Given an oriented point cloud data set S = {pi , ni}, we want to retrieve the surfce:
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- -The implicit representation of the 3D model has constrains
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- -Point cloud data might contain noise and outliers
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- -For any S in F(x) where F(x) is the implicit function S must be guaranteed to be a manifold
Step-1 Constrains Satisfaction : We construct an implicit function F(x) with following conditions 1- For point x on the surface, the value of F(x) = 0 2- for any x = s+α for a point s on the surface , F(x) = s+α
Step-2 Implicit Surface: Given our implicit surface constraints , we can now build a system of linear equations for any point on the surface. this model can be represented as the sum over c =1 , wc⋅ϕ(∥x−c∥^2)
ϕ is a kernel function. The weights w are used in a weighted least squares fitting process, where the contributions of nearby points are emphasized while those of more distant points are diminished. in our case ϕ is the windland function with center c. Thus the further points are for the center c the less dominant their entry. Thus our system is diagonally dominant
Step 3 - Iso Surfacing: Using the marching cube algorithm , we can retrieve the mesh
