Radial Persistence Transform in Graph & Manifold Analysis

• Radial Persistence Transform is a method that indexes persistence data by angular or center-based parameters, linking traditional PHT to practical graph and manifold analyses.
• It leverages a radial binary multi-search to efficiently reconstruct graph edges, substantially reducing computational complexity compared to conventional edge scans.
• The approach extends to radial filtrations in extended persistence frameworks, suggesting robust applications in both sensitivity and output-driven topological recovery.

Searching arXiv for the cited papers to ground the article. “Radial Persistence Transform” is not a formal term used in the cited papers, but the phrase accurately describes two closely related constructions in topological shape analysis. In the graph setting, it denotes a radial organization of directions and associated augmented persistence information around each vertex, together with a radial binary multi-search that reconstructs edges from a finite subset of augmented persistence diagrams (Fasy et al., 2022). In the manifold-with-boundary setting, the Extended Persistent Homology Transform (XPHT) is defined for height functions indexed by directions, and the paper explicitly states that “similar results could hold for other kinds of functions, such as radial functions,” which suggests a radial analogue built from extended persistence of distance-type filtrations (Turner et al., 2022). Under this usage, the radial perspective is not a separate transform with a single canonical definition, but a structured way of indexing and exploiting persistence information by angular or center-based parameters.

1. Conceptual scope and relation to established transforms

The underlying framework is the Persistent Homology Transform (PHT) and its augmented and extended variants. For a simplicial complex $K$ embedded in $\mathbb{R}^d$, the PHT/APHT assigns to each direction $s \in \mathbb{S}^{d-1}$ the persistence data of the lower-star filtration induced by the height function in that direction (Fasy et al., 2022). In the notation of the graph reconstruction paper, the augmented transform is

$X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$

A faithful discretization is defined there as “a finite subset of $X$ from which all other elements of $X$ can be deduced” (Fasy et al., 2022).

The radial interpretation arises because the graph reconstruction algorithm does not merely sample arbitrary directions. It organizes directions by angular position in a fixed projection plane, radially around each vertex, and then uses those directions to localize adjacency information (Fasy et al., 2022). The paper states that the improvement in edge reconstruction comes from a “radial binary (multi-)search” exploiting the fact that graph edges can be ordered radially with respect to a reference plane (Fasy et al., 2022). This suggests a graph-focused radial version of APHT rather than a distinct transform with new invariance properties.

In the XPHT setting, the indexing parameter remains directional: each unit vector is assigned the extended persistence module of the corresponding height function (Turner et al., 2022). However, the same paper states that similar results could hold for “other kinds of functions, such as radial functions” (Turner et al., 2022). A plausible implication is that a radial persistence transform can be understood as a direct analogue of XPHT in which directions are replaced by centers or radial parameters, while the persistence object is extended persistence rather than ordinary persistence.

2. Persistent-homological foundations

For a simplicial complex $K$ and a filter function $f$, the sublevel sets

$$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$

form an increasing filtration, and persistent homology records the birth and death of homological features as $t$ varies (Fasy et al., 2022). The resulting persistence diagram is a multiset of birth–death pairs $\mathbb{R}^d$0 (Fasy et al., 2022).

The graph reconstruction paper uses the augmented persistence diagram (APD), in which every simplex is assigned to exactly one birth–death pair and some pairs may lie on the diagonal $\mathbb{R}^d$1 (Fasy et al., 2022). For graphs, the augmentation has two consequences stated explicitly in the paper: every vertex appears in the diagram, and every edge corresponds either to a $\mathbb{R}^d$2-dimensional death or to a $\mathbb{R}^d$3-dimensional birth (Fasy et al., 2022). This simplex-level encoding is what allows the algorithm to count edges and compute indegrees from persistence data.

For a graph $\mathbb{R}^d$4 GP-immersed in $\mathbb{R}^d$5, and a direction $\mathbb{R}^d$6, the lower-star filtration is defined by

$\mathbb{R}^d$7

for vertices $\mathbb{R}^d$8, and

$\mathbb{R}^d$9

for edges $s \in \mathbb{S}^{d-1}$0 (Fasy et al., 2022). The paper denotes the $s \in \mathbb{S}^{d-1}$1-dimensional APD by $s \in \mathbb{S}^{d-1}$2 and the union over all dimensions by

$s \in \mathbb{S}^{d-1}$3

The XPHT paper enlarges this directional paradigm from ordinary persistence to extended persistence. For a bounded function $s \in \mathbb{S}^{d-1}$4, it defines the extended persistence module over the parameter space

$s \in \mathbb{S}^{d-1}$5

where

$s \in \mathbb{S}^{d-1}$6

and the order is

$s \in \mathbb{S}^{d-1}$7

The module assigns

$s \in \mathbb{S}^{d-1}$8

with

$s \in \mathbb{S}^{d-1}$9

(Turner et al., 2022). Extended persistence differs from ordinary persistence in that essential classes become finite intervals spanning the ordinary and relative parts of the filtration, which yields finite Wasserstein distances even when two shapes have different Betti numbers (Turner et al., 2022).

3. Radial organization in graph reconstruction

The graph reconstruction paper improves a prior finite-discretization result for APHT on graphs by replacing a linear scan over candidate edges with a radial binary multi-search (Fasy et al., 2022). Previous work by Belton et al. reconstructed a GP-immersed graph with $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$0 using $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$1 augmented diagrams in

$X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$2

time, with edge reconstruction as the bottleneck because each pair of vertices is tested for adjacency (Fasy et al., 2022). The new method reduces edge reconstruction to $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$3 diagrams, where $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$4, rather than $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$5, and the time improves accordingly from the previous $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$6 bottleneck (Fasy et al., 2022).

The radial step begins by fixing the orthogonal projection $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$7 onto the $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$8-plane and ordering projected vertices around a chosen center vertex $X = \{\,(\dgm{}{s}, s)\,\}_{s\in \mathbb{S}^{d-1}}.$9 by angle around $X$0 (Fasy et al., 2022). The paper’s general position assumption requires: every set of $X$1 points is affinely independent; no three points are colinear after projection into the $X$2-plane; and every point has a unique height with respect to $X$3 (Fasy et al., 2022). These conditions guarantee that each vertex has a well-defined cyclic order of all other vertices around it in the projection plane and that indegree computations are unambiguous.

The central counting quantity is the indegree of a vertex in a direction. The paper defines

$X$4

as the number of edges incident to $X$5 with height $X$6 (Fasy et al., 2022). It then proves an edge-counting lemma: for a filter function $X$7, the edges with function value $X$8 are in one-to-one correspondence with

$X$9

and if $X$0 is unique among vertices, the cardinality of this multiset is exactly $X$1 (Fasy et al., 2022). The paper further states that if $X$2 has unique height in direction $X$3, then $X$4 can be computed from $X$5 in $X$6 time, using $X$7 diagrams (Fasy et al., 2022).

This is the mechanism by which radial localization becomes possible. By choosing a direction perpendicular to a selected angular boundary, one obtains an indegree count at the height of $X$8; by subtracting contributions from already identified edges outside the target sector, one recovers the number of edges from $X$9 into that angular region (Fasy et al., 2022).

4. Edge arcs, radial binary multi-search, and sweep reconstruction

To formalize angular sectors, the paper defines an edge arc $K$0 centered at $K$1 with start and stop angles $K$2, a radially ordered vertex array $K$3, and an edge count $K$4 equal to the number of edges between $K$5 and vertices in $K$6 (Fasy et al., 2022). Geometrically, $K$7 represents a sector in the upper half-plane above $K$8 between angles $K$9 and $f$0 (Fasy et al., 2022).

For such an arc, the paper proves an arc-count lemma. Let $f$1 be the direction perpendicular to $f$2 so that the arc is entirely below $f$3, and let $f$4 be the set of edges with height $f$5 that are not in $f$6. If no other vertex has height $f$7, then

$f$8

(Fasy et al., 2022). This is the key relation that converts persistence information into an exact edge count for an angular sector.

The binary-search step is implemented by Algorithm 3.1 in the paper, $f$9, which splits an edge arc into two subarcs of approximately equal cardinality in the candidate vertex list (Fasy et al., 2022). The split is chosen by selecting an angle between consecutive rays, using the minimal angular separation $$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$0, and setting a perpendicular direction

$$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$1

in the $$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$2-plane (Fasy et al., 2022). Theorem 3.2 states that the algorithm uses $$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$3 diagrams and time $$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$4, while producing left and right subarcs whose vertex lists partition the original list in clockwise order and satisfy

$$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$5

(Fasy et al., 2022).

Outgoing edges of a vertex $$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$6 are then found by Algorithm 3.2, $$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$7 (Fasy et al., 2022). The paper describes the procedure as follows. One computes

$$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$8

the number of edges from vertices above $$K_t := \{ \sigma \in K \mid f(\sigma) \le t\}, \quad t \in \mathbb{R},$$9, initializes a stack with a single edge arc covering the entire upper half-plane, and repeatedly applies three rules: discard arcs with $t$0; if $t$1, declare every candidate vertex in the arc adjacent to $t$2; otherwise split the arc and continue (Fasy et al., 2022). Theorem 3.3 states that this finds all outgoing edges of $t$3, in clockwise order, using $t$4 diagrams and $t$5 time (Fasy et al., 2022). Because each split halves the candidate set up to rounding, each recovered outgoing edge requires $t$6 angular bisections (Fasy et al., 2022).

Full reconstruction is obtained by sweeping vertices in increasing $t$7-height. Algorithm 3.3, $t$8, first obtains the APD in direction $t$9, precomputes clockwise orderings of vertices above each vertex, computes the minimal angular separation $\mathbb{R}^d$00, and then processes vertices bottom-to-top, using already reconstructed edges from below as the incoming set $\mathbb{R}^d$01 when recovering outgoing edges (Fasy et al., 2022). Theorem 3.4 states that $\mathbb{R}^d$02 reconstructs $\mathbb{R}^d$03 using $\mathbb{R}^d$04 diagrams and $\mathbb{R}^d$05 time (Fasy et al., 2022). The paper summarizes the improvement qualitatively as fewer diagrams, output sensitivity with dependence on the actual number of edges $\mathbb{R}^d$06, and dimension-friendly behavior because the radial structure lives in a two-dimensional projection (Fasy et al., 2022).

5. Extended persistence and radial-function analogues

The XPHT paper defines the Extended Persistent Homology Transform by assigning to each direction $\mathbb{R}^d$07 the tuple

$\mathbb{R}^d$08

(Turner et al., 2022). It then defines a distance between shapes by integrating Wasserstein distances between their extended persistence modules: $\mathbb{R}^d$09 (Turner et al., 2022). Because extended persistence turns essential classes into finite intervals, this distance remains finite even when shapes have different Betti numbers (Turner et al., 2022).

The paper’s principal theoretical contribution for manifolds with boundary is that the extended persistence of a manifold $\mathbb{R}^d$10 for a height function can be deduced from the extended persistence of the boundary $\mathbb{R}^d$11, together with the sign labels of boundary critical points (Turner et al., 2022). Theorem 3.2 states that if

$\mathbb{R}^d$12

then

$\mathbb{R}^d$13

where $\mathbb{R}^d$14 selects intervals according to whether their births correspond to positive or negative critical points of $\mathbb{R}^d$15 with the appropriate Morse indices (Turner et al., 2022). The paper further gives explicit formulas for essential classes in terms of minima and maxima of the height function on connected boundary components (Turner et al., 2022).

These results are directional rather than radial, but the paper explicitly states that “it is reasonable to expect that similar results could hold for other kinds of functions, such as radial functions” (Turner et al., 2022). It also presents a consistent radial substitution in notation: $\mathbb{R}^d$16 as the candidate radial function centered at $\mathbb{R}^d$17 (Turner et al., 2022). This suggests a radial extended persistent homology transform in which the parameter space is a family of centers $\mathbb{R}^d$18 rather than a sphere of directions, and the persistence modules are those of radial filtrations instead of linear height filtrations. The paper even writes the corresponding transform and distance formulas as a natural extension: $\mathbb{R}^d$19 and

$\mathbb{R}^d$20

(Turner et al., 2022). Since the paper does not formalize these as theorems, they are best read as a technically grounded extension rather than an established definition.

6. Assumptions, computational practice, and limitations

Both lines of work rely on strong regularity assumptions. In the graph case, correctness depends on the general position conditions in Assumption 2.1: affine independence, no three projected vertices colinear, and unique heights in the reference direction $\mathbb{R}^d$21 (Fasy et al., 2022). The paper notes that a version of the unique-height condition can be enforced algorithmically by basis tilting (Fasy et al., 2022). It also assumes exact augmented persistence diagrams returned by an oracle, and its implementation model uses efficient lookup in the diagram by birth and death values (Fasy et al., 2022). The method is tailored to graphs and one-skeleta; the paper explicitly states that “Radially ordering higher dimensional simplices is not well-defined, and this issue prevents the methods presented here from being immediately transferrable” (Fasy et al., 2022).

The XPHT framework likewise assumes Morse-theoretic regularity for the function under study, either in smooth or PL form (Turner et al., 2022). For binary images, the paper builds a PL manifold with boundary from 8-adjacent foreground and 4-adjacent background pixels, proves that the resulting boundary is a disjoint union of simple closed PL curves, and then computes $\mathbb{R}^d$22 of boundary filtrations by union–find (Turner et al., 2022). It handles non-generic height ties by identifying $\mathbb{R}^d$23-critical vertices and flat critical segments, and classifies positivity of critical points via explicit geometric tests involving determinants and rotated edge vectors (Turner et al., 2022). Directions are sampled as

$\mathbb{R}^d$24

and antipodal duality reduces the number of required height filtrations by a factor of two when $\mathbb{R}^d$25 is even (Turner et al., 2022).

The principal limitation of the graph-based radial method is scope: it reconstructs one-skeleta, not higher-dimensional simplicial structure (Fasy et al., 2022). The principal limitation of the radial extension suggested by XPHT is that it remains a proposal rather than a completed theory in the cited paper (Turner et al., 2022). The paper states that the Morse-theoretic and boundary-based decomposition principles “are not specific to linear functions” and that radial functions are a natural candidate (Turner et al., 2022). This suggests that a fully developed radial persistence transform would require proofs of injectivity, stability, and computational reductions comparable to those already established for PHT, APHT, and XPHT.

Taken together, the two papers locate the radial persistence idea at the intersection of directional topological transforms and structured parameterization. In one case, the radial structure is explicit and algorithmic: directions are organized by angle around vertices to enable output-sensitive graph reconstruction from APHT data (Fasy et al., 2022). In the other, the radial structure is prospective but theoretically motivated: extended persistence for radial functions is presented as a natural generalization of the height-based XPHT framework (Turner et al., 2022).