Introduction to Curvelets

1
Introduction to Curvelets

• Authors Emmanuel Candès and David Donoho
• Story-teller ? Julia Dobrosotskaya

2
Edges discontinuities across curves

• Synthesis edge location is known in advance,
  representation adapted respectively
• Analysis edge location is unknown two
  approaches arise
• Adaptive (Lagrangian representation constructed
  using the full knowledge of the structure and
  adapting to the structure perfectly)
• Non-adaptive (Eulerian fixed, constructed once
  and for all).

3
2D edge-adapted schemes (unpublished attempts)

• Adaptive triangulation aims to represent a
  function by partitioning the plane into a
  sequence of triangular meshes, refining them from
  stage to stage, and using piecewise linear
  functions with such triangular supports as
  building elements
• Adaptively warped wavelet transformation- here
  the object is deformed to obtain only horizontal
  and vertical edges, then tensor-product wavelets
  are used.
• Both good for image synthesis. Analysis-?

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Quantifying rates of approximation
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Surprise Despite common belief that adaptive
representation is essentially more powerful than
fixed non-adaptive, it turns out that there is a
fixed non-adaptive technique essentially as good
as adaptive representation from the point of view
of asymptotic m-term approximation errors.
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An Important Precedent a case where a
non-adaptive scheme is roughly competitive with
an ideal adaptive scheme

• f - piecewise polynomial P pieces, deg ?D.
• Adaptive piecewise polynomial approximation
• Need to keep P (D1) coefficients and P-1
  breakpoints for exact reconstruction.
• Non-adaptive wavelet approximation by Daubechies
  compactly supported wavelets
• View f as a digital signal f(i/N), i1,..,N
  (information kept up to scale 1/N)
• C log2N P (D1) coefficients needed.

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Failure of Wavelets on Edges

• f smooth on 0,12 away from a discontinuity
  along a C2 curve ?
• Grid of squares 2-j ? 2-j has O(2j) squares
  intersecting ?
• At level j each wavelet is localized near the
  corresponding square
• Wavelet coefficient is controlled by
• Therefore, there are about 2j coefficients of
  size about 2-j
• Nth largest wavelet coefficient is of size about
  1/N.

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Point and Curve Discontinuities

• A discontinuity point affects all the Fourier
  coefficients in the domain.
• Hence the FT doesnt handle points
  discontinuities well.
• Using wavelets, it affects only a limited number
  of coefficients.
• Hence the WT handles point discontinuities well.
• Discontinuities across a simple curve affect all
  the wavelets coefficients on the curve.
• Hence the WT doesnt handle curves
  discontinuities well.
• Curvelets are designed to handle curves using
  only a small number of coefficients.
• Hence the CvT handles curve discontinuities well.

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Curvelet Transform

• The Curvelet Transform includes four stages
• Sub-band decomposition
• Smooth partitioning
• Renormalization
• Ridgelet analysis

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Sub-band Decomposition

• Dividing the image into resolution layers.
• Each layer contains details of different
  frequencies
• P0 low-pass filter.
• ?1, ?2, band-pass (high-pass) filters.
• The original image can be reconstructed from the
  sub-bands
• Energy preservation

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Sub-band Decomposition
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Sub-band Decomposition

• Low-pass filter ?0 deals with low frequencies
  near ??1.
• Band-pass filters ?2s deals with frequencies near
  domain ??22s, 22s2.
• Recursive construction
• ?2s(x) 24s? (22sx).
• The sub-band decomposition is simply applying a
  convolution operator

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Sub-band Decomposition

• The sub-band decomposition can be approximated
  using the well known wavelet transform
• Using wavelet transform, f is decomposed into
  S0, D1, D2, D3, etc.
• P0 f is partially constructed from S0 and D1,
  and may include also D2 and D3.
• ?s f is constructed from D2s and D2s1.

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Sub-band Decomposition

• P0 f is smooth (low-pass), and can be
  efficiently represented using wavelet base.
• The discontinuity curves effect the high-pass
  layers ?s f. Can they be represented efficiently?
• Looking at a small fragment of the curve, it
  appears as a relatively straight ridge.
• We will dissect the layer into small partitions.

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Smooth Partitioning
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Smooth Partitioning

• A grid of dyadic squares is defined
• Qs all the dyadic squares of the grid.
• Let w be a smooth windowing function with main
  support of size 2-s?2-s.
• For each square, wQ is a displacement of w
  localized near Q.
• Multiplying ?s f with wQ (?Q?Qs) produces a
  smooth dissection of the function into squares.

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Smooth Partitioning

• The windowing function w is a nonnegative smooth
  function.
• Partition of the energy
• The energy of certain pixel (x1,x2) is divided
  between all sampling windows of the grid.
• Example
• An indicator of the dyadic square (but not
  smooth!!).
• Smooth window function with an extended compact
  support
• Expands the number of coefficients.

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Smooth Partitioning

• Partition of the energyReconstruction
• Parserval relation

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Renormalization

• Renormalization is centering each dyadic square
  to the unit square 0,1?0,1.
• For each Q, the operator TQ is defined as
• Each square is renormalized

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Before the Ridgelet Transform

• The ?s f layer contains objects with frequencies
  near domain ??22s, 22s2.
• We expect to find ridges with width ? 2-2s.
• Windowing creates ridges of width ? 2-2s and
  length ? 2-s.
• The renormalized ridge has an aspect ratio of
  width ? length2.
• We would like to encode those ridges efficiently
• Using the Ridgelet Transform.

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Ridgelets

• Developed by E. Candes in PhD thesis, 1998
• System of analysis is based on ridge functions
• Continuous ridgelet transform
• with a reproducing formula and Parceval relation
• Frames giving stable series expansions (special
  discrete collections of ridge functions)

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The Ridgelet Transform

• (Ortho-)ridgelets are an orthonormal set ?? for
  L2(R2) (introduced by D. Donoho, 1998).

• Divides the frequency domain to dyadic coronae
  ??2s, 2s1.
• In the angular direction, samples the s-th corona
  at least 2s times.
• In the radial direction, samples using local
  wavelets.

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The Ridgelet Transform

• The ortho-ridgelet element has a formula in the
  frequency domainwhere,
• ?i,l are periodic wavelets for -?, ? ).
• i is the angular scale and l?0, 2i-11 is the
  angular location.
• ?j,k are Meyer wavelets for ?.
• j is the ridgelet scale and k is the ridgelet
  location.

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Ridgelet Sampling Motivation

• Behavior of FT of functions with
  singularities along lines
• FT decays slowly along associated lines through
  the origin in the frequency domain
• Constant radius arc in the frequency domain
  encounters a Fourier ridge when crossing the
  line of slow decay
• Ridgelet sampling uses wavelets in angular
  direction in order to capture the ridge by few
  wavelets
• In the radial direction the Fourier ridge is
  oscillatory, this is captured by local cosines.

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Ridgelet Analysis

• Each normalized square is analyzed in the
  ridgelet system
• The ridge fragment has an aspect ratio of
  2-2s?2-s.
• After the renormalization, it has localized
  frequency in band ??2s, 2s1.
• A ridge fragment needs only a very few ridgelet
  coefficients to represent it.

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Digital Ridgelet Transform (DRT)

• Unfortunately, the (current) DRT is not truly
  orthonormal.
• An array of n?n elements cannot be fully
  reconstructed from n?n coefficients.
• The DRT uses n?2n coefficients for almost perfect
  reconstruction
• Still some research can be done

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Curvelet Transform

• The four stages of the Curvelet Transform were
• Sub-band decomposition
• Smooth partitioning
• Renormalization
• Ridgelet analysis

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Image Reconstruction

• The Inverse of the Curvelet Transform
• Ridgelet Synthesis
• Renormalization
• Smooth Integration
• Sub-band Recomposition

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Example
Roy Lichtenstein In The Car 1963
Original Image (256?256)
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Example
Adding Gaussian Noise
Original
Noise Reduction using Curvelet transform.
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Example
Adding Gaussian Noise
Original
Noise Reduction using Curvelet transform.
Curvelet Transform
WT Thresholding
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References

• 1 E.J. Candès and D.L. Donoho. Curvelets A
  Surprisingly Effective Non-adaptive
  Representation for Objects with Edges Curve and
  Surface Fitting Saint Malo 1999
• 2 D.L. Donoho and M.R. Duncan. Digital Curvelet
  Transform Strategy, Implementation and
  Experiments Technical Report, Stanford
  University 1999
• 3 Graphical examples fromJean Luc Starks
  homepage http//www.cs.technion.ac.il/zdevir/
• and Zvi Devirs homepage
• http//www-stat.stanford.edu/jstarck/