In computer graphics, smooth data reconstruction on 2D or 3D manifolds
usually refers to subdivision problems. Such a method is only valid based on
dense sample points. The manifold usually needs to be triangulated into meshes
(or patches) and each node on the mesh will have an initial value. While the
mesh is refined the algorithm will provide a smooth function on the redefined
manifolds. However, when data points are not dense and the original mesh is not
allowed to be changed, how is the "continuous and/or smooth" reconstruction
possible? This paper will present a new method using harmonic functions to
solve the problem. Our method contains the following steps: (1) Partition the
boundary surfaces of the 3D manifold based on sample points so that each sample
point is on the edge of the partition. (2) Use gradually varied interpolation
on the edges so that each point on edge will be assigned a value. In addition,
all values on the edge are gradually varied. (3) Use discrete harmonic function
to fit the unknown points, i.e. the points inside each partition patch.
The fitted function will be a harmonic or a local harmonic function in each
partitioned area. The function on edge will be "near" continuous (or "near"
gradually varied). If we need a smoothed surface on the manifold, we can apply
subdivision algorithms. This paper has also a philosophical advantage over
triangulation meshes. People usually use triangulation for data reconstruction.
This paper employs harmonic functions, a generalization of triangulation
because linearity is a form of harmonic. Therefore, local harmonic
initialization is more sophisticated then triangulation. This paper is a
conceptual and methodological paper. This paper does not focus on detailed
mathematical analysis nor fine algorithm design.