have been considered for similarity estimation and tracking between merge trees [4, 56, 62, 73, 81], extremum graphs [15], or Reeb graphs [1, 3, 79]. These methods enable the tracking of complex structural patterns, linking the features of interest together. However, our application setup (Sec. 2) only requires a binary characterization of objects: pixels are either debris or background. The debris cloud does not exhibit specific local structures between the individual debris fragments, that would need to be characterized and tracked with advanced topological descriptors, such as merge trees. Debris fragments simply need to be isolated and tracked. For this reason, we focus in our work on a simpler approach based on extremum extraction and tracking [72]. Specifically, it matches extrema based on an assignment optimization, irrespective of the structural patterns they may form locally or globally.

This allows for a straightforward and less constrained matching strategy. Then, this paper describes how to extend this topology tracking framework to incorporate the necessary domain knowledge about the acquisitions (Sec. 4) and physical assumptions (Sec. 5) to improve its accuracy, trustworthiness and interpretability for domain experts. 1.2

Contributions

This paper makes the following new contributions: 1. Domain-tailored approach for debris tracking: We document how to extend an off-the-shelf topology tracking framework [72] to incorporate domain knowledge and physical assumptions to improve tracking accuracy and usability. We document the resulting improvements both in terms of debris characterization and physical validation over established domain tools [23, 76]. While our approach builds on an existing framework [72], we followed a general methodology to adapt it to our application constraints, based on: (i) a formalization of observations, assumptions and physical hypotheses (Sec. 2.3), (ii) a specific modification of the existing framework [72] to incorporate domain knowledge (Sec. 4) and (iii) a novel, tailored post-processing of the tracking output to integrate physical assumptions (Sec. 5). We believe that this generic, three-step workflow may be applied to other domain-specific applications, particularly those involving feature tracking. In that regard, our work provides an example of how to implement this methodology, which may be useful for other researchers applying topological methods to domain sciences. 2. Case studies: The tracking performance of our approach allows for novel capabilities for the reliable visual analysis of debris, in terms of mass and velocity distributions. We illustrate these features on several use cases, for varying impact angles and physics. Our analysis corroborates and refines prior hypotheses from domain experts regarding the main trends in debris distributions. This hypothesis refinement was made possible by the visual representations (angle/velocity scatterplots, mass/velocity distributions) generated by our approach, illustrating a concrete case of applied end-to-end data transformation for visual reasoning. 3. Implementation: We provide a C++ implementation of our approach that can be used for reproducibility purposes. 4. Database: We contribute our database of hypervelocity impact acquisitions, i.e., 8 two-dimensional time-varying scalar fields (about 2.5 gigabytes), counting between 256 and 1050 frames, capturing different dynamic fragmentation modalities (projectilebased and laser-based). This database may constitute a benchmark for future research in feature tracking. 2

A PPLICATION CONTEXT

This section documents the technical context of HVI fast imaging. 2.1

Impact physics

We briefly recall here the physical phenomena occurring when a small projectile impacts a large target at hypervelocity (Fig. 2), since they will be considered later, when formulating initial hypotheses (Sec. 2.3). (i) Phenomenology: If the impact velocity is high enough (i.e., typically higher than a few km/s), both materials undergo a shock compression [33]. In the vicinity of the impact point, large deformations occur. In the case of brittle targets (such as rocks or ceramics), these deformations are the result of crack formation and propagation, leading to many separate debris fragments [12]. Moreover, when the compression is realeased, significant velocity can be imparted to the fragments, resulting in material (or debris) ejection [43]. During its propagation in the target, the shock amplitude decreases. The balance between the initial shock pressure, its attenuation during propagation, the strength and the microstructure of the target, yields a finite-size crater, whose dimension and shape result from this complex and transient process [86]. (ii) Debris ejection processes: Given the transient nature of cratering, simulations have allowed a few insights in the mechanisms leading to debris ejection. Jetting [35] is a high-pressure, early-time phenomenon

caused by extreme pressure gradients at impact, leading to the ejection of very high-velocity, fine debris. Spallation [36] results from the shockwave, causing the detachment of larger, slower fragments. Below the projectile impact area, the fragmentation may be more likely due to shear stress under confining pressure [12]. (iii) Analogy with lasers: As discussed above, debris ejection is mostly driven by the shock and release waves generated by the HVI, and it is not only due to the projectile. Thus, it is also possible to investigate cratering with high intensity lasers (Sec. 6.2), that allow similar dynamical loading [2], along with easier diagnostic implementation [68]. 2.2

Ejecta video recording

HVI monitoring typically involves the recording by a high-frame-rate camera of the impact of the projectile against the target under study and the subsequent debris ejection (Fig. 2). The projectile (here an aluminum ball of 2 mm radius) simulates an orbital debris hitting the shielding system of a satellite, represented in the experiment by a porous and isotropic graphite target. In our setup, the projectile is launched at 4.79 km/s by a two-stage light-gas gun (2SLGG HERMES), a device capable of accelerating a millimeter size projectile at velocity ranging from 1 to 10 km/s. The impact is captured by a ultra-high-speed camera, at a typical frame-rate of 200,000 images per second. (Fig. 2). As discussed in Sec. 2.1, debris ejection can also be generated with laser-based HVI. Our database also includes acquisitions obtained with this modality, with shock waves induced by a high-power pulsed laser, and filmed by a specific, intensified, multi-channel ultra-high-speed camera, at a typical frame-rate of 2 million images per second. Specifically, our database contains 7 acquisitions of projectile launches, including 5 and 2 with respective impact angles of 90 and 45°. These consist in 2D grayscale videos, counting from 400 to 1,050 frames, with a resolution from 384 × 176 to 384 × 256 pixels. Our database also contains an acquisition of impact generated via laser pulse (90° impact angle), for a resolution of 400 × 250 over 256 frames. For each experiment, each video frame undergoes a basic presmoothing step based on 1-neighborhood pixel averaging prior to file export. Moreover, a depth scan of the target is conducted after impact, to acquire the crater’s shape, often analyzed via sectional profiles (Fig. 2). This also enables the measure of the ejected mass. 2.3

Observations, assumptions and hypotheses

The visualization of the HVI acquisitions by the domain experts triggered several observations (regarding the acquisition process itself), accompanied by specific physical assumptions, as well as initial hypotheses regarding the behavior of the debris. The observations and assumptions will be exploited to incorporate domain knowledge (Sec. 4) and physical priors (Sec. 5) into our tracking approach, while our case study will evaluate the corroboration of the initial hypotheses (Sec. 6.2). Observation O1: Important vignetting. Given the high-frame-rate camera setups, hypervelocity impact images are typically affected by strong vignetting artifacts, resulting in large variations in pixel intensity and contrast throughout the image. As shown in Fig. 3, this challenges methods based on intensity thresholding for debris characterization. Observation O2: Debris shape and intensity variation. Important shape and size variations can be observed among the debris population. Also, since each debris fragment travels in 3D with a spin motion, its 2D projection on the image yields a shape that varies through time. This, in combination with vignetting artifacts, generates debris fragments that, taken individually, vary in shape, size and pixel intensity over time, sometimes drastically. This challenges debris characterization. Observation O3: Insufficient capture frequency. Even at very high frame-rates, a single debris fragment can travel so fast that its projections in consecutive frames do not overlap (Fig. 3). This challenges methods based on object overlap estimation for debris tracking. Observation O4: Important occlusion. Since the acquisition captures with a 2D video a 3D phenomenon, important occlusions inevitably occur. This effect is particularly important at the beginning of the recording, when the majority of the debris are captured in the form of a large blob, that progressively spreads into individual pieces. Occlusion is also clearly noticeable later in the ejection, when multiple debris

Fig. 4: Multiscale topological analysis pipeline used in our work: the topological features of a scalar field (a) can be captured by pairs of critical points (i.e., persistence pairs) in the persistence diagram (b). In the presence of noise (c), spurious features are contained in the diagram in the vicinity of the diagonal (d). This enables their visual identification and removal via topological simplification (e)-(f). In this terrain dataset, the shape of each hill can be nicely captured by the merge tree (f), in particular by considering the regions associated to its leaf arcs (g).

may occlude each other. While these objects overlap in the 2D image, they should still be characterized as distinct debris fragments. This also challenges methods based on overlap estimation for debris tracking. In addition to these observations, several physical assumptions can be made given the order of velocities captured by these experiments. Physical assumption PA1: Constant individual velocity. Given the recording duration (< 10 ms) and the debris velocities (> 1 m/s), the effect of gravity can be neglected. Then, each debris fragment can be safely assumed to travel according to an individual velocity vector that is constant through the acquisition. Physical assumption PA2: Constant individual mass. Given the debris velocities, it can be safely assumed that there is no significant interaction among debris. It follows that debris fragments do not split into multiple pieces after the initial impact. Physical assumption PA3: Uniform density. Given the target material, all the debris can be considered to have the same volume density. After observing the acquisitions, the domain experts formulated several hypotheses that they would like to investigate visually. Hypothesis H1: Multi-phase ejection. The debris ejection seems to occur in several phases. For projectile-based HVI, it is assumed that these different phases correspond to the different fragmentation processes discussed in Sec. 2.1 (e.g., jetting and spallation). Hypothesis H2: Velocity disparity correlated to angle trajectory. According to previous studies [29, 60, 83], a signature of the different processes (e.g., jetting and spallation, Sec. 2.1) can be found in debris statistics, especially angle-velocity and mass-velocity distributions. Our work provides a reliable framework specifically tailored for debris tracking, supporting the computation of statistical summaries that enable the visual investigation of the above hypotheses (Sec. 6.2). 3

BACKGROUND

This section presents the technical background to our work. We refer to textbooks [17] for an introduction to computational topology. 3.1

Input representation

Each input acquisition is given as a floating-point time-varing scalar field F = { f1 , f2 , . . . , fnt }, where nt is the number of time steps. Each time step is modeled as a piecewise linear (PL) scalar field ft : K → R, with t ∈ {1, 2, . . . , nt }, which encodes, for convenience, the opposite of the pixel intensity It at time t, on the triangulation K of the input 2D pixel grid, i.e., ∀v ∈ K, ft (v) = −It (v). For brevity, we will omit the notation t when non-ambiguously focusing on a single time step. For a given time step, f values are provided on the vertices of K and f is interpolated with barycentric coordinates on the other simplices of K. f is enforced to be injective on the vertices with a variant of simulation of simplicity [20]. In practice, we represent each time step as an elevated 3D terrain (Fig. 4), where the elevation coordinate (i.e., Z) is given by the f value of each vertex. −1 The sub-level set of f for the isovalue i, noted f−∞ (i), is defined as the subset of K valued below the isovalue i. The super-level set, noted −1 f+∞ (i), is defined symmetrically (subset of K above i). As i continu−1 ously evolves from −∞ to +∞, the topology of f−∞ (i) changes at precise locations, called critical points [17]. New connected components −1 of f−∞ (i) emerge on minima of f . On the saddle points of f , either two

−1 connected components of f−∞ (i) merge, or a single component loops −1 back onto itself to form a topological handle in f−∞ (i) (saddles can be enforced to be non-degenerate via saddle unfolding [17]). Finally, topological handles are completely filled on maxima. As shown in Fig. 4, the critical points capture the topological features of the scalar field: minima, saddles and maxima encode pits, valleys and peaks.

3.2 Persistence diagrams Real-life data is often affected by noise. This is particularly the case with our hypervelocity impact acquisitions. As shown in Fig. 4, in the presence of noise, the slightest oscillation in data values yields spurious critical points. Topological persistence [17, 18] addresses this issue. It is an established framework for characterizing the importance of critical points. In our setup, it will be instrumental for the detection of the most salient critical points, irrespective of vignetting effects (brightness and contrast variations throughout the image). In the domain, each topological feature (i.e., connected component, −1 handle) of f−∞ (i) can be associated with a unique pair of critical points, called a persistence pair (c, c′ ), corresponding to its birth and death. The Elder rule [17] provides a mechanism for arranging the set of critical points into persistence pairs. For instance, if two connected −1 components of f−∞ (i) merge at a saddle point c′ , the younger component (created last, in c) dies, in favor of the older one (created first). In practice, the life-span of connected components can be tracked with a Union-Find data structure [13]. In 2D, handle creation (on saddles) and destruction (on maxima) can also be tracked with a Union-Find data structure by duality [18, 26], by considering the super-level sets of f . Persistence pairs can be represented visually in the persistence diagram, noted D ( f ). Specifically, each persistence pair (c, c′ ) is embedded as a vertical bar in 2D, such that the x-coordinate of the bar is given by f (c) and that the y-coordinates of its bottom and top are given by f (c) and f (c′ ) respectively. Then, the life-span of the corresponding feature can be directly read from the diagram as the length of the bar, | f (c′ ) − f (c)|, called topological persistence [18]. As shown in Fig. 4, the most prominent features in the data are captured with large bars in the diagram, which can be easily distinguished from the noise, characterized by short bars near the diagonal. In contrast to isovalue thresholding approaches which are challenged by vignetting effects (Fig. 3), the salience evaluation provided by topological persistence (based on the capture of large intensity variations) overcomes this issue and it will be instrumental in our framework to robustly identify debris. Persistence diagrams are often simplified, by discarding the bars shorter than the expected noise level. Topological simplification [19,40] is a process that enables the estimation of a new function fb, which is close to the input function f , and whose topology is exactly described by the simplified diagram. Topological simplification is an established pre-processing step in practice, as it enables the access to simplified versions of the input, for multi-scale topological analysis. 3.3 Merge trees As described next, debris are characterized and tracked in our work via the critical points of the time-varying scalar field. Once a debris is identified and tracked, its geometry needs to be evaluated in order to estimate its mass (Sec. 5.4). For that, we rely on the merge tree [9, 25] to extract the geometry of the hills captured by the maxima (Fig. 4).

Fig. 5: Debris characterization by persistence thresholding: each time step is represented as a 3D terrain. Noise and vignetting artifacts result in the presence of many spurious features, for diverse intensity values (a). Maxima, in yellow, are represented with spheres whose radius is a function of topological persistence. The persistence curve (b) leads to the visual identification (inflection point) of a persistence threshold separating noise from features. This enables the characterization of the debris as the most persistent maxima in this terrain representation (c).

The merge tree of f is a 1-dimensional simplicial complex defined as the quotient space M( f ) = K/ ∼ by the equivalence relation ∼, which states that two points p1 and p2 are equivalent if f (p1 ) = f (p2 ) and if p1 and p2 belong to the same connected component of super-level
−1 set f+∞ f (p1 ) . In short, this tree tracks the creation of connected −1 components of f+∞ (i) at its leaves, and merge events at its interior nodes (Fig. 4). This tree can also be computed by maintaining a Union-Find while sweeping the data from top to bottom values [9, 25]. Interestingly, the construction of these trees maintains the relationship that assigns each vertex of K to its corresponding arc in the tree. Then, after computation, K can be precisely partitioned into the regions corresponding to its arcs. In particular, the regions corresponding to its leaf arcs nicely capture the hills associated to each maximum (Fig. 4). 4

TOPOLOGY TRACKING

This section describes how to extend an off-the-shelf topology tracking framework [72], to incorporate domain knowledge about the acquisition, in particular regarding the observations documented in Sec. 2.3. 4.1

4.2 Debris tracking Soler et al. [72] presented a feature tracking approach based on the optimal assignment of the extrema between two consecutive time-steps fi and f j of a time-varying scalar field. In our work, debris can be followed by tracking persistent maxima over time. Thus, we focus on this off-the-shelf tracking approach and we document here adjustments to make it fit our application context (Sec. 2). Off-the-shelf tracking framework. Here we recap the approach by Soler et al. [72], in a way that eases the formalization of the modifications we considered in our work. First, each input saddle-maximum persistence pair p = (c, c′ ) of a diagram D ( f ) is represented as a point in R5 , where the first three coordinates of p are the 3D coordinates of the maximum c′ in the domain and the remaining two coordinates are the birth and death values of p, i.e., f (c) and f (c′ ) respectively. Second, to account for a domain-dependent disparity between 3D distances and function values, diagrams undergo a scaling procedure. Specifically, each resulting 5-dimensional point p is scaled, with the element-wise multiplication of its coordinates by those of a scaling vector s ∈ R5 . Third, in order to match the saddle-maximum pairs of a first diagram D ( fi ) to a second D ( f j ), both diagrams undergo an augmentation procedure. This process aims at enabling the modeling of feature creation or destruction between time steps. For that, for each saddle-maximum pair pi ∈ D ( fi ), a corresponding dummy pair ∆pi is created. It is also modeled as a 5-dimensional point, whose first three coordinates correspond to the 3D coordinates of the mid-point between the critical points c and c′ , and whose last two coordinates,
modeling birth and death, are given identical values: f (c) + f (c′ ) /2. Then, the diagrams D ( fi ) and D ( f j ) are augmented with all the dummy pairs of the other:

D ′ ( fi ) = D ( fi ) ∪ {∆(p j ) | p j ∈ D ( f j )} D ′ ( f j ) = D ( f j ) ∪ {∆(pi ) | pi ∈ D ( fi )}. In short, this augmentation procedure inserts dummy features with zero persistence allowing a pair pi to either be matched with a pair in D ( f j ) or to its dummy pair ∆pi . After augmentation, both diagrams have the same size by construction (i.e., |D ′ ( fi )| = |D ′ ( f j )|). The weight w(pi , p j ) associated to the matching a persistence pair pi ∈ D ′ ( fi ) to another one p j ∈ D ′ ( f j ) is then given by the L2 -norm between the corresponding 5-dimensional vectors:

Debris characterization

This section describes the characterization of debris in our work. While the pipeline described in this section is typical of the state of the art [17], we emphasize here details that are specific to our application context (Sec. 2) and which motivate the design of our framework. (i) Feature specification. As described in Sec. 3.1, the scalar field f considered at each time step is the opposite of pixel intensity values. Moreover, each time step is represented by an elevated terrain, where the elevation coordinate (i.e., Z) is given by f . Then, in this framework, debris are characterized by prominent hills in the terrain. (ii) Topological characterization. Peaks in the elevated terrains of f are captured by the maxima of f . However, as illustrated in Fig. 5, the acquired images are subject to noise, leading to the identification of many spurious maxima associated with persistence pairs of low persistence (Sec. 3.2). Thus, we only consider in the following the maxima involved in persistence pairs with a persistence larger than a fraction εmin of the overall function range, Fig. 5(c). This fraction can be visually identified at an inflection point, Fig. 5(b), in the persistence curve (complementary cumulative function of persistence pairs). As described in Sec. 6, this parameter is set once for all (see Appendix A for a description of our parameter setup protocol). After this initial persistence thresholding, topological simplification (Sec. 3.2) will be performed such that our subsequent analysis (Sec. 5.4) also discards spurious maxima consistently. Note that, since topological persistence is defined as a difference between the function values of paired critical points (Sec. 3.2), it naturally addresses the vignetting artifacts (observation O1 from Sec. 2.3). Indeed, while pixel intensity progressively decreases away from the center of the image, debris persistence remains nearly constant over the domain, easing their identification.

w(pi , p j ) = ∥pi − p j ∥2 . In the special case where both pi and p j are dummy pairs, w(pi , p j ) is set to zero. Then, the optimal matching φ ∗ between the maxima of fi and those of f j is a bijection from D ′ ( fi ) to D ′ ( f j ), computed as the minimizer of the following assignment functional E(φ ):
E(φ ) = ∑ w pi , φ (pi ) . pi ∈ D ′ ( f i )

In short, φ ∗ is a matching which minimizes the sum of the costs of mapping the maxima of fi to those of f j according to the weight w. Modification M1: Pre-rotation. In our application, debris mostly travel along a direction that is orthogonal to the target surface (Sec. 2). Thus, in case of oblique projectile launches (e.g., 45° angle), we first rotate the data in the XY plane, such that the X-axis becomes orthogonal to the target surface (and thus aligned with the main direction of travel). Modification M2: Pre-scaling. Since each time step is represented as a 3D terrain (Sec. 3.1), the Z-coordinate of each maximum already captures its scalar value. Thus, we consider the following scaling vector s = (sX , sY , sZ , 0, 0) (i.e., with 0 scaling for birth and death values, the death value being already encoded along the Z-coordinate). With this modification, the optimization of the matching between maxima (described above) becomes a simple assignment problem between point clouds in 3D, the simplest form of optimal transport [55]. In particular, in order to mitigate the insufficient capture frequency of the employed cameras (observation O3, Sec. 2.3), we will typically use of a low value for sX (as documented in Appendix A), since debris can undergo large displacements along this direction from one time step to the next.

Fig. 6: Each trajectory path computed by the topology tracking (a) is transformed into a trajectory segment (b) via linear regression, given the physical assumption PA1 of per-fragment constant velocity vector.

Modification M3: Maximum appearance and disappearance. At the beginning of the acquisition, due to important occlusion (observation O4, Sec. 2.3), the majority of the debris are initially captured by a large blob, which progressively spreads into individual pieces, leading to the appearance of more and more persistent maxima with time. Symmetrically, towards the end of the acquisition, debris progressively exit the image, leading to a decrease in the number of persistent maxima. Therefore, for these two reasons, maximum appearance and disappearance still need to be modeled in our assignment problem. Originally, given a saddle-maximum pair pi = (c, c′ ), the selected tracking framework [72] considered a 3D embedding for its dummy pair ∆pi at the midpoint between the critical points c and c′ . However, in our applications, saddles tend to be located in the background of the image, which tends to be a flat plateau. As a consequence, the saddle c associated to a maximum c′ can be arbitrarily far from it, leading to arbitrary destruction weights in our setup. To address this, we simply re-define the weight for matching a persistence pair pi to its dummy pair ∆pi to a constant value: w(pi , ∆pi ) = ω. This parameter is stable across multiple acquisitions and thus, we set it once for all to a conservative value (see Appendix A for our parameter setup protocol). This means that if no pair p j ∈ D ′ ( f j ) is located within a ball of radius 2ω around pi , pi will have to be considered as destroyed from time step i to j. This accounts for debris leaving the image. Symmetrically, if no pair pi ∈ D ′ ( fi ) is located within a ball of radius 2ω around p j , this means that the debris captured by p j just emerged a time step j as an individual piece, e.g., from the initial debris blob (observation O4, Sec. 2.3). 5

P HYSICS - DRIVEN POST- PROCESSING

This section describes how to extend the tracking approach presented above, in order to incorporate the physical assumptions provided by domain experts (documented in Sec. 2.3). Specifically, it presents how to post-process the raw trajectories produced by the initial tracking into physically plausible trajectories, that can be directly exploited for the reliable analysis of debris ejection in hypervelocity impacts. 5.1

From trajectory paths to trajectory segments

The debris tracking algorithm described in the previous section produces a set P of nT trajectory paths P = {P1 , P2 , . . . PnT }, each path Pi being an ordered set of nPi points, representing the maxima matched between consecutive time steps: Pi = {mi1 , mi2 , . . . , minP }, each maximum i

mik being associated with its time step through the map t(mik ). As documented in Sec. 2.3, taken individually, each debris fragment can be assumed to travel along a constant velocity vector (physical assumption PA1). Thus, each debris trajectory should eventually be modeled with a unique line segment (Fig. 6). For that, we consider for each trajectory Pi a unique 2D line li (t) = ai t + bi , where bi is the debris (X,Y ) position extrapolated at temporal origin and where ai is the estimated 2D velocity vector of the corresponding debris. Specifically, the constants ai and bi are estimated via standard least squares linear regression [32, 67]. Also, for each line li , we store the trajectory segment Si = [li (tsi ), li (tei )] over which the debris was actually followed by the trajectory path Pi , i.e., tsi = t(mi1 ) and tei = t(minP ). i Finally, we maintain a map between each trajectory segment Si and its initial trajectory path Pi .

Fig. 7: Gaps in trajectory paths can occur when a debris fragment, identified by a persistent maximum (arrow) at time step i (a), is associated at the next time step j with a maximum below the persistence cutoff (b). At a later time step k (c), the corresponding maximum can become persistent again, yielding a tracking gap between the time steps i and k.

Fig. 8: Consecutive trajectory segments (a), in yellow, are concatenated into a single trajectory line (b) under the physical assumption PA2 of per-fragment constant mass, by condering space-time criteria (Sec. 5.2).

5.2

From trajectory segments to trajectory lines

As documented in Sec. 2.3, there is no significant secondary impact among debris and, consequently, debris do not merge or split (physical assumption PA2). However, the practical setup of the acquisitions challenges this hypothesis. As reported in observation O4 (Sec. 2.3), since the acquisition captures in 2D a phenomenon that is intrinsically 3D, important occlusion inevitably occurs. Specifically, two debris, characterized by two independent maxima at time step i can overlap in 2D at the following time step j, resulting in a single, persistent maximum. In that scenario, the trajectory segment (Sec. 5.1) of one of the two debris will artificially terminate at time step j. Moreover, this debris may be tracked again by a new, independent segment at a later time step k, resulting overall in a gap in its tracking (between time steps j and k). Furthermore, as reported in observation O2 (Sec. 2.3), the shape and intensity of each debris can vary significantly through time. Consequently, a debris characterized by a maximum that is persistent enough at time step i may be characterized at time step j by a maximum whose persistence falls below the cutoff threshold (Sec. 4.1). In that case, the trajectory segment will also artificially terminate at step j. Then, the debris may be tracked again by a new, independent segment at a later time step k, as soon as the associated maximum becomes again persistent enough, also resulting in gap in the tracking (between time steps j and k, Fig. 7). In both cases, the gaps in the tracking will result in an over-estimation of the number of trajectories, since a given debris may be tracked by multiple, consecutive trajectory segments. To address this, a trajectory continuation procedure is introduced, to concatenate the trajectory segments corresponding to a single fragment (Fig. 8). To evaluate if two consecutive segments Si and S j correspond to the same debris, we consider three complementary criteria. (i) Temporal proximity. The temporal gap between the two segments j δi j = ts − tei must be in the interval [0, δmax ]. (ii) Directional consistency. The absolute value of the angle θi j between the velocity vectors ai and a j must be in the interval [0, θmax ]. (iii) Spatial continuity. The fragment position extrapolated from Si at j j time ts must coincide with the start of S j : the distance di j = ||li (ts ) − j l j (ts )||2 must be smaller than a threshold dmax (close to 0). Then, given a segment Si , we first evaluate its available candidates for concatenation based on the above thresholds (δmax , θmax , dmax ). Then, Si is greedily assigned to the candidate segment S j which minimizes the

Fig. 9: The regions associated to the leaf-arcs of the merge tree provide a first estimation of the debris shapes (a). Since saddles can be located arbitrarily far in the flat background, this segmentation tends to overestimate debris shape. For each of these regions, we apply Otsu’s algorithm [53] to automatically identify an optimal cutoff (dashed line) in its pixel intensity distribution (c), resulting in regions which individually better capture the debris shape, despite vignetting artifacts (b).

sum of the above penalties. At this point, each debris is now represented by an ordered set of ni trajectory segments Si = {S1i , S2i , . . . , Sni i }, for which we estimate a unique trajectory line Li , also via least squares linear regression (Sec. 5.1). Then, the output of this stage is a collection L of n trajectory lines L = {L1 , L2 , . . . , Ln }, where we maintain, for each trajectory line Li , a map to its set of trajectory segments Si . 5.3 Physically plausible trajectory lines At this point, each trajectory line Li ∈ L represents a single debris fragment. Given the numerous challenges related to the initial time steps of the acquisition (insufficient capture frequency, large blob representing the entire population of debris, c.f. Sec. 2.3), several trajectory lines of L are outliers (with few, short, misoriented segments), in particular for the debris identified early in the sequence. Thus, we introduce a last thresholding procedure to remove outliers which are physically aberrant, enabling a focus on the reliable trajectories only. For that, we consider a geometrical cone of physically plausible trajectories. Specifically, we only maintain in the remainder a trajectory line Li if (i) it intersects the target at X = 0, (ii) does so at a positive time value, (iii) with an angle with the horizontal axis that has an absolute value smaller than Θmax , and (iv) whose X-component of the velocity vector ai is larger than a lower bound Xmin . The latter constraint on the X-component enables the removal of stationary objects (e.g., the target), while the angle constraint filters out miroriented outliers. This last thresholding produces the final set of trajectory lines L∗ considered in the rest of our work, and for which the physical validity is guaranteed (plausible cone of trajectories with target intersection). 5.4 Debris mass estimation The tracking approach described above identified a list of physically plausible trajectory lines modeling the displacement of each debris. This first output will enable the reliable estimation of the distribution of debris velocities (by considering the velocity vector ai of each trajectory line Li ). As described in Sec. 2, in addition to this distribution, engineers are also interested in the debris mass distribution. For this, we rely in this work on a spherical debris shape hypothesis as well as an approximation of the average size of each debris, described below. For each trajectory line Li , we consider its set of trajectory segments Si (Sec. 5.2). Each trajectory segment S j ∈ Si is itself associated to a trajectory path Pj (Sec. 5.1), modeling the sequence of maxima j tracked by the approach described in Sec. 4. Given a maximum mk ∈ Pj , we can retrieve the corresponding time step by considering the map j t(mk ) (Sec. 5.1). In the scalar field ft(m j ) , let R be the region of the k

j image corresponding to the leaf arc of M( ft(m j ) ) associated to mk , as k

discussed in Sec. 3.3. Then, the shape of the corresponding debris can by nicely captured by the region R. However, in certain cases, since the background of the image tends to be a flat plateau, the saddle associated j to mk in the merge tree can be located arbitrarily far in the background,

Fig. 10: Due to occlusion, independent debris fragments at frame i (a) may momentarily overlap at frame j (b). These configurations are detected in our approach (in white) by identifying, for each tracking gap (Sec. 5.4), the merge tree leaf-arc region it belongs to in the image.

resulting in an over-estimation of the debris size by the region R (Fig. 9). To address this issue, we rely on the Otsu segmentation algorithm [53] which we restrict to R. Specifically, this statistical approach optimizes a cutoff value for ft(m j ) , to maximize the variance between the bright k and dark pixels, resulting in an improved approximation of the shape j j Dk of the debris fragment associated to the maximum mk (Fig. 9). Next, we evaluate the size of the debris fragment on the image by j j considering the area ak of Dk . Then, for the trajectory line Li , we j consider its average debris area Ai , by averaging the values ak over all the possible values of j (segments) and k (maxima). Under the hypothesis of a spherical debris shape, the average area Ai enables the estimation of the volume Vi of the debris fragment associated to the trajectory line Li . From this volume estimation Vi , the mass Mi of the debris is finally evaluated based on the hypothesis of uniform volume density (physical assumption PA3, Sec. 2.3). Note that, for a given trajectory line Li , its set Si of trajectory segments may not completely cover it, due to tracking gaps derived from occlusion or temporary low persistences (pink segments in Fig. 8). For each frame t where a line Li has a tracking gap (i.e., it does not have any associated segment in Si ), we perform an inclusion test to determine the merge tree leaf arc region Li (t) belongs to. This alternative fallback approach enables the identification of overlapping debris in the image, when a given region is claimed by several trajectory lines, given the above inclusion test (Fig. 10). When the inclusion test does not identify any leaf-arc region, we simply represent the trajectory line Li with an isolated, colored sphere on the corresponding frame. 6

R ESULTS

This section presents experimental results obtained with our approach, implemented in C++ within TTK [6, 75]. Our implementation and our experimental datasets (Sec. 2.2) are available at this address: https://github.com/tloloum/DebrisTracer. Appendix A documents the parameters of our approach and the protocol we followed to adjust them, once for all, based on our reference dataset (Fig. 1). It also documents our parameter setting protocol for TrackMate [23, 76], the established tool used by domain experts, to which we will compare. Three domain experts (one junior and two senior scientists, all co-authors of this paper) were involved in our work. They contributed to the problem definition (Sec. 2.1), the bibliographic study (Sec. 1.1), the data acquisition (Sec. 2.2), the formalization of the observations, assumptions and hypotheses (Sec. 2.3), the physics-driven post-processing (Sec. 5), and the interpretation of the case studies (Sec. 6.2). 6.1

Validation

We evaluate the accuracy of our approach based on several criteria, including its ability to predict derived experimental measures (ejected mass and crater profiles, Sec. 2), hence assessing its physical relevance. (i) Debris count: Fig. 11 provides a qualitative and quantitative comparison between TrackMate and DebrisTracer in terms of debris identification. As it is based on persistence, DebrisTracer manages to correctly identify debris of a variety of sizes, including very small ones

Fig. 11: Qualitative and quantitative comparison of debris characterization between an established domain tool (TrackMate [23, 76], purple circles) and DebrisTracer (yellow spheres) at time step 50 (top) and 200 (bottom) of our reference dataset (Fig. 1). Visually, our approach makes fewer misses in debris detection (a)-(b). This is confirmed quantitatively with a larger number of identified fragments throughout the sequence (c).

Fig. 11(a)-(b). This is confirmed quantitatively, Fig. 11(c), with a larger number of identified debris fragments through the entire sequence. (ii) Trajectory path linearity: We evaluate the improvement of our tracking, in comparison to the original off-the-shelf topology tracking framework [72] considered in this work. For that, we focus on the trajectory paths (Sec. 5.1) extracted by both approaches (after assignment). Specifically, we evaluate their relevance with regard to the constant velocity assumption (PA1, Sec. 2.3). For that, we estimate how well these paths already align with a straight line, by considering the coefficient of determination (in [0, 1]) associated to their linear regression [32, 67]. Specifically, the original tracking framework obtains an average coefficient of 0.755, while our modifications (Sec. 4.2) yield an average coefficient of 0.910, a 20.5% improvement. (iii) Ejected mass estimation: For our reference dataset (Fig. 1), the total ejected mass is evaluated experimentally to 324 mg from the crater volume (Sec. 2.2). To assess the physical relevance of the tracking frameworks considered in our work, we will use them to predict this mass, based on their tracking information. TrackMate does not provide direct methods for particle mass estimation and thus cannot be used directly for this quantitative evaluation. In order to still enable a baseline comparison, we suggest instead the following, easily implementable strategy, based on simple transformations of TrackMate’s outputs. With the selected detector backend (Appendix A), each debris is represented in TrackMate by a disk of constant area, whose size is user parameterized (in our setup, 7-pixel diameter). From this area information, we can estimate the volume of each debris under the hypothesis of spherical debris shape (similarly to Sec. 5.4), and hence its mass (given the uniform volume density physical assumption PA3, Sec. 2.3). By summing the computed mass of all the identified trajectories, this yields an estimated total mass of 1, 453 mg. This +348% over-estimation can be explained by the fact that the trajectories provided by TrackMate exhibit many temporal gaps. Then, a single debris fragment is tracked by several trajectories at several points in time, resulting in an over-estimation of its contribution to the total ejected mass. This issue might be addressed in the future by extending TrackMate with a tailored post-processing of trajectories (similar to our approach, Sec. 5). In contrast, while DebrisTracer correctly identifies more fragments (Fig. 11), it also tracks them individually better over time, resulting in an estimated total ejected mass of 296 mg. This −9% under-estimation is much closer to the experimental data (error reduction by a factor of 47.25), hence assessing the physical relevance of our approach. (iv) Crater profile reconstruction: We further evaluate the physical accuracy of the generated debris trajectories by estimating, from the tracking information, the crater profiles measured experimentally (Fig. 2) on our reference dataset (Fig. 1). For that, we estimate the ejected mass distribution on the target (i.e., at X = 0) by binning the target segment on the image, and estimating the contribution of each debris to the bins, given its computed trajectory and estimated mass, according to a spherical debris shape hypothesis (Fig. 12). This mass distribution yields a profile similar in shape to the experimental crater profiles. For visual comparison, we display these profiles in the same reference frame, Fig. 12(c), by scaling the Y -coordinate of the mass

Fig. 12: The debris trajectories (a) and shapes (b) computed by DebrisTracer enable the estimation of the distribution of ejected mass along the target (c). This distribution (yellow) nicely coincides with the experimental crater depth profile (purple), while a similar evaluation based on TrackMate (blue) yields a global over-estimation of the ejected mass.

Fig. 13: Distributions computed by DebrisTracer (reference dataset, Fig. 1). The angle/velocity scatterplot (a) reveals three distinct modes: thin high-speed (yellow), broad intermediate-speed (purple) and wide slow (blue) ejections. This split is confirmed by 2 discontinuities (b) in the overall complementary cumulative mass/velocity distribution (black).

distribution according to the estimated total mass (c.f., previous paragraph): the global profile minima are scaled at 91% for DebrisTracer and 448% for TrackMate, with regard to the global minimum of the crater profile measured experimentally. Fig. 12 shows that our approach provides an improved estimation of the crater profile, in particular by better capturing the edges of the crater. This validates the physical accuracy of the trajectories and masses estimated by DebrisTracer, enabling its reliable use for the interpretation of case studies (Sec. 6.2). 6.2 Experimental case studies To evaluate the application relevance of our framework, we proceed to its usage for the analysis of two dynamic fragmentation experiments. Appendices B, C and our companion video include further datasets, illustrating the stability and versatility of our approach. (i) Reference dataset (Fig. 1): This use case focuses on the setup described in Sec. 2.2, with a 90° impact angle against a target surface. This dataset is typical of such HVI recordings and we have used it as a reference for setting up the parameters of our approach (Appendix A). Our approach enables an accurate tracking of the debris fragments (one color per fragment, Fig. 1), and a precise estimation of their shape (enabling their mass estimation, Sec. 5.4). Although the initial time steps reveal a large blob containing most of the debris, DebrisTracer still enables a reliable estimation of the debris trajectories throughout the sequence, based on the linear regression (Sec. 5.1) implementing the physical assumption PA1 on constant individual velocities (Sec. 2.3). Then, the identification of debris at a later stage in the sequence still enables the reconstruction of their early trajectory (thanks to this physical input), even when most debris overlap (a challenging configuration for traditional topological approaches). This physics-driven trajectory estimation enables a reliable statistical analysis of the debris trajectories. Fig. 13(a) shows the scatterplot of debris ejection angles as a function of their axial velocities. In particular, this scatterplot exhibits a very specific pattern, enabling the visual identification of two axial speed cutoffs, delimiting three distinctive regimes. A first regime (yellow) involves few high-speed debris (> 400 m/s) within a thin ejection cone ([−10°, 10°]). This corroborates the hypothesis H1 (Sec. 2.3), describing a multi-phase ejection. Specifically, the yellow regime coincides with debris that could be expected from jetting (Sec. 2.1). A slower regime, starts for axial velocities below

Fig. 14: Analyzing an oblique launch (45° impact angle). In a pre-process, the dataset is rotated to align the target surface with the vertical axis (X = 0). In a post-process, the dataset is rotated back to its original angle for visualization purposes. Despite the variability in impact angle, DebrisTracer still provides an accurate debris trajectory and shape estimation (a)-(c). The derived angle/velocity scatterplot (d) suggests the presence of two modes, with a strong bias for the slower debris (blue) towards negative ejection angles (towards the projectile trajectory). This mode split is confirmed visually by a discontinuity (e) in the overall complementary cumulative mass/velocity distribution (black).

400 m/s. However, unexpectedly, this second regime seems to be split in two modes (purple and blue). In the first mode (purple), a significant number of intermediate-speed debris fragments ([120 m/s, 400 m/s]) travel within an ejection cone of intermediate width ([−35°, 35°]). In the second mode (blue), a much larger number of fragments travel within a larger cone ([−45°, 45°]) more slowly (< 120 m/s), at speeds which decrease for increasing ejection angles (triangle shape, blue point cloud in Fig. 13(a)). This denotes the energy absorption capabilities of the target’s porous material, characteristic of spallation (Sec. 2.1). This corroborates the hypothesis H2 (Sec. 2.3). The identification of a first ejection regime (yellow) and the split of the second (purple and blue) is confirmed by the complementary cumulative mass distribution, as a function of axial velocity, shown in Fig. 13(b). In particular, this plot shows clear discontinuities in the global distribution (black), matching the speed cutoffs delimiting the regimes, confirming the split of the debris population into three types (yellow, purple and blue). Overall, DebrisTracer confirms the experts’ hypotheses, while providing new insights (the second ejection regime exhibits distinct sub-processes). (ii) 45° projectile impact (Fig. 14): This use case focuses on the same acquisition setup (Sec. 2.2), but this time with a 45° impact angle. This experiment shows that the parameters we have selected in Appendix A are stable across datasets, as the trajectories and the shapes of the debris are also accurately estimated for this dataset. The statistical analysis of the extracted trajectories also corroborates the hypothesis H2 (Sec. 2.3) as the debris with the largest ejection angle are also the slowest. It also corroborates the hypothesis H1 (Sec. 2.3), as it reveals a split in two regimes in this case (blue and purple), whose transition is also marked by a discontinuity in the complementary cumulative mass distribution. The scatterplot of ejection angles as a function of axial speeds shows a clear distortion of the scatterplot towards negative angles for debris under 100 m/s. This reveals that a non-negligible portion of the debris follow a reflective ejection, which is not orthogonal to the surface target, but biased towards the projectile’s trajectory. This original, visual insight can be accurately quantified thanks to our approach, via statistical moments of the ejection angle (i.e., −2.24° on average). 6.3

Computational aspects

The processing of an input dataset by our approach involves several stages, for which we report here the time complexity and the timings obtained on a commodity desktop computer. First, for each time step, the persistence diagram is computed for
debris detection (Sec. 4.1). In 2D, this can be achieved in O nK α(nK ) steps after data sorting, where nK is the number of vertices in K and α is the inverse of the Ackermann function. Second, persistent maxima are selected in linear time, in at most O (nK ) steps. Third, persistent maxima are matched between consecutive frames via optimal assignment (Sec. 4.2), implemented with the Auction algorithm [5] in O (n2m ) steps in practice, where nm ≪ nK is the number of maxima. Third, trajectories are refined through physicsbased post-processing (Sec. 5). Linear regression is applied in linear

time (Sec. 5.1), while segment matching (Sec. 5.2) is quadratic in the number of segments in the worst case. Outlier filtering (Sec. 5.3) is linear. Overall, the three above phases require 3.5 s. of computation for our reference dataset (Fig. 1). For comparison, TrackMate, for a compatible output, took 9.7 s. to process our reference dataset (Fig. 1), resulting in a ×2.77 speedup for our approach. To additionally estimate the debris
mass distribution (Sec. 5.4), topological simplification (O nK log(nK ) steps) must be performed right before merge tree com
putation (O nK α(nK ) steps), so that the extracted merge-tree leaf regions do coincide with the selected persistent maxima. Overall, this mass estimation requires another 1.8 s. of computation with TTK. 6.4 Limitations In comparison to TrackMate, whose setting relies mostly on the adjustment of one important parameter (the estimated debris size), our approach involves more parameters. However, we documented a protocol for their adjustment (Appendix A) and we showed that the resulting parameters values were stable across datasets, in paricular across modalities (projectile launch versus laser pulse). Given their considerable speeds with regard to the capture frequency, the early debris are difficult to track reliably. Moreover, in certain cases, a non-negligible portion of the early debris blob may leave the image before splitting into multiple connected components, hence preventing the identification of its individual debris fragments. However, this latter issue is a limitation of the recording process, which captures in 2D a phenomenon that is intrinsically 3D, hence resulting in inevitable occlusion. Also, low persistence maxima are discarded from our analysis and thus, not tracked. Finally, our debris tracking (Sec. 4.2) is based on an assignement optimization, which may still produce few incorrect matchings, yielding physically implausible trajectories (later on filtered out by our post-processing, Sec. 5), leading to certain fragments not being tracked during certain time intervals. Despite this, DebrisTacer still manages to identify and track a sufficient portion of debris fragments, to enable their reliable visual analysis (Sec. 6.2). Finally, our debris volume estimation could be improved. We currently rely on a spherical debris shape assumption and more accurate estimations might be obtained by considering advanced shape hypotheses, possibly relying on a pre-documented dictionary of debris shapes. 7 C ONCLUSION This application paper presented DebrisTracer, a tailored framework for the reliable tracking of debris in HyperVelocity Impact (HVI) fast imaging. Extensive experiments demonstrated the accuracy improvement provided by DebrisTracer over established tools used by domain experts [23,76]. For instance, DebrisTracer reduces by a factor of 47.25 the error on the predicted ejected mass. Our improved tracking enables the trustworthy visual analysis of this complex space-time phenomenon. Our detailed analyses corroborated the initial hypotheses formulated by domain experts (multi-phase ejection, slow lateral debris), while providing new visual insights which were not directly visible from the experimental recordings. Specifically: • For frontal launches, the angle/velocity distributions revealed two sub-modes for the slow debris, the slowest mode clearly characterized by decreasing speeds for increasing ejection angles, denoting the energy absorption capabilities of the target material. • For oblique launches, the angle/velocity distributions revealed a partial reflective ejection towards the projectile’s trajectory. • Without a projectile (laser-based HVI), no initial high-speed debris ejection was observed in the early stages of the phenomenon. These finer characterizations of the material’s response to the impact enable an improved assessment of its usability in the applications. We believe our work opens several research avenues for the experimental analysis of HyperVelocity Impacts (HVI). In the future, we will proceed to detailed investigations of additional experimental campaigns, given the new analysis capabilities of DebrisTracer. To improve its estimation of ejected mass, and overcome the spherical debris shape assumption, we will consider more advanced strategies, for instance based on shape/mass databases of experimentally collected debris.

ACKNOWLEDGMENTS This work is partially supported by the European Commission grant ERC-2019COG “TORI” (ref. 863464, https://erc-tori.github.io/).

R EFERENCES [1] Time-varying reeb graphs for continuous space–time data. Computational Geometry, 2008. doi: 10.1016/j.comgeo.2007.11.001 2 [2] B. Aubert, D. Hébert, J. Rullier, E. Lescoute, L. Videau, and L. Berthe. Laser simulation of hypervelocity impact into porous graphite. Int. Jour. of Impact Engineering, 2025. doi: 10.1016/j.ijimpeng.2025.105474 3 [3] U. Bauer, B. D. Fabio, and C. Landi. An Edit Distance for Reeb Graphs. In Eurographics Workshop on 3D Object Retrieval, 2016. doi: 10.2312/3dor. 20161084 2 [4] K. Beketayev, D. Yeliussizov, D. Morozov, G. H. Weber, and B. Hamann. Measuring the distance between merge trees. In Proc. of TopoInVis. 2014. doi: 10.1007/978-3-319-04099-8_10 2 [5] D. P. Bertsekas and D. Castañon. Parallel synchronous and asynchronous implementations of the auction algorithm. Parallel Computing, 1991. doi: 10.1016/S0167-8191(05)80062-6 9 [6] T. Bin Masood, J. Budin, M. Falk, G. Favelier, C. Garth, C. Gueunet, P. Guillou, L. Hofmann, P. Hristov, A. Kamakshidasan, C. Kappe, P. Klacansky, P. Laurin, J. Levine, J. Lukasczyk, D. Sakurai, M. Soler, P. Steneteg, J. Tierny, W. Usher, J. Vidal, and M. Wozniak. An Overview of the Topology ToolKit. In TopoInVis, 2019. doi: 10.1007/978-3-030 -83500-2_16 7 [7] A. Bock, H. Doraiswamy, A. Summers, and C. T. Silva. TopoAngler: Interactive Topology-Based Extraction of Fishes. IEEE TVCG, 24(1):812– 821, 2018. doi: 10.1109/TVCG.2017.2743980 2 [8] P.-T. Bremer, G. H. Weber, J. Tierny, V. Pascucci, M. S. Day, and J. B. Bell. A topological framework for the interactive exploration of large scale turbulent combustion. In Proc. of e-Science, 2009. doi: 10.1109/e-Science .2009.42 2 [9] H. Carr, J. Snoeyink, and U. Axen. Computing contour trees in all dimensions. In Symp. on Dis. Alg., pp. 918–926. SIAM, Philadelphia, 2000. doi: 10.1016/S0925-7721(02)00093-7 2, 4, 5 [10] A. F. Cheng, A. S. Rivkin, P. Michel, J. Atchison, O. Barnouin, L. Benner, N. L. Chabot, C. Ernst, E. G. Fahnestock, M. Kueppers, P. Pravec, E. Rainey, D. C. Richardson, A. M. Stickle, and C. Thomas. Aida dart asteroid deflection test: Planetary defense and science objectives. Planetary and Space Science, 2018. doi: 10.1016/j.pss.2018.02.015 1 [11] E. L. Christiansen, K. Nagy, D. M. Lear, and T. G. Prior. Space station mmod shielding. Acta Astronautica, 65(7):921–929, 2009. doi: 10.1016/j. actaastro.2008.01.046 1 [12] G. S. Collins, H. J. Melosh, and B. A. Ivanov. Modeling damage and deformation in impact simulations. Meteoritics & Planetary Science, 39(2):217–231, 2004. doi: 10.1111/j.1945-5100.2004.tb00337.x 3 [13] T. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. 2009. 4 [14] T. Daniel, M. Olejniczak, and J. Tierny. Bondmatcher: H-bond stability analysis in molecular systems. IEEE TVCG, 2026. doi: 10.1109/TVCG. 2025.3634636 2 [15] S. Das, R. Sridharamurthy, and V. Natarajan. Time-varying extremum graphs. CGF, 2024. doi: 10.1111/cgf.15162 2 [16] H. Doraiswamy, V. Natarajan, and R. S. Nanjundiah. An exploration framework to identify and track movement of cloud systems. IEEE TVCG, 2013. doi: 10.1109/TVCG.2013.131 2 [17] H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. American Mathematical Society, 2009. 1, 2, 4, 5 [18] H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological Persistence and Simplification. DCG, 2002. doi: 10.1007/S00454-002-2885-2 2, 4 [19] H. Edelsbrunner, D. Morozov, and V. Pascucci. Persistence-sensitive simplification functions on 2-manifolds. In SoCG, 2006. doi: 10.1145/ 1137856.1137878 4 [20] H. Edelsbrunner and E. P. Mucke. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. on Graphics, 1990. doi: 10.1145/77635.77639 4 [21] T. Eidevåg, E. S. Thomson, S. Sollén, J. Casselgren, and A. Rasmuson. Collisional damping of spherical ice particles. Powder Technology, 2021. doi: 10.1016/j.powtec.2021.01.025 2 [22] W. Engelke, T. B. Masood, J. Beran, R. Caballero, and I. Hotz. Topologybased feature design and tracking for multi-center cyclones. In Proc. of TopoInVis, 2021. doi: 10.1007/978-3-030-83500-2_5 2

[23] D. Ershov, M.-S. Phan, J. W. Pylvänäinen, S. U. Rigaud, L. Le Blanc, A. Charles-Orszag, J. R. W. Conway, R. F. Laine, N. H. Roy, D. Bonazzi, G. Duménil, G. Jacquemet, and J.-Y. Tinevez. Trackmate 7: integrating state-of-the-art segmentation algorithms into tracking pipelines. Nature Methods, 2022. doi: 10.1038/s41592-022-01507-1 1, 2, 3, 7, 8, 9, 12 [24] S. Ghosh, Z. Deng, G. Kim, C. Goodman, J. Moreno, R. Nunez, M. A. Foster, and R. C. Hurley. Quantifying 3d ejecta velocities during highvelocity impact experiments into concrete. International Journal of Impact Engineering, 2025. doi: 10.1016/j.ijimpeng.2025.105543 2, 12 [25] C. Gueunet, P. Fortin, J. Jomier, and J. Tierny. Task-Based Augmented Contour Trees with Fibonacci Heaps. IEEE TPDS, 2019. doi: 10.1109/ TPDS.2019.2898436 4, 5 [26] P. Guillou, J. Vidal, and J. Tierny. Discrete Morse Sandwich: Fast Computation of Persistence Diagrams for Scalar Data – An Algorithm and A Benchmark. IEEE TVCG, 2023. doi: 10.1109/TVCG.2023.3238008 4 [27] H. Guo, D. Lenz, J. Xu, X. Liang, W. He, I. R. Grindeanu, H.-W. Shen, T. Peterka, T. Munson, and I. Foster. Ftk: A simplicial spacetime meshing framework for robust and scalable feature tracking. IEEE TVCG, 2021. doi: 10.1109/TVCG.2021.3073399 2 [28] C. Heine, H. Leitte, M. Hlawitschka, F. Iuricich, L. De Floriani, G. Scheuermann, H. Hagen, and C. Garth. A survey of topology-based methods in visualization. CGF, 2016. doi: 10.1111/CGF.12933 2 [29] D. Hébert, B. Jodar, B. Aubert, N. Teyssier, J.-L. Rullier, E. Ando, and I. Bertron. Comparison of two models for the simulation of hypervelocity impacts on porous graphite. AIP Conference Proceedings, 2023. doi: 10. 1063/12.0020464 1, 4 [30] D. Janches and J. Chau. Observed diurnal and seasonal behavior of the micrometeor flux using the arecibo and jicamarca radars. Journal of Atmospheric and Solar-Terrestrial Physics, 2005. doi: 10.1016/j.jastp. 2005.06.011 1 [31] M. Jutzi and P. Michel. Hypervelocity impacts on asteroids and momentum transfer. I. Numerical simulations using porous targets. Icarus, 2014. doi: 10.1016/j.icarus.2013.11.020 1 [32] R. E. Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 1960. doi: 10.1115/1.3662552 6, 8 [33] R. Kinslow. High-Velocity Impact Phenomena. Academic Press, New York, 1970. 2, 3 [34] A. E. Koniges, C. S. Debonnel, J. Andrew, D. Eder, D. Kalantar, N. Masters, A. Fisher, R. Anderson, B. Gunney, B. Brown, K. Sain, F. Bonneau, J.-L. Bourgade, J.-M. Chevalier, P. Combis, A. Geille, J.-P. Jadaud, Maroni, D. Raffestin, J.-L. Ulmer, J. Vierne, A. M. Tobin, M. Meyers, and H. Jarmakani. Experiments for the validation of debris and shrapnel calculations. Journal of Physics: Conference Series, 2008. doi: 10.1088/1742-6596/ 112/3/032072 1 [35] K. Kurosawa, Y. Nagaoka, H. Senshu, K. Wada, S. Hasegawa, S. Sugita, and T. Matsui. Dynamics of hypervelocity jetting during oblique impacts of spherical projectiles investigated via ultrafast imaging. Journal of Geophysical Research: Planets, 2015. doi: 10.1002/2014JE004730 3 [36] K. Kurosawa, T. Okamoto, and H. Genda. Hydrocode modeling of the spallation process during hypervelocity impacts: Implications for the ejection of martian meteorites. Icarus, 2018. doi: 10.1016/j.icarus.2017. 09.015 3 [37] S. Le Thanh and T. Weinkauf. A comparative study of different edit distance-based methods for feature tracking using merge trees on timevarying scalar fields. In Proc. of TopoInVis, 2025. doi: 10.1109/ TopoInVis68599.2025.00005 1, 2 [38] E. Lescoute, T. De Resseguier, and J. M. Chevalier. Gel versus aerogel to collect high velocity ejectas from laser shock-loaded metallic targets for post-recovery analyses. AIP, 2012. doi: 10.1063/1.3686236 2 [39] M. Li, D. Chatterjee, F. Glassmeier, F. Senf, and B. Wang. Tracking low-level cloud systems with topology. In Proc. of TopoInVis, 2025. doi: 10.1109/TopoInVis68599.2025.00013 2 [40] J. Lukasczyk, C. Garth, R. Maciejewski, and J. Tierny. Localized topological simplification of scalar data. IEEE TVCG, 2020. doi: 10.1109/TVCG. 2020.3030353 4 [41] J. Lukasczyk, C. Garth, G. H. Weber, T. Biedert, R. Maciejewski, and H. Leitte. Dynamic nested tracking graphs. IEEE TVCG, 2020. doi: 10. 1109/TVCG.2019.2934368 2 [42] P. Matura, S. Signetti, S. Moser, D. Gebhardt, E. Watson, A. SchindlerTyka, J. L. S. Murillo, N. Durr, M. Büttner, S. Jachmich, M. Lehnen, and U. Kruezi. Pellet shattering process for the iter disruption mitigation system – part ii: Synthetic-diagnostics-based fragment tracking, calibration, and validation of simulation models. HVI Symp., 2024. doi: 10.

1115/HVIS2024-063 2 [43] H. J. Melosh. Ejection of rock fragments from planetary bodies. Geology, 1985. doi: 10.1130/0091-7613(1985)13<144:EORFFP>2.0.CO;2 3 [44] T. Michikami, K. Moriguchi, S. Hasegawa, and A. Fujiwara. Ejecta velocity distribution for impact cratering experiments on porous and low strength targets. Planetary and Space Science, 2007. doi: 10.1016/j.pss. 2006.05.002 2 [45] M. C. Miller, J. R. Celeste, M. A. Stoyer, L. J. Suter, M. T. Tobin, J. Grun, J. F. Davis, C. W. Barnes, and D. C. Wilson. Debris characterization diagnostic for the nif. Review of Scientific Instruments, 2001. doi: 10. 1063/1.1310587 1 [46] A. M. Nakamura, K. Nomura, and S. Hasegawa. Simultaneous detection of the size and velocity of the largest ejecta particles with velocities exceeding 1 km s-1. PLANETARY SCIENCE JOURNAL, 2025. doi: 10. 3847/PSJ/ae1d76 2 [47] F. Nauleau, F. Vivodtzev, T. Bridel-Bertomeu, H. Beaugendre, and J. Tierny. Topological Analysis of Ensembles of Hydrodynamic Turbulent Flows – An Experimental Study. In IEEE LDAV, 2022. doi: 10. 1109/LDAV57265.2022.9966403 2 [48] E. Nilsson, J. Lukasczyk, W. Engelke, T. B. Masood, G. Svensson, R. Caballero, C. Garth, and I. Hotz. Exploring cyclone evolution with hierarchical features. In Proc. of TopoInVis, 2022. doi: 10.1109/TopoInVis57755. 2022.00016 2 [49] E. Nilsson, J. Lukasczyk, T. B. Masood, C. Garth, and I. Hotz. Towards benchmark data generation for feature tracking in scalar fields. In Proc. of TopoInVis, 2022. doi: 10.1109/TopoInVis57755.2022.00017 1, 2 [50] E. Nilsson, J. Lukasczyk, T. B. Masood, C. Garth, and I. Hotz. Probabilistic gradient-based extrema tracking. In Proc. of TopoInVis, 2023. doi: 10. 1109/TopoInVis60193.2023.00014 2 [51] M. Nishida, K. Kuzuya, K. Hayashi, and S. Hasegawa. Effects of alloy type and heat treatment on ejecta and crater sizes in aluminum alloys subjected to hypervelocity impacts. International Journal of Impact Engineering, 2013. doi: 10.1016/j.ijimpeng.2012.11.004 2 [52] P. Oesterling, C. Heine, G. H. Weber, D. Morozov, and G. Scheuermann. Computing and visualizing time-varying merge trees for high-dimensional data. In Proc. of TopoInVis, 2017. doi: 10.1007/978-3-319-44684-4_5 2 [53] N. Otsu. A threshold selection method from gray-level histograms. IEEE Transactions on Systems, Man, and Cybernetics, 1979. doi: 10.1109/ TSMC.1979.4310076 7 [54] S. Parsa. A deterministic o(m log m) time algorithm for the reeb graph. In SoCG, 2012. doi: 10.1145/2261250.2261289 2 [55] G. Peyré and M. Cuturi. Computational optimal transport: With applications to data science. Found. Trends Mach. Learn., 2019. 5 [56] M. Pont, J. Vidal, J. Delon, and J. Tierny. Wasserstein Distances, Geodesics and Barycenters of Merge Trees. IEEE TVCG, 2022. doi: 10.1109/TVCG. 2021.3114839 2 [57] G. Prudhomme, P. Mercier, L. Berthe, J. Bénier, and P.-A. Frugier. Frontal and tilted pdv probes for measuring velocity history of laser-shock induced calibrated particles. Journal of Physics: Conference Series, 2014. doi: 10. 1088/1742-6596/500/14/142022 2 [58] N. Ravi, V. Gabeur, Y.-T. Hu, R. Hu, C. Ryali, T. Ma, H. Khedr, R. Rädle, C. Rolland, L. Gustafson, E. Mintun, J. Pan, K. V. Alwala, N. Carion, C.-Y. Wu, R. Girshick, P. Dollár, and C. Feichtenhofer. Sam 2: Segment anything in images and videos. In ICLR, 2025. doi: 10.48550/arXiv.2408. 00714 1, 2 [59] J. Reininghaus, J. Kasten, T. Weinkauf, and I. Hotz. Efficient computation of combinatorial feature flow fields. IEEE TVCG, 2012. doi: 10.1109/ TVCG.2011.269 2 [60] B. Reynier, B. Jodar, T. Geral, B. Aubert, J. L. Rullier, E. Lescoute, C. Le Bras, L. Taddei, J. M. Chevalier, D. Hébert, and M. Arrigoni. Laserdriven cratering into porous graphite: Experimental investigation on ejecta distribution. International Journal of Impact Engineering, 2026. doi: 10. 1016/j.ijimpeng.2025.105476 2, 4 [61] B. Reynier, R. M. Mircioaga, J. Le Clanche, L. Taddei, J. M. Chevalier, D. Hébert, and M. Arrigoni. High-velocity laser-driven flyer impact on paraffin gel. International Journal of Impact Engineering, 2025. doi: 10. 1016/j.ijimpeng.2025.105311 2 [62] H. Saikia, H.-P. Seidel, and T. Weinkauf. Extended branch decomposition graphs: Structural comparison of scalar data. CGF, 2014. doi: 10.1111/cgf .12360 2 [63] H. Saikia and T. Weinkauf. Global Feature Tracking and Similarity Estimation in Time-Dependent Scalar Fields. CGF, 2017. doi: 10.1111/cgf. 13163 2

[64] H. Saikia and T. Weinkauf. Fast topology-based feature tracking using a directed acyclic graph. In Proc. of TopoInVis, 2020. doi: 10.1007/978-3 -030-43036-8_10 2 [65] J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, S. Saalfeld, B. Schmid, J.-Y. Tinevez, D. J. White, V. Hartenstein, K. Eliceiri, P. Tomancak, and A. Cardona. Fiji: an open-source platform for biological-image analysis. Nature Methods, 2012. doi: 10.1038/nmeth.2019 2 [66] A. Schnorr, D. N. Helmrich, D. Denker, T. W. Kuhlen, and B. Hentschel. Feature tracking by two-step optimization. IEEE TVCG, 2020. doi: 10. 1109/TVCG.2018.2883630 2 [67] G. A. F. Seber and A. J. Lee. Linear Regression Analysis. John Wiley & Sons, 2 ed., 2003. doi: 10.1002/9780471722199 6, 8 [68] G. Seisson, G. Prudhomme, P.-A. Frugier, D. Hébert, E. Lescoute, A. Sollier, L. Videau, P. Mercier, M. Boustie, and L. Berthe. Dynamic fragmentation of graphite under laser-driven shocks: Identification of four damage regimes. International Journal of Impact Engineering, 2016. doi: 10.1016/j.ijimpeng.2015.12.012 3 [69] N. Shivashankar and V. Natarajan. Parallel Computation of 3D MorseSmale Complexes. CGF, 2012. doi: 10.1111/J.1467-8659.2012.03089.X 2 [70] N. Shivashankar, P. Pranav, V. Natarajan, R. van de Weygaert, E. P. Bos, and S. Rieder. Felix: A topology based framework for visual exploration of cosmic filaments. IEEE TVCG, 2016. doi: 10.1109/TVCG.2015.2452919 2 [71] M. Soler, M. Petitfrere, G. Darche, M. Plainchault, B. Conche, and J. Tierny. Ranking Viscous Finger Simulations to an Acquired Ground Truth with Topology-Aware Matchings. In IEEE LDAV, 2019. doi: 10. 1109/LDAV48142.2019.8944365 2 [72] M. Soler, M. Plainchault, B. Conche, and J. Tierny. Lifted Wasserstein matcher for fast and robust topology tracking. In IEEE LDAV, 2018. doi: 10.1109/LDAV.2018.8739196 1, 2, 3, 5, 6, 8 [73] R. Sridharamurthy, T. B. Masood, A. Kamakshidasan, and V. Natarajan. Edit Distance between Merge Trees. IEEE TVCG, 2020. doi: 10.1109/ TVCG.2018.2873612 2 [74] H. Theisel and H.-P. Seidel. Feature flow fields. In Proc. of Symposium on Data Visualisation, 2003. doi: 10.5555/769922.769938 2 [75] J. Tierny, G. Favelier, J. A. Levine, C. Gueunet, and M. Michaux. The Topology ToolKit. IEEE TVCG, 2017. doi: 10.1109/TVCG.2017.2743938 7 [76] J.-Y. Tinevez, N. Perry, J. Schindelin, G. M. Hoopes, G. D. Reynolds, E. Laplantine, S. Y. Bednarek, S. L. Shorte, and K. W. Eliceiri. Trackmate: An open and extensible platform for single-particle tracking. Methods, 2017. doi: 10.1016/j.ymeth.2016.09.016 1, 2, 3, 7, 8, 9, 12 [77] A. A. Valsangkar, J. M. Monteiro, V. Narayanan, I. Hotz, and V. Natarajan. An exploratory framework for cyclone identification and tracking. IEEE TVCG, 2019. doi: 10.1109/TVCG.2018.2810068 2 [78] E. Watson, M. Gulde, and S. Hiermaier. Fragment tracking in hypervelocity impact experiments. Procedia Engineering, 2017. doi: 10.1016/j. proeng.2017.09.770 2 [79] G. Weber, P.-T. Bremer, M. Day, J. Bell, and V. Pascucci. Feature tracking using reeb graphs. In Proc. of TopoInVis, 2011. doi: 10.1007/978-3-642 -15014-2_20 2 [80] T. Weinkauf, H. Theisel, A. Van Gelder, and A. Pang. Stable feature flow fields. IEEE TVCG, 2011. doi: 10.1109/TVCG.2010.93 2 [81] F. Wetzels, M. Pont, J. Tierny, and C. Garth. Merge Tree Geodesics and Barycenters with Path Mappings . IEEE TVCG, 2024. doi: 10.1109/TVCG .2023.3326601 2 [82] W. Widanagamaachchi, C. Christensen, V. Pascucci, and P.-T. Bremer. Interactive exploration of large-scale time-varying data using dynamic tracking graphs. In LDAV, 2012. doi: 10.1109/LDAV.2012.6378962 2 [83] Y. Yamaguchi, K. Matsubara, and A. M. Nakamura. Hypervelocity impact experiments on polycarbonate targets: Cratering efficiency and crater shape. In HVI Symp., 2025. doi: 10.1115/HVIS2024-046 2, 4 [84] Y. Yamaguchi and A. M. Nakamura. Experimental study on the effect of target properties on high-velocity ejecta from rocks. ICARUS, 451, JUN 2026. doi: 10.1016/j.icarus.2026.117005 2 [85] L. Yan, T. B. Masood, R. Sridharamurthy, F. Rasheed, V. Natarajan, I. Hotz, and B. Wang. Scalar field comparison with topological descriptors: Properties and applications for scientific visualization. CGF, 2021. doi: 10. 1111/cgf.14331 1, 2 [86] J. A. Zukas, T. Nicholas, H. F. Swift, L. B. Greszczuk, and D. R. Curran. Impact Dynamics. John Wiley & Sons, New York, 1982. 3

B

Fig. 15: Analyzing a laser-pulse based HVI acquired with ultra-highspeed cameras. Despite this change in physics and acquisition modality, DebrisTracer still provides an accurate debris trajectory and shape estimation (a)-(c). The derived angle/velocity scatterplot (d) also suggests the presence of two groups of debris (purple: fast, blue: slow), also reported in the complementary cumulative mass/velocity distribution (e).

A

PARAMETER SETTING

This appendix describes our protocol for adjusting the parameters of our approach (DebrisTracer) and those of TrackMate [23, 76], the established tool used by domain experts, to which we will compare. (i) DebrisTracer: The trajectory post-processing of our approach (Sec. 5) relies on several thresholds. However, this step aims mostly at discarding physical outliers. Then, these parameters have been set once and for all, to conservative values. Specifically, given two consecutive segments (Sec. 5.2), their temporal proximity threshold (δmax ) has been set to 30 time steps, the maximum absolute value of their angle (θmax ) to 20°, their spatial continuity (dmax ) to the size of the largest debris we have observed (i.e., 15-pixel diameter). Finally, we selected a conservative value (80°) for the absolute value of the maximum angle Θmax between a trajectory line and the X-axis (Sec. 5.3). Earlier in our pipeline, the tracking of debris identified as maxima of f (Sec. 4) relies on the following list of parameters: 1. a persistence threshold (εmin , Sec. 4.1), expressed as a fraction of the function span, enables, on a per time step basis, the identification of the debris as the most salient maxima of f . Since the datasets are provided as gray scale images and since the target is present on all time steps, the function span is stable across time. Hence, this parameter is set to a common value for all time steps. 2. a scaling vector (sx , sy , sz , Sec. 4.2) enables the assignment to favor a horizontal displacement in the debris tracking. 3. a dedicated destruction weight (ω, Sec. 4.2), expressed as a ratio of the bounding box, prevents the assignment between consecutive time steps of debris which are too far from each other. We adjusted these 5 parameters as follows, by restricting our analysis to the dataset presented in Fig. 1, which is a typical example from our database. After an initial visual inspection, we identified relevant exploration intervals for these parameters (εmin ∈ [0.05, 0.10], sx ∈ [0, 1], sy ∈ [0, 1], sz ∈ [0, 1] and ω ∈ [0.01, 0.05]), which we uniformly sampled (10 samples each). Next, we ran our entire analysis (Secs. 4 and 5) for each resulting combination of parameter samples and we selected as optimal parameters the values which resulted in the largest number of maintained trajectory lines for the time step 50, which is particularly challenging given the small size and large speeds of its debris. Specifically, this exploration yielded the following optimal parameter values: εmin = 0.075, ω = 0.02, sx = 0.1, sy = 0.9, sz = 0.1. (ii) TrackMate: this tool supports a number of algorithms for particle detection, all relying on a fundamental parameter, the diameter of the particle to track. For that, we considered the average of the debris sizes we have observed (i.e., 7-pixel diameter). For our experiments, we selected the algorithm Laplacian of Gaussian, as it provided the largest number of detections for the time step 50 of our reference dataset (Fig. 1). Note that this algorithm has also been used in recent papers on debris tracking [24]. Moreover, this approach is a relevant geometrical alternative to our topological strategy (Sec. 4.1) as it estimates salient maxima, not by persistence, but via a geometrical characterization (based on a Gaussian smoothing followed by Laplacian computation). Regarding tracking, TrackMate supports several approaches, including assignment and overlap based approaches. Specifically, for each method we only maintained the extracted trajectories whose linear regression was intersecting the target surface at X = 0. Then, we selected the tracking approach which maximized the number of maintained trajectories at the time step 50 of our reference dataset (Fig. 1).

L ASER PULSE IMPACT

This use case (Fig. 15) covers a different dynamic fragmentation modality (based on laser pulse instead of projectile launch) to evaluate the versatility of our approach. In this experiment, specific cameras have been used (Sec. 2.2), yielding distinct gray scale levels from our previous use cases based on projectile launches. Despite these changes, DebrisTracer still identifies accurately the trajectories and the shapes of the debris, in particular the early high-speed debris, given the high capture frequency. This validates the versatility of DebrisTracer as well as the stability of its parameters (Appendix A). In the early images, Fig. 15(a), the debris can be visually separated into two groups, the left one corresponding to slow debris and the right one to faster debris. In Fig. 15(b), most of the fast debris have left the image. This observation is also illustrated by the analysis of the scatterplot of ejection angles as a function of axial velocity, Fig. 15(c), where two groups of debris can be identified, corroborating hypothesis H1 (Sec. 2.3). Our analysis also enables further original insights: in contrast to a projectile-based HVI, the initial high-speed ejection in a laser-based HVI does not seem restricted to a thin cone, as found in our first use case (yellow, Fig. 13).

C

R ESULT GALLERY

Fig. 16 provides a gallery of additional tracking results, obtained with DebrisTracer, on several hypervelocity impact acquisitions based on projectile launches (90° and 45° impact angles). These additional results illustrate the versatility and stability of our approach.

Fig. 16: Gallery of tracking results (colored regions) obtained with DebrisTracer on several hypervelocity impact acquisitions ((a)-(d), time is represented with the horizontal axis) based on projectile launches (90° and 45° impact angles). DebrisTracer enables an accurate and physically plausible tracking of debris trajectories, enabling a trustworthy statistical analysis of the debris population.