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isosurfacing

# isosurfacing - Advanced Iso-Surfacing Algorithms Jian Huang...

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Advanced Iso-Surfacing Algorithms Jian Huang, CS594, Spring 2002 This set of slides are developed and used by Prof. Han-Wei Shen at Ohio State University.

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Iso-contour/surface Extractions 2D Iso-contour 3D Iso-surface
Iso-contour (0) Remember bi-linear interpolation p2 p3 p0 p1 P =? p4 p5 To know the value of P, we can first compute p4 and P5 and then linearly interpolate P

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Iso-contour (1) Consider a simple case: one cell data set The problem of extracting an iso-contour is an inverse of value interpolation. That is: p2 p3 p0 p1 Given f(p0)=v0, f(p1)=v1, f(p2)=v2, f(p3)=v3 Find the point(s) P within the cell that have values F( p ) = C
Iso-contour (2) p2 p3 p0 p1 We can solve the problem based on linear interpolation (1) Identify edges that contain points P that have value f(P) = C (2) Calculate the positions of P (3) Connect the points with lines

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Iso-contouring – Step 1 (1) Identify edges that contain points P that have value f(P) = C v1 v2 If v1 < C < v2 then the edge contains such a point
Iso-contouring – Step 2 (2) Calculate the position of P Use linear interpolation: P = P1 + (C-v1)/(v2-v1) * (P2 – P1) v1 v2 P p1 p2 C

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Iso-contouring – Step 3 p2 p3 p0 p1 Connect the points with line(s) Based on the principle of linear variation, all the points on the line have values equal C
Inside or Outside? Just a naming convention 1. If a value is smaller than the iso-value, we call it “Inside” 2. If a value is greater than the iso-value, we call it “Outside” p2 p3 p0 p1 - + outside cell p2 p3 p0 p1 - inside cell

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Extend the same divide-and-conquer algorithm to three dimension 3D cells Look at one cell at a time Let’s only focus on voxel Iso-surface Extraction
Divide and Conquer _ + + + + _ _ _ + + + + _ _ _ _ (2 triangles)

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How many cases? Now we have 8 vertices So it is: 2 = 256 8 How many unique topological cases?
Case Reduction (1) Value Symmetry + + _ _ _ _ _ _ + + _ _ + + + +

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Case Reduction (2) Rotation Symmetry + + _ _ _ _ _ _ _ _ + + _ _ _ _ By inspection, we can reduce 256 14
Iso-surface Cases Total number of cases: 14 + 3

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Marching Cubes Algorithm A Divide-and-Conquer Algorithm v1 v2 v3 v4 v5 v6 v7 v8 Vi is ‘1’ or ‘0’ (one bit) 1: > C; 0: <C (C= iso-value) Each cell has an index mapped to a value ranged [0,255] Index = v8 v7 v6 v5 v4 v3 v2 v1
Marching Cubes (2) Given the index for each cell, a table lookup is performed to identify the edges that has intersections with the iso-surface 0 1 2 3 14 e1, e3, e5 Index intersection edges e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12

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Marching Cubes (3) + + + + _ _ _ _ Perform linear interpolations at the edges to calculate the intersection points Connect the points
Why is it called marching cubes?

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isosurfacing - Advanced Iso-Surfacing Algorithms Jian Huang...

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