PICO: Procedural Iterative Constrained Optimizer for Geometric Modeling.

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.