Fourier-Based Granularity Encoder

Updated 16 December 2025

Fourier-Based Granularity Encoder is a method that quantifies data scales by decomposing signals into Fourier modes to provide coarse-to-fine representations.
It leverages mathematical foundations like Fourier and Bessel expansions to isolate and parameterize scale-specific features, enabling efficient spectral manipulation and interpretability.
Practical implementations, such as Quantized Fourier Features and FINE, demonstrate its effectiveness in neural imaging, physics-informed autoencoding, and optical encoding despite computational trade-offs.

A Fourier-Based Granularity Encoder is any transformation or representation scheme that leverages Fourier analysis to quantify, encode, or control the scale-wise structure (“granularity”) of data. Such encoders explicitly exploit the frequency localization properties of the Fourier domain, enabling structured access to coarse-to-fine features, efficient spectral manipulation, and interpretable multiresolution representations across various application domains. The notion of granularity in this context refers to how information at different spatial or frequency scales is isolated, parameterized, or compressed via spectral coefficients or basis functions.

1. Mathematical Foundation: Fourier Expansion and Multiscale Encoding

A defining property of Fourier-based granularity encoders is their use of frequency-basis decompositions to represent data with explicit control or interpretation across scales. In classical settings, this involves expanding a signal or field in terms of orthogonal Fourier modes. For example, the “2D Fourier granularity encoder” for the characterization of initial states in heavy ion collisions applies a Bessel–Fourier expansion on the disk, representing a function $E(r,\phi)$ as: $E(r,\phi) = \sum_{m=-M}^M \sum_{n=1}^N a_{m,n} f_{m,n}(r,\phi)$ with basis functions

$f_{m,n}(r,\phi) = \frac{1}{J_{|m|+1}(\lambda_{m,n})} J_m\left(\frac{r}{r_0} \lambda_{m,n}\right) e^{im\phi}$

where $J_m$ are Bessel functions and $\lambda_{m,n}$ labels the zeros. The $a_{m,n}$ serve as a quantification of energy fluctuations at precise angular and radial scales, encoding the “granularity” at those frequencies (Coleman-Smith et al., 2012).

Granularity can be further analyzed through norms and invariants over the spectral coefficients, such as the $L_2$ norm (total fluctuation power), the $H_1$ (Sobolev) norm (incorporating gradients), and the angular roughness $M_1$ . The scale-invariant roughness ratio $R^2$ is defined as: $R^2 = \frac{H_1^2}{L_2^2} - 1$ providing a rescaling-independent measure of the prevalence of high-frequency structure.

2. Algorithmic Realizations in Machine Learning

Fourier-based granularity encoders are prominent in deep learning for implicit representations, neural field models, and imaging. Key algorithmic instantiations include:

Quantized Fourier Features (QFF): QFFs discretize each Fourier (sinusoidal) feature along its value—implementing a granularity encoder directly in the frequency space. Each sinusoidal channel is quantized into $M$ uniform bins, and a small learnable vector per bin parameterizes the frequency-dependent behavior. QFF can be implemented efficiently by linearly interpolating between bin features and adding the original sinusoid, ensuring smoothness and multiscale access without the discontinuities of spatial bins. This yields a compact, continuous, and multiresolution encoding suitable for neural radiance fields, implicit shape models, and image-level MLPs. Fast convergence on high frequencies and flexible memory scaling are key advantages (Lee et al., 2022).
Fourier Domain Encoding for Images: The Fourier Image Transformer (FIT) represents an image as a sequential list of its (radially ordered) Fourier coefficients. Each prefix of this sequence corresponds to including all coefficients inside a Fourier “ring,” making the encoding inherently granular and multi-scale. Transformers conditioned on low-frequency prefixes can be trained to autoregressively or via encoder–decoder architectures fill in finer-scale coefficients, yielding coarse-to-fine or arbitrarily sampled reconstructions, with direct handling of irregular spectral samples in CT reconstruction tasks (Buchholz et al., 2021).
Fourier-Invertible Neural Encoder (FINE): FINE employs only invertible transformations up to a bottleneck, where dimensionality reduction is performed by retaining only the lowest $2k+1$ Fourier modes, explicitly enforcing a spectral granularity constraint. The architecture is strictly shift-equivariant: all transformations, including invertible pointwise nonlinearities and spectral convolutions, commute with circular translations. Reconstruction fidelity versus compression is governed directly by the truncation level $k$ , providing a tunable “granularity knob.” FINE outperforms both classical DFT truncation and CNN autoencoders in physical interpretability and parameter efficiency for homogeneous flows and turbulence data (Ouyang et al., 21 May 2025).

3. Fourier Granularity in Physical and Optical Encoding

Fourier-based granularity encoding extends beyond signal processing and machine learning to the design of encoding mechanisms in physical systems, notably optics:

Non-local Optical Encoders in Computational Imaging: Programmable phase masks and point spread functions (PSF) in microscopes can be optimized in the Fourier domain to achieve a desired “granularity” of spatial encoding, as seen in snapshot microscopy. The encoder is parameterized by a high-dimensional Fourier-phase mask, whose effect is propagated through a wave-optical model to the image. A corresponding “global” FourierNet decoder processes the measured data using global Fourier convolutional layers as the first stage, leveraging full non-local context to decode structured information (Deb et al., 2021).
Granularity-Field of View Trade-Offs: Physically, the granularity of encoding can refer to the axial or lateral sampling density, determined by degree and placement of Fourier features in PSF design. Matching the optical encoder’s spectral properties to the region of interest is critical for optimal inversion quality, with clear trade-offs between fine granularity (narrow $z$ range with dense sampling) and coarse granularity (wider $z$ with sparser sampling).

4. Practical Implementations and Algorithmic Pseudocode

Several concise algorithmic realizations for Fourier-based granularity encoders were proposed:

Quantized Fourier Features (QFF-Lite) Implementation:

def QFF_Lite(x, Θ, L=6, M=128):
    # x: [..., K] in [-bound, bound]
    # Θ: [K, 2L, M, N] learnable
    freqs = 2**torch.arange(L) * π # [L]
    φ = []
    for k in range(K):
        scaled = x[...,k,None] * freqs
        φk = torch.cat([torch.sin(scaled), torch.cos(scaled)], dim=-1)
        φ.append(φk)
    φ = torch.stack(φ, dim=-2) # [...,K,2L]
    u = (φ + 1) / 2 * (M - 1)
    t0 = u.floor().long().clamp(0, M-1)
    t1 = (t0+1).clamp(0, M-1)
    α  = (u - t0.float())[...,None]
    Θ0 = Θ.gather(2, t0[...,None].expand(...,N))
    Θ1 = Θ.gather(2, t1[...,None].expand(...,N))
    τ  = Θ0*(1-α) + Θ1*α
    φ_exp = φ[...,None]
    E = (τ + φ_exp).reshape(..., K*2L*N)
    return E

As shown, each frequency channel is independently quantized, interpolated, and combined with the base sinusoid signal (Lee et al., 2022).

2D Fourier-Granularity Expansion for Fluctuation Encoding:

for m in range(-M, M+1):
    for n in range(1, N+1):
        λ = nth_zero_of_bessel_function(m, n)
        for each (r_i, φ_j):
            f[m,n][i,j] = (1 / J_{|m|+1}(λ)) * J_m((r_i/r0)*λ) * exp(i*m*φ_j)
            weight[i,j] = (r_i * Δr * Δφ) / (π*r0^2)
        a[m,n] = sum_over_ij(E_grid[i,j] * conj(f[m,n][i,j]) * weight[i,j])

This expansion provides both a high-dimensional signature and scalar, scale-invariant surrogates for granularity (Coleman-Smith et al., 2012).

5. Empirical Performance and Application Domains

Fourier-based granularity encoders have been empirically validated across applications:

Implicit Neural Field Representations & View Synthesis: QFF enables faster high-fidelity fitting of natural images, neural radiance fields, and signed distance functions compared to standard positional encoding or spatial-grid-based schemes. For instance, QFF-3D achieves similar or superior PSNR in NeRF tasks with reduced model size and accelerated convergence (e.g., reaching final PSNR in ~5K iters versus ~50K for standard MLPs) (Lee et al., 2022).
2D Density Fluctuation Classification: The Bessel–Fourier ‘granularity encoder’ provides a discriminative summary of event shapes in heavy ion physics. The scale-invariant roughness $R^2$ enables distinguishing between different models (e.g., MC-Glauber vs. MC-KLN/CGC) based on their scale-specific fluctuation power, with higher $R^2$ indicating fine-scale structure (Coleman-Smith et al., 2012).
Physics-Informed Autoencoding: FINE realizes compact, interpretable, and symmetry-preserving dimensionality reduction in turbulent flow data, with much lower reconstruction error than CNN autoencoders and classical DFT or proper orthogonal decomposition; latent modes remain physically interpretable (Ouyang et al., 21 May 2025).
Computational Imaging and Optics: Optimized phase profile encoders and global FourierNet decoders enable end-to-end scalable, non-local encoding/decoding for 3D microscopy, with empirical gains up to 7372 $\times$ larger encoded volumes than previous techniques (Deb et al., 2021).

6. Properties, Benefits, and Limitations

Key characteristics and operational trade-offs include:

Scale-Tunability and Interpretability: The explicit granularity knob (such as the truncation level $k$ , number of bins $M$ , or number of modes retained) allows a direct trade-off between detail retention and compactness. Latent dimensions can be mapped to interpretable frequency bands or physical modes (in FINE, QFF).
Continuity and Regularization: Unlike spatial binning, Fourier-based encoders maintain continuity, avoiding block artifacts. Interpolation in QFF ensures outputs are piecewise linear in the sinusoid domain.
Computation and Scalability: Sequence or coefficient length is dictated by target resolution (signal length, image size), which can become computationally expensive for full-resolution encodings; linearized attention mechanisms, interpolation, and spectral parameterizations are used to mitigate this.
Domain Limitations: Direct FFT encodings assume periodicity and are less effective for highly localized phenomena. In high-complexity images, naive spectral decoders generate speckle noise, motivating hybrid strategies (FIT+conv-block).
Symmetry Preservation: Encoders such as FINE exactly preserve translation or rotation equivariance when spectral truncation is the only non-invertible layer.

7. Summary Table: Representative Fourier-Based Granularity Encoders

Technique	Core Mechanism	Primary Application Areas
QFF	Quantized Fourier feature binning & interp	Neural fields, NeRF, SDF, images
FDE/FIT	Ordered Fourier coefficient sequencing	Image completion, sparse CT recon
FINE	Invertible net + Fourier truncation	Homogeneous flows, physics learning
2D Fourier-Disk	Bessel-Fourier expansion, norm statistics	Heavy-ion initial state analysis
FourierNet Optics	Global Fourier conv., spectral PSF design	Computational imaging, microscopy

Each method offers a rigorously defined, multiscale representation or encoder, enabling scalable, interpretable, and application-specific control of granularity in the Fourier domain.

For further technical and empirical details, see (Lee et al., 2022, Buchholz et al., 2021, Ouyang et al., 21 May 2025, Deb et al., 2021), and (Coleman-Smith et al., 2012).