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
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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
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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
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Parallelogram Prediction
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K-way Prediction is better
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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
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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
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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
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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
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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
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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
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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
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Quantizing differential coordinates
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Quantizing differential coordinates
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Quantization error A coarsely-sampled sphere
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Quantization error After quantization to 8 bits/coordinate
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Quantization error A finely-sampled sphere:
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Quantization error After (the same) quantization to 8 bits/coordinate…
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Quantizing differential coordinates
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Quantizing differential coordinates
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Differential Coordinates - Encoding Meshes in Differential...

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