Cubic Mean Value Coordinates

¹Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing

²Department of Computer Science and Engineering, Washington University in St. Louis

Applications using cubic mean value coordinates. Left: shape deformation using curved cage networks, (a,b): input images, (c,d): deformed results. Right: an adaptive gradient mesh (g) created from a given gradient mesh (e), and (f,h) are the rasterized images of (e,g).

We present a new method for interpolating both boundary values and gradients over a 2D polygonal domain. Despite various previous efforts, it remains challenging to define a closed-form interpolant that produces natural-looking functions while allowing flexible control of boundary constraints. Our method builds on an existing transfinite interpolant over a continuous domain, which in turn extends the classical mean value interpolant. We re-derive the interpolant from the mean value property of biharmonic functions, and prove that the interpolant indeed matches the gradient constraints when the boundary is piece-wise linear. We then give closed-form formula (as generalized barycentric coordinates) for boundary constraints represented as polynomials up to degree 3 (for values) and 1 (for normal derivatives) over each polygon edge. We demonstrate the flexibility and efficiency of our coordinates in two novel applications, smooth image deformation using curved cage networks and adaptive simplification of gradient meshes.

Cubic Mean Value Coordinates [35.8M Paper] [2.9M Paper] [7.1M Video]

Xian-Ying Li, Tao Ju, and Shi-Min Hu.

ACM Transactions on Graphics (Proceedings of SIGGRAPH 2013), 32(4): article 126.

@article {li2013cubic,
  author = {Li, Xian-Ying and Ju, Tao and Hu, Shi-Min},
  title = {Cubic Mean Value Coordinates},
  journal = {ACM Transactions on Graphics},
  year = {2013},
  volume = {32},
  number = {4},
  pages = {126:1-10},
}

Cubic mean value interpolant was first introduced by Michael S. Floater and Christian Schulz in their paper "Pointwise radial minimization: Hermite interpolation on arbitrary domains". In this work we discover a new connection between this important interpolant and the classical mean value interpolant by re-deriving it using certain mean value properties of biharmonic functions. We also give closed-form expressions for cubic mean value coordinates in 2D discrete case.