Edge-Avoiding Wavelets and their Applications

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Edge-Avoiding Wavelets and their Applications

• Raanan Fattal
• Hebrew University of Jerusalem, Israel

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Introduction

• Many image operations require scale separation
• Noise removal, DR compression, resolution
  reduction, detail enhancement, image compression,
  etc.
• Linear translation-invariant (LTI) is the most
  common one

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Introduction

• Current affairs in LTI filtering
• Tradeoffs are well understood
• Transparent view in Fourier space
• Fast computing,
• Fast-Fourier Transform O(nlogn)
• Multi-scale computation O(n)
• Fast filter transforms for image processing,
  Burt 1981
• Efficient Multi-scale versions (DWT, Laplacian
  Pyramid, Dyadic WT)
• Efficient O(n) solvers for implicit formulation,
• Multi-grid method (geometric)
• Multi-scale Preconditioned Conjugate Gradients
  method

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Introduction

• LTI filtering separates small variations based on
  their scale, yet fails on large variations

Inter-scale correlations
band-independent dynamic range compression
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Introduction

• Solution edge-preserving filtering or robust
  smoothing
• Bilateral filter weights chage in space

Combined
Result
Input
Graphs taken from Durand Dorsey 2002
Anisotropic Diffusion Perona and Malik 1990,
Black et al. 1998 Bilateral Filter Tomasi and
Manduchi, 1998 Digital TV filter Chan et al.
2001
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Introduction

• Current affairs in data-dependent filtering
• Many possibilities for edge-stopping function
• Costly
• Non-linear anisotropic diffusion requires
  multiple iterations
• Bilateral filter additional dimensions Paris and
  Durand 2005
• Costly multi-scale constructions
• Multi-scale Bilateral filter O(nlogn) Fattal et
  al. 2007
• Weighted least squares O(nlogn) Farbman et al.
  2008
• Implicit formulation yields poorly-conditioned
  systems
• Hierarchical preconditioning Szeliski 2006

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Current Work

• A new family of data-dependent second-generation
  wavelets allowing
• Edge-preserving multi-scale construction in O(n)
  time
• Avoiding implicit formulations and
  poorly-conditioned systems by explicit O(n)
  transformation

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Second-Generation Wavelets

• Classic wavelets theory
• Uniform notion of smoothness a single pair of
  wavelet and scaling functions
• Second-generation wavelets change in space
• Irregular sampling
• Boundary wavelets
• Domains with complicated geometry
• Image compression
• Constructed via data-prediction Harten 1993 and
  lifting Sweldens 1993 schemes

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Lifting Scheme

• Spatial biorthogonal wavelets generation scheme
• Three steps
• Split pick coarse and fine nodes
• Predict estimate fine from coarse
• Update preserve averages (smooth coarse nodes)

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Example Constructing the (2,2)-CDF

• Define prediction by linear interpolation

CDF Wavelets are translation invariant No use of
Fourier, allowing easy spatial adaptations
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Edge-Avoiding Wavelets

• Instead of TI regression formulae we use
  posteriori robust predictions
• with
• And also

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Edge-Avoiding Wavelets

• Enjoy flexibility of lifting scheme to easily
  handle boundaries
• Computing weights loses the in-place (double data)

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Inter-Scale Decorrelation
Edge-avoiding scaling func.
translation-invariant trans.
Edge-avoiding trans.
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Applications
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Dynamic Range Compression

• Switch to YUV color space

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Dynamic Range Compression
Explicit computation no need to solve
inhomogeneous Poisson equations 12ms for 1Mpixel
image x5 faster than single-scale Durand and
Dorsey 2002 x200 faster than Farbman et al.
2008,
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Dynamic Range Compression
Band-independent processing, no gain control
needed
Gain Control Li et al. 2005
WRB EA wavelets
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Detail Enhancement
Input
WRB EA wavelets
Farbman et al. 2008
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Detail Enhancement
Input
WRB EA wavelets
Farbman et al. 2008
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Edge-Preserving Smoothing
Input
subsampling reduces O(nlogn) computation to
O(n) no quality costs
WRB EA wavelets
Farbman et al. 2008
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Edge-Aware Interpolation

• EA Scaling functions contain input edge data
• Pull-push scheme Gortler et al. 19931996

edge
1D example
pull
push
Computed in linear-time O(n)
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Edge-Aware Interpolation Colorization
Input
Levin et al. 2004
WRB EAW
Explicit computation avoids solving
poorly-conditioned inhomogeneous Poisson
equations (x200 faster than Levin et al. 2004
using Szeliski 2006)
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Conclusions

• Combined two proven methodologies
• Enjoy the FAST performance of the cascade
  algorithm
• Decompose to robust decorrelated image
  coordinates
• Multi-scale edge-preserving decomposition in
  O(N)
• Match computational lightness of LTI/MS-LTI
• Accelerate data-dependent implicit formulations
  by explicit computations

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• Thank you!
• Thanks to Mike Werman and Dani Lischinski

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Edge-Aware Interpolation De-colorization
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WCDF Compared to WRB
WRB
WCDF
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Amiram Harten 1947-1994

• Tel Aviv University, app. math. dept.
• Hyperbolic solvers
• TVD schemes and ENO schemes
• Data-prediction schemes
• Multiresolution representation of data a
  general framework
• Wikipedia the article on ENO, titled, Uniformly
  High Order Accurate Essentially Non-oscillatory
  Schemes, III was published in Journal of
  Computational Physics, in 1987 and is one of the
  most cited papers in the field of scientific
  computing

Stanley Osher
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Two 2D Constructions

• Weighted (2,2)-CDF each axis separately
• Weighted Red-Black (WRB) quincunx lattice

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Dynamic Range Compression
Rainhard et al. 2002
WRB EA wavelets
Durand and Dorsey 2002
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Detail Enhancement
Farbman et al. 2008
Input
Fattal et al. 2007
WRB EA wavelets