1. introduction
   1. rich variation over the surface
      1. weathered objects
         1. dirt
            1. tends to accumulate in areas of low accessibility
         2. corrosion
            1. often starts at exposed areas
   2. empirically model the variation with statistical correlation
   3. our goal
      1. not to simulate the physics of weathering processes
      2. to reproduce the rich visual appearance of a textured object.
   4. example
      1. Given a target mesh, we synthesize a new texture that reproduces the variation and geometry correlation from the source
      2. Rather than seeking physical accuracy, we aim at reproducing a visually consistent transfer of the texture.
      3. texture by numbers
         1. a guidance field of scalar or vector values drives a texture synthesis algorithm.
   5. challenges
      1. First, we need to compile a list of geometric properties or features that are relevant to texture variation
      2. Second, the number of possible features is too large and needs to be reduced to prevent the curse of dimensionality at the texture synthesis step.
      3. In addition, irrelevant elements in the guidance field could lead to spurious variations in the transferred texture.
      4. it can be challenging to transfer to a target mesh bearing little similarity with the source.
   6. contribution
      1. We introduce a dimensionality-reduction step
      2. we introduce a feature matching technique
         1. does not strictly respect the input geometry
         2. greatly increases the realism of the synthesized texture
         3. also addresses missing data problems
      3. We also introduce a parametric variant based on the Heeger and Bergen model
         1. lacks the ability to represent structured textures
         2. superior in producing detailed variations in case of strong geometry correlation.
         3. computationally less expensive and does not require parameter tweaking.
2. Related Work
   1. Texture Synthesis over meshes.
      1. PFH00, Tur01, WL01, YHBZ01, SCA02, TZL∗ 02, WGMY05
      2. Gorla et al. [GIS03]
   2. Spatially Varying Texture Synthesis
      1. 2D
         1. Wei01, MZD05, ZFCG05
      2. 3D
         1. ZZV∗ 03
      3. Texture-by-numbers
         1. Ash01,HJO∗ 01,EF01
   3. User-Driven Decoration.
      1. Zhou et al. [ZWT∗ 05]
      2. Zelinka et al. [ZG04]
   4. Weathering
      1. Mil94
      2. Wong et al. [WNH97]
      3. Dorsey and Hanrahan [DH96]
      4. Lu et al. [LGR∗ 05] and Giorghiades et al. [GLX∗ 05]
         1. propose suitable guidance fields for specific phenomena
            1. drying [LGR∗ 05]
            2. paint crackling and patina formation [GLX∗ 05]
3. Method Overview
   1. inputs
      1. a (source) triangular mesh
      2. a target mesh
      3. a texture map
         1. obtained from calibrated photographs
      4. acquired using a commercial scanner
   2. output
      1. generate a texture map exhibiting similar texture variation and correlation
   3. compile geometric features
   4. characterize
   5. a feature matching step
   6. The guidance field
4. Geometric Correlation
   1. Data Representation
      1. Turk [Tur01] and Wei et al. [WL01].
         1. uniformly distributed set of surface points
      2. [LPRM02]
         1. a texture atlas can be employed
      3. Turk [Tur01]
         1. build our representation
      4. [Tur91]
         1. is sampled uniformly at multiple resolutions using point repulsion
      5. each point
         1. a vector x of geometric features
         2. a texel y (RGB triplet).
            1. use superscripts s and t to denote the source and target features (texels)
            2. can be evaluated at any surface location using scattered data interpolation [She68]
      6. construct a Gaussian pyramid over the mesh
   2. Geometric Features
      1. 4 features we need
         1. Normalized Height
         2. Surface Normal
         3. Multi-Scale Solid Angle Curvature
            1. Connolly [Con86]
            2. Given a sphere centered at the point of interest, the solid angle curvature is equal to the solid angle subtended by the intersection of the surface with this sphere.
            3. 2D analogy of the multi-scale solid angle curvature.
            4. empirically chose four different radii
            5. advantage
            6. provides information for both concave and convex parts of the surface
            7. can be evaluated easily at multiple scales, unlike discrete mean curvature [MDSB02] and accessibility.
         4. Directional Occlusion.
            1. We introduce a novel measure which can be seen as a directional extension of ambient occlusion [ZIK98]
            2. We capture the large-scale directional variation of visibility
         5. The total dimensionality of our feature set is 12.
            1. Importance of our feature set.
      2. example
         1. Two geometric features: solid angle curvature and directional occlusion
   3. Correlation Analysis
      1. we need to reduce the dimensionality of the former (12D) to make synthesis tractable.
      2. PCA only characterizes the extent of a dataset in one space.
      3. Canonical Correlation Analysis(CCA) [Hot36, MKB00, Bor98]
         1. finds an affine low-rank transformation such that the source feature vectors x^s and the texel RGB vectors y^s have maximal correlation.
         2. returns two matrices W_x and W_y and a diagonal matrix of respective correlations \sqrt{Lambda}
            1. W_x
            2. transform the features
            3. W_y
            4. transform the texture
         3. \[\sqrt{Lambda}W^{\intercal _{x}x^s \approx W^{\intercal}_{y}y^{s}\]
         4. x^{s\bigstar}
            1. x^{s\bigstar}=W_{x}{\bigstar \intercal }x^{s}
         5. d
            1. the lowest dimensionality among the two datasets
            2. equal to three(cf. RGB)
         6. W_{x}^{\bigstar \intercal }
            1. basically tell us which features are relevant
         7. x^{t\bigstar} = W_x^{\bigstar \intercal}x^{t}
5. Feature Matching
   1. Marginals
      1. Matching the features is similar to matching color distributions [RAGS01, HB95]
   2. Multiscale Feature Content
      1. the feature distribution must be matched at different scales
      2. decompose the features into a Laplacian pyramid
         1. computed from the Gaussian pyramid of features
      3. match the pyramid coefficients in addition to the actual feature values
6. Texture Transfer
   1. non-parametric Synthesis
      1. guidance field can be readily applied to the previ- ously introduced non-parametric constrained synthesis tech- niques [HJO∗ 01, EF01].
   2. Parametric Synthesis
      1. Constrained Heeger and Bergen(H&B)
         1. relies on global histograms
         2. segmentation of the object through k-means clustering [Llo82]
         3. For each of these segments, we compute a set of histograms for the (decorrelated [HB95]) intensity and Laplacian pyramid levels.
         4. The same clustering algorithm is performed on the target guidance field.
      2. Feathering
         1. feather the result to avoid seams
         2. using a Gaussian with a standard deviation equal to that of the corresponding cluster’s distribution.
         3. The feathering is used on both the source and target.
      3. Initialization
         1. Although clustering already enforces much of the structure, additional measures should be taken to obtain a crisp result.
         2. The Heeger and Bergen model is too weak to synthesize structure such as crevices.
         3. the initialization should contain most of the structure, which will be refined further by H&B
            1. apply a simple linear regression model
            2. a fixed amount of white noise is added
      4. Discussion
7. Results