summer seminar ruizhen hu. spectral sampling of manifolds (siggraph asia 2010) accurate...

Summer SeminarRuizhen Hu

• Spectral Sampling of Manifolds (Siggraph Asia 2010)

• Accurate Multidimensional Poisson-Disk Sampling (TOG)

• Efficient Maximal Poisson-Disk Sampling

• Blue-Noise Point Sampling using Kernel Density Model

• Differential Domain Analysis for Non-uniform Sampling

Sampling

Noise & filtering

• Filtering Solid Gabor Noise

• Accelerating Spatially Varying Gaussian

Spectral Sampling of Manifolds

A. Cengiz Öztireli Marc Alexa Markus Gross ETH Zürich TU Berlin ETH Zürich

Authors

CENGİZ ÖZTİRELİ

PhD CandidateComputer Graphics LaboratoryETH Zürich

Research interests: Reconstruction, sampling and processing of surfaces, and sketch based modeling.

Marc Alexa

Professor Electrical Engineering & Computer Science TU Berlin

Markus Gross

ProfessorDepartment of Computer Science ETH Zürich

Research interests: Computer graphics, image generation, geometric modeling, computer animation, and scientific visualization

Motivation

• Goal: finding optimal sampling conditions for a given surface representation

• Work: propose a new method to solve this problem based on spectral analysis of manifolds, kernel methods and matrix perturbation theory

Contributions

• Efficient, simple to implement, easy to control through intuitive parameters, feature sensitive

• Result in accurate reconstructions with kernel based approximation methods and high quality isotropic samplings

• A discrete spectral analysis of manifolds using results from kernel methods and matrix perturbation theory

Main Algorithm

• Input: a set of points lying near a manifold with normals + a kernel function definition = a continuous surface

Algorithms for Sampling

• Subsampling: measuring the effect of a point on the manifold using the Laplace-Beltrami spectrum

measures the contribution of a point to the manifold definition

measures the contribution of a point to the manifold definition

Algorithms for Sampling

• Resampling: maximizing and equalizing s (x) for all points– use local operations and move points in a simple gradient

ascent

Results

Results

Results

Conclusions

• New algorithms for the simplification and resampling of manifolds depending on a measure that restricts changes to the Laplace-Beltrami spectrum

• Limitations: the algorithms are greedy and thus not theoretically guaranteed to give the optimal sampling

• Future Directions:– texture on a surface– isotropic adaptive remeshing

Accurate Multi-Dimensional Poisson-Disk Sampling

Manuel N. Gamito Steve Maddock Lightwork Design Ltd The University of Sheffield

Authors

Manuel Noronha Gamito

software engineerLightwork Design Ltd

Steve Maddock

Senior Lecturer The University of Sheffield

Research interests: character animation, specifically modelling and animating faces

Poisson-Disk Sampling

• Definition:– Each sample is placed with uniform probability density– No two samples are closer than , where is some chosen

distribution radius– A distribution is maximal if no more samples can be inserted

• Poisson-Disk sampling is useful for:– Distributed ray tracing [Cook 1986; Hachisuka et al. 2008]– Object placement and texturing [Lagae and Dutré 2006; Cline et al. 2009]– Stippling and dithering [Deussen et al. 2000; Secord et al. 2002] – Global Illumination [Lehtinen et al. 2008]

Previous Methods

• Approximate Methods– Relax at least one of the sampling conditions

• Accurate Methods– Brute force

• Dart Throwing [Dippé and Wold 1985]– Assisted by a spatial data structure

• Voronoi diagram [Jones 2006]• Scalloped sectors [Dunbar and Humphreys 2006]• Uniform grid [Bridson 2007]• Simplified subdivision tree and uniform grid [White et al. 2007]

Main Algorithm

Radius vs. Number of Samples

• A distribution can be specified by supplying either– The distribution radius r– The desired number of samples N

• When the number of samples is specified– The algorithm uses a radius r based on N and on the measured

packing density of sample disks• The packing density was obtained by averaging the packing

densities measured from 100 distributions generated by our algorithm

• The number of samples of the resulting maximal distribution is approximately equal to the desired number N (error<5%)

Results

• Number of samples• Sampling time• Samples per second

Results

Conclusions

• A Poisson-Disk Sampling Algorithm that– Is statistically correct (see proof in paper)– Is efficient through the use of a subdivision tree– Works in any number of dimensions

• Subject to available physical memory– Generates maximal distributions– Allows approximate control over the number of samples– Can enforce periodic or wall boundary conditions on the

boundaries of the domain

Future Work

• Make it multi-threaded– Distant parts of the domain can be sampled in parallel with

different threads– Some synchronisation between threads is still required

• Generate non-uniform distributions– Have the distribution radius be a function of the position in the

domain

• Work over irregular domains– Discard subdivided tree nodes that fall outside the domain

Efficient Maximal Poisson-Disk Sampling

AuthorsMohamed S. Ebeida

post-doctorCarnegie Mellon university

Anjul Patney

PhDCarnegie University of California, Davis

Scott A. Mitchell

Principal Member of Technical StaffSandia National Laboratories

Andrew Davidson

PhDCarnegie University of California, Davis

Patrick M. Knupp

Distinguished Member Technical StaffSandia National Laboratories

John D. Owens

Associate ProfessorCarnegie University of California, Davis

Work

• generating a uniform Poisson-disk sampling that is both maximal and unbiased over bounded non-convex domains

Motivation

• Maximal Poisson-disk sampling distributions:– Avoid aliasing– Have blue noise property

• Bias-free:– Crucial in fracture propagation simulations

Conditions

• Maximal : the sample disks overlap cover the whole domain leaving no room to insert an additional point

• Bias-free the likelihood of a sample being inside any subdomain is proportional to the area of the subdomain, provided the subdomain is completely outside all prior samples’ disks

•

Previous methods

• relax the unbiased or maximal conditions, or require potentially unbounded time or space

• Dart-throwing– unbiased but also not maximal

• Tile-based– biased and require relatively large storage.

Main Algorithm

• First phase:– an unbiased, near-maximal covering– voids: the part of a grid cell outside all circles

• Second phase:– completes the maximal covering– darts are thrown directly into the voids, maintaining the bias-

free condition– A maximal distribution is achieved when the domain is

completely covered, leaving no room for new points to be selected

Sequential Sampling

1. Generate a background grid; mark interior and boundary cells

2. Phase I. Throw darts into square cells; remove hit cells

3. Generate polygonal approximations to the remaining voids

4. Phase II. Throw darts into voids; update remaining areas

Algorithm through Phase I

Voids

• Polygonal approximations to arc-voids

Results

Implementation Performance

Conclusions

• An efficient algorithm for maximal Poisson-disk sampling in two-dimensions– the final result is provably maximal– the sampling is unbiased– it is O(n log n) in expected time– it is O(n) in deterministic memory required– not limited to convex domains– efficiently implemented in both sequential and parallel forms

• Future work: 3D maximal Poisson-disk sampling algorithm

Blue-Noise Point Sampling using Kernel Density Model

Raanan FattalHebrew University of Jerusalem, Israel

Author

Raanan Fattal

Alon faculty memberSchool of Computer Science and EngineeringThe Hebrew University of Jerusalem

Work

• A new approach for generating point sets with high-quality blue noise properties that formulates the problem using a statistical mechanics interacting particle model

Contributions

• present a new approach that formulates the problem using a statistical mechanics interacting particle model

• derive a highly efficient multi-scale sampling scheme for drawing random point distributions

• avoids the critical slowing down phenomena that plagues this type of models

Previous work

• Dart throwing– constrain a minimal distance between every pair of points

• Relaxation– follow a greedy strategy that maximizes this distance– two main shortcomings:

• teriminatin• impreciseness : apparent blur

New Approach

• model the target density as a sum of nonnegative radially-symmetric kernels

The j-th kernel centered around the point xj:

New Approach

• The error of this approximation

• Minimizing E, with respect to the kernel centers– equivalent to the one obtained by converged Lloyd’s iterations– has the ability to achieve spectral enhancement

• We unify error minimization and randomness by defining a statistical mechanics particle model using E

New Approach

• Assigning each configuration a probability density according to the following Boltzmann-Gibbs distribution:

Drawing samples

• Markov-chain Monte Carlo(MCMC)• Gibbs sampler• Langevin method• Metropolis-Hastings(MH) test

Critical slowing down

Multi-scale sampling

Results

Results

Differential Domain Analysis for Non-uniform Sampling

Li-Yi Wei Rui WangMicrosoft Research University of Massachusetts Amherst

Authors

Li-Yi Wei

ResearcherMicrosoft Research

Rui Wang

Assistant Professor Department of Computer ScienceUniversity of Massachusetts

Work

• new methods for analyzing non-uniform sample distributions

Previous work

• Two common methodologies exist for evaluating the quality of samples– spatial uniformity: discrepancy & relative radius– Power spectrum analysis, including radial mean and anisotropy

• However, existing methods are primarily designed for uniform Euclidean domains and can not be easily extended to general non-uniform scenarios, such as– adaptive– anisotropic– non-Euclidean domains

Contributions

• A reformulation of standard Fourier spectrum analysis into a form that depends on sample location differentials

• A generalization of this basic formulation, including different distance transformations for various domains, and range selection for better control of quality and speed

• Applications in spectral and spatial analysis for non-uniform sample distributions

Core idea

• Fourier power spectrum– Fourier transform:

– power spectrum:

• Differential representation:

Core idea

• Integral form:

• General kernel:

– Range selection (for computational reasons):

Where is a local distance measure of s’ with respect to the local frame centered at s. In uniform Euclidean domains,

Core idea

• Non-uniform domain：

Where is a differential domain transformation function that locally warps each d from a non-uniform to a (hypothetical) uniform

Kernel selection

• a cos kernel may amplify some information while obscure others

• A Gaussian kernel, in contrast, displays the main peak clearly without undulations. it can manifest the distribution properties more clearly, e.g. more apparent characteristic structures

Spectral Analysis

• Exact – Isotropic Euclidean domain– Anisotropic Euclidean domain

• Range selection• Radial measures

– We can compute the circular average and variance of p(d)– The former gives the radial mean, indicating the overall

distance-based property of p(d)– The latter gives the anisotropy, which reveals if there is any

directional bias/structure in the distribution

Comparisons

• How our method relates and compares to traditional Fourier spectrum analysis?

Results

Results

Filtering Solid Gabor Noise

Ares Lagae George DrettakisKatholieke Universiteit Leuven REVES/INRIA Sophia-Antipolis

Authors

Ares Lagae

Postdoctoral FellowComputer Graphics Research GroupKatholieke Universiteit Leuven

George Drettakis

Group LeaderREVES/Inria Sophia-Antipolis

Work

• we show that a slicing approach is required to preserve continuity across sharp edges, and we present a new noise function that supports anisotropic filtering of sliced solid noise

Filtering

• Filtering the noise on the surface using frequency clamping works better if the power spectrum of the noise on the surface is bandpass.

• The Fourier Slice Theorem: projecting in the spatial domain corresponds to slicing in the frequency domain, and vice versa

Previous works

• Perlin noise[Perlin 1985]: – use slicing and frequency clamping– Filtering introduces an aliasing vs. detail loss trade-off since the

power spectrum of Perlin noise is not band-pass

• Wavelet noise [Cook and DeRose 2005]:– use projection and frequency clamping– Even though the power spectrum of the noise on the surface is

band-pass, filtering does not fully solve the aliasing vs. detail loss trade-off

• Gabor noise [Lagae et al. 2009]: – use projection and a filtering approach specific to Gabor noise– This results in high-quality anisotropic filtering, however,

introduces discontinuities at sharp edges

New noise

• solid random-phase Gabor noise:– use slice (preserves continuity at sharp edges)– Since filtering is inherently a 2D operation, we have to explicitly

model the slicing of the 3D Gabor kernels to be able to filter the resulting 2D Gabor kernels. This requires the introduction of

• a new Gabor kernel, the phase-augmented Gabor kernel• a new Gabor noise, random-phase Gabor noise

Slicing Solid Gabor Noise

• The Gabor kernel of Lagae et al. [2009] is not closed under slicing:

Phase-augmented Gabor kernel

• The phase-augmented Gabor kernel is closed under slicing:

• solid random-phase Gabor noise (also closed under slicing):

Slicing Random-Phase Gabor Noise

• solid random-phase Gabor noise is also closed under slicing

• the statistical properties of a sliced solid random-phase Gabor noise are obtained using the analytical expressions for the statistical properties of a 2D random-phase Gabor noise

Filtering Sliced Solid Gabor Noise

Conclusion

• A new procedural noise function, random-phase Gabor noise, that supports – continuity across sharp edges– high-quality anisotropic filtering– Anisotropy

• Future work: – exploring volumetric filtering of solid Gabor noise– further exploring anisotropy in the context of solid noise– designing user interfaces for interacting with anisotropic solid

noise

Results

Results

Accelerating Spatially Varying Gaussian Filters

Jongmin Baek David E. JacobsStanford University

Jongmin Baek

Ph.D. studentStanford University

David E. Jacobs

PhD candidateStanford University

Authors

Motivation

Input

GaussianFilter

SpatiallyVaryingGaussianFilter

1) Accelerating Spatially Varying Gaussian Filters

2) Accelerating Spatially Varying Gaussian Filters

3) Accelerating Spatially Varying Gaussian Filters

4) Applications

Roadmap

Gaussian Filters

Position Value

Gaussian Filters

Each output value …

Gaussian Filters

… is a weighted sum of input values …

Gaussian Filters

… whose weight is a Gaussian …

Gaussian Filters

… in the space of the associated positions.

Gaussian Blur

Gaussian Filters: Uses

Bilateral Filter

Gaussian Filters: Uses

Non-local Means Filter

Gaussian Filters: Uses

Applications Denoising images and meshes Data fusion and upsampling Abstraction / Stylization Tone-mapping ...

Gaussian Filters: Summary

Previous work on fast Gaussian Filters Bilateral Grid (Chen, Paris, Durand; 2007) Gaussian KD-Tree (Adams et al.; 2009) Permutohedral Lattice (Adams, Baek, Davis; 2010)

Summary of Previous Implementations:

A separable blur flanked by resampling operations.Exploit the separability of the Gaussian kernel.

Gaussian Filters: Implementations

Spatially Varying Gaussian Filters

Spatially varying covariance matrix

Spatially Invariant

Trilateral Filter (Choudhury and Tumblin, 2003)

Tilt the kernel of a bilateral filter along the image gradient.

“Piecewise linear”instead of“Piecewise constant”model.

Spatial Variance in Previous Work

Spatially Varying Gaussian Filters: Tradeoff

Benefits: Can adapt the kernel spatially. Better filtering performance.

Cost: No longer separable. No existing acceleration schemes.

Input Bilateral-filtered Trilateral-filtered

Problem: Spatially varying (thus non-separable) Gaussian filter

Existing Tool: Fast algorithms for spatially invariant Gaussian filters

Solution: Re-formulate the problem to fit the tool. Need to obey the “piecewise-constant” assumption

Acceleration

Naïve Approach (Toy Example)I LOST THE GAME

Input Signal

Desired Kernel1 1 12 3 4

filtered w/ 1filtered w/ 2filtered w/ 3filtered w/ 4

1 1 1

2

3Output Signal4

In practice, the # of kernels can be very large.

Challenge #1

Pixel Location x

Desired Kernel K(x)

Range ofKernels needed

Sample a few kernels and interpolate.

Solution #1

Desired Kernel K(x)

Sampledkernels

Interpolate result!

Pixel Location x

K1

K2

K3

Interpolation needs an extra assumption to work:

The covariance matrix Ʃi is either piecewise-constant,

or smoothly varying.

Kernel is spatially varying, but locally spatially invariant.

Assumptions

Runtime scales with the # of sampled kernels.

Challenge #2

Desired Kernel K(x)

Filter only some regions of the image with each kernel.

(“support”)

Pixel Location x

Sampledkernels

K1

K2

K3

In this example, x needs to be in the support of K1 & K2.

Defining the Support

Desired Kernel K(x)

Pixel Location x

K1

K2

K3

Dilating the Support

Desired Kernel K(x)

Pixel Location x

K1

K2

K3

Algorithm

1) Identify kernels to sample.2) For each kernel, compute the support needed.3) Dilate each support.4) Filter each dilated support with its kernel.5) Interpolate from the filtered results.

Algorithm

1) Identify kernels to sample.2) For each kernel, compute the support needed.3) Dilate each support.4) Filter each dilated support with its kernel.5) Interpolate from the filtered results.

K1

K2

K3

Algorithm

1) Identify kernels to sample.2) For each kernel, compute the support needed.3) Dilate each support.4) Filter each dilated support with its kernel.5) Interpolate from the filtered results.

K1

K2

K3

Algorithm

1) Identify kernels to sample.2) For each kernel, compute the support needed.3) Dilate each support.4) Filter each dilated support with its kernel.5) Interpolate from the filtered results.

K1

K2

K3

Algorithm

1) Identify kernels to sample.2) For each kernel, compute the support needed.3) Dilate each support.4) Filter each dilated support with its kernel.5) Interpolate from the filtered results.

K1

K2

K3

Algorithm

1) Identify kernels to sample.2) For each kernel, compute the support needed.3) Dilate each support.4) Filter each dilated support with its kernel.5) Interpolate from the filtered results.

K1

K2

K3

Applications

HDR Tone-mapping

Joint Range Data Upsampling

Application #1: HDR Tone-mapping

Input HDR

Detail

BaseFi

lter

Output

Attenuate

Tone-mapping Example

Bilateral Filter Kernel Sampling

Application #2: Joint Range Data Upsampling

Range Finder Data

Sparse Unstructured Noisy

Scene Image

Output

Filter

Synthetic Example

Scene Image Ground Truth Depth

Synthetic Example

Scene Image Simulated Sensor Data

Synthetic Example : Result

Kernel SamplingBilateral Filter

Synthetic Example : Relative Error

Bilateral Filter Kernel Sampling

2.41% Mean Relative Error 0.95% Mean Relative Error

Real-World Example

Scene Image Range Finder Data

*Dataset courtesy of Jennifer Dolson, Stanford University

Real-World Example: Result

Input

Bilateral

Naive

KernelSampling

Performance

Kernel Sampling

Choudhury and Tumblin (2003) Naïve

Tonemap1 5.10 s 41.54 s 312.70 sTonemap2 6.30 s 88.08 s 528.99 s

Kernel Sampling

Kernel Sampling(No segmentation)

Depth1 3.71 s 57.90 sDepth2 9.18 s 131.68 s

1. A generalization of Gaussian filters• Spatially varying kernels• Lose the piecewise-constant assumption.

2. Acceleration via Kernel Sampling• Filter only necessary pixels (and their support)

and interpolate.

3. Applications

Conclusion