By: Hikmet Dursun

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Wavelets for Computer Graphics A PrimerEric J.
Stollnitz Tony D. DeRose David H. Salesin

• By Hikmet Dursun
• HIGH PERFORMANCE COMPUTING AND SIMULATIONS
  (Spring 06)

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Some wavelet applications

• approximation theory
• signal processing
• computer graphics
• image editing
• image compression
• automatic level-of-detail control for editing and
  rendering curves and surfaces
• surface reconstruction from contours
• fast methods for solving simulation problems in
  animation

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One-dimensional Haar wavelet transform

• Scaling functions

• Wavelet functions

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Haar Basis Functions
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In preparation for image compression

• The goal of compression is to express an initial
  set of data using some smaller set of data,
  either with or without loss of information.

gt
For orthonormal basis The square of the L2 error
in this approximation is
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Wavelets in two dimensions

• How to perform a wavelet decomposition of the
  pixel values in a two-dimensional image?
• The standard decomposition of an image
• First apply the one-dimensional wavelet transform
  to each row of pixel values.
• Treat these transformed rows as if they were
  themselves an image and apply the one-dimensional
  transform to each column.
• The nonstandard decomposition alternates between
  operations on rows.
• Apply one step of horizontal pairwise averaging
  and differencing on the pixel values in each row
  of the image.
• Apply vertical pairwise averaging and
  differencing to each column of the result.
• gt describe the scaling functions and wavelets
  that form a two-dimensional wavelet basis.

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Standard vs NonStandard Decomposition
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Image Decomposition
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Image compression

• Wavelet image compression using the L2 norm
• Compute coefficients c1, , cm representing an
  image in a normalized two-dimensional Haar basis.
• Sort the coefficients in order of decreasing
  magnitude to produce
• the sequence c(1), , c(m).
• 3. Find the smallest so that

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Generalization to other Lp norms
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Multiresolution analysis

• A nested set of vector spaces
• The resolution of functions in Vj increases.
• The basis functions for the space Vj are known as
  scaling functions.

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Multiresolution analysis

• define wavelet spaces
• Wj orthogonal complement of Vj in Vj1.
• Wj includes all the functions in Vj1 that are
  orthogonal to all those in Vj under some chosen
  inner product. The functions choosen as a basis
  for Wj are called wavelets.

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Multiresolution analysis

• Define

the subspaces Vj are nested gt scaling functions
are refinable. For all j 1, 2, There must
exist matrices of constants Pj and Qj such that
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Example
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Multiresolution analysis

• orthogonality condition
• gt

gtthere are many different wavelet bases for a
given wavelet spaceWj
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The filter bank analysis - decomposition

• a function in some approximation space Vj
• gt coefficients of this function in terms of some
  scaling function basis

• Decomposition To create a low-resolution
  version version Cj-1 of Cj with a smaller number
  of coefficients mj-1
• linear filtering and down-sampling on the mj
  entries of Cj

• detail matrix

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gtsynthesis/reconstruction In Haar Basis
matrices represent the averaging and differencing
operations
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The filter bank
The procedure for splitting Cj into a
low-resolution part Cj-1 and a detail part Dj-1
can be applied recursively to the low-resolution
version Cj-1. This recursive process is known as
a filter bank.
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Designing a multiresolution analysis

• Select the scaling functions for each j
  0, 1, .
• Select an inner product defined on the functions
  in V0, V1,
• Select a set of wavelets that span Wj for
  each j 0, 1, .

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B-spline scaling functions
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The scaling functions are chosen, here comes the
second step of designing a multiresolution
analysisSelection of an inner product rule
To complete multiresolution analysis Functions
for the spaces Wj that are orthogonal complements
to the spacesVj. Recall
Additional requirement each column of Qj is to
have a minimal number of consecutive nonzeros.
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Where to use B-spline wavelets

• Editing the sweep of the curve
• Multiresolution surfaces

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Thanks for listening