Mathematics of Information Technology and Complex Systems


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Research

The overall goal of this project is to develop new mathematical representations or adapt existing ones, to capture the essence of shape and motion as perceived by designers.  The project objectives from a broader perspective listed in order of a design workflow are:

            To evaluate the ways in which designers express their ideas and develop new digital interaction paradigms for conceptual design and animation, that are unencumbered by the mathematics of surface representation.

            Higher order surface derivatives such as normal and curvature are as important as surface position in the way designers perceive and express shape.  Aggregates of these attributes are referred to in the design world as surface- features. We will investigate new geometric representations that are conducive to interactive shape manipulation, directly through surface-features and other complex surface attributes. We will also undertake research on making current surface representation techniques more amenable to direct manipulation through feature editing.

            To develop a signal-processing framework for geometry allowing the detection, analysis, filtering and alteration of the various surface and temporal attributes or features just described.

            To develop the necessary mathematical associations that would allow our new digital styling tools to be used in conjunction with traditional design techniques of sculpting and sketching. In particular we hope to develop the necessary mathematical constructs to automate the transition between digital and physical versions of a design prototype.

            One particularly important class of surface-features are 3D curves or character-lines that provide a sparse but suggestive representation of both the objectís overall shape and the relative importance and flow of geometric detail in parts of the object. Character-lines have a direct correlation to object sketches and we will develop techniques to automatically generate a correspondence between them.   

            To extend statistical modeling techniques to the area of shape reconstruction from sparse and incomplete information such as a few surface-features or character-lines. The project will use model-based knowledge and shape statistics along with interactive data-mining techniques to make appropriate inferences about the overall shape of an object.

Sub-projects

  •  Hybrid mathematical surface modeling

  • Signal processing of surface geometry

  • Inferring 3D shape from character-lines

  • Geometric and temporal restructuring

  • Feature extraction and editing using constraint optimization

  • 3D device interaction with surface and time constraints