폴리곤 왜곡 #1

Complex Barycentric Coordinates with Applications to Planar Shape Deformation

Ofir Weber       Mirela Ben-Chen       Craig Gotsman

Eurographics 2009 - Computer Graphics Forum 28, 2 (2009) Gunter Enderle Best Paper Award

Image deformation using Point-to-Point complex barycentric coordinates. Click on the image for hi-res version

Abstract:

Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data values. Traditionally, the coordinates are required to satisfy a number of key properties, the first being that they are real and positive. In this paper we relax this requirement, allowing the barycentric coordinates to be complex numbers. This allows us to generate new families of barycentric coordinates, which have some powerful advantages over traditional ones. Applying complex barycentric coordinates to data which is itself complex-valued allows to manipulate functions from the complex plane to itself, which may be interpreted as planar mappings. These mappings are useful in shape and image deformation applications. We use Cauchy뭩 theorem from complex analysis to construct complex barycentric coordinates on (not necessarily convex) polygons, which are shown to be equivalent to planar Green coordinates. These generate conformal mappings from a given source region to a given target region, such that the image of the source region is close to the target region. We then show how to improve the Green coordinates in two ways. The first provides a much better fit to the polygonal target region, and the second allows to generate deformations based on positional constraints, which provide a more intuitive user interface than the conventional cage-based approach. These define two new types of complex barycentric coordinates, which are shown to be very effective in interactive deformation and animation scenarios.

Paper - PDF:

Supplementary Material - PDF:

Video - QuickTime (H264):

BibTeX entry:

@article{Complex_Coordinates:2009, author = {Ofir Weber and Mirela Ben-Chen and Craig Gotsman}, title = {Complex Barycentric Coordinates with Applications to Planar Shape Deformation}, journal = {Computer Graphics Forum (Proceedings of Eurographics)}, volume = {28}, number = {2}, year = {2009}, }

More examples:

Comparison of the Point-to-Point Cauchy-Green complex barycentric coordinates and the MLS similarity coordinates. (left to right) Original image; Deformation using Point-to-Point coordinates; Deformation using MLS coordinates. The Point-to-Point coordinates better handle control points whose Euclidean distance is small, yet their geodesic distance within the cage is large. Click on the image for hi-res version

Deformation of a bird using Szego coordinates. The cage has 21 vertices. Click on the image for hi-res version

A giraffe (left), and its deformation (right) using Point-to-point Cauchy-Green coordinates, with 16 control points. The cage has 113 vertices. Click on the image for hi-res version

Color-coded visualization of one P2P coordinate function. (left to right) Real part; Imaginary part; Absolute value. Click on the image for hi-res version