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Generalized Maps for the Analysis of 3D Shapes and Shape Collections

Final Report Summary - GEN-MAPS-3D-SHAPES (Generalized Maps for the Analysis of 3D Shapes and Shape Collections)

The goal of this project is to create informative, structure-preserving maps between 3D shapes, to investigate generalized representation of such maps, and to develop novel applications of such maps to large datasets of geometric shapes.

Since the beginning of the project we have explored various directions, both theoretical and applicative towards this goal. First, we studied the novel Functional Maps representation as an alternative for classical point-to-point maps. The functional map representation describes a map between two shapes as a correspondence between functions on the shapes, instead of a correspondence between points. This allows to consider various algebraic operators on maps, such as eigen-decomposition, for their analysis and processing.
Next, we have investigated the applicability of functional maps for the visualization of maps between shapes and collections of maps. This approach filled a large gap in the area of evaluating maps between shapes, which was insofar limited to comparison to the ground truth, which is in most cases not available.

Later we have extended this approach, and defined a functional operator which describes the difference between two shapes as an object in itself instead of a number. This allowed us to perform operations such as computing differences of differences, computing shape analogies, parameterizing the intrinsic variability in a shape collection and exploring shape collections.

In addition, we have used the shape difference operator for aligning two shape collections, by considering them as points clouds representing two shape spaces. The shape difference operator allowed us, on the one hand, to describe shapes in an intrinsic coordinate invariant way, which paved the way to using standard dimensionality reduction techniques for representing the collection as a point cloud. In addition, using the shape difference operator we defined shape analogy constraints between the two collections, which allowed us to align them by solving a single linear system.

In a parallel line of research, the idea of designing linear operators which describe geometric objects has led us to the development of Functional Vector Fields, which are linear operators representing tangent vector fields on surfaces. This representation is highly useful, as it allows to design and manipulate tangent vector fields easily, with the additional benefit of using algebraic operations for geometric tasks. For example, divergence-free tangent vector fields are closely related to volume preserving self maps of the surface. Using this relationship allows us to leverage concepts from Lie group theory, which we applied to design a simple yet highly stable and efficient method for the simulation of incompressible fluids on curved surfaces.

In the third year of the project we have proceeded in two directions. On the one hand we have extended our toolbox of functional operators to tangent vector fields, exploring challenging geometric constructions such as the covariant derivative of vector fields and parallel transport, using a spectral operator-based approach. Furthermore, we extended our work on shape variability to consider extrinsic information, and applied it for advanced analysis of shape collections as well as shape synthesis from operators. On the other hand, we have explored additional applications of the functional vector field representation to simulating viscous thin films on surfaces, including the generated intricate fingering effects (e.g. honey or chocolate dripping on a surface). Our simulation is robust and efficient, and based on a novel formulation in terms of mass transport.

In the fourth year of the project we continued working on both aspects: vector fields and maps. We explored alternative representation of vector fields on bounded planar domains, simulating the Hele-Shaw flow on the GPU, and allowing user control. We further computed conformal maps between planar domains while allowing the boundary to slide. The research branch of maps has led us to develop novel ways for extracting pointwise maps from the computed functional maps (which are now widely accepted in the community, and used by additional research groups). Finally, relating maps and tangent vector fields, we have derived a solution to the difficult problem of finding a tangent vector field advecting between a source and target function. This problem has deep connections to optimal transport theory, which is a promising direction of future research.

The details on these projects are available at:

During the course of the project the PI has disseminated the results through various courses, tutorials and workshops. Based on the results obtained in this project, the PI has secured additional funding for the next 5 years, which will allow her to push the functional and operator-based representation to new frontiers. The PI has additionally supported postdocs, PhDs and MSc students, which all contributed to the research done in the project. The CIG grant has been of paramount importance to the PI’s career, by supporting the scientific research, collaborations and dissemination.