Shape Gradient Domain (V4.5)

Shape Gradient Domain (Version 4.5)

links description executable notes examples changes

LINKS
[Chuang et al. 2009], [Prada et al. 2016], [Chuang et al. 2016]
Windows (x64) Executables
Source Code
GitHub Repository
(Older Versions: V4.0, V3.0, V2.0, V1.0)

DESCRIPTION

This code performs gradient domain processing [Chuang et al. 2009, Chuang et al. 2016] on signals defined on a mesh, where the signal can be either a color-field represented as a color per vertex or is the position of the vertices themselves. The code supports both sharpening and smoothing of the signals through the solution of a screened-Poisson equation. Specifically, given an input signal F, it solves for the signal G minimizing:
E(G) = ||F-G||² + β⋅||λ⋅∇F - ∇G||²
where β is the gradient-fitting weight and λ is the gradient scale factor.
The code supports inhomogenous processing by allowing the user to replace the Riemannian metric, g, given by the embedding, with a metric that adjusts to the curvature of the surface. Specifically, given orthonormal principal curvature directions, the (idenity) metric is replaced with:
g ← Id. + ε⋅Κ² where Id. is the identity matrix, Κ² is the diagonal matrix whose entries are the squares of the principal curature values, and ε is the curvature weight. (When ε=0, this reproduces the Riemannian metric given by the embedding.)
Curvatures are estimated using the surface normals. If none are provided, the vertex normals are estimated as the area-weighted sum of adjacent triangle normals.
For robustness, the implementation supports pre-smoothing the normals before estimating curvatures. This can either be done directly by smoothing the individual channels or by performing the harmonic smoothing described in [Prada et al. 2016]. In either case, this amounts to minimizing an energy of the form:
E(G) = ||F-G||² + γ⋅||∇G||²
where γ is the normal smoothing weight.
In the case that the signal processed describes the vertices' normals, we also support fitting the original geometry to the processed normals using the approach of [Yu et al. 2004].
Given input positions F and a target normal field N This amounts to minimizing:
E(G) = ||F-G||² + δ⋅||(∇F - N⋅⟨∇F,N⟩) - ∇G||²
where δ is the projected gradient fitting weight weight.

EXECUTABLE

--in <input geometry>: This string specifies the name of the input geometry, represented in PLY format.
[--out <ouput geometry>]: This string specifies the name of the output geometry, represented in PLY format.
[--nIters <normal smoothing iterations>]: This integer value gives the number of iterations of harmonic normal smoothing to be performed before estimating curvatures. (If normal smoothing is desired, we have found that a single iteration suffices.)
The default value for this parameter is 0.
[--nWeight <normal smoothing time>]: This floating point value gives the normal smoothing weight (γ).
The default value for this parameter is 10^-4.
[--gWeight <gradient interpolation weight>]: This floating point value gives the weight for gradient interpolation (β).
The default value for this parameter is 10^-4.
[--gScale <gradient scale>]: This floating point value gives the scale factor for the target gradient field (λ).
The default value for this parameter is 1.0.
[--kWeight <curvature weight>]: This floating point value gives the curvature weight for adjusting the metric (ε).
The default value for this parameter is 0.0.
[--npWeight <normal projection weight>]: This floating point value gives the normal projection weight (δ).
The default value for this parameter is 10².
[--signal]: This inter value describes which signal is to be processed:

0: Vertex positions
1: Vertex normals (will be assigned if none are give)
2: Vertex colors
3: Vertex normals used to fit the geometry.
The default value for this parameter is 0.
[--anisotropic]: If this flag is enabled, the metric is scaled anisotropically along the two principal curvature directions -- Κ=Diag(κ₁,κ₂). Otherwise, the metric is scaled by the same amount in both diretions -- Κ² ← tr(Κ²)⋅Id.
[--hNormals]: If this flag is enabled, the harmonic smoothing of [Prada et al. 2016] is used to pre-process the normals before estimating curvatures. (If this flag is enabled, the default value of --nWeight may need to be increased.)
[--verbose]: If this flag is enabled, the code will output processing information.

NOTES

The code requires Eigen as a numerical solver.
If you are using Eigen and your implementation is backed by Intel's Math Kernel Library (see discussion here), enable the EIGEN_USE_MKL_ALL macro by defining it in the file PreProcessor.h. (The two versions of the Windows executables are compiled with and without MKL support.)

EXAMPLES

Top row: Isotropic sharpening:

     ShapeGradientDomain --in armadillo.ply --out armadillo.sharp.ply --gScale 2 --gWeight <gradient weight>

Middle row: Isotropic smoothing:

     ShapeGradientDomain --in armadillo.ply --out armadillo.smooth.ply --gScale 0 --gWeight <gradient weight>

Bottom row: Anisotropic smoothing:

     ShapeGradientDomain --in armadillo.ply --out armadillo.smooth.ply --gScale 0 --gWeight <gradient weight> --kWeight 0.02 --nIters 1

<gradient weight>

1e-3

1e-4

1e-5

HISTORY OF CHANGES

Version 2.0:

Added options to weight both value and gradient interpolation terms.
Changed default values to correspond to diffusion time.

Version 3.0:

Integrated normal smoothing within the gradient-domain processing application.

Version 4.0:

Fixed default value weight to 1.0.
Added support for fitting the geometry to prescribed normals.

Version 4.5:

Enabled symbolic matrix factorization in pre-processing.
Added support (and made default) normal smoothing through per-channel diffusion.

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