Procedural modeling has produced amazing results, yet fundamental issues such as controllability and limited user guidance persist. We introduce a novel procedural model called PICO (Procedural Iterative Constrained Optimizer) and PICO-Graph that is the underlying procedural model designed with optimization in mind. The key novelty of PICO is that it enables the exploration of generative designs by combining both user and environmental constraints into a single framework by using optimization without the need to write procedural rules. The PICO-Graph procedural model consists of a set of geometry generating operations and a set of axioms connected in a directed cyclic graph. The forward generation is initiated by a set of axioms that use the connections to send coordinate systems and geometric objects through the PICO-Graph, which in turn generates more objects. This allows for fast generation of complex and varied geometries. Moreover, we combine PICO-Graph with efficient optimization that allows for quick exploration of the generated models and the generation of variants. The user defines the rules, the axioms, and the set of constraints; for example, whether an existing object should be supported by the generated model, whether symmetries exist, whether the object should spin, etc. PICO then generates a class of geometric models and optimizes them so that they fulfill the constraints. The generation and the optimization in our implementation provides interactive user control during model execution providing continuous feedback. For example, the user can sketch the constraints and guide the geometry to meet these specified goals. We show PICO on a variety of examples such as the generation of procedural chairs with multiple supports, generation of support structures for 3D printing, generation of spinning objects, or generation of procedural terrains matching a given input. Our framework could be used as a component in a larger design workflow; its strongest application is in the early rapid ideation and prototyping phases.

中文翻译：

---

PICO：用于几何建模的程序迭代约束优化器

程序建模产生了惊人的结果，但诸如可控性和有限的用户指导等基本问题仍然存在。我们引入了一种新的程序模型，称为 PICO（程序迭代约束优化器）和 PICO-Graph，它是在设计时考虑到优化的底层程序模型。PICO 的关键新颖之处在于，它通过使用优化将用户和环境约束组合到一个框架中，而无需编写程序规则，从而能够探索生成式设计。PICO-Graph 过程模型由一组几何生成操作和一组连接在有向循环图中的公理组成。前向生成由一组公理发起，这些公理使用连接通过 PICO-Graph 发送坐标系和几何对象，这反过来又会生成更多的对象。这允许快速生成复杂多样的几何图形。此外，我们将 PICO-Graph 与高效优化相结合，可以快速探索生成的模型和变体的生成。用户定义规则、公理和约束集；例如，生成的模型是否应支持现有对象，是否存在对称性，对象是否应自旋等。然后 PICO 生成一类几何模型并对其进行优化，使其满足约束条件。我们实现中的生成和优化在模型执行期间提供交互式用户控制，提供持续反馈。例如，用户可以绘制约束并引导几何图形以满足这些指定的目标。我们在各种示例中展示了 PICO，例如生成具有多个支撑的程序椅、生成用于 3D 打印的支撑结构、生成旋转对象或生成与给定输入匹配的程序地形。我们的框架可以用作更大设计工作流程中的一个组件；它最强大的应用是在早期的快速构思和原型设计阶段。