Analysis of shape grammars: continuity of rules

The rules in a shape grammar apply in terms of embedding to take advantage of the parts that emerge visually in the appearance of shapes. While the shapes are kept unanalyzed throughout a computation, their descriptions can be defined retrospectively based on how the rules are applied. An important outcome of this is that continuity for rules is not built-in but it is "fabricated" retrospectively to explain a computation as a continuous process. An aspect of continuity analysis that has not been addressed in the literature is how to decide which mapping forms to use to study the continuity of rule applications. This is addressed in this paper in a new approach to continuity analysis, which uses recent results on shape topology and continuous mappings. A characterization is provided that distinguishes the suitable mapping forms from those that are inherently discontinuous or practically inconsequential for continuity analysis. It is also shown that certain inherent properties of shape topologies and continuous mappings provide an effective method of computing topologies algorithmically.Comment: 23 pages, 6 Figures, 6 Tables. Research Report, 2020, MIT. Preprint of Journal Article (2021

SHREC’21: Quantifying shape complexity

This paper presents the results of SHREC’21 track: Quantifying Shape Complexity. Our goal is to investigate how good the submitted shape complexity measures are (i.e. with respect to ground truth) and investigate the relationships between these complexity measures (i.e. with respect to correlations). The dataset consists of three collections: 1800 perturbed cube and sphere models classified into 4 categories, 50 shapes inspired from the fields of architecture and design classified into 2 categories, and the data from the Princeton Segmentation Benchmark, which consists of 19 natural object categories. We evaluate the performances of the methods by computing Kendall rank correlation coefficients both between the orders produced by each complexity measure and the ground truth and between the pair of orders produced by each pair of complexity measures. Our work, being a quantitative and reproducible analysis with justified ground truths, presents an improved means and methodology for the evaluation of shape complexity