Jinc Kernel in 3D Reconstruction

• Jinc Kernel is the spatial-domain representation of the ideal 3D low-pass filter, characterized by a brick-wall frequency response that achieves perfect anti-aliasing.
• Its slow spatial decay creates computational challenges and visual artifacts, making it impractical for high-resolution applications.
• Modulated variants, such as modulated Gaussian and Student’s t, balance spectral fidelity with spatial compactness to enhance 3D scene rendering.

The Jinc kernel is the spatial-domain form of the ideal low-pass filter (ILPF) for 3D signal reconstruction, arising as the inverse Fourier transform of an indicator function bounded by a spherical cutoff frequency. It is characterized by a frequency response with unit magnitude within a cutoff radius $f_c$ and zero beyond, constituting a "brick-wall" spectrum that achieves perfect anti-aliasing. While the Jinc kernel's frequency selectivity is theoretically optimal, its spatial decay is slow, leading to computational inefficiencies and practical artifacts. Recent research has proposed modulated variants that balance the conflicting demands of spectral fidelity and spatial compactness, providing improved performance for 3D scene representations (Zhang et al., 25 Jan 2026).

1. Mathematical Formulation of the Jinc Kernel

The ideal 3D low-pass filter in the frequency domain is given by a hard threshold: $$H(\mathbf{f}) = \begin{cases} 1, & \|\mathbf{f}\| \leq f_c \ 0, & \|\mathbf{f}\| > f_c \end{cases}$$ where $f_c$ is the cutoff frequency and $\mathbf{f} \in \mathbb{R}^3$ is the frequency vector. Its inverse Fourier transform, under radial symmetry, produces the spatial-domain Jinc kernel: $h(r) = \frac{2f_c^2}{r}\, j_1(2\pi f_c r)$ with $r = \|\mathbf{r}\|$ and $$j_1(\alpha) = \frac{\sin\alpha - \alpha\cos\alpha}{\alpha^2}$$ being the spherical Bessel function of the first kind.

For more general support, the anisotropic form is: $$h(\mathbf{x}) = \frac{j_1\left(\sqrt{(\mathbf{x}-\mu)^\top \Sigma^{-1}(\mathbf{x}-\mu)}\right)}{\sqrt{(\mathbf{x}-\mu)^\top \Sigma^{-1}(\mathbf{x}-\mu)}}$$ where $\mu$ denotes the center and $\Sigma$ the covariance matrix. These spatial forms define the "Jinc" family.

The corresponding frequency-domain property is: $$\mathcal{F}\{h\}(\mathbf{f}) = H(\mathbf{f}) = \mathbf{1}_{\|\mathbf{f}\| \leq f_c}$$ implying zero frequency leakage and an instantaneous drop to zero at $\|\mathbf{f}\| = f_c$.

2. Jinc Kernel and Aliasing Suppression

In sampling theory, discretizing a continuous 3D radiance field with a pixel grid induces periodic spectrum replication. Without appropriate low-pass filtering, high-frequency replicas fold into the baseband, introducing aliasing. The application of a reconstruction kernel $h$ with Fourier transform $\mathcal{F}\{h\}$ is intended to suppress these artifacts.

The Jinc kernel provides perfect anti-aliasing:

• For $$\mathcal{F}\{h\}(\mathbf{f}) = \mathbf{1}_{\|\mathbf{f}\| \leq f_c}$$, only the baseband is preserved.
• For kernels such as Gaussians, exponentials, and Student’s t, the frequency roll-off is non-ideal, resulting in persistent, nonzero values beyond $f_c$. This allows high-frequency leakage and overlapping of spectral replicas, producing aliasing artifacts.

These distinctions are critical in 3D scene reconstruction methods such as Gaussian Splatting [Kerbl et al. 2023] and Student's t Splatting [Zhu et al. 2025], where kernel choice impacts rendering fidelity (Zhang et al., 25 Jan 2026).

3. Spatial Inefficiency of the Jinc Kernel

Although the Jinc kernel achieves ideal frequency selectivity, its spatial decay is asymptotically slow ($h(r) \propto r^{-1}$). Table 1 in (Zhang et al., 25 Jan 2026) reports a 95% spatial energy radius of $r_{95\%} \approx 5.6$ (normalized units), much larger than that of a Gaussian kernel ($\approx 2.8$). This entails:

• Large spatial support, increasing both computational load and memory requirements.
• Necessity for each rendered pixel to integrate contributions from many spatial primitives.
• Truncation within finite render windows, which disrupts continuity across tiles and produces visible artifacts, such as rectangle boundaries.

A plausible implication is that the pure Jinc kernel, despite its spectral advantage, is impractical for high-resolution or resource-constrained applications.

4. Modulated Kernels: Compactness–Fidelity Trade-off

To address the spatial inefficiency of the Jinc kernel, a family of modulated kernels superimposes frequency sidebands on standard spatial kernels:

• Modulated Gaussian: For $g(x) = \exp(-x^2/2\sigma^2)$ and Fourier transform $G(f)$,

$h_g(x) = g(x) [\,\omega + (1-\omega)\cos(f_0 x)]$

$$\mathcal{F}\{h_g\}(f) = \omega G(f) + \tfrac{1-\omega}{2}[G(f-f_0) + G(f+f_0)]$$

with $f_0$ set to half the full-width at half-maximum of the base kernel and $\omega \in [0,1]$, typically $\omega=0.5$.

• Modulated Student’s t: Analogous construction using a radially symmetric Student’s t kernel $t(r)$ and its Fourier transform $T(f)$.

This modulation sharpens the spectral transition at the cutoff, reducing frequency leakage with minimal impact on spatial compactness. Empirical results (Table 1 in (Zhang et al., 25 Jan 2026)) indicate that modulated Gaussians maintain $r_{95\%} \approx 2.8$ while materially improving frequency cut-off sharpness relative to unmodulated forms.

5. Empirical Performance and Comparative Evaluation

Experiments conducted using NeRF-Synthetic, Mip-NeRF360, Tanks & Temples, and Deep Blending datasets demonstrate key differences:

• Low-Resolution View Synthesis: For $64\times64$ images, Jinc splatting attains a PSNR of 29.87 dB, outperforming Student’s t (29.17 dB) and Gaussian (24.01 dB). At $128\times128$, Jinc achieves 31.40 dB vs. 30.24 dB (Student’s t) and 22.30 dB (Gaussian). These results validate the elimination of aliasing due to the ideal frequency property.
• Full-Resolution Rendering: Substituting the baseline kernels in 3DGS and SSS with modulated versions yields up to $+0.72$ dB PSNR and $+0.012$ SSIM improvements for modulated Gaussians, with consistent but smaller gains ($\approx +0.06$ dB PSNR) for modulated Student’s t. Qualitative results indicate reduced color-bleeding and sharper texture retention in fine structures.
• Aliasing and Perceptual Quality: Pure Jinc and its modulated variants uniformly reduce aliasing, as observed by improved LPIPS and fewer visual artifacts (Zhang et al., 25 Jan 2026).

6. Practical Guidelines for Kernel Deployment

Table: Guidelines for Jinc and Modulated Kernel Selection

Application Regime	Recommended Kernel	Key Parameter Settings
Low-resolution, high-aliasing	Pure Jinc	$\alpha\approx 30$ for support, tile and overlap to avoid border artifacts
High-resolution, compute-bound	Modulated Gaussian/Student’s t	$\omega\in[0.4,0.6]$, $f_0\approx 0.5\times\text{FWHM}$, truncation at $3\sigma$
Cutoff frequency ($f_c$)	—	Set $f_c\approx 1/(2\Delta x)$ (Nyquist)
Jinc window size	—	Truncate at $r_{\max} \approx 5/f_c$ to ensure $j_1(2\pi f_c r_{\max})/r_{\max}<10^{-3}$

Proper parameter selection is crucial for balancing spatial efficiency and frequency-domain behavior. For practical use, the Jinc kernel is optimal where aliasing is dominant, but modulated kernels are preferable when computational resources are constrained.

7. Broader Significance and Context

Reframing 3D reconstruction kernels as low-pass filters unifies the analysis of explicit scene representations (e.g., 3D Gaussian Splatting [Kerbl et al. 2023], Student’s t Splatting [Zhu et al. 2025]) under signal processing principles. The Jinc kernel establishes the theoretical performance bound for anti-aliasing, while modulated variants present a viable compromise for practical deployment. This perspective highlights the interplay between spatial support, spectral sharpness, and computational feasibility—fundamental considerations in contemporary 3D view synthesis pipelines (Zhang et al., 25 Jan 2026).