Infinite Homography Warping

• Infinite homography warping is a projective image transform that uses the limiting case of a plane at infinity to encode pure camera rotation and preserve global projectivity.
• It addresses parallax challenges by combining the infinite homography with an Epipolar Displacement Field, ensuring robust stitching and faithful high-frequency detail reproduction.
• The technique underpins learning-based frameworks and group-theoretic approaches, enabling continuous, artifact-reduced synthesis in both image and video generation.

Infinite homography warping refers to a class of projective image warping techniques characterized by the use of the so-called "infinite homography," a planar projective transform derived from the limiting case where the reference plane resides at infinity. This concept has become integral to modern approaches in image stitching, camera control in video synthesis, and learning-based image warping, providing a foundation for warping methods that maintain global projectivity, precisely encode 3D camera rotations in 2D feature spaces, and achieve theoretically infinite output resolution. Solutions built atop infinite homography warping address challenges such as parallax, viewpoint unification, and high-frequency fidelity in warping-based generation and registration pipelines.

1. Mathematical Definition and Derivation

The infinite homography, denoted $H_\infty$, arises from the plane-induced homography between two views with camera projection matrices $P = K [I \mid 0]$ and $P' = K' [R \mid t]$, where $K$, $K'$ are camera intrinsics and $(R, t)$ the relative pose. For a 3D plane $\pi: n^T X + d = 0$, the standard homography mapping image $I$ to image $J$ is

$H = K' (R - t n^T/d) K^{-1}.$

As the plane recedes to infinity ($d \to \infty$), the term $t n^T/d$ vanishes, yielding the infinite homography

$H_\infty = K' R K^{-1}.$

This mapping captures a pure rotation (with no translation or depth dependency) and preserves projective structure globally. For two cameras with identical intrinsics ($K' = K$), it simplifies to $H_\infty = K R K^{-1}$. When applied to a pixel $p = (u, v, 1)^T$, the warped coordinate in the target frame is given in homogeneous coordinates as $\tilde{x}_\infty = H_\infty p$ and normalized to $(x_\infty, y_\infty)$ by dividing by the third component. This mapping ensures that the warped point and its true correspondence lie on the same epipolar line, complying with fundamental epipolar constraints (Yu et al., 2023, Kim et al., 18 Dec 2025).

2. Epipolar Geometry and Parallax Compensation

While $H_\infty$ enforces global projectivity and corrects for camera rotation, it does not model parallax effects arising from scene depth variation. To address this, an additional displacement field is introduced, specifically along epipolar lines. In the context of image stitching with parallax, this field—referred to as the Epipolar Displacement Field (EDF)—is estimated using thin-plate splines (TPS) based on matched feature correspondences. Formally, the EDF is defined as a corrective vector $\Delta p(p_\infty) = (g_x, g_y)^T$, modeled by TPS as

$$g_x(u, v) = \sum_{i=1}^N \omega_i \phi(\| (u, v) - x_{H, i} \|) + \alpha_1 u + \alpha_2 v + \alpha_3,$$

where $\{x_{H, i}\}$ are the $H_\infty$-warped anchor points, $\omega$ are TPS weights, and $\phi(r) = r^2 \ln r$ is the radial basis. TPS parameters are obtained by minimizing a regularized energy functional subject to side conditions enforcing zero net translation and affine consistency. The full warping map becomes

$p \mapsto x_\infty(p) + \Delta p(p),$

preserving epipolar alignment and allowing the EDF to absorb multi-plane parallax. The stitched panorama is obtained by inverse mapping, typically via backward sampling at the composite map (Yu et al., 2023).

3. Infinite Homography Warping in Learning-based Warping Frameworks

Infinite homography warping underpins several recent neural methods for continuous, artifact-reduced image warping and generation. In implicit neural representation pipelines, homography warping is extended to infinite-resolution image synthesis by modeling the residual high-frequency content at any target location with local Fourier features, where the homography’s local Jacobian encodes local frequency deformation. The core workflow involves:

• An encoder producing local feature maps.
• Per-location estimates of amplitude, frequency, and phase for a local Fourier expansion.
• Warping modeled via the homography Jacobian:

$$J_H(x,y) = \frac{1}{w^2} \begin{bmatrix} w h_{11} - u h_{31} & w h_{12} - u h_{32} \ w h_{21} - v h_{31} & w h_{22} - v h_{32} \end{bmatrix}$$

• Infinite-resolution warping achieved by query-based synthesis and local neighborhood interpolation.

On standard datasets, such architectures, when combined with explicit handling of the homography’s differential structure, achieve both state-of-the-art PSNR and visually superior preservation of fine detail and edges under strong perspective transformations. In practical pipelines such as LTEW, the Jacobian is either used directly or learned implicitly, allowing for artifacts-free upscaling and generic warp generalization (Lee et al., 2022).

4. Infinite Homography Warping for Video Generation and Camera Control

In generative video diffusion frameworks, infinite homography warping encodes camera rotations directly on the latent features, decoupling rotational motion from depth-induced translation (parallax). Given source and target camera parameters, infinite homography warping yields a "noise-free" latent rotation:

$L' = W(H_\infty, L)$

where $W$ applies $H_\infty$ with differentiable backward warping and bilinear sampling. To complete the full view synthesis, a residual-parallax predictor module learns to provide the translation- and depth-dependent correction at each location:

$x' = H_\infty x + \frac{1}{Z} K_t t.$

The predicted residual is incorporated as a learned correction within the network. Homography-guided self-attention then conditions the model on both the rotated and target latents along with camera parameters, facilitating high-fidelity, pose-consistent video generation without explicit depth estimation. This approach substantially outperforms classical reprojection and trajectory-conditioned baselines on synthetic and real-world video benchmarks regarding pose accuracy and visual fidelity (Kim et al., 18 Dec 2025).

5. Group-Theoretic and Lie-Algebraic Formulation

Infinite homography warping admits a group-theoretic interpretation, with the full homography group corresponding to $\text{SL}(3)$ and the infinite homography capturing the pure rotation subgroup. Warped Convolutional Networks (WCNs) leverage this algebraic structure by decomposing the homography into commuting subgroups (translation, rotation+scale, aspect ratio, shear, perspective). Each subgroup's action is interpreted as an elementary warp, and group convolution is implemented via translation-invariant convolution on the warped domain. This yields a cascade where parameters of each subgroup are estimated as pseudo-translation regressions, and the full homography is assembled from their exponential maps. The group structure supports efficient, equivariant learning and accurate, interpretable homography regression (Zhan et al., 2022).

Subgroup	Generator(s)	Example Action
Translation	$A_1, A_2$	Image shift
Rotation + isotropic scale	$A_3, A_4$	Rotation, global scaling
Aspect ratio	$A_5$	Anisotropic scaling
Shear	$A_6$	Horizontal shear
Perspective x, y	$A_7$, $A_8$	Perspective effects

WCNs demonstrate superior empirical performance on benchmarks for homography estimation, planar object tracking, and classification, confirming that algebraic decomposition and infinite homography warping yield architectures with both mathematical fidelity and strong empirical results.

6. Applications, Advantages, and Limitations

Infinite homography warping is utilized across diverse tasks:

• Parallax-tolerant panoramic stitching, where it achieves projectivity preservation and robust alignment under large parallax, provided the scene is rigid and cameras are calibrated (Yu et al., 2023).
• Infinite-resolution image warping, enabling artifact-free synthesis under arbitrary projective transforms (Lee et al., 2022).
• Camera-controlled novel-view video generation, attaining high pose-fidelity and visual realism in diffusion-based generation without explicit depth supervision (Kim et al., 18 Dec 2025).
• Homography-invariant deep learning via SL(3)-equivariant networks (Zhan et al., 2022).

Principal advantages include:

• Global projective structure preservation.
• Exact encoding of camera rotation in 2D, facilitating decoupled treatment of translation/parallax effects.
• Compatibility with differentiable, end-to-end learning and continuous-resolution synthesis.
• Support for aggressive data augmentation strategies involving rotation, scaling, and focal-length variation, critical for camera control learning.

Key limitations are tied to assumptions about static, rigid scenes and known or estimable camera intrinsics and rotation. Infinite homography warping alone is insufficient for scenes with severe occlusion, FOV $>180^\circ$, or dominant nonrigid deformation. Furthermore, extensions to extreme nonlinear warps or highly curved mapping domains may require higher-order encodings or explicit geometric modeling. Empirical limitations observed in LTEW include degradation of high-frequency fidelity under out-of-distribution deformations (Lee et al., 2022).

7. Experimental Validation

Quantitative and qualitative evaluation across multiple domains provides compelling evidence of the effectiveness of infinite homography warping modules:

• In panoramic stitching, qualitative comparisons show reduction in alignment artifacts and distortion under large parallax (Yu et al., 2023).
• LTEW achieves mean PSNR of 31.10 dB (in-scale) and 26.92 dB (out-of-scale) for challenging homography warps, outperforming strong SR baselines (Lee et al., 2022).
• In video generation, infinite homography warping yields the best camera error (3.16° rotation, 0.44 m translation) and lowest perceptual metrics (FID 29.7, FVD 286.9) among tested approaches (Kim et al., 18 Dec 2025).
• On classification, planar tracking, and homography estimation benchmarks, WCN leveraging infinite homography-based decomposition achieves state-of-the-art accuracy and robustness under large projective transformations (Zhan et al., 2022).

These results demonstrate that infinite homography warping, when coupled with mechanisms for parallax compensation and differential encoding, forms a robust foundation for both classical and learning-based image and video transformation systems.