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Differential Coordinates

# Differential Coordinates - Encoding Meshes in Differential...

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Encoding Meshes in Differential Coordinates Daniel Cohen-Or Tel Aviv University

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Outline Differential surface representation Compact shape representation Mesh editing and manipulation …about surface reconstruction
Irregular meshes In graphics, shapes are mostly represented by triangle meshes

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Irregular meshes In graphics, shapes are mostly represented by triangle meshes
Irregular meshes Geometry: Vertex coordinates (x 1 , y 1 , z 1 ) (x 2 , y 2 , z 2 ) . . . (x n , y n , z n ) Connectivity (the graph) List of triangles (i 1 , j 1 , k 1 ) (i 2 , j 2 , k 2 ) . . .

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Parallelogram Prediction
Parallelogram Prediction

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K-way Prediction is better
k-way prediction is like predicting that a vertex is in the average of its adjacent neighbors

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Motivation Meshes are great, but: Topology is explicit, thus hard to handle Geometry is represented in a global coordinate system Single Cartesian coordinate of a vertex doesn’t say much about the shape
Differential coordinates Represent local detail at each surface point better describe the shape Linear transition from global to differential Useful for operations on surfaces where surface details are important

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Differential coordinates Detail = surface – smooth (surface) Smoothing = averaging () 1 ii j jNi i d =− δ vv 1 j i d δ
Laplacian matrix The transition between the δ and xyz is linear: ⎛⎞ ⎜⎟ ⎝⎠ L 1 2 n x x x M M () 1 2 x x x n δ = M M 0 i ij di j D otherwise = = = otherwise j N i A ij 0 ) ( 1 1 LI D A = −

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Laplacian matrix The transition between the δ and xyz is linear: = L v x δ x = L = L ( ) () ii j i j jNi w =− δ vv v y v z δ y δ z
Basic properties Rank(L) = n-c (n-1 for connected meshes) We can reconstruct the xyz geometry from delta up to translation L = x δ 1 L = x δ

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Quantizing differential coordinates Quantization is one of the major methods to reduce storage space of geometry data What happens if we quantize the δ - coordinates? Can we still go back to xyz ? How does the reconstruction error behave? L = x δ ε →= + δδ δ “High-pass Quantization for Mesh Encoding”, Sorkine et al. 03
Quantizing differential coordinates 11 () LL ε −− ′′ = =+ x δδ How does the reconstruction error behave?

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Find the differences between the horses… Quantizing differential coordinates
This one is the original horse model Quantizing differential coordinates

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This is the model after quantizing δ to 8 bits/coordinate There is one anchor point (front left leg) Quantizing differential coordinates
Original model Quantizing differential coordinates

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This is the model after quantizing δ to 7 bits/coordinate, one anchor Quantizing differential coordinates
Quantizing differential coordinates

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Quantizing differential coordinates
Quantization error A coarsely-sampled sphere

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Quantization error After quantization to 8 bits/coordinate
Quantization error A finely-sampled sphere:

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Quantization error After (the same) quantization to 8 bits/coordinate…
Quantizing differential coordinates

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Quantizing differential coordinates
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Differential Coordinates - Encoding Meshes in Differential...

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