SAD: Soft Anisotropic Diagrams

TL;DR

SAD represents an image as a soft, anisotropic, differentiable diagram over learnable sites. Each pixel is a softmax blend over its top-K nearby sites under a site-dependent distance, yielding a differentiable partition of unity with explicit ownership and content-aligned boundaries. A GPU-friendly top-K propagation scheme keeps cost constant per pixel, enabling fast fitting at matched or better quality.

Fitting Process

Each clip is a triple: RGB reconstruction, diagram view, and tau-position heatmap. It shows how sites migrate, temperatures sharpen, and the representation converges to content-aligned regions during optimization.

Method

We scatter many small sites across the image. Each site has a position, a color, a reach radius, an oriented anisotropic shape, and a temperature that controls boundary sharpness.

\[ s_i(\mathbf{x}) = d_{\mathbf{A}_i}\!\left(\mathbf{x}, \mathbf{p}_i\right) - r_i \]

The pixel color is a soft blend of nearby sites. The temperature \(\tau_i\) controls how abruptly ownership changes between adjacent sites.

\[ \hat{\mathbf{c}}(\mathbf{x}) = \sum_{i \in \mathcal{N}_K(\mathbf{x})} w_i(\mathbf{x})\,\mathbf{c}_i \] \[ w_i(\mathbf{x}) = \frac{\exp\!\left(-\tau_i s_i(\mathbf{x})\right)} {\sum_{j \in \mathcal{N}_K(\mathbf{x})} \exp\!\left(-\tau_j s_j(\mathbf{x})\right)} \]

Because each pixel only depends on a fixed-size nearby-site list, both rendering and fitting remain GPU-friendly.

Comparisons

Reconstruction quality on the Image-GS benchmark. Average metrics over 45 images at varying bitrates.

Method	Metric	0.2 BPP	0.3 BPP	0.4 BPP	0.5 BPP
Image-GS	PSNR up	31.32	32.79	33.80	34.57
	SSIM up	0.8923	0.9112	0.9228	0.9307
	LPIPS down	0.1309	0.1033	0.0873	0.0769
Instant-NGP	PSNR up	26.66	29.41	29.86	30.69
	SSIM up	0.7703	0.8253	0.8304	0.8461
	LPIPS down	0.2472	0.1701	0.1656	0.1463
SAD	PSNR up	33.87	35.72	36.97	37.86
	SSIM up	0.8983	0.9202	0.9334	0.9422
	LPIPS down	0.0914	0.0678	0.0546	0.0458

Image compression on Kodak. Average reconstruction quality and training time over 24 images at N = 50,000 primitives.

Method	PSNR up	SSIM up	LPIPS down	Time (s) down
Image-GS	36.90	0.9521	0.0272	28
Instant-NGP	37.72	0.9494	0.0249	8.2
Fast 2DGS	43.13	-	-	10
SAD	46.00	0.9871	0.0032	2.2

Reconstruction quality on DIV2K. Average metrics over 100 images at varying bitrates and variable resolution.

Method	Metric	0.5 BPP	2.0 BPP
Image-GS	PSNR up	28.48	32.15
	SSIM up	0.7914	0.8820
	LPIPS down	0.2515	0.1480
Instant-NGP	PSNR up	26.44	29.24
	SSIM up	0.7045	0.7940
	LPIPS down	0.2778	0.1755
VBNF	PSNR up	27.13	31.28
	SSIM up	0.7495	0.8737
	LPIPS down	0.4321	0.2765
Instant-GI	PSNR up	-	38.01
Fast 2DGS	PSNR up	-	37.81
SAD	PSNR up	30.00	34.73
	SSIM up	0.7982	0.9115
	LPIPS down	0.1995	0.0844

Reconstruction quality on CLIC. Average metrics over 41 images at varying bitrates.

Method	Metric	0.5 BPP	2.0 BPP
Image-GS	PSNR up	30.65	34.15
	SSIM up	0.8223	0.8907
	LPIPS down	0.2280	0.1449
Instant-NGP	PSNR up	28.59	32.67
	SSIM up	0.7559	0.8475
	LPIPS down	0.2351	0.1287
SAD	PSNR up	31.82	36.13
	SSIM up	0.8176	0.9112
	LPIPS down	0.1870	0.0884

BibTeX

@article{iinbor2026sad,
  title   = {Soft Anisotropic Diagrams for Differentiable Image Representation},
  author  = {Iinbor, Laki and Dou, Zhiyang and Matusik, Wojciech},
  journal = {ACM Transactions on Graphics},
  year    = {2026},
  note    = {SIGGRAPH 2026}
}