Optimizing 4D Wires for Sparse 3D Abstraction

TL;DR

This paper introduces a unified 4D spline-based framework for 3D geometric abstraction that enhances structural coherence and aesthetic quality, enabling gradient-based optimization with modern guidance signals.

Contribution

It proposes a single continuous 4D B-spline representation for 3D shapes, transforming sketching into a global routing problem and supporting differentiable rendering for optimization.

Findings

Unified 4D spline captures complex volumetric forms.

Method improves structural coherence over discrete curve collections.

Supports optimization with guidance signals like SDS and CLIP.

Abstract

We present a unified framework for 3D geometric abstraction using a single continuous 4D wire, parameterized as a B-spline with spatial coordinates and variable width $(x,y,z,w)$. Existing approaches typically represent shapes as collections of many independent curve segments, which often leads to fragmented structures and limited physical realizability. In contrast, we show that a single continuous spline is sufficiently expressive to capture complex volumetric forms while enforcing global topological coherence. By imposing continuity, our method transforms 3D sketching from a local density-accumulation process into a global routing problem, providing a strong inductive bias toward cleaner aesthetics and improved structural coherence. To enable gradient-based optimization, we introduce a differentiable rendering pipeline that efficiently rasterizes variable-width curves with bounded projection error. This formulation supports robust optimization using modern guidance signals such as Score Distillation Sampling (SDS) or CLIP. We demonstrate applications including image-to-3D abstraction, multi-view wire art generation, and differentiable stylized surface filling. Experiments show that our unified representation produces structures with higher semantic fidelity and improved structural coherence compared to approaches based on collections of discrete curves.