Sweeping-Plane Filtration in Zebrafish Patterns

• Sweeping-plane filtration is a method that constructs a nested family of simplicial complexes using a linear height function to reveal directional persistent homology features.
• The approach decodes barcode signatures to quantify zebrafish stripes, breaks, and spots by exploiting directional sweeps and periodic boundary conditions.
• It integrates agent-based simulations with binary image construction to derive robust quantitative metrics for stripe counts, widths, and interruption positions.

Searching arXiv for the cited papers to ground the article in current records. arxiv_search(query="(Khoudari et al., 14 Sep 2025)", max_results=5) arxiv_search(query="sweeping-plane filtration zebrafish (Khoudari et al., 14 Sep 2025)", max_results=10) Sweeping-plane filtration is a nested family of complexes obtained by revealing data according to a linear height function, typically of the form $h(x)=v\cdot x+c$, so that simplices enter the filtration when all of their vertices have height at most a threshold. In the zebrafish-pattern study "Quantifying topological features and irregularities in zebrafish patterns using the sweeping-plane filtration" (Khoudari et al., 14 Sep 2025), this construction is used as an upper-star filtration on a simplicial complex built from binary images derived from agent-based simulations, and its directional dependence is exploited to decode stripe count, stripe width, interruptions, spot count, and spot size from persistence barcodes. The same broad sweep idea also appears in distinct forms in computational geometry and plane-sweep reconstruction, but the zebrafish study makes the filtration itself the central inferential device for persistent homology and biological interpretation (Khoudari et al., 14 Sep 2025).

1. Formal definition and mathematical setting

In the zebrafish application, the domain is a two-dimensional image plane $\Omega\subset \mathbb{R}^2$, and the sweep is specified by a linear height function

$h(x)=v\cdot x+c,$

where $v\in\mathbb{R}^2$ is the chosen sweep direction and $c\in\mathbb{R}$ is an offset. Given a simplicial complex $K$ built on data in $\Omega$, the sweeping-plane filtration is the nested family $\{K_t\}_{t\in\mathbb{R}}$ defined by

$$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$

In this implementation, a simplex is included once all of its vertices have been revealed, so the construction is an upper-star filtration with respect to the vertex function $h$ (Khoudari et al., 14 Sep 2025).

The underlying complex is not built directly from the raw point cloud. Instead, agent-based point sets are transformed into a binary image with voxel width $\Omega\subset \mathbb{R}^2$0, a pixel is assigned value $\Omega\subset \mathbb{R}^2$1 if it contains at least one focal cell and $\Omega\subset \mathbb{R}^2$2 otherwise, vertices are the centroids of $\Omega\subset \mathbb{R}^2$3-valued pixels, edges connect vertices whose pixels lie in the Moore neighborhood, and triangles are added by clique or flag completion whenever three vertices are pairwise connected by edges. Operationally, at each threshold one includes the revealed vertices and all edges and triangles allowed by that fixed adjacency rule. This yields a height-filtered simplicial complex on the centroids of the currently revealed $\Omega\subset \mathbb{R}^2$4-pixels (Khoudari et al., 14 Sep 2025).

A central feature of the construction is that direction is not incidental. Four sweep directions are used: top-to-bottom, bottom-to-top, left-to-right, and right-to-left. For top-to-bottom and bottom-to-top sweeps, periodicity is enforced in the horizontal direction so that the left and right boundaries are glued to a cylinder; for left-to-right and right-to-left sweeps, that periodicity is not imposed. This makes the filtration explicitly anisotropic, which is essential for patterns whose organization is strongly directional, such as horizontal stripes (Khoudari et al., 14 Sep 2025).

2. Construction pipeline in the zebrafish study

The biological data come from the off-lattice agent-based model of Volkening and Sandstede (2018), which simulates five pigment cell types on a growing two-dimensional domain with rules for migration, differentiation, division, competition, and phenotypic transitions. Growth is uniform and deterministic, whereas stochasticity arises from discrete birth, death, and transition events. The analysis uses 1000 wild-type simulations and 1000 pfeffer mutant simulations at 66 days post fertilization on domains of height $\Omega\subset \mathbb{R}^2$5 mm and length $\Omega\subset \mathbb{R}^2$6 mm, periodic in $\Omega\subset \mathbb{R}^2$7 (Khoudari et al., 14 Sep 2025).

The point-set to image conversion depends on phenotype. For wild-type patterns, melanophores are the focal cells. For pfeffer, loose iridophores are used because melanophores are sparse and noisy. No additional smoothing or segmentation is applied. Resolution is chosen by a Betti-CROCKER-style insensitivity analysis: the study computes $\Omega\subset \mathbb{R}^2$8 heatmaps across filtration step $\Omega\subset \mathbb{R}^2$9 and candidate $h(x)=v\cdot x+c,$0 values for 25 unbroken striped snapshots, then selects $h(x)=v\cdot x+c,$1 in a range where $h(x)=v\cdot x+c,$2 is stable across $h(x)=v\cdot x+c,$3. The chosen value is $h(x)=v\cdot x+c,$4m, which gives image height $h(x)=v\cdot x+c,$5 pixels $h(x)=v\cdot x+c,$6 and length $h(x)=v\cdot x+c,$7 pixels $h(x)=v\cdot x+c,$8 (Khoudari et al., 14 Sep 2025).

For a fixed direction $h(x)=v\cdot x+c,$9, the implementation orders $v\in\mathbb{R}^2$0-pixels by height, adds vertices once revealed, connects Moore neighbors, adds triangles to fill cliques, enforces periodicity in $v\in\mathbb{R}^2$1 for top-to-bottom and bottom-to-top only, and records homology in dimensions $v\in\mathbb{R}^2$2. The reported persistent-homology summaries are barcodes in dimensions $v\in\mathbb{R}^2$3 and $v\in\mathbb{R}^2$4, persistence diagrams for illustration, Betti curves $v\in\mathbb{R}^2$5 and $v\in\mathbb{R}^2$6, and the $v\in\mathbb{R}^2$7 heatmaps used for resolution selection. The zebrafish analysis does not use bottleneck distances, Wasserstein distances, persistence landscapes, or persistence images (Khoudari et al., 14 Sep 2025).

The notation used to organize barcode features is also directional and persistence-aware. Features are partitioned into persistent and nonpersistent, according to whether death time is at least $v\in\mathbb{R}^2$8, and into zero-born and nonzero-born, according to whether birth time is $v\in\mathbb{R}^2$9 or positive. Thus a symbol such as $c\in\mathbb{R}$0 denotes the $c\in\mathbb{R}$1th persistent, nonzero-born $c\in\mathbb{R}$2 feature under direction $c\in\mathbb{R}$3 (Khoudari et al., 14 Sep 2025).

3. Barcode semantics and directional decoding

The methodological significance of the filtration lies in its barcode semantics. Under top-to-bottom and bottom-to-top sweeps, which are perpendicular to horizontal stripes and use horizontal periodicity, each uninterrupted horizontal stripe yields a persistent $c\in\mathbb{R}$4 loop. The birth time of that loop records when the stripe first spans the periodic direction. Before loop birth, jumps in $c\in\mathbb{R}$5 reflect the exposure of jagged boundary fragments, and many short nonpersistent $c\in\mathbb{R}$6 bars encode boundary roughness. Under left-to-right and right-to-left sweeps, which are parallel to stripe orientation and do not use horizontal periodicity, interruptions are exposed instead: a stripe break yields a persistent, nonzero-born $c\in\mathbb{R}$7 bar, and an interstripe break yields a zero-born, nonpersistent $c\in\mathbb{R}$8 bar whose death marks the merger of two stripe components (Khoudari et al., 14 Sep 2025).

For spotted pfeffer patterns, the mapping changes. Under top-to-bottom and bottom-to-top with $c\in\mathbb{R}$9-periodicity, each spot corresponds to a persistent $K$0 class and the spot count equals $K$1. Vertical spot height is estimated from paired birth times of associated persistent $K$2 classes in the two opposite sweep directions. Horizontal spot length can be estimated analogously from left-to-right and right-to-left pairings, although the study reports higher error for that quantity because of stripe-wise alignment in pfeffer (Khoudari et al., 14 Sep 2025).

The paper turns these signatures into explicit metrics. For example, the number of spots is

$K$3

and the number of stripe breaks is

$K$4

For unbroken stripes, the maximum width of stripe $K$5 is estimated by

$K$6

while the minimum width uses persistent $K$7 births instead of persistent $K$8 births. Break widths are also estimated directionally; for stripe break $K$9 the width is

$\Omega$0

and these values are lower bounds because of pixelation (Khoudari et al., 14 Sep 2025).

The directional logic can be summarized briefly as follows.

Sweep direction	Dominant barcode signature	Interpreted feature
TB / BT	Persistent $\Omega$1 loops	Uninterrupted horizontal stripes
LR / RL	Persistent nonzero-born $\Omega$2	Stripe breaks
LR / RL	Zero-born nonpersistent $\Omega$3	Interstripe breaks

Break position is resolved in two coordinates. The horizontal position is the center of the measured break width for stripe breaks and is derived from left-to-right death time for interstripe breaks. The dorsal-ventral position is assigned by $\Omega$4-means clustering of top-to-bottom and bottom-to-top feature birth times, with clusters ordered bottom-to-top to map to labeled stripe positions such as $\Omega$5, $\Omega$6, $\Omega$7, and $\Omega$8 (Khoudari et al., 14 Sep 2025).

4. Quantitative findings and biological predictions

The full uncropped wild-type analysis reports that 997 of 1000 simulations are classified as striped, including patterns with breaks, while 3 have breaks in all blue stripes and are classified as spotted by barcode logic, though they are reported within the “both broken” category in summary statistics. Among striped wild-type patterns, only $\Omega$9 are unbroken, more than half are “broken stripe(s),” and the remainder are “broken interstripe(s)” or “both.” A manual audit of 100 patterns suggests approximately $\{K_t\}_{t\in\mathbb{R}}$0 classification error, with rare misclassifications arising from stray pixels near lateral boundaries or from complex adjacent breaks (Khoudari et al., 14 Sep 2025).

Stripe number is reported as highly robust in the model at this stage. Counting stripes as intended biological stripes rather than fragments, the mean stripe count across the 997 striped wild-type patterns is $\{K_t\}_{t\in\mathbb{R}}$1, and only approximately $\{K_t\}_{t\in\mathbb{R}}$2 of patterns have 3 or 5 stripes. For unbroken patterns only, the mean maximum stripe width is approximately $\{K_t\}_{t\in\mathbb{R}}$3m and the mean interstripe width is approximately $\{K_t\}_{t\in\mathbb{R}}$4m, with standard deviations approximately $\{K_t\}_{t\in\mathbb{R}}$5m (Khoudari et al., 14 Sep 2025).

Interruptions are frequent but typically narrow. The mean number of stripe breaks per striped pattern is approximately $\{K_t\}_{t\in\mathbb{R}}$6. Most breaks are one pixel wide, corresponding to $\{K_t\}_{t\in\mathbb{R}}$7m under the chosen discretization, and the mean measured width is approximately $\{K_t\}_{t\in\mathbb{R}}$8m, explicitly interpreted as a lower bound because of discretization. Their dorsal-ventral distribution is strongly nonuniform: approximately $\{K_t\}_{t\in\mathbb{R}}$9 of breaks occur in the peripheral stripes, $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$0 or $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$1, consistent with sequential formation and accumulated stochasticity away from $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$2. By contrast, the empirical distribution of break centers along the domain length is consistent with approximately uniform random, matching the absence of an imposed anterior-posterior bias in the model (Khoudari et al., 14 Sep 2025).

For pfeffer patterns, the mean number of spots at 66 days post fertilization is approximately $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$3. Spot-length and spot-height distributions are reported, but length estimates are noisier than height estimates. This suggests that the same directional barcode logic is more reliable for some geometric quantities than for others, even within one phenotype class (Khoudari et al., 14 Sep 2025).

5. Comparison with Vietoris–Rips and related sweep constructions

The zebrafish study frames the sweeping-plane filtration as complementary to Vietoris–Rips. Vietoris–Rips on point clouds grows a proximity scale and emphasizes local pairwise distances; it is isotropic and does not use orientation unless orientation is explicitly engineered. By contrast, the sweeping-plane or height filtration reveals data by global directional order under a fixed adjacency rule, so direction directly conditions the barcode. In the striped setting, perpendicular sweeps expose complete stripes, while parallel sweeps expose interruptions. The study therefore associates Vietoris–Rips more naturally with spot size and spot number, and the sweeping-plane construction with striped patterns and defects (Khoudari et al., 14 Sep 2025).

The empirical comparison on cropped wild-type patterns supports that division of labor. On the central band between $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$4 and $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$5, the fraction of unbroken versus broken patterns is very similar between methods: $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$6 unbroken for sweeping-plane and $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$7 unbroken for Vietoris–Rips, with comparable percentages for broken categories. The paper emphasizes, however, that sweeping-plane directly detects and counts interruptions and locates them along $$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$8 and by stripe index, whereas earlier Vietoris–Rips work on zebrafish patterns was chiefly used to estimate numbers of stripes or spots and stripe widths and required a priori pattern-type labeling and cropping (Khoudari et al., 14 Sep 2025).

Related but distinct sweep-based constructions appear in other areas. In "Plane-Sweep Incremental Algorithm: Computing Delaunay Tessellations of Large Datasets" (Trencséni et al., 2012), sorting sites along the first principal component induces a monotone filtration on the final Delaunay tessellation,

$$K_t=\{\sigma\in K:\max_{x\in Vertices(\sigma)} h(x)\le t\}.$$9

where a simplex enters once its circumsphere lies wholly behind the sweep plane. In "SonarSweep: Fusing Sonar and Vision for Robust 3D Reconstruction via Plane Sweeping" (Chen et al., 1 Nov 2025), the “filtration over planes” is a discretized sequence of hypothesized planes aligned with sonar geometry, and evidence is accumulated across plane index in a 4D cost volume rather than in homology. This suggests that “sweeping-plane filtration” names a broader family of ordered geometric constructions, although the filtered objects, the governing certificates, and the inferential outputs differ substantially across persistent homology, Delaunay offlining, and cross-modal depth estimation.

6. Assumptions, limitations, and extensions

The zebrafish application is conditioned by the assumptions of the underlying agent-based model: uniform epithelial growth, periodic $h$0-boundary, a fixed initial interstripe $h$1 associated with horizontal myoseptum guidance, and stochastic local rules for cell events. Within that modeling framework, stripe formation proceeds sequentially from center to periphery, and accumulated noise is expected dorsally and ventrally. The barcode interpretations are therefore not purely image-theoretic; they are tied to the developmental geometry and boundary conditions of the simulations (Khoudari et al., 14 Sep 2025).

The method also has technical limitations. It is direction-dependent and tuned for horizontal stripes; vertical stripes would require rotating the sweep directions or adding an orientation-detection step. Because left-to-right and right-to-left sweeps do not use horizontal periodicity, breaks at the left or right image edges need special handling. The method is sensitive to stray pixels, which can generate spurious $h$2 bars that mimic breaks. The authors mitigate this with the $h$3-insensitivity analysis and simple post-hoc rules, including discarding estimated breaks with negative width and flagging pathologies with break widths greater than $h$4 of image length, but sub-percent to several-percent error rates remain depending on the failure mode considered (Khoudari et al., 14 Sep 2025).

Spot-size estimation is a further limitation. Matching spot-associated persistent $h$5 bars across directions is ambiguous, and a manual audit reports approximately $h$6 mismatches for heights and $h$7 for lengths. The study also notes that length estimates are noticeably noisier than height estimates. These observations constrain how far the directional pairing logic can be pushed without additional geometric priors or preprocessing (Khoudari et al., 14 Sep 2025).

Several extensions are explicitly proposed. Multi-directional aggregation is already used, but extended persistent homology could reduce the number of directions while retaining information. Multi-parameter persistence in $h$8 would formalize the resolution-insensitivity criterion. Combining sweeping-plane with Vietoris–Rips is suggested as a way to use sweeping-plane for stripe defects and Vietoris–Rips for spot size and count. The paper also identifies image cleaning steps such as morphological denoising as a possible way to reduce stray-pixel artifacts, and it notes that repeating the pipeline with gold, namely dense iridophores, as the signal would provide direct interstripe metrics including counts, widths, and breaks (Khoudari et al., 14 Sep 2025).

Within these constraints, the sweeping-plane filtration functions as an automated and interpretable map from directional persistent-homology signatures to biologically meaningful quantities. In the zebrafish study, that map yields two principal predictions: stripe count is highly robust, whereas peripheral stripe interruptions are comparatively susceptible to variation (Khoudari et al., 14 Sep 2025).