Yin-Yang Rasterizer for Structured Omnidirectional Imaging

• The technique applies the unique Fermat spiral to partition a disk into symmetric, equal-area lobes with exact overlap properties.
• It extends to omnidirectional imaging through two near-rectangular tiles, enabling quasi-uniform mapping and efficient CNN-based processing for 360-degree view synthesis.
• Integration with 3D Gaussian splatting achieves high-fidelity rendering with faster inference and improved PSNR, SSIM, and LPIPS metrics over baseline methods.

A Yin-Yang Rasterizer is a computational technique for structured image decomposition and rendering that leverages mathematically principled grid partitioning, originating from the unique axisymmetric properties of the Fermat spiral and generalized to the domain of omnidirectional image synthesis and 3D scene rendering. The method exploits the congruence properties of the Yin-Yang symbol—specifically, the splitting of a disk or sphere into two "perfect" regions with maximal symmetry-breaking—as well as a pair of rectangularly parameterized tiles on the sphere, to enable quasi-uniform rasterization, efficient convolutional processing, and high-fidelity Gaussian splatting without per-scene optimization (0902.1556, Lee et al., 2024).

1. Mathematical Foundations: Perfect Sets and the Fermat Spiral

The original Yin-Yang partition for a disk was rigorously characterized by Banakh, Verbitsky, and Vorobets: the only smooth curve, algebraic in polar coordinates, that (i) bisects a unit-area disk into two congruent sets, (ii) meets each $r=\text{const}$ circle twice, (iii) meets each radius exactly once, and (iv) splits the disk into two "perfect" sets—so that any reflection of one half overlaps its preimage in area at most $1/4$—is the one-turn Fermat spiral,

$\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$

In Cartesian coordinates, this is

$$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$

and the resulting halves each have area $1/2$. Analytical integration verifies that the overlap of a lobe and its reflected copy is exactly $1/4$ (0902.1556).

2. Rasterization Algorithms for the Planar Yin-Yang Disk

Rasterizing the Yin-Yang disk at $N \times N$ pixel resolution involves:

• Sampling the Fermat spiral at sufficiently fine angular increments ($\Delta\theta \sim \pi/(10N)$ for $N \geq 512$), with $(x_k, y_k)$ points mapped into pixel indices.
• Forming two chains (each corresponding to a lobe), drawing anti-aliased lines or a closed polygon.
• Performing fill operations (winding-number or parity-based flood fill) to color the interior of each lobe.
• Rendering the two small "eye" disks at centers $1/4$0 of each lobe, using a solid-disk raster with radius $1/4$1.
• Applying anti-aliasing by pixel supersampling (such as $1/4$2 subpixels) or direct polygon-pixel intersection.
• Ensuring numerical precision through double-precision arithmetic, clipping points outside the bounding circle, and selecting $1/4$3 so that the chord-to-arc discrepancy stays below $1/4$4 px.

This canonical algorithm ensures equal-area lobes and exact overlaps in the continuous case, providing a mathematically optimal symbol rasterization (0902.1556).

3. The Yin-Yang Grid for Omnidirectional Images

In omnidirectional imaging, the Yin-Yang grid decomposes the sphere into two rectangular tiles with low internal distortion, allowing standard convolutional processing. An omnidirectional image is modeled as a function $1/4$5, where $1/4$6 (elevation) and $1/4$7 (azimuth). The Yin tile,

$1/4$8

is near-rectangular and minimally distorted. The Yang tile is a $1/4$9-rotated copy: if $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$0 is a unit-sphere point, the Yang tile applies $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$1 (rotation matrix) to $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$2. This ensures that each tile covers a quasi-uniform region in spherical coordinates (Lee et al., 2024).

For pixel coordinates $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$3 in a Yin tile of size $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$4:

$\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$5

with analogous mapping for Yang.

4. Decomposition and Interpolation in Yin-Yang Rasterization

Converting an equirectangular omnidirectional image to Yin or Yang tiles involves:

• Precomputing a dense $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$6 grid.
• For each Yin tile pixel $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$7, inverting the mapping to retrieve $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$8, then bilinear interpolating $\pi^2 r^2 = \phi, \quad 0 \leq \phi \leq \pi.$9.
• For Yang, rotating each sphere point back by $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$0, performing the spherical-to-map conversion, and bilinear interpolation.

Image tiles $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$1 (Yin) and $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$2 (Yang) are thus extracted as nearly perspective-like, rectangular images, suitable for standard CNN and transformer architectures (Lee et al., 2024).

5. Feed-Forward 3D Gaussian Splatting with Yin-Yang Tiles

OmniSplat utilizes the Yin-Yang rasterizer to adapt feed-forward 3D Gaussian Splatting (3DGS) to omnidirectional data. The pipeline comprises:

• Extracting multiple Yin/Yang tiles (e.g., $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$3) and corresponding camera extrinsics.
• Processing each tile through a shared CNN+transformer backbone to generate per-tile feature maps $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$4.
• Building a cost volume using plane-sweeping across depths $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$5, warping and aggregating features from different tiles and origins (Yin-to-Yin, Yang-to-Yin).
• Regressing a set of $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$6 3D Gaussians (mean $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$7, covariance $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$8, opacity $$x(\phi) = \frac{1}{\pi}\sqrt{\phi}\cos\phi, \qquad y(\phi) = \frac{1}{\pi}\sqrt{\phi}\sin\phi,$$9, SH color $1/2$0) via a lightweight MLP on the aggregated cost volume.
• Rendering for each output pixel by mapping to a spherical ray $1/2$1, projecting each Gaussian along $1/2$2, computing the projected covariance, and compositing via depth-ordered alpha blending.

No per-scene fine-tuning occurs; the procedure remains entirely feed-forward (Lee et al., 2024).

6. Rendering and Novel-View Synthesis

Yin-Yang rasterization proceeds by:

• For a Yin tile, converting $1/2$3 grid points to spherical directions (latitude and longitude), and calculating for each Gaussian $1/2$4 the depth $1/2$5 and projected ellipse $1/2$6 in the local image plane.
• Evaluating the elliptical Gaussian weight $1/2$7 for each point and assembling the final color $1/2$8 as a depth-sorted transmittance sum.
• Applying the same method to Yang tiles (with inverse rotation), followed by remapping these two planar images back to equirectangular or spherical format.

This approach supports faithful 360-degree novel-view synthesis, extensible to high-resolution omnidirectional datasets (Lee et al., 2024).

7. Quasi-Uniform Grid Structure and Performance Properties

Key technical advantages of the Yin-Yang rasterizer are:

• The restricted angular range ($1/2$9 elevation, $1/4$0 azimuth) for each tile ensures nearly uniform pixel spacing, minimizing spherical distortion.
• Standard CNN kernels can be applied directly without loss of translational equivariance, permitting use of off-the-shelf, perspective-trained architectures.
• The region of overlap between Yin and Yang tiles is small; geometric consistency is preserved via cross-blending as described in cost volume aggregation.
• Quantitative evaluation on established datasets (OmniBlender, Ricoh360, OmniPhotos, 360Roam, OmniScenes, 360VO) demonstrates that OmniSplat’s Yin-Yang rasterizer achieves $1/4$1 dB higher PSNR, $1/4$2 higher SSIM, and $1/4$3 lower LPIPS compared to PixelSplat(P), while yielding $1/4$4–$1/4$5 faster inference than optimization-based omnidirectional 3DGS. Performance on Ricoh360: $1/4$6 s inference time, PSNR $1/4$7 dB, SSIM $1/4$8, LPIPS $1/4$9, compared to $N \times N$0 s and $N \times N$1 dB for MVSplat(P) (cubemap) (Lee et al., 2024).

Summary Table: Yin-Yang Rasterizer Key Properties

Property	Planar (Disk) (0902.1556)	Omnidirectional (Spherical) (Lee et al., 2024)
Partition curve	Fermat spiral (unique by congruence)	Two tiles via orthogonal spherical rectangles
Region shape	“Perfect” lobes, equal area, 1/4 overlap	Quasi-uniform, rectangular tiles, small overlap
Algorithmic focus	Polygonal chain, fill, anti-aliasing	CNN-friendly, fast tile decomposition, splatting
Rendering artifact	None (mathematically exact)	Minor interpolation from tiling, low distortion
Application focus	Symbol rasterization, image motifs	Omnidirectional view synthesis, 3DGS acceleration

The Yin-Yang rasterizer unifies rigorous symmetry-aware disk partitioning with practical, distortion-minimizing spherical image tiling. This duality enables the exploitation of standard convolutional priors and feed-forward 3DGS pipelines for omnidirectional images, delivering both analytical guarantees and empirical performance gains (0902.1556, Lee et al., 2024).