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In computer graphics, smooth data reconstruction on 2D or 3D manifoldsusually refers to subdivision problems. Such a method is only valid based ondense 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 themesh is refined the algorithm will provide a smooth function on the redefinedmanifolds. However, when data points are not dense and the original mesh is notallowed to be changed, how is the "continuous and/or smooth" reconstructionpossible? This paper will present a new method using harmonic functions tosolve the problem. Our method contains the following steps: (1) Partition theboundary surfaces of the 3D manifold based on sample points so that each samplepoint is on the edge of the partition. (2) Use gradually varied interpolationon 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 functionto fit the unknown points, i.e. the points inside each partition patch. Thefitted function will be a harmonic or a local harmonic function in eachpartitioned area. The function on edge will be "near" continuous (or "near"gradually varied). If we need a smoothed surface on the manifold, we can applysubdivision algorithms.

Harmonic Functions for Data Reconstruction on 3D Manifolds