MAP-curvature: Map-Based Geometric Analysis

• MAP-curvature is a framework that extracts geometric curvature from maps, using tools like distance map Hessians, PCA-based regressions, and moment-map techniques to reveal structural features.
• In porous media, MAP-curvature employs Hessian eigenvalue analysis of distance maps to accurately locate pore bodies and throats while avoiding saddle-induced over-partitioning.
• In point-cloud geometry and differential geometry, it offers closed-form estimators for curvature metrics, enhancing robustness and efficiency over traditional approaches.

MAP-curvature is a term used in several distinct research contexts to denote curvature quantities or curvature-driven constructions associated with maps. In the most explicit computational sense, it denotes a pore-space characterization framework based on the curvature of the distance map of a binary porous-media image, using the Hessian of the distance map to locate pore bodies and throats and to partition pore space (Ben-Noah et al., 2024). In point-cloud geometry, the term is also used for a statistically grounded, closed-form local regression estimator of the Weingarten map from sampled manifold data, from which principal, mean, Gaussian, and sectional curvatures are computed (Cao et al., 2019). In differential and generalized Kähler geometry, related usages identify curvature through a moment-map framework or through the curvature induced by canonical maps such as the Abel–Jacobi map (Goto, 2021, Biswas, 2020). The common thread is that curvature is not treated as an isolated scalar invariant, but as an object extracted from a map, a pullback metric, a Hessian field, or an operator-valued geometric datum.

1. Terminological scope and main usages

The term has no single universal definition across the literature. The most direct and fully named occurrence is the pore-space framework introduced as MAP-curvature, which uses the curvature of the distance map (DM) of a binary porous-media image to identify pore bodies and throats and then partition the void space into distinct pores (Ben-Noah et al., 2024). That formulation is algorithmic and image-based.

A second concrete usage appears in point-cloud geometry, where the paper describes an efficient WME/MAP-curvature estimator for the Weingarten map $A_\xi$ on an embedded manifold sampled in Euclidean space (Cao et al., 2019). There, curvature is obtained by first estimating shape operators from local normal-field variation and then recovering extrinsic invariants algebraically.

Other entries in the literature use “MAP-curvature” in a broader or analogical way. In generalized Kähler geometry, the generalized scalar curvature is presented as the moment map for a generalized Hamiltonian action, so “MAP-curvature” refers to curvature realized through a moment-map construction (Goto, 2016, Goto, 2021). In geometric flow and submanifold theory, the expression is also used informally for curvature quantities attached to Gauss maps, classifying maps, or ambient map-coupled flows (Álvarez-Vizoso, 2022, Alarcon et al., 2024, Gomes et al., 27 Oct 2025, Schick et al., 2020). This suggests that the term functions more as a research label than as a standardized invariant.

2. Distance-map MAP-curvature in porous media

In porous-media analysis, MAP-curvature is defined through the signed Euclidean distance map from each voxel to the nearest solid–pore interface. If the binary image defines the pore domain $\mathbb P$ and the solid domain $\mathbb S$, with $\partial \mathbb S$ the solid–pore interface, the distance map is

$$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$

Positive values correspond to the pore phase and negative values to the solid phase (Ben-Noah et al., 2024). The assignment of negative values in the solid is used to reduce irrelevant inflection points at grain surfaces and to reduce numerical-gradient errors.

The curvature map is built from the Hessian matrix of the distance map,

$$H_{ij} = \frac{\partial^2 f(\mathbf x)}{\partial x_i \partial x_j},$$

computed from numerical derivatives. Because the Hessian is symmetric, its eigenvalues are real, and the determinant of the Hessian is related to Gaussian curvature, while the eigenvalues are the principal curvatures of the distance map (Ben-Noah et al., 2024).

This framework separates two tasks that are often conflated in classical distance-map watershed workflows. Critical-point localization is performed with the distance map and the determinant of the Hessian; partitioning is performed on a curvature-derived map built from Hessian eigenvalues, especially the smallest eigenvalue $\lambda_1$. That separation is the main mechanism by which the method avoids saddle-induced over-partitioning (Ben-Noah et al., 2024).

3. Critical-point localization: pore bodies and throats

The locating stage begins with local extrema of the distance map. Pore bodies are identified from local maxima of the distance map, using an 8-connected neighborhood in 2D and a 26-connected neighborhood in 3D (Ben-Noah et al., 2024). The pore-body radius is the distance-map value at that maximum.

Throat centers are associated with saddle points of the distance map, but the method does not search for saddles directly in the distance map. Instead, it uses the determinant of the Hessian. At saddle points, the determinant of the Hessian is negative, and the paper explicitly uses the result from Lakemond et al. that the local minima of the determinant map identify saddle points of the distance map (Ben-Noah et al., 2024). The resulting workflow is:

1. compute the distance map;
2. compute the Hessian $H$;
3. compute $\det(H)$;
4. locate local minima of $\det(H)$ as throat centers.

For throats, the throat radius is the distance-map value at the saddle location, and throat orientation can be estimated from the maximal gradient direction of the distance map at the saddle, or from geometry between nearby solid–pore interfaces (Ben-Noah et al., 2024).

The eigenvalue structure of the Hessian is also used diagnostically. In 3D, at saddle points $\mathbb P$0 and $\mathbb P$1, while the sign of $\mathbb P$2 indicates saddle index (Ben-Noah et al., 2024). This makes Hessian eigenvalues useful for screening out false saddle points.

4. Partitioning via Hessian eigenvalues

The partitioning stage is not performed on the distance map directly, but on a curvature-derived map built from the smallest Hessian eigenvalue $\mathbb P$3. The binary indicator

$\mathbb P$4

marks regions where the curvature is negative (Ben-Noah et al., 2024).

Two partitioning strategies are described. In $\mathbb P$5-based medial axis partitioning, one thresholds the $\mathbb P$6-map, skeletonizes the resulting binary image, and combines the skeleton with the original void space to separate pores. The skeleton passes through the pore throats and traces the medial structure of the pore space (Ben-Noah et al., 2024). The paper notes that intersections of ridges and valleys correspond to saddle points and hence throat centers. A stated limitation is that with large throats, some pores may merge; the threshold at $\mathbb P$7 can be adjusted, but too aggressive a threshold can over-split large pores.

In $\mathbb P$8-based watershed partitioning, watershed flooding is applied to the $\mathbb P$9 map rather than to the distance map. The stated reason is that in the distance map, saddles are positive and can be misread as pore markers, whereas in the $\mathbb S$0 map, saddle points are negative like grains and are therefore not mistakenly treated as pore seeds (Ben-Noah et al., 2024). If some pores are left without a pore-body marker, the algorithm annexes them to the nearest pore center.

The stated advantage is the elimination of the common problem of saddle-induced over-partitioning shared by all traditional marker-based watershed methods, without the need for morphological reconstruction (Ben-Noah et al., 2024).

5. Structural outputs and reported behavior

The locating step yields pore-body centers and radii, throat centers and radii, throat directions, pore-body size distributions, throat size distributions, and mean coordination number (Ben-Noah et al., 2024). The partitioning step yields distinct pore labels, pore volumes or areas, pore surface areas or perimeters, coordination-number distributions, skeleton extraction of the pore space, and pore-network models without morphological reconstruction (Ben-Noah et al., 2024).

The reported validation includes regular sphere packings, a Finney random packing, 2D granular media, and Berea sandstone. For simple cubic and BCC packings, the method finds exact body and throat locations, with coordination numbers matching theory: SC has $\mathbb S$1, and BCC has $\mathbb S$2 (Ben-Noah et al., 2024). In the Finney medium, throats neighboring more than two pores occur about twice as often in the traditional distance-map watershed as in the proposed $\mathbb S$3-based medial-axis method (Ben-Noah et al., 2024). For Berea sandstone, the locating method yields a mean coordination number $\mathbb S$4, while the partitioned pore count from the $\mathbb S$5-based medial axis is close to the number of located critical bodies (Ben-Noah et al., 2024).

Real images require false-critical-point handling. The paper uses single-layer Gaussian filtering and Gaussian pyramid filtering, which reduce false maxima and saddles without creating new extrema, together with a complementary dilution step that removes overlapping pore bodies, overlapping saddle points, and saddle points whose sphere is fully embedded inside a pore-body sphere (Ben-Noah et al., 2024).

6. Point-cloud MAP-curvature and the Weingarten map

A second major meaning of MAP-curvature is the closed-form estimation of the Weingarten map from point-cloud data sampled from a manifold embedded in Euclidean space (Cao et al., 2019). Let $\mathbb S$6 be an embedded submanifold, $\mathbb S$7, and $\mathbb S$8. The Weingarten map is defined by

$\mathbb S$9

and the key first-order expansion is

$\partial \mathbb S$0

The estimator proceeds in two steps. First, local PCA estimates an orthonormal tangent basis and a normal basis. Second, for each normal basis vector, a local least-squares problem estimates the corresponding shape operator. In matrix form, the closed-form estimator is

$\partial \mathbb S$1

The paper identifies this as the efficient WME/MAP-curvature estimator because it avoids nonlinear optimization and directly returns the shape operator in the estimated tangent basis (Cao et al., 2019).

Curvature is then derived from the estimated Weingarten maps using standard differential-geometric identities. For a surface in $\partial \mathbb S$2,

$\partial \mathbb S$3

and in a general embedded $\partial \mathbb S$4-manifold, the mean curvature vector is

$\partial \mathbb S$5

The statistical analysis yields

$\partial \mathbb S$6

and the bandwidth choice $\partial \mathbb S$7 gives the optimal rate

$\partial \mathbb S$8

(Cao et al., 2019). Empirically, the method is reported to achieve lower MSE and better robustness than local quadratic surface fitting, especially under Gaussian noise, and to produce stable Gaussian and mean curvature estimates on a brain cortical surface point cloud (Cao et al., 2019).

This suggests a useful distinction. In porous media, MAP-curvature refers to a Hessian-of-distance-map framework. In point-cloud geometry, it refers to an estimator of the shape operator from discrete manifold samples.

7. Related geometric and moment-map formulations

Several mathematically distinct constructions are adjacent to the MAP-curvature label because they define curvature through maps or pullbacks rather than directly from a background metric.

For the Abel–Jacobi map $\partial \mathbb S$9, when $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$0 the map is an embedding, so the pullback of the flat metric on $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$1 becomes a Kähler metric on $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$2 (Biswas, 2020). When $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$3, the curvature of the pullback metric is always nonpositive; it is strictly negative everywhere if $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$4 is not hyperelliptic, and in the hyperelliptic case it vanishes exactly at the fixed points of the hyperelliptic involution (Biswas, 2020). Here the curvature is controlled by the differential of an associated Grassmannian map.

For normally flat submanifolds in a space form, the Gauss image carries the third fundamental form

$$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$5

and the curvature of the Gauss image is explicitly determined by the original curvature and Weingarten operators of the immersion (Álvarez-Vizoso, 2022). The paper presents this as a Gauss-map analogue of theorema egregium.

In generalized Kähler geometry, the generalized scalar curvature is realized as a moment map. For a generalized Kähler structure of symplectic type, the generalized scalar curvature $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$6 is the moment map for the generalized Hamiltonian action on the space of compatible generalized complex structures (Goto, 2016). In the twisted setting, the scalar curvature $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$7 is likewise identified as the moment map for a modified action of generalized Hamiltonian diffeomorphisms (Goto, 2021). These constructions provide the strongest purely geometric justification for the phrase “moment-map curvature.”

Other nearby uses are more contextual. In a coupled Ricci flow–harmonic map heat flow background, mean curvature flow acquires correction terms involving $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$8, the potential $$f(\mathbf x_k) = \frac{\text{sign}[\mathbb{I}(\mathbf x_k \in \mathbb P)-0.5]}{\Delta x}\min_{\mathbf x_j \in \partial \mathbb S}\left(\lVert \mathbf x_k - \mathbf x_j \rVert \right).$$9, and boundary contributions, leading to a boundary Harnack-type quantity and Huisken-type monotonicity formulas in a “MAP-curvature” setting (Gomes et al., 27 Oct 2025). In positive scalar curvature theory, the phrase is used informally for the curvature theory associated with the canonical classifying map $$H_{ij} = \frac{\partial^2 f(\mathbf x)}{\partial x_i \partial x_j},$$0, where lower bounds on the rank of groups of positive scalar curvature metrics are expressed in terms of the cokernel and kernel of the induced map on rational homology or $$H_{ij} = \frac{\partial^2 f(\mathbf x)}{\partial x_i \partial x_j},$$1-theory (Schick et al., 2020).

8. Conceptual synthesis

Across these usages, MAP-curvature does not denote a single invariant. It denotes a family of constructions in which curvature is derived from a map-dependent object: the Hessian of a distance map, the Weingarten map estimated from normal-field variation, the pullback of a flat metric by the Abel–Jacobi map, the curvature of a Gauss image, or the moment map associated with generalized scalar curvature (Ben-Noah et al., 2024, Cao et al., 2019, Biswas, 2020, Álvarez-Vizoso, 2022, Goto, 2016, Goto, 2021).

Two features recur. First, curvature is operationalized through an auxiliary structure that is easier to compute or analyze than the original geometry: a distance map, a local regression system, a tautological bundle over a Grassmannian, or pure-spinor data. Second, the resulting curvature quantity is used not only descriptively but structurally: to locate throats and pore bodies, to estimate principal and mean curvatures from discrete data, to characterize negativity or degeneracy loci of pullback metrics, or to formulate moment-map and rigidity statements.

A plausible implication is that the term persists because it names a productive strategy rather than a fixed formula: encode geometry through a map, differentiate or pull back that encoding, and recover curvature from the resulting operator, tensor, or scalar quantity.