Wavelets in Physics

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Wavelets in Physics

• Paul Nerenberg

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Why Wavelets?

• Fourier analysis is inadequate it loses time
  localization for a given frequency.
• Fourier analysis is uneconomical still need
  infinite series even if function is almost flat.
• Fourier transform is unstable with respect to
  perturbation small changes in original function
  can completely modify the Fourier spectrum.
• Because theyre fun, no doubt.

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General Areas of Application

• Turbulence analysis looking at fluid flows.
• Astrophysics image analysis and processing.
• Systems of differential or polynomial equations
  (all physics) wavelets can make solving them
  easier (its a deal, its a steal).
• All types of things I probably cant even
  conceive of, but if it involves a signal, it can
  probably be analyzed with wavelets.

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Multi-dimensional Wavelets

• Lets take a brief look at the 2-D CWTbecause
  Im a physicist, this isnt mathematically
  rigorous, but its still cool.
• First, start out with s and s-hat for our 2-D
  space

Now, lets move on to the CWT
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CWT in 2-D

• Only need to satisfy admissibility condition

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Choices of Wavelets

• Isotropic wavelets good for pointwise analysis
  pick a wavelet that is invariant under rotation
  e.g. 2-D Mexican hat wavelet.
• Anisotropic wavelets detect directional
  features in an image/signal support is contained
  in a convex cone e.g. 2-D Morlet wavelet and
  Cauchy wavelets.

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CWT in 3-D

• Lets have some fun in 3-D

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Wavelets in Space-Time (dont ask)

• Necessary for analysis of time-dependent
  signalstranslations and dilations in space and
  time independently.
• The interesting possibility here relativistic
  wavelets, e.g. Poincaré wavelets, constrained
  within light cone, so useful for situations with
  EM fields.

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Modeling Maxwells Equations and Magnetotellurics
A Specific Example

• Looking at solar wind-induced EM currents in the
  Earths magnetic core.
• Core problem a set of differential equations
  with boundary conditions (a ?12...different
  media) that needs to be solved with an iterative
  process.

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Beyond our basic equations and boundary
conditions

• Make some approximations (physicists love
  them!)so now we get two simple equations

• Only need the transverse mode of the wave, so
  approximate again, and get a nice 2nd order eq.

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Looking at secondary component of the electric
field

• So electric field has two components we want
  the secondary one

• Begin to break down some omegasdomain
  decomposition starts here

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Yet another formulation

• Now we have

• Bring on the Lagrangians

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Formulating it with Lagrangiansand the final
iteration

• Almost theredefine a space too

• Finite elements enter the picture and we get our
  final iterative algorithm

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The Good News

• This result does convergeits actually worth
  something.

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So what does this all meanand why do we care at
all?

• We took a set of differential problems, performed
  a domain decomposition, and then using a hybrid
  finite element method, found an iterative
  algorithm for solving these problems.
• This reveals a core nature of wavelets they can
  simplify extremely complex models this is great
  for physics (or any area of applied science).