SBIR-STTR Award

This Small Business Innovation Research Phase I project proposes to investigate applications of Combinatorial Morse Theory in Reverse Engineering. It relies on a single mathematical approach: the definition of a continuous function on a polygonal model and the decomposition of the surface based on the gradient flow of that function. Variants of this theory will be used to solve several important problems, including the automatic conversion of triangulations into a set of smooth NURBS surfaces, locally smoothing polygonal models, and identifying feature lines. The major advantage of this over earlier approaches to the conversion problem is the capacity to base the algorithm on a number of different criteria for surface analysis and to achieve the best result by intermingling these different criteria. Morse theory is the key to computing patch layouts that naturally adapt to and follow the shape of the surface, a property that is difficult to achieve but necessary to automatically construct high-quality NURBS surfaces of scanned or triangulated CAD models. Applications for the proposed project can be classified into two usages: Duplication: referencing a physical part creates a digital model. o Remodeling: a designed digital model is kept to be consistent with a physical reference. The proposed innovation may have significant impact on customers who are not using a digital process today in their design and manufacturing process; this covers 90% of all products being made in the United States. Worldwide manufacturing, only 1% of all products made uses CAD/CAM systems. There are many applications that can benefit from a successful implementation of this proposal, such as CFD analysis on actual parts, digital inventory for legacy parts, historical preservation, and custom-made consumer products including medical devices, shoes, clothes, and wearable computers

This Small Business Innovation Research (SBIR) Phase II project will investigate applications of Combinatorial Morse Theory in Reverse Engineering, a field that focuses on converting physical objects into a digital representation suitable for CAD, CAM, and CAE. The biggest challenge in this field is to automate the conversion process while producing a model that meets all the requirements of downstream applications. These requirements include both an accurate representation of features and a high degree of smoothness. Combinatorial Morse Theory relies on a single mathematical approach: the definition of a continuous function on a polygonal model and the decomposition of the surface based on the gradient flow of that function. One advantage of this over earlier approaches to the conversion problem is its flexibility obtained by adapting to and combining different analysis criteria. Morse theory is the key to computing patch layouts that naturally adapt to and follow the shape of the surface, a property that cause it enables product differentiations and existing processes to be carried out efficiently, cost-effectively, and automatically. The societal impact of this technology includes the improvement of work environments due to reduction of dust, noise, and work-related injuries associated with traditional processes, prevention of loss of lives and equipment by enabling sampling based inspections as well as improvement of the quality life through customized medical devices, and apparel that conform perfectly to the wearer.