Spherical Epipolar Module

• Spherical Epipolar Module is a framework that replaces planar epipolar geometry with spherical mappings to accurately enforce cross-view correspondences in panoramic and fisheye imaging.
• It leverages spherical projections and rectification techniques to reduce distortions, enabling robust feature matching and enhanced 3D reconstruction across multiple views.
• Applications span panoramic diffusion, video generation, multi-view stereo, and fisheye motion analysis, offering measurable improvements in accuracy and point correspondence over traditional methods.

A spherical epipolar module is a geometry-aware mechanism for omnidirectional, fisheye, or panoramic vision that enforces epipolar constraints in a spherical or equirectangular domain rather than in a rectified pinhole image plane. In the literature, the term is used explicitly for cross-view attention in panoramic diffusion and video generation, where attention is restricted to spherical epipolar lines, and closely related formulations appear in multi-view stereo, 360 stereo rectification, and fisheye motion analysis, where correspondences are constrained on the unit sphere or transformed into single-axis disparity through spherical mappings (Ye et al., 2024, Ji et al., 24 Sep 2025, Elhashash et al., 2022, Obeidavi et al., 23 Jun 2026).

1. Terminological scope and conceptual role

In the strictest sense, a spherical epipolar module refers to a component that computes or exploits spherical epipolar geometry for correspondence, fusion, or consistency enforcement across views. In DiffPano, this takes the form of a Spherical Epipolar-Aware Attention Module for text-driven panoramic generation, designed because existing epipolar attention mechanisms are for perspective images and “do not work with the unique geometry of spherical/equirectangular panoramic images” (Ye et al., 2024). In CamPVG, the term denotes a module that “enforces geometric constraints through adaptive attention masking along epipolar lines,” enabling cross-view feature aggregation in panoramic video generation (Ji et al., 24 Sep 2025).

The broader research area includes methods that do not always use the same label but implement the same geometric principle. “Investigating Spherical Epipolar Rectification for Multi-View Stereo 3D Reconstruction” replaces planar rectification with projection onto a sphere to reduce distortions caused by differences in principal rays (Elhashash et al., 2022). “Spherical-to-ERP Epipolar Rectification for Single-Axis Disparity in 360 Stereo” uses spherical-to-equirectangular projection to straighten curved epipolar correspondences and recover one-dimensional disparity (Obeidavi et al., 23 Jun 2026). For fisheye motion analysis, the epipolar constraint is reformulated directly on the unit sphere, where epipolar loci are great circles rather than planar lines (Mariotti et al., 2021, Mariotti et al., 2020).

This suggests that “spherical epipolar module” is best understood as a family of operators rather than a single canonical architecture. Across applications, the common function is to replace unconstrained 2D matching or attention with geometry restricted to a lower-dimensional locus induced by calibrated multi-view spherical imaging.

2. Spherical epipolar geometry and coordinate mappings

The geometric basis is the replacement of planar image coordinates with spherical directions. In wide-FoV and fisheye settings, planar epipolar lines become curved image loci, whereas on the sphere they are represented as great circles or as intersections of epipolar planes with the viewing sphere (Mariotti et al., 2021, Mariotti et al., 2020). For static points in fisheye motion segmentation, the spherical epipolar plane normal is given by

$$\mathbf{n}^\prime = \frac{\mathbf{p} \times \mathbf{e}^\prime}{|\mathbf{p} \times \mathbf{e}^\prime|}$$

and the epipolar deviation by

$$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$

with the geodesic alternative

$$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$

(Mariotti et al., 2021).

In spherical rectification for MVS, planar-rectified pixels are mapped from image coordinates to camera coordinates and then to spherical coordinates:

$$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$

and

$$\begin{aligned} \lambda &= \arctan2(X_c, Z_c) \ \varphi &= \arctan2(-Y_c, \sqrt{X_c^2 + Z_c^2}). \end{aligned}$$

To ensure epipolar-line coincidence as rows in the spherical image, the method swaps $X_c$ and $Y_c$ before applying the angular formulas, and then defines spherical image coordinates by

$u = s\lambda, \qquad v = s\varphi$

(Elhashash et al., 2022).

For 360 stereo, spherical coordinates are mapped to an ERP grid through

$u = S_u \cdot \theta, \qquad v = S_v \cdot \phi,$

so that left–right rigs exhibit disparities only along $u$ and top–bottom rigs only along $$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$0 after rectification (Obeidavi et al., 23 Jun 2026). In panoramic diffusion, DiffPano maps panorama pixels $$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$1 to spherical angles by

$$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$2

and then to camera Cartesian coordinates by

$$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$3

CamPVG uses a related spherical direction parameterization

$$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$4

and

$$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$5

for equirectangular panoramas (Ye et al., 2024, Ji et al., 24 Sep 2025).

A recurring consequence is that the module must respect spherical projection explicitly. Straight-line epipolar assumptions from perspective geometry are insufficient in equirectangular or fisheye imagery because the corresponding loci are curved in image space but geometrically simple on the sphere.

3. Rectification-driven formulations for stereo and multi-view reconstruction

One major instantiation of spherical epipolar reasoning is spherical rectification. In multi-view stereo reconstruction, the standard workflow is epipolar rectification followed by dense matching, but large differences in principal ray directions introduce substantial distortions when images are rectified to a common plane, especially for oblique images and multi-camera arrays (Elhashash et al., 2022). The proposed remedy is to perform planar rectification first and then project the rectified images onto a sphere, with dense matching carried out in the spherical image domain.

The implementation sequence reported for this method is: estimate rectification homographies $$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$6 and $$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$7; rectify both images; project each rectified pixel to the sphere after computing $$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$8 and swapping $$\xi_e = | \mathbf{n}^\prime \cdot \mathbf{p}^\prime |,$$9; map to spherical image coordinates $$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$0; rotate the image by $$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$1 as needed so epipolar lines are horizontal; perform hierarchical semi-global matching on the spherical images; and triangulate using known camera parameters (Elhashash et al., 2022). The paper states that this approach “minimize[s] distortions caused by differences in principal rays” and reports improvements over frame-based epipolar correction on two aerial-based datasets with multi-camera head systems (Elhashash et al., 2022).

The quantitative comparison is specific. For completeness, the spherical model increases point count by 4.05% on the Dortmund dataset and by 3.70% on the Bordeaux dataset over frame-based rectification. For accuracy, measured as mean absolute cloud-to-cloud distance to LiDAR, the paper reports the following values (Elhashash et al., 2022):

Dataset	Frame-based	Spherical
Dortmund mean (m)	0.879	0.789
Dortmund std (m)	0.769	0.748
Bordeaux mean (m)	0.309	0.285
Bordeaux std (m)	0.233	0.234

The corresponding improvements are described as up to 10.23% in accuracy for Dortmund and 7.6% for Bordeaux, using LiDAR as ground truth (Elhashash et al., 2022).

A related but distinct rectification strategy appears in 360 stereo. There, omnidirectional stereo images violate the single-axis disparity assumption because epipolar correspondences follow curved great-circle paths and induce two-dimensional displacements (Obeidavi et al., 23 Jun 2026). The proposed preprocessing converts spherical imagery to ERP, which “straightens epipolar curves and restores a one-dimensional disparity structure.” RAFT then computes dense 2D optical flow on the ERP pair, and Epipolar-Aligned Channel Selection retains only the baseline-aligned component:

$$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$2

for left–right rigs, or

$$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$3

for top–bottom rigs (Obeidavi et al., 23 Jun 2026). The paper states that the resulting spherical-to-ERP-to-RAFT+EACS pipeline yields “accurate, smooth, and structurally consistent disparity maps at real-time speed” on synthetic fisheye stereo datasets (Obeidavi et al., 23 Jun 2026).

These rectification-based approaches differ from attention modules architecturally, but they embody the same computational principle: spherical epipolar structure is used to restore a constrained search geometry that conventional dense matching can exploit.

4. Attention-based spherical epipolar modules in panoramic diffusion and video generation

The most explicit use of the term appears in generative modeling for panoramas. DiffPano introduces a “spherical epipolar-aware multi-view diffusion model” whose core operation is to enforce cross-view attention only along spherical epipolar lines in reference panoramas (Ye et al., 2024). For a target-view pixel, the method samples points on the corresponding 3D ray, projects those samples to the $$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$4 nearest reference panoramas, gathers features at the reprojected locations, and performs cross-attention with the target feature as query and the sampled reference features as keys and values (Ye et al., 2024).

The attention primitive is standard:

$$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$5

but the eligible key/value locations are constrained by spherical epipolar geometry rather than a dense frame-wide search (Ye et al., 2024). To encode 3D geometry, the module concatenates Plücker coordinates of the ray and harmonically transformed depth information:

$$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$6

The module is inserted at multiple locations in the UNet, including the MidBlock and several upsampling and downsampling blocks (Ye et al., 2024).

DiffPano also gives an explicit closed-form expression for the spherical epipolar line in ERP source panoramas:

$$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$7

with coefficients $$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$8 determined by the involved geometry (Ye et al., 2024). The reported role of this design is to ensure multi-view consistency for generated panoramic images across camera poses.

CamPVG develops a closely related module for camera-controlled panoramic video generation, but formulates it as adaptive attention masking along spherical epipolar lines (Ji et al., 24 Sep 2025). Given relative pose

$$\xi_{\mathbf{S}^2} = \left| \arcsin\left( \mathbf{n}^\prime \cdot \mathbf{p}^\prime \right) \right|$$9

the module constructs the epipolar plane in the target panoramic view and derives the epipolar line in equirectangular pixel coordinates as

$$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$0

where $$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$1 and $$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$2 are coefficients derived from 3D camera geometry (Ji et al., 24 Sep 2025).

Its distinctive feature is mask construction. For each query pixel, $$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$3 points are uniformly sampled along the epipolar line. For any candidate key pixel $$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$4, the minimum spherical distance to the sampled set $$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$5 is computed as

$$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$6

Pixels within a threshold are marked valid, forming a binary mask $$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$7 (Ji et al., 24 Sep 2025). Cross-view aggregation is then performed with masked attention:

$$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$8

The paper states that this layer is applied before temporal attention in the video diffusion UNet, so that cross-view geometric consistency is learned before temporal dynamics (Ji et al., 24 Sep 2025).

Across both systems, the module is neither a generic transformer block nor a mere coordinate conversion. Its defining property is that geometric admissibility is computed from spherical epipolar relations and then used to restrict feature aggregation to plausible cross-view correspondences.

5. Extensions to fisheye constraints, pose estimation, and related modules

Outside panoramic generation, spherical epipolar formulations have been used as direct geometric constraints. In monocular fisheye moving-object detection, the epipolar constraint is reformulated in spherical coordinates in a way that is “invariant to specific camera configuration once the calibration is known” (Mariotti et al., 2020). The same paper states that the great-circle formulation is integrated with positive depth, positive height, and an anti-parallel constraint to address motion-parallax ambiguity for parallel moving objects (Mariotti et al., 2020). A later paper on motion segmentation in fisheye cameras reiterates the same spherical epipolar construction and notes that the approach is model-agnostic for any central camera with known calibration (Mariotti et al., 2021).

For essential-matrix estimation in 360-FoV imagery, “Robust 360-8PA” defines the spherical epipolar constraint directly in terms of unit bearing vectors:

$$\begin{aligned} X_c &= \frac{(x - c_x)}{f} \ Y_c &= -\frac{(y - c_y)}{f} \ Z_c &= 1 \end{aligned}$$9

It then introduces a preconditioning strategy with diagonal normalization

$$\begin{aligned} \lambda &= \arctan2(X_c, Z_c) \ \varphi &= \arctan2(-Y_c, \sqrt{X_c^2 + Z_c^2}). \end{aligned}$$0

and normalized bearings

$$\begin{aligned} \lambda &= \arctan2(X_c, Z_c) \ \varphi &= \arctan2(-Y_c, \sqrt{X_c^2 + Z_c^2}). \end{aligned}$$1

leading to

$$\begin{aligned} \lambda &= \arctan2(X_c, Z_c) \ \varphi &= \arctan2(-Y_c, \sqrt{X_c^2 + Z_c^2}). \end{aligned}$$2

The paper reports that this normalization can increase camera pose accuracy by about 20% without significantly increasing computation time, and presents further gains when combined with constant weighted least-square optimization and relaxed RANSAC iteration counts (Solarte et al., 2021).

A different but conceptually adjacent use of epipolar constraints appears in cross-view attention for industrial anomaly detection. The paper introduces an epipolar geometry-constrained attention module that masks cross-view attention using the fundamental matrix in a planar setting (Liu et al., 14 Mar 2025). It explicitly remarks that for “Spherical Epipolar Modules,” the epipolar mask would be adapted from planar lines to spherical epipolar curves or great circles (Liu et al., 14 Mar 2025). This does not itself present a spherical implementation, but it clarifies how the module abstraction generalizes from perspective to spherical multi-camera systems.

These examples show that spherical epipolar modules can act as rectifiers, constraints, masks, or attention selectors. What remains invariant is the use of calibrated spherical geometry to narrow correspondence search and regularize estimation.

6. Design trade-offs, limitations, and open directions

The literature identifies several limitations. In spherical MVS rectification, coordinate conversion, spherical warping, and potentially larger epipolar search domains introduce computational overhead; residual distortions may persist at very high latitudes or with large radial distortions; and geometric correction does not directly resolve photometric or radiometric inconsistencies (Elhashash et al., 2022). In 360 stereo ERP rectification, equirectangular projection introduces area distortion near the poles, and the method is most effective for horizontal or vertical stereo rigs rather than arbitrary baseline orientations (Obeidavi et al., 23 Jun 2026).

For attention-based panoramic generation, the need to compute spherical epipolar loci, sample along them, and aggregate multi-view features implies a stronger dependence on accurate pose and projection geometry than in geometry-agnostic generation. DiffPano frames the module as necessary because perspective epipolar attention does not transfer directly to spherical or equirectangular imagery (Ye et al., 2024). CamPVG similarly motivates its module by the fact that perspective formulations break down in panoramic projection and by the difficulty of panoramic pose representation (Ji et al., 24 Sep 2025).

Several open directions are named explicitly in the source material. For 360 stereo, future work includes “direct spherical-domain matching” without projection, exploration of projection alternatives such as “cube-map, icosahedral, etc.,” and more reliable detection of epipolar geometry from arbitrary omnidirectional rigs (Obeidavi et al., 23 Jun 2026). For spherical rectification in MVS, potential directions include integration with learning-based or PatchMatch variants, generalization to omnidirectional or multi-fisheye arrangements, automated scale selection for the spherical image parameter $$\begin{aligned} \lambda &= \arctan2(X_c, Z_c) \ \varphi &= \arctan2(-Y_c, \sqrt{X_c^2 + Z_c^2}). \end{aligned}$$3, and end-to-end learning that jointly optimizes rectification and matching in a spherical domain (Elhashash et al., 2022).

A plausible implication is that future spherical epipolar modules will increasingly combine exact geometry with learned correspondence or generation networks, while differing mainly in where the geometric restriction is inserted: before matching through rectification, during feature aggregation through masking or attention, or after matching through geometry-aware refinement. The surveyed papers collectively support that interpretation, but they do not yet establish a single unified standard architecture for the term (Ye et al., 2024, Ji et al., 24 Sep 2025, Elhashash et al., 2022, Obeidavi et al., 23 Jun 2026).