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arxiv: 2606.30490 · v1 · pith:5PZDCDXXnew · submitted 2026-06-29 · 🧮 math.AT

From Frames to Features: Scalable Zigzag Persistence for Binary Video

David Lanners This is my paper

Pith reviewed 2026-06-30 02:50 UTC · model grok-4.3

classification 🧮 math.AT
keywords zigzag persistencebinary videoconnected componentstopological data analysisbarcode computationreal-time processingH0 homologyH1 homology
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The pith

H0 and H1 barcodes for binary video zigzag persistence are extracted directly from connected-component dynamics encoded in a graph.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the H0 and H1 barcodes in zigzag persistence for binary videos can be obtained directly from the dynamics of connected components rather than from full cubical complexes. Encoding these dynamics in a graph allows use of a near-linear time barcode algorithm, with total runtime scaling linearly in the number of pixels and parallelizable across frames. This makes high-resolution video analysis feasible in real time on consumer hardware. A sympathetic reader cares because it removes the computational bottleneck that previously made topological feature tracking in video impractical at scale.

Core claim

The H0 and H1 barcodes can be extracted directly from connected-component dynamics. By encoding these dynamics in a graph, cubical complexes are bypassed entirely, and the near-linear time barcode decomposition algorithm by Dey and Hou can be leveraged, leading to significant speedups and real-time performance on 4K video.

What carries the argument

Graph encoding of connected-component dynamics, which carries the zigzag persistence information for H0 and H1.

If this is right

  • Runtime is dominated by graph construction which scales linearly with pixel count.
  • Processing is embarrassingly parallel across frames.
  • Real-time zigzag persistence becomes possible on 4K video on consumer hardware.
  • Significant speedups are achieved compared to methods using cubical complexes and Gaussian elimination.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This graph encoding might extend to higher-dimensional or non-binary spatio-temporal data if the information preservation holds.
  • The linear scaling opens the door to processing longer sequences or higher frame rates without proportional cost increases.
  • Integration with existing video pipelines could make topological invariants a standard feature in real-time analysis tools.

Load-bearing premise

That the connected-component dynamics encoded as a graph preserve exactly the same H0 and H1 zigzag persistence information as the full cubical filtration.

What would settle it

Running both the graph method and a standard cubical complex method on the same small binary video sequence and finding differing barcodes would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.30490 by David Lanners.

Figure 1
Figure 1. Figure 1: Pipeline overview. Intermediate frames are generated by taking the union of white regions between original video frames. Connected components in each frame form the vertices of a formigram, while directed edges represent their evolution through intermediate frames. Finally, we compute the corresponding zigzag barcode in near-linear time. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Cubical complex constructions induced by binary image I: The T-construction on the foreground TI (1) ⊆ K realises 8-connectivity, while the V -construction VI (1) ⊆ K∗ realises 4-connectivity. Note that TI (1) is homotopy equivalent to a circle, whereas VI (1) consists of four discrete points. Both the T-construction and the V -construction will serve as models for the topology of the foreground and backgr… view at source ↗
Figure 3
Figure 3. Figure 3: Dual topology pairings induced by a binary image I. Subfigures (b) and (c) correspond to the complementary connectivity choices (κI , κ¯I ) = (8, 4) and (4, 8), respectively. In each case, the foreground and background are shown via the geometric realisations of the corresponding subcomplexes of K and K∗ , yielding distinct but consistent topologies. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Pointwise logic operations for binary images. The arrows represent inclusions of the foregrounds. Lemma 2.28 shows that unions and intersections of consecutive coloured regions are realised via interpolation frames. This leads us to the following definition. Definition 2.30. Let V = (Ii) n i=1 be a binary video. The insertion of interpolation frames of the form Ii ∨ Ii+1 defines the topological diagrams F … view at source ↗
Figure 5
Figure 5. Figure 5: The curse of resolution: Transition between two frames F1 and F2 of resolution 1000 × 1000 via their union. Pixels in F2 \ F1 (cyan) are inserted going from F1 to F1 ∪ F2, while pixels in F1 \ F2 (red) are deleted going from F1 ∪ F2 to F2. Due to the high resolution, even a small displacement leads to a large symmetric difference; in this example, a total of |F1∆F2| ≈ 4 · 104 pixels change, each inducing c… view at source ↗
Figure 6
Figure 6. Figure 6: Merge graphs of partial formigrams S1, S2 : Pτ → setpar. Note that the edge sets of S1 arise as graphs of total functions, hence S1 is in fact a formigram, not just a partial formigram. For a topological diagram X, the merge graph G(π0(X)) thus encodes the evolution of connected components across X in a graph. Before turning to their efficient construction in the setting of binary videos, we introduce the … view at source ↗
Figure 7
Figure 7. Figure 7: Merge Graph M: Nodes with two colours represent dual-type states. Arrows are consistently oriented with the zigzag type τ , and each vertex is the tail of at most one edge in each direction. Next, we introduce some common constructions. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Disjoint union of two formigrams S1 and S2: G(S1 ⊔ S2) = G(S1) ⊔ G(S2). Moreover, they form the two connected components of this graph. Definition 3.18. Let M be a merge graph. A connected component of M is a connected component of its underlying undirected graph, obtained by forgetting the directions of the edges. Remark 3.19. Each connected component C of a merge graph M determines a subfunctor of F(M) b… view at source ↗
Figure 9
Figure 9. Figure 9: Linearisation k[M] of a Merge graph M. Note that an ordering on the fibres (top to bottom) was chosen to get the matrix representations. The linearisation behaves nicely with respect to disjoint unions. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Connected component labeling (CCL) of a binary image I. The two choices κ ∈ {4, 8} lead to distinct κ-connected components Cκ(I). By Lemma 2.28, these maps are induced by inclusions. Hence each connected component of Ii or Ii+1 is contained in a unique connected component of Ii ∨ Ii+1, so it suffices to track a single representative pixel per component. Let PIi = {p1, . . . , pmIi } and PIi+1 = {q1, . . .… view at source ↗
Figure 11
Figure 11. Figure 11: Merge graph obtained from a binary video V: Interpolation frames Ii ∨ Ii+1 = max(Ii , Ii+1) are inserted, inducing the topological diagram F ∪ V . Vertices correspond to connected components; edges encode their evolution under inclusion. 4. Interval Decomposition Algorithm Given a topological diagram X : Pτ → top, we turn to the problem of efficiently comput￾ing the barcode of its associated H0-zigzag mod… view at source ↗
Figure 12
Figure 12. Figure 12: Left: Merge graph G(π0(X)) with merging and splitting vertices highlighted, obstructing a barcode structure. Right: The goal: a barcode basis of the same zigzag module. Given an elementary formigram S, the algorithm of Dey and Hou constructs a barcode graph Mbar from G(S) such that k[S] ∼= k[Mbar] in near-linear time in the number of vertices. It proceeds by iteratively simplifying the merge graph G(S). W… view at source ↗
Figure 13
Figure 13. Figure 13: 4-Frontier Forest. The truncation M[1, 4] contains no merging vertices and decomposes into bars and rooted trees. To better understand the relation between the vertices of merge graphs, we introduce the following definition. From this point onward, we denote the discrete interval {i ∈ Z | a ≤ i ≤ b} by Ja, bK. Definition 4.6. Let M be a merge graph over τ . Let 1 ≤ tstart ≤ tend ≤ |τ |, a level path from … view at source ↗
Figure 14
Figure 14. Figure 14: H0- and H1-barcode computation. Runtime comparison with the cubical-complex pipeline on synthetic videos consisting of T = 10 frames. 5 runs are performed for each method, and the mean is plotted. The shaded area corresponds to the standard error. For each resolution, we perform five runs of both ImageZigzag and ZigVid, computing the H0- and H1-zigzag barcodes obtained from the union construction. The tim… view at source ↗
Figure 15
Figure 15. Figure 15: Runtime scaling with image resolution. A synthetic video of resolution 3000 × 3000 is downsampled to varying resolutions, and the H0- and H1-zigzag barcodes of the resulting 900-frame videos are computed. The plotted values show the mean over 5 runs. The barcode decomposition stage contributes negligibly to the total runtime and exhibits little dependence on image resolution. through parallelisation. Real… view at source ↗
Figure 16
Figure 16. Figure 16: Parallel scaling on 4K video. Runtime for computing the H0- and H1-zigzag barcodes of a 3840 × 2160, 900-frame video (30 seconds at 30 fps) as a function of thread count. Using 6 threads, the full barcode computation completes in approximately 30 seconds, demonstrating real-time zigzag persistence on 4K video. The dashed curve indicates ideal linear scaling relative to the single-thread runtime. A. Proof … view at source ↗
read the original abstract

Zigzag persistence tracks topological features in spatio-temporal data through combinatorial invariants called barcodes. For binary videos, existing methods are bottlenecked by the construction of prohibitively large cubical complexes and performing Gaussian elimination on large boundary matrices, rendering high-resolution videos out of reach. We show that the $H_0$ and $H_1$ barcodes can be extracted directly from connected-component dynamics. By encoding these dynamics in a graph, we bypass cubical complexes entirely and are able to leverage the near-linear time barcode decomposition algorithm by Dey and Hou, leading to significant speedups. The total runtime of our pipeline is dominated by the construction of the underlying graph structures, which scales linearly with pixel count and is embarrassingly parallel across frames, ensuring excellent scalability. We demonstrate how this approach enables zigzag persistence on 4k video at real-time rates on consumer hardware.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that H0 and H1 zigzag persistence barcodes for binary video can be extracted directly from connected-component dynamics encoded as a graph. This bypasses construction of large cubical complexes on the space-time grid, allows use of the near-linear Dey-Hou barcode algorithm, and yields linear scaling in pixel count with frame-wise parallelism, enabling real-time 4K video processing.

Significance. If the graph encoding is shown to induce identical H0/H1 zigzag modules, the result would be a substantial practical advance in computational algebraic topology, removing the main bottleneck for spatio-temporal data and opening real-time topological video analysis. The explicit linear-time and parallel claims, together with the avoidance of Gaussian elimination on large boundary matrices, would be the primary strengths.

major comments (2)
  1. [Abstract] Abstract: the central claim that the component-dynamics graph preserves exactly the same H1 zigzag module as the cubical filtration is load-bearing yet unsupported by any proof sketch, boundary-map verification, or numerical check; H1 arises from 2-cycles in the cubical boundary operator, and adjacency recording of components alone does not automatically guarantee preservation of the relevant 1-cycles and their relations across frames.
  2. [Graph construction] Graph-construction section (inferred from abstract): no explicit statement or lemma shows that the inclusion maps or dual cycle information between consecutive frames are faithfully encoded, so it remains possible for the induced persistence diagrams to differ on H1 even when H0 matches.
minor comments (1)
  1. [Abstract] Abstract should cite the Dey-Hou reference explicitly rather than naming the authors only.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for explicit justification of the H1 equivalence. We address the points below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the component-dynamics graph preserves exactly the same H1 zigzag module as the cubical filtration is load-bearing yet unsupported by any proof sketch, boundary-map verification, or numerical check; H1 arises from 2-cycles in the cubical boundary operator, and adjacency recording of components alone does not automatically guarantee preservation of the relevant 1-cycles and their relations across frames.

    Authors: We agree that the abstract states the central claim without a supporting lemma or verification. The graph construction records component adjacencies and their evolution in a manner intended to encode the cycle relations that appear as 1-cycles in the cubical setting. To address the concern directly, the revised manuscript will contain an explicit lemma establishing that the induced H1 zigzag module is identical, together with a short numerical check on a controlled example confirming that the barcodes match. revision: yes

  2. Referee: [Graph construction] Graph-construction section (inferred from abstract): no explicit statement or lemma shows that the inclusion maps or dual cycle information between consecutive frames are faithfully encoded, so it remains possible for the induced persistence diagrams to differ on H1 even when H0 matches.

    Authors: The current text does not contain a formal lemma on the preservation of inclusion maps or cycle information. The graph is built so that edges between component nodes across frames capture the spatial overlaps that determine when 1-cycles form or break; this is why H1 is claimed to be recovered. We will add a precise statement and short proof of this equivalence in the revision, making the encoding of the relevant maps explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; extraction method presented as independent computational reduction

full rationale

The abstract and description present a direct extraction of H0/H1 barcodes from connected-component dynamics encoded as a graph, bypassing cubical complexes and using an external algorithm (Dey and Hou). No equations, fitted parameters, self-citations, or ansatzes appear that would make any claimed result equivalent to its inputs by construction. The equivalence to cubical zigzag is asserted as a derived property of the encoding rather than defined circularly; the derivation chain remains self-contained with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

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