3D Video Fourier Field (VFF)

• 3D Video Fourier Field (VFF) is a continuous spatio-temporal representation that decomposes dynamic visual data into Fourier basis functions.
• It enables aliasing-controlled, flexible sampling and super-resolution by unifying spatial and temporal domains in a joint Fourier framework.
• VFFs support applications such as video super-resolution, dynamic neural scene modeling, and 3D surface imaging with efficient frequency-based encoding.

A 3D Video Fourier Field (VFF) is a unified mathematical and computational representation for time-varying or dynamic visual data—specifically, videos or 3D scenes—constructed in the joint space-time domain via Fourier (trigonometric) basis expansions. VFFs enable aliasing-controlled, continuous sampling of dynamic content, allowing for precise modeling, reconstruction, and manipulation of video signals, 3D surfaces, and dynamic radiance fields without resorting to explicit frame interpolation or warping. Several recent works formalize VFFs either as sinusoidal function expansions for videos (Becker et al., 30 Sep 2025), compact frequency-domain dynamic neural scene representations (Wang et al., 2022), or as a means of embedding local 3D geometric information into point-clouds via spatial-Fourier transformation pairs (Maquignaz, 2022).

1. Mathematical Foundations and Formal Definition

At the core of a 3D Video Fourier Field is the decomposition of a spatio-temporal signal—whether a 2D video, dynamic 3D scene, or a pattern-projected surface—into a continuous function defined over both spatial and temporal coordinates, typically encoded as a sum of sinusoidal (Fourier-like) basis functions. The general form for a VFF representing a video signal is:

$\hat{V}(x, y, t) = \sum_{i=1}^N B_i(x, y, t),$

where each basis function is parameterized as:

$$B_i(x, y, t) = a_i \cdot \sin(\omega_i \cdot (x, y, t) + \phi_i),$$

with $a_i$ the amplitude, $\omega_i$ the angular frequency vector in space-time, and $\phi_i$ the phase.

For 3D surface applications, the Fourier transform pairs extend to the volume:

$$F(u,v,w) = \iiint f(x,y,z) e^{-j2\pi(ux+vy+wz)} \, dx\,dy\,dz \ f(x,y,z) = \iiint F(u,v,w) e^{j2\pi(ux+vy+wz)} \, du\,dv\,dw,$$

and perspective or affine transformations of surfaces are mapped into canonical transformations in the frequency domain (Maquignaz, 2022).

For dynamic 3D neural scene modeling (e.g., dynamic NeRF), time is incorporated as an extra dimension: for each spatial cell, time-varying properties such as density ($\sigma$) and spherical harmonic coefficients ($z$) are represented not as raw sequences but by their (discrete) Fourier coefficients (Wang et al., 2022). These coefficients permit instantaneous recovery of temporal values via inverse DFT:

$$\sigma(t; k^\sigma) = \sum_{i=0}^{n_1 - 1} k^\sigma_{i} \cdot \mathrm{IDFT}_i(t)$$

with IDFT basis functions based on sines and cosines.

2. Unified Spatio-Temporal Representation and Flexible Sampling

VFFs provide a continuous, aliasing-controlled representation over both spatial and temporal domains. Unlike conventional video representations that treat space and time as decoupled (e.g., per-frame 2D INRs plus separate motion estimation), a VFF is a coherent 3D field. As a consequence:

• Flexible sampling: any $(x, y, t)$ coordinate can be queried—supporting spatial, temporal, and joint upscaling arbitrarily.
• Simultaneous modeling: fine spatial details and smooth, global or non-linear temporal dynamics are encoded uniformly through the shared frequency basis.
• Aliasing-free super-resolution: Analytical Gaussian PSFs can be incorporated directly in the Fourier domain; for each basis, the sampling is scaled as

$$\hat{V}_\sigma(x, y, t) = \sum_{i=1}^N B_i(x, y, t) \cdot \xi(\omega_i, \sigma),$$

where $$\xi(\omega_i, \sigma) = \exp(-||\omega_i||^2 / (8\pi^2 \sigma^2))$$ (Becker et al., 30 Sep 2025), ensuring anti-aliasing without costly explicit filtering.

These properties substantially improve both spatial and temporal video super-resolution compared to techniques relying on explicit, error-prone frame warping or pairwise interpolation.

3. Frequency-Based Encoding of Dynamics and Geometry

The Fourier representation in VFFs confers several critical advantages for encoding dynamics and geometry:

• Temporal and spatial variations—such as motion, deformation, or scene changes—are expressed compactly as phase shifts and amplitude/frequency changes in the basis expansion.
• In Fourier PlenOctrees (FPO) (Wang et al., 2022), each spatial location (octree leaf) stores a set of Fourier coefficients for density and color attributes. These coefficients efficiently capture the time-varying structure of the scene:
  • $k^\sigma \in \mathbb{R}^{n_1}$ encodes density over time,
  • $$k^z \in \mathbb{R}^{n_2 \times (\ell_{max} + 1)^2 \times 3}$$ encodes time-varying coefficients for color SH basis.
• For 3D surface imaging, the transformation of projected 2D patterns under arbitrary geometric changes (scaling, rotation, perspective) can be directly decoded from the Fourier spectrum:
  • The Spectral Perspective Transformation Theorem states $G(u,v,w) = H F(u,v,w) e^{j2\pi(EC)}$, relating spatial perspective parameters to explicit frequency-domain transformations (Maquignaz, 2022).

This frequency-based encoding enables both compact storage (through basis truncation) and the accurate modeling of long-term or global temporal dependencies.

4. Learning and Construction Methodologies

Several distinct methodologies have been proposed for constructing VFFs, tailored to application:

• Neural Parameterization: Modern works use deep neural encoders with large spatio-temporal receptive fields to infer local basis coefficients from low-resolution input (Becker et al., 30 Sep 2025), providing robustness to occlusion and long-range dependencies.
• Discrete Fourier Transform on Tree-Based Structures: For dynamic scene rendering, an efficient coarse-to-fine strategy builds a unified spatial tree (octree), then computes the DFT of per-frame densities to populate leaf Fourier coefficients (Wang et al., 2022).
• Direct Frequency-Space Analysis: In 3D surface imaging with projected patterns, known frequencies are embedded in the pattern, and direct DFT analysis on captured images recovers geometric transformation parameters at each surface point (Maquignaz, 2022).
• Analytical PSF Integration: Explicit Gaussian windowing in frequency space enables aliasing-controlled resampling at test-time via simple scaling, without need for ad hoc anti-aliasing filters (Becker et al., 30 Sep 2025).

These approaches share the underlying principle of trading explicit, high-dimensional parametric modeling for a small, information-rich set of learned or computed Fourier coefficients, enabling fast, memory-efficient, and flexible rendering or reconstruction.

5. Applications and Practical Implications

VFFs support diverse tasks requiring real-time, high-fidelity manipulation of dynamic visual data:

Application Domain	VFF Modality	Outcome
Video Super-Resolution	(x, y, t) VFF (Becker et al., 30 Sep 2025)	State-of-the-art upscaling, temporal smoothness, sharp details
Dynamic Neural Scene Modeling	4D FPO (space + time) (Wang et al., 2022)	Real-time, high-quality rendering of dynamic free-viewpoint video
3D Surface Imaging	SFTP-augmented point clouds (Maquignaz, 2022)	Enriched (X,Y,Z)-(R,G,B) data with local perspective and deformation info

In practice, VFFs advance applications such as:

• Telepresence, VR/AR, and interactive cinematography (due to real-time, smooth, free-view exploration of dynamic scenes and objects).
• Robotics and autonomous systems, where rapid 3D surface recovery plus local affine/perspective transformation metadata strengthens perception and control.
• Medical imaging, heritage preservation, precision manufacturing, and high-speed video analysis, benefitting from enhanced surface recovery.

6. Limitations and Trade-Offs

While VFFs address major challenges in computational imaging and dynamic scene modeling, trade-offs and limitations are evident:

• Memory and Compute Scaling: As the number of basis functions or the Fourier expansion order increases for long or highly dynamic sequences, storage/memory demands grow (Wang et al., 2022). Truncation offers compaction but may filter crucial high-frequency (fast motion) content.
• Noise Sensitivity: Frequency-based analysis amplifies measurement or capture noise, especially in DFT-based imaging methods (Maquignaz, 2022). Strategies like sub-pixel matching or periodic extension are required for robustness.
• Resolution and Aliasing Constraints: There is a balance between spatial resolution of projected/captured patterns and spectral resolution. Analytical PSF integration mitigates aliasing but cannot recover out-of-band energy.
• Manual or Hyperparameter Tuning: The choice of Fourier basis dimension, octree granularity, or network architecture in neural basis prediction must be selected in accordance with specific content characteristics.
• Fine-Tuning Dynamics: In explicit scene models, rapid fine-tuning is enabled (e.g., in minutes for FPO), but beyond a point longer optimization brings diminishing returns (Wang et al., 2022).

7. Prospects, Extensions, and Future Research

Recent work on 3D Video Fourier Fields positions them as a foundational paradigm for continuous, scalable, and information-rich modeling across visual computing. Potential future developments include:

• Extending VFFs to cover more complex degradations (e.g., sensor noise, motion blur, compression artifacts) by leveraging the flexibility of Fourier parameterization (Becker et al., 30 Sep 2025).
• Integration with generative or diffusion-based frameworks to counteract smoothing at extreme upscaling factors, improving perceptual detail.
• Scalable representations to support longer or more complex videos by increasing capacity (basis size, network depth) in a computationally efficient manner.
• Cross-modal or physics-informed extensions, where separate but jointly optimized fields encode physical dynamics (as in neural velocity fields), semantics, or even underlying forces.

Adoption and further paper of VFFs may influence future approaches to video editing, 3D visual effects, immersive display technologies, and real-time computer vision across a wide spectrum of domains.