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Multiscale Transforms: Wavelets, Ridgelets, Curvelets and Their Applications, Slides of Electronics engineering

Bharat Ratna Dr. B. R. Ambedkar UniversityElectronics engineering

An overview of multiscale transforms, focusing on wavelets, ridgelets, and curvelets. Topics covered include the fourier transform, time-frequency analysis, cauchy schwartz inequality, continuous wavelet transform, 2d wavelet transform, anisotropic frames, and their applications in data representation, sparsity, and edge detection. The document also discusses the heisenberg uncertainty principle and various transforms such as the mexican hat wavelet, undecimated wavelet transform, and curvelet transform.

Typology: Slides

2012/2013

Uploaded on 03/23/2013

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Multiscale transforms : wavelets, ridgelets,
curvelets, etc.
Outline :
•The Fourier transform
•Time-frequency analysis and the Heisenberg principle
•Cauchy Schwartz inequality
•The continuous wavelet transform
•2D wavelet transform
•Anisotropic frames : Ridgelets, curvelets, etc.
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Multiscale transforms : wavelets, ridgelets,

curvelets, etc.

Outline :

  • The Fourier transform
  • Time-frequency analysis and the Heisenberg principle
  • Cauchy Schwartz inequality
  • The continuous wavelet transform
  • 2D wavelet transform
  • Anisotropic frames : Ridgelets, curvelets, etc.

The Fourier transform (1)

  • Diagonal representation of shift invariant linear transforms.
  • Truncated Fourier series give very good approximations to smooth functions.
  • Limitations :
  • Provides poor representation of non stationary signals or image.
  • Provides poor representations of discontinuous objects (Gibbs effect)

What is good representation for data?

  • Computational harmonic analysis seeks representations of s signal as linear combinations of basis, frame, dictionary, element :
  • Analyze the signal through the statistical properties of the coefficients
  • The analyzing functions (frame elements) should extract features of interest.
  • Approximation theory wants to exploit the sparsity of the coefficients.

coefficients basis, frame

Seeking sparse and generic representations

  • Sparsity
  • Why do we need sparsity?
    • data compression
    • Feature extraction, detection
    • Image restoration

sorted index

few big

many small

Heisenberg uncertainty principle

  • Different tilings in time frequency space :
  • Localization in time and frequency requires a compromise

Windowed/Short term Fourier transform

  • Invertibility condition :
  • Reconstruction :
  • Decomposition :

with

( with a gaussian window w, this is the Gabor transform)

Continuous Wavelet Transform

  • Example : The mexican hat wavelet

2D Continuous Wavelet transform

  • either a genuine 2D wavelet function (e.g. mexican hat) or a separable wavelet i.e. tensor product of two 1D wavelets.
  • example :

Images obtained using the nearly isotropic undecimated wavelet transform obtained with the a trous algorithm.

Continuous Ridgelet Transform

Ridgelet function:
The function is constant along lines.Transverse to these ridges, it is a wavelet.
Ridgelet Transform (Candes, 1998):

Rf  a , b ,    a , b ,   x f x   dx

 a , b ,    x  a

1

2  x^1 cos(^ )^ ^ x^2 sin(^ )^ ^ b

a

The ridgelet coefficients of an object f are given by analysis
of the Radon transform via:



Rf ( a , b , )  Rf ( , t ) (

tb

a

 ) dt

  • SNR = 0.
Ridgelet Filtering (5sigma)

Local Ridgelet Transform

The ridgelet transform is optimal to find only lines of the size of the image.
To detect line segments, a partitioning must be introduced. The image is
decomposed into blocks, and the ridgelet transform is applied on each block.
Image
Partitioning
Ridgelet
transform