The Story of Wavelets Theory and Engineering Applications

1
The Story of WaveletsTheory and Engineering
Applications
Jamboree 12 April 11, 2001

• In todays show
• Stationary discrete wavelet transform
• Two-dimensional wavelet transform
• 2D-DWT using MATLAB
• Implementation issues
• Image compressing using 2D-DWT

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Is DWT an LTI Transformation???

• Recall week 1 of your DSP class Linearity,
  memory, time invariance, causality, etc.
• Is DWT
• Linear?
• Time-invariant?

How can we find out?
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Stationary Wavelet Transfporm(SWT)

• DWT is not time invariant Not Good !
• What makes DWT time varying? Decimation (down
  sampling)
• DWT can be made time invariant, however, the
  transform must be redundant!!!
• Stationary wavelet transform
• ?-decimated DWT
• ?-decimated DWT???
• At any given level you can have two different
  DWT, due to choice in discarding even or odd
  indexed elements during subsampling
• At J levels, you can have N2J different DWTs.
  The particular DWT chosen can be denoted by ??1
  ?2 ?N, ?j1, if odd indexed elements are
  chosen, ?j0, if even indexed elements are chosen

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SWT

• SWT is defined as the average of all ?-decimated
  DWTs
• For 6 level DWT?64 DWTs
• For 10 level DWT ?1024 DWTs
• .
• An efficient algorithm

where
Hj
Hj-1
Gj
Gj-1
Note No subsampling is involved!!!.
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SWT

• Does it work? MATLAB DEMO

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Applications of SWT

• Denoising denoisingdenoising
• MATLAB demo noisy doppler noisy quadchirp
• Interval dependent thresholds

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1D-DWT?2D-DWT

• Recall fundamental concepts of 1D-DWT
• Provides time-scale (frequency) representation of
  non-stationary signals
• Based on multiresolution approximation (MRA)
• Approximate a function at various resolutions
  using a scaling function, ?(t)
• Keep track of details lost using wavelet
  functions, ?(t)
• Reconstruct the original signal by adding
  approximation and detail coeff.
• Implemented by using a series of lowpass and
  highpass filters
• Lowpass filters are associated with the scaling
  function and provide approximation
• Highpass filters are associated with the wavelet
  function and provide detail lost in approximating
  the signal

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2-D DWT

• How do we generalize these concepts to 2D?
• 2D functions ?? images f(x,y) ?? Im,n
  intensity function
• What does it mean to take 2D-DWT of an image? How
  do we interpret?
• How can we represent an image as a function?
• How do we define low frequency / high frequency
  in an image?
• How to we compute it?
• Why would we want to take 2D-DWT of an image
  anyway?
• Compression
• Denoising
• Feature extraction

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2D Scaling/Wavelet Functions

• We start by defining a two-dimensional scaling
  and wavelet functions
• If ?(t) is orthogonal to its own translates,
• is also orthogonal to its own translates.
  Then, if fo(x,y) is the projection of f(x,y) on
  the space Vo generated by s??(x,y)

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2D-DWT

• Just like in 1D we generated an approximation of
  the 2D function f(x,y). Now, how do we compute
  the detail lost in approximating this function?
• Unlike 1D case there will be three functions
  representing the details lost
• Details lost along the horizontal direction
• Details lost along the vertical direction
• Details lost along the diagonal direction
• 1D ? Two sets of coeff. a(k,n) d (k,n)
• 2D? Four sets of coefficients a(k,n), b(k, n),
  c(k, n) d(k,n)

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Four Faces of 2D-DWT

• One level of 2D DWT reconstruction

Approximation coefficients
Detail coefficients along the horizontal
direction
Detail coefficients along the vertical direction
Detail coefficients along the diagonal direction
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Implementation of 2D-DWT
COLUMNS
LL

ROWS
H
2 1
COLUMNS

LH
INPUT IMAGE
COLUMNS
ROWS

HL

G
2 1
ROWS
HH
COLUMNS
INPUT IMAGE
LLH
LH
LL
LH
LL
LHL
LHH
HH
HH
HL
HL
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Up and Down Up and Down
Downsample columns along the rows For each row,
keep the even indexed columns, discard the odd
indexed columns
2 1
Downsample columns along the rows For each
column, keep the even indexed rows, discard the
odd indexed rows
Upsample columns along the rows For each row,
insert zeros at between every other sample
(column)
Upsample rows along the columns For each column,
insert zeros at between every other sample (row)
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Implementing 2D-DWT
Decomposition
ROW i
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Reconstruction
LL
H
H
LH
G
ORIGINAL IMAGE
HL
H
G
G
HH
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2-D DWT ON MATLAB
Load Image (must be .mat file)
Choose wavelet type
Hit Analyze
Choose display options