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Direct Topology Tracking in Continuous Implicit Models

Guanqun Ma, David Lenz, Kaiyuan Tang, Hanqi Guo, Chaoli Wang, Tom Peterka, Bei Wang

arXiv:2609.12157v1cs.CGcs.LG

TL;DR

Grid-based topology analysis can discard the smoothness of continuous implicit models and introduce discretization artifacts. This paper tracks critical points by querying models and derivatives directly, and reports robust, accurate trajectories across analytic, MFA, and INR representations, including bifurcations.

  • Problem

    Topology analysis of continuous implicit models remains largely unexplored, while grid-based methods require sampling that discards smoothness and differentiability and may introduce aliasing or resolution-dependent artifacts.

    Source: §1 · p. 1

  • Method

    The framework tracks critical points in continuous implicit models by querying model derivatives, correcting ODE predictions to restore the zero-gradient condition, and reconstructing branches at degenerate events.

    Source: §1 · p. 2

  • Results

    Experiments across analytic functions, MFA models, and INR representations demonstrate robust, accurate critical-point trajectories, including bifurcations.

    Source: §8 · p. 8

  • Takeaways & Limitations

    The work establishes a foundation for topological data analysis on continuous implicit models and supports topology-aware operations in model representations and learning pipelines.

    Source: §8 · p. 8

    The 3D case requires substantially more boundary initialization, degenerate-point detection, and trajectory tracing, resulting in longer runtime; the goal is feasibility, not real-time optimization.

    Source: §7.3 · p. 8

Abstract

from arXiv · show

We present a framework for tracking topological features directly within continuous implicit models. Such models, including implicit neural representations (INRs) and multivariate functional approximations (MFAs), are increasingly adopted to represent scientific data without the resolution constraints of discrete grids. They offer compact, smooth, and differentiable representations of complex fields, enabling new opportunities for high-performance data storage, reconstruction, and analysis. Given a continuous implicit model, our method tracks the evolution of critical points by querying the model and its derivatives, thereby eliminating the need to resample onto a grid. This approach enables faithful feature tracking while avoiding discretization-induced artifacts such as aliasing. We demonstrate the generality of our framework across a range of implicit representations, including analytic functions, MFAs, and INRs, and show that it produces smooth, coherent critical point trajectories. By enabling feature tracking directly on continuous representations, our method supports a new class of feature-driven visualization workflows centered on implicit models.

1 INTRODUCTION

Continuous implicit models offer smooth, differentiable, resolution-independent representations, but topology analysis commonly relies on grids that can discard these properties. This work tracks critical points directly in analytic functions, MFAs, and INRs using model queries and derivatives.

1 INTRODUCTION

  • Grid-based topology analysis can discard implicit models’ smoothness and differentiability and introduce aliasing or resolution-dependent artifacts.The section frames direct tracking as a way to avoid discretizing the domain.Source: §1 · p. 1
  • The framework tracks critical points directly in continuous implicit models, preserving differentiability and eliminating the need for discretization.Earlier work extracted critical points from MFAs [25] and later extracted contours, Jacobi sets, and ridge-valley graphs [24].Source: §1 · p. 2
  • For time-varying scalar fields, the pipeline queries models and derivatives directly, combines ODE prediction with correction of the zero-gradient condition, and reconstructs branches at degenerate events.The reconstruction handles trajectory appearance, disappearance, merging, and splitting.Source: §1 · p. 2
  • The method is evaluated on analytic functions, MFAs, and INRs, without constructing a full discrete grid representation.The evaluation covers 2D and 3D time-varying models.Source: §1 · p. 2

2 RELATED WORK

Prior work established continuous MFA and INR representations and several approaches to topology extraction and tracking. This work adapts derivative-based continuation to continuous implicit models, explicitly correcting criticality and reconstructing branches at degenerate events.

2 RELATED WORK

  • MFA uses smooth B-splines that support function and derivative evaluation, and prior work extracted critical points [25], contours, Jacobi sets, and ridge-valley graphs [24] directly from MFA.These results motivate MFA as a substrate for continuous topology analysis.Source: §2 · p. 2
  • INRs learn continuous mappings from coordinates to signal values; CoordNet is used here as a representative architecture for evaluating the framework.INR research spans images, videos, volumes, and scientific data.Source: §2 · p. 2
  • FFF represents feature trajectories as streamlines of a derived space-time vector field, providing derivative-based continuation rather than independent matching across time steps.The related-work discussion also surveys combinatorial tracking, merge-tree regions, persistence metrics, and partial feature matching [47] [48] [37].Source: §2 · p. 2
  • Unlike FFF’s drift-prone continuation, this method refines each ODE prediction until the zero-gradient condition is met and reconstructs branches at degenerate events.It localizes degenerate points and forms trajectory graphs for multiple merging or splitting branches, rather than representing pair creation or annihilation only as a fold tangent to a time slice.Source: §2 · pp. 2–3
  • The formulation applies to d-dimensional time-varying scalar fields and is evaluated in 2D and 3D, while computational cost increases with dimension.The cited SFFF critical-point formulation is limited to 2D time-dependent vector fields; its 3D application addresses parallel-vector lines.Source: §2 · p. 3

3 TECHNICAL BACKGROUND

The background introduces continuous MFA and CoordNet representations, then reviews particle tracing as numerical integration of a velocity field. These foundations supply model queries and trajectory-integration concepts for the paper’s tracking method.

3.1 Multivariate Functional Approximation

  • MFA encodes data as a continuous function using compactly represented tensor-product B-spline coefficients, with analytic differentiation and arbitrary-dimensional queries.A degree-p spline is piecewise polynomial with p −1 continuous derivatives; knots partition its domain and control points shape the curve or surface.Source: §3.1 · p. 3
  • The 1D B-spline formulation generalizes to multidimensional domains through tensor-product basis construction, illustrated by a curve and a surface.The figure distinguishes control points, control mesh, and the resulting curve or surface.Source: §3.1 · p. 3, Figure 2 · p. 3

3.2 Implicit Neural Representation: CoordNet

  • CoordNet takes spatial-temporal coordinates as input and predicts data values, providing a continuous field representation after training.Its fully connected architecture uses SIREN-based residual blocks with sinusoidal activations; the paper converts learned parameters to float64 for inference.Source: §3.2 · p. 3

3.3 Particle Tracing

  • Particle tracing numerically integrates an initial value problem in which a particle’s position evolves under a time-dependent velocity field.Classical RK4 advances the time-parameterized solution; arc-length-parameterized RK4 instead uses spatial step size to control displacement, with temporal change determined accordingly.Source: §3.3 · pp. 3–4

4 METHOD

The method tracks critical-point trajectories directly in continuous implicit models, querying model derivatives and combining trajectory integration with boundary initialization, degenerate-event handling, and correction. It applies to representations supporting function evaluation and higher-order derivatives.

4.1 Critical Point Tracking

  • Critical-point paths satisfy the zero-spatial-gradient condition and are advanced using a velocity derived by differentiating that condition over time.The velocity formulation assumes a nonsingular spatial Hessian along regular trajectory portions; degenerate cases are handled separately.Source: §4.1 · p. 4

4.2 Initialization

  • Boundary critical points seed tracing; candidate samples initialize Newton root finding rather than define a discretized scalar field.The approach does not guarantee formal coverage, so highly localized or closely spaced structures may need finer or adaptive seeding.Source: §4.2 · pp. 4–5

4.3 Degenerate Cases

  • The pipeline detects degenerate events, where the spatial Hessian is singular, using Newton’s method and traces paths outward from critical points on a surrounding hypercube.It assumes these events are isolated in space-time; higher-order degeneracies are outside the formulation’s scope.Source: §4.3 · p. 5

4.4 Particle Tracing

  • Particle tracing combines time- and arc-length-parameterized RK4 steps, corrects predictions with fixed-time Newton iterations, and splits trajectories at detected degenerate crossings.It removes duplicate trajectories and connects nearby endpoints to degenerate points using spatial and temporal step-size tolerances.Source: §4.4 · pp. 5–6

5 EXPERIMENTAL CONFIGURATION

Experiments evaluate critical-point tracking on time-varying continuous models, comparing against discrete baselines in 2D and assessing scalability in 3D. The inputs include analytic functions and scientific datasets represented with MFA or INR models.

5.1 Evaluation Protocol

  • Evaluation uses ground-truth trajectories for analytic functions and critical-point-condition checks for MFA and INR models, with 2D comparisons against SFFF and LWM.Reported measures include gradient norm, connected components, analytic-function branch counts, and a scale-normalized Hessian determinant for MFA and INR degenerate points.Source: §5.1 · p. 6

5.2 Continuous Models as Input

  • The 2D inputs include analytic functions and Vortex Street and Heated Cylinder datasets represented with MFA and CoordNet models.The scientific datasets use selected time intervals intended to focus on developed flow structures.Source: §5.2 · p. 6
  • Scalability tests include a 3D Quartic Potential and a 3D Vortex dataset represented using MFA.The 3D Vortex evaluation uses vorticity magnitude as its scalar field.Source: §5.2 · p. 6

6 CRITICAL POINT TRACKING FROM ANALYTIC FUNCTIONS

Across three analytical scalar fields, the method tracks critical-point trajectories against known ground truth and identifies degenerate points.

6 CRITICAL POINT TRACKING FROM ANALYTIC FUNCTIONS

  • The analytic experiments use s = r/40, selected because the number of connected components remains stable across step sizes.The parameter selection is reported for all three functions.Source: §6 · p. 6
  • The extracted trajectories match ground truth exactly, and the spatial gradient norm stays below the prescribed threshold.The analytic-function evaluation is summarized in Table 1.Source: §6 · p. 6, Table 1 · p. 7
  • All three analytic cases are tracked accurately, including bifurcations and all four degenerate points in each quartic potential.The 2D Quartic Rotation case also tracks nine rotating trajectories.Source: §6 · p. 7, Figure 6 · p. 7

7 CRITICAL POINT TRACKING FROM MFA AND INR MODELS

Across MFA and INR datasets, implicit extraction satisfies the critical-point condition of the continuous models and recovers trajectory structures with local differences from sampled methods.

7 CRITICAL POINT TRACKING FROM MFA AND INR MODELS

  • For every MFA and INR model, implicit extraction keeps maximum spatial gradient norms below ϵ; its degenerate-point determinant values are smaller than SFFF and LWM's.These residual comparisons test adherence to the continuous model's critical-point and Hessian-singularity conditions, not complete trajectory correctness.Source: §7 · p. 7

7.1 Tracking Results for 2D MFA Models

  • In both 2D MFA models, implicit extraction and SFFF have broadly similar global structures, but zooms expose local loop and trajectory differences; LWM zigzags.Implicit extraction recovers a closed loop in Vortex Street and a small loop absent from SFFF in Heated Cylinder.Source: §7.1 · p. 7, Figure 8 · p. 8, Figure 9 · p. 8

7.2 Tracking Results from 2D INR Models

  • In the two 2D INR models, implicit extraction yields smooth trajectories and resolves a four-trajectory junction that SFFF misses, while SFFF and LWM show local artifacts.An equivalent discrete method at s = r/960 would require a 7680 × 960 × 1440 grid exceeding 84.9 GB; INR tracking also takes longer than MFA tracking.Source: §7.2 · pp. 7–8, Table 3 · p. 7

7.3 Tracking Results for 3D MFA Model

  • Implicit and discrete extraction recover comparable large-scale critical-point structures, while implicit extraction directly queries the continuous MFA and avoids constructing the 17.7 GB sampled representation required by the discrete pipeline.Source: §7.3 · p. 8

8 CONCLUSION

The framework tracks critical points directly on continuous implicit models, with experiments across analytic, MFA, and INR representations. Its step-size procedure selects parameters by refining them until connected-component counts stabilize.

8 CONCLUSION

  • The framework reports robust, accurate critical-point trajectory tracking, including bifurcations, across analytic functions, MFAs, and INRs.The conclusion notes limitations from costly INR derivatives, nearly constant fields, parameter selection, seed coverage, and numerical tolerance.Source: §8 · p. 8
  • The conclusion figures compare implicit extraction with SFFF and LWM for INR models, and with discrete extraction for the 3D Vortex MFA.The INR figures show trajectory views with zoomed regions; the 3D MFA figure includes magnified trajectories and locations across four time steps.Source: Figure 10 · p. 9, Figure 11 · p. 9, Figure 12 · p. 9

A STEP-SIZE PARAMETERIZATION

  • Step sizes are refined through convergence studies until the number of connected components stabilizes; MFA uses separate spatial and temporal steps, while analytic and INR cases use equal steps when scales are comparable.The candidates are ss = ls/k and st = lt/k for MFA, and ss = st = r/k for analytic and INR models.Source: A STEP-SIZE PARAMETERIZATION · p. 12

B HARDWARE AND IMPLEMENTATION

The implementation combines parallelized analytic and MFA evaluation with LibTorch-based INR derivatives. The supporting sections describe derivative approximations, complexity, and parameter-selection procedures.

B HARDWARE AND IMPLEMENTATION

  • Analytic and MFA models use TBB with one thread per physical core; INR models use LibTorch and MPI, with autograd for first derivatives and central finite differences for higher orders.Performance measurements use an eight-core Intel i9 CPU with 32 GB DDR4 memory.Source: B HARDWARE AND IMPLEMENTATION · p. 12

C CENTRAL FINITE-DIFFERENCE FORMULA

  • The hybrid derivative strategy retains autograd for accurate first-order gradients and uses central finite differences for higher-order derivatives to avoid higher-order autograd's memory and computational overhead.The second-order finite-difference approximations incur O(h^2) truncation error; CoordNet inputs normalized to [−1, 1]^3 use h = e−4.Source: C CENTRAL FINITE-DIFFERENCE FORMULA · p. 12

D COMPUTATIONAL COMPLEXITY

  • The pipeline complexity is O((Ninitial+Nstart/s)E), where E is one model value-and-gradient evaluation; MFA has E = O(pd), while INR has E = O(LW^2).INR evaluation cost is dominated by dense matrix multiplications during forward–backward evaluation.Source: D COMPUTATIONAL COMPLEXITY · p. 12

E PARAMETER JUSTIFICATION FOR ANALYTIC FUNCTIONS

F PARAMETER JUSTIFICATION FOR SCIENTIFIC DATASETS

  • The scientific-dataset parameter study evaluates gradient norm, loop count, and connected-component count, focusing primarily on obstacle-free regions when applicable.The obstacle-free region is identified as the meaningful domain.Source: F PARAMETER JUSTIFICATION FOR… · p. 13

F.1 Parameter Selection of Step Size for INR Models

  • For both INR models, the selected step size is s = r/80, based on stability in the number of connected components.The stability plots are shown in Fig. 14.Source: §F.1 · p. 13, Figure 14 · p. 13

F.2 Ablation Study on the Number of Seeds

Analytic functions are robust to seed-count choice, while the experiments use 40 seeds for analytic functions and 80 samples for INR models.

F.2 Ablation Study on the Number of Seeds

  • Analytic functions are robust to the choice of seed count.The assessment reports initial-point counts at t = 0.Source: §F.2 · p. 13
  • The experiments use 40 seeds along the shortest domain dimension for analytic functions and 80 samples for INR models once results stabilize.Sampling density is scaled proportionally along the remaining dimensions.Source: §F.2 · p. 13
  • At t = 0, Fig. 15 relates initial-point counts to seed counts for analytic functions and INR models.The figure places analytic functions on the left and INR models on the right.Source: Figure 15 · p. 13

F.3 Ablation Study on ϵ

Across the model ablations, connected-component counts generally stabilize as step size decreases, while loop counts can remain sensitive; reported spatial gradient norms stay below e−10.

F.3 Ablation Study on ϵ

  • The ϵ ablation finds initial-point counts largely stable across a wide threshold range, motivating ϵ = e−10 for all experiments.ϵ defines the threshold ||∇fx|| < ϵ.Source: Figure 16 · p. 13, §F.3 · p. 13

F.4 Ablation Study on ϵk

  • The ϵk ablation plots degenerate-point counts against ϵk, the threshold on the scale-normalized Hessian.The supplied results identify the plotted quantity and threshold but do not report the trend.Source: Figure 17 · p. 13, §F.4 · p. 13

F.5 Vortex Street MFA Model

  • For the Vortex Street MFA, connected-component counts stabilize as step size decreases, but loop counts may rise as marginal cycles separate; gradient norms stay below e−10.The connected-component count is used to select step size because it captures large-scale structure and converges reliably.Source: Figure 18 · p. 13, Figure 19 · p. 13, §F.5 · p. 13

F.6 Heated Cylinder MFA Model

  • For the Heated Cylinder MFA, connected-component counts converge as step size decreases, while smaller steps can separate near-touching trajectories into additional short-lived loops; gradient norms stay below e−10.Source: Figure 20 · p. 14, Figure 21 · p. 14, §F.6 · p. 14

F.7 Vortex MFA Model

  • For the Vortex MFA, connected-component counts converge as step size decreases, while separating marginal cycles can add loops after component counts stabilize; gradient norms stay below e−10.Source: Figure 22 · p. 14, Figure 23 · p. 14, §F.7 · p. 14

F.8 Vortex Street INR Model

  • For the Vortex Street INR, connected-component counts remain relatively stable across step sizes, loop counts stay between 1 and 3, and gradient norms remain below e−10.The loop-count range is reported for the obstacle-free setting.Source: Figure 24 · p. 14, Figure 25 · p. 14, §F.8 · p. 14

F.9 Heated Cylinder INR Model

G ADDITIONAL EXPERIMENTAL RESULTS

With obstacle regions retained, implicit extraction yields smooth trajectories that agree closely with SFFF, while LWM trajectories are more zigzag. The obstacle region is excluded because its zero-velocity points form a non-isolated set.

G ADDITIONAL EXPERIMENTAL RESULTS

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