We introduce the problem of knot-based inverse perceptual art. Given multiple target images and their corresponding viewing configurations, the objective is to find a 3D knot-based tubular structure whose appearance resembles the target images when viewed from the specified viewing configurations. To solve this problem, we first design a differentiable rendering algorithm for rendering tubular knots embedded in 3D for arbitrary perspective camera configurations. Utilizing this differentiable rendering algorithm, we search over the space of knot configurations to find the ideal knot embedding. We represent the knot embeddings via homeomorphisms of the desired template knot, where the homeomorphisms are parametrized by the weights of an invertible neural network. Our approach is fully differentiable, making it possible to find the ideal 3D tubular structure for the desired perceptual art using gradient-based optimization. We propose several loss functions that impose additional physical constraints, enforcing that the tube is free of self-intersection, lies within a predefined region in space, satisfies the physical bending limits of the tube material and the material cost is within a specified budget. We demonstrate through results that our knot representation is highly expressive and gives impressive results even for challenging target images in both single view as well as multiple view constraints. Through extensive ablation study we show that each of the proposed loss function is effective in ensuring physical realizability. We construct a real world 3D-printed object to demonstrate the practical utility of our approach. To the best of our knowledge, we are the first to propose a fully differentiable optimization framework for knot-based inverse perceptual art.

翻译：我们提出基于绳结的逆向视错觉艺术问题。给定多张目标图像及其对应的视角配置，目标是寻找一种三维绳结管状结构，使其在指定视角下呈现与目标图像相似的视觉效果。为解决该问题，我们首先设计了一种可微渲染算法，用于渲染嵌入三维空间中的管状绳结，并支持任意透视相机配置。借助该可微渲染算法，我们在绳结配置空间中搜索理想结嵌入。我们通过目标模板结的同胚映射来表示结嵌入，其中同胚由可逆神经网络的权重参数化。该方法完全可微，使得可通过梯度优化找到满足视错觉需求的三维理想管状结构。我们提出若干损失函数以施加额外物理约束，确保管状结构无自交、位于预定义空间区域、满足管材物理弯曲极限，且材料成本在指定预算内。实验结果表明，我们的结表示具有高度表达能力，即使在具有挑战性的单视图及多视图约束目标图像上也能取得显著效果。通过广泛消融研究，我们证明各损失函数在确保物理可实现性方面的有效性。我们构建了实际3D打印物体以验证该方法的实用价值。据我们所知，这是首个提出基于绳结逆向视错觉艺术的完全可微优化框架的方法。