Variegated and multifaceted shapes fill our world. Currently, there is no universally accepted vocabulary for shape analysis which inhibits the emergence and deployment of shape comparison technologies. This project draws its inspiration from an old idea usually attributed to Christiaan Huygens: when waves (usually represented as complex exponentials) emanate orthogonally from each point on the shape boundary, they travel in open space until they meet other waves emanating from different shape boundary locations. Each point in space is then "owned" by a wavefront thereby giving rise to the complex wave representation (CWR) of shape. When shape alignment is required (usually as part of a larger shape recognition goal), the CWRs of the two shapes can be more robustly aligned than the original shapes, since in the CWR, every point in space (in a bounded region) now carries an imprint of the shape. Shape analysis often raises the need for performing shape statistics: shape averages (usually referred to as shape atlases) and deviations from the mean are required. Shape statistics carried out in the CWR are arguably simpler since the vocabulary shifts to averages and standard deviations of frequencies (of the waves in the shape representation) which is rather straightforward. The significance of this work lies in the introduction of complex wave representations into the shape analysis lexicon. This research is expected to impact all areas of shape representation and analysis: the matching, registration, indexing and recognition of shapes with open source code dissemination facilitating re-use and vertical integration.

Project Report

The primary focus of my present research is the construction and application of complex wave representations (CWR) for shape analysis. The motivation and need for this new representation is now described. Representations inspired by probability theory and statistics—placed under the same rubric here—have thrived and prospered in shape analysis. Especially when shapes are parametrized by point-sets, probabilistic representations have become quite popular due to the relative ease of estimating densities (for example via histogramming) in lower dimensions. In the past twenty years, shape correspondence, non-rigid deformable matching, shape dictionaries etc. have all seen considerable progress since the robustness afforded by the representations has allowed for outlier detection, incomplete shape matching and so on. The principal drawback of probabilistic shape representations is that they have thus far failed to adequately represent relational (curve and surface) information present in shapes. While distance transforms can be used to implicitly represent a set of planar 2D curves, they do not carry any probabilistic information. The complex wave representation of shape is introduced precisely to address this lack. To set the stage in an informal manner, we turn to Huygens’ principle: we consider waves emanating from every point on a shape boundary. The influence of each wave isotropically decreases as you move away from its source location and the directionality of the wave depends on the normal vector at the source location. From a computational perspective, we cannot place a wave source at each point on the shape boundary—instead we consider planar curves with a discrete number of source locations. We then allow the magnitude of the wave function to reflect probabilistic information---the further we are from the shape boundary the lesser the probability and vice versa. But it is the phase of the wave which carries the new ingredient. We embed the curve information into the phase of the wave function. Essentially, the phase encodes curve tangent information---the directionality of the wave. The entire shape boundary can be reconstructed from the zero level set of the phase of the wave function. In this way, we retain the advantages of the probabilistic information but include new information regarding curve (or sets of curves) in the phase of the wave. To the best of our knowledge, this is a new contribution. While the mag- nitude of the wave function can be expected to behave similarly to the point-set density estimators mentiond above, curve representation via the phase of the wave function—obtained by merely turning curve normals into spatial frequency attributes of a wave—is new. It remains to be seen if the complex wave function shape representation can be leveraged for shape atlases, registration and indexing—and we plan to pursue these topics in the near future.

Project Start
Project End
Budget Start
2011-08-15
Budget End
2013-07-31
Support Year
Fiscal Year
2011
Total Cost
$117,304
Indirect Cost
Name
University of Florida
Department
Type
DUNS #
City
Gainesville
State
FL
Country
United States
Zip Code
32611