High-Pass Quantization as long as Mesh Encoding Olga Sorkine, Daniel Cohen-Or, Sivan To

High-Pass Quantization as long as Mesh Encoding Olga Sorkine, Daniel Cohen-Or, Sivan To www.phwiki.com

High-Pass Quantization as long as Mesh Encoding Olga Sorkine, Daniel Cohen-Or, Sivan To

Uncle Buck,, Host has reference to this Academic Journal, PHwiki organized this Journal High-Pass Quantization as long as Mesh Encoding Olga Sorkine, Daniel Cohen-Or, Sivan Toledo Eurographics Symposium on Geometry Processing, Aachen 2003 Overview Geometry quantization Visual quality Connection to spectral properties Geometry quantization – introduction Each mesh vertex is represented by Cartesian coordinates, in floating-point. Geometry compression requires quantization, normally 10-16 bits/coordinate (xi, yi, zi)

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Geometry quantization – introduction Using smoothness assumptions, the quantized coordinates are predicted in addition to the prediction errors are entropy-coded [Touma in addition to Gotsman 98] Quantization error Quantization necessarily introduces errors. The finer the sampling, the more it suffers. Quantization error An example: coarsely-sampled sphere original Quantized to 8 bits/coordinate

Quantization error A finely-sampled sphere with the same quantization original Same quantization to 8 bits/coordinate Quantization error – discussion Quantization of the Cartesian coordinates introduces high-frequency errors to the surface. High-frequency errors alter the visual appearance of the surface – affect normals in addition to lighting. Only conservative quantization (usually 12-16 bits) avoids these visual artifacts. Quantization – our approach Trans as long as m the Cartesian coordinates to another space using the Laplacian matrix of the mesh. Quantize the trans as long as med coordinates. The quantization error in the regular Cartesian space will have low frequency. Low-frequency errors are less apparent to a human observer.

Relative (laplacian) coordinates Represent each vertex relatively to its neighbours average of the neighbours the relative coordinate vector Laplacian matrix A Matnn(R) the adjacency matrix D Matnn(R) the degree-diagonal matrix: Then, the Laplacian matrix L Matnn(R) is: Laplacian matrix The previous as long as m is not symmetric. We will use the symmetric Laplacian:

Properties of L Sort the eigenvalues of L in accending order: We can represent the geometry in L’s eigenbasis: eigenvectors “frequencies” low frequency components high frequency components Previous usages of Laplacian matrix [Karni in addition to Gotsman 00] – progressive geometry compression Use the eigenvectors of L as a new basis of Rn Transmit the coordinates according to this spectral basis First transmit the lower-eigenvalue coefficients (low frequency components), then gradually add finer details by transmitting more coefficients. Previous usages of Laplacian matrix [Taubin 95] – surface smoothing “Push” every vertex towards the centroid of its neighbours, i.e. v’ = (I – L)v Iterate, with positive in addition to negative values of (to reduce shrinkage effect)

Previous usages of Laplacian matrix [Ohbuchi et al. 01] – mesh watermarking Embed a bitstring in the low-frequency coefficients Changes in low-frequency components are not visible [Alexa 02] – morphing using relative coordinates Produces locally smoother morphs [Gotsman et al. 03] A more general class of Laplacian matrices Mesh embedding on a sphere using eigenvectors Quantizing the – coordinates Trans as long as m Cartesian to -coordinates: Quantize -coordinates To get back Cartesian coordinates: (fixed-point quantization) Discussion of the linear system The matrix L is singular, so L1 doesn’t exist. Adding one anchor point fixes this problem (substitute one vertex (x, y, z) – removes translation degrees of freedom)

Discussion of the linear system By quantizing the , we put high-frequency error into . L has very small eigenvalues, so L1 has very large eigenvalues ( 1/) Thus, L1 amplifies small errors in addition to reverses the frequencies. Small quantization error as long as , high frequency NOT so small !! low frequency Spectrum of quantization error Write x as: x = a1 e1 + a2 e2 + + an en There as long as e, = Lx = 1a1 e1 + 2a2 e2 + + n-1an-1 en-1 + nan en Quantization error as long as ( + q ) is: q = c1 e1 + c2 e2 + + cn-1 en-1 + cn en Resulting error in x: qx = L1 q = (1/1)c1 e1 + (1/2)c2 e2 + + (1/n-1)cn-1 en-1 + (1/n)cn en large i – high frequencies high frequency error – here ci are large (1/i) is small – attenuates high-frequency errors (1/i) is large – amplifies low-frequency errors Small i – low frequencies low frequencies – small ci Thus, the error in x will contain strong low-frequency components but weak high-frequency components. Discussion of the linear system Example of low-frequency error: Find the differences between the horses

Discussion of the linear system Example of low-frequency error: This one is the original horse model Discussion of the linear system Example of low-frequency error: This is the model after quantizing to 8 bits/coordinate There is one anchor point (front left leg) Making the error lower We add more anchor points, whose Cartesian coordinates are known, as well as the – coordinates. This “nails” the geometry in place, reducing the low-frequency error

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Rectangular Laplacian We add equations as long as the anchor points By adding anchors the matrix becomes rectangular, so we solve the system in least-squares sense: L constrained anchor points Choosing the anchor points A greedy scheme. Add one anchor point at a time. Each time nail down the vertex that achieved the maximal error after reconstruction. This process is slow, but it is done only by the encoder. Only a small number of anchors is needed. We experiment with 0.1%, which gives very good results. The effect of anchors on the error Positive error – vertex moves outside of the surface Negative error – vertex moves inside the surface 0 – Cartesian quantization 8b/c -quantization 7b/c 4 anchors -quantization 7b/c 20 anchors -quantization 7b/c 2 anchors

Visual error metric Euclidean distance between in addition to does not faithfully represent the visual error (Cartesian quantization errors are small but the normals change a lot ) Karni in addition to Gotsman [2000] propose a “visual metric”: x – x’vis = x – x’2 + (1 – )GL(x) – GL(x’)2 = 0.5 We are not sure that should be 0.5 Visual error metric We measured the two error components separately: Mq = x – x’2 Sq = GL(x) – GL(x’)2 Evis = Mq + (1 – ) Sq Rate-distortion curves

Discussion of the linear system This is the model after quantizing to 7 bits/coordinate, one anchor Invertible square Laplacian We could simply eliminate the anchors from the system, erasing the rows in addition to the columns of the anchor vertices Use this “reduced” Laplacian instead of L in addition to remember the anchors’ (x, y, z) positions separately Invertible Laplacian artifacts Produces bad results when we quantize , because no smoothness constraints are posed on the anchors

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