Wavelets and compression

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Wavelets and compression

• Dr Mike Spann

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Contents

• Scale and image compression
• Signal (image) approximation/prediction simple
  wavelet construction
• Statistical dependencies in wavelet coefficients
  why wavelet compression works
• State-of-the-art wavelet compression algorithms

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Image at different scales
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Correlation between features at different scales
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Wavelet construction a simplified approach

• Traditional approaches to wavelets have used a
  filterbank interpretation
• Fourier techniques required to get synthesis
  (reconstruction) filters from analysis filters
• Not easy to generalize

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Wavelet construction lifting

• 3 steps
• Split
• Predict (P step)
• Update (U step)

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Example the Haar wavelet

• S step
• Splits the signal into odd and even samples

even samples
odd samples
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Example the Haar wavelet

• P step
• Predict the odd samples from the even samples

For the Haar wavelet, the prediction for the odd
sample is the previous even sample
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Example the Haar wavelet
Detail signal
l
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Example the Haar wavelet

• U step
• Update the even samples to produce the next
  coarser scale approximation

The signal average is maintained
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Summary of the Haar wavelet decomposition
Can be computed in place
..
..
-1
-1
P step
1/2
U step
1/2
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Inverse Haar wavelet transform

• Simply run the forward Haar wavelet transform
  backwards!

Then merge even and odd samples
Merge
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General lifting stage of wavelet decomposition

U
P
Split
-
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Multi-level wavelet decomposition

• We can produce a multi-level decomposition by
  cascading lifting stages

lift
lift
lift
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General lifting stage of inverse wavelet
synthesis
-
P
Merge
U

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Multi-level inverse wavelet synthesis

• We can produce a multi-level inverse wavelet
  synthesis by cascading lifting stages

lift
...
lift
lift
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Advantages of the lifting implementation

• Inverse transform
• Inverse transform is trivial just run the code
  backwards
• No need for Fourier techniques
• Generality
• The design of the transform is performed without
  reference to particular forms for the predict and
  update operators
• Can even include non-linearities (for integer
  wavelets)

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Example 2 the linear spline wavelet

• A more sophisticated wavelet uses slightly more
  complex P and U operators
• Uses linear prediction to determine odd samples
  from even samples

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The linear spline wavelet

• P-step linear prediction

Linear prediction at odd samples
Detail signal (prediction error at odd samples)
Original signal
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The linear spline wavelet

• The prediction for the odd samples is based on
  the two even samples either side

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The linear spline wavelet

• The U step use current and previous detail
  signal sample

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The linear spline wavelet

• Preserves signal average and first-order moment
  (signal position)

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The linear spline wavelet

• Can still implement in place

-1/2
P step
-1/2
-1/2
-1/2
U step
1/4
1/4
1/4
1/4
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Summary of linear spline wavelet decomposition
Computing the inverse is trivial
The even and odd samples are then merged as before
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Wavelet decomposition applied to a 2D image
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Wavelet decomposition applied to a 2D image
approx
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Why is wavelet-based compression effective?

• Allows for intra-scale prediction (like many
  other compression methods) equivalently the
  wavelet transform is a decorrelating transform
  just like the DCT as used by JPEG
• Allows for inter-scale (coarse-fine scale)
  prediction

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Why is wavelet-based compression effective?
1 level Haar
Original
1 level linear spline
2 level Haar
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Why is wavelet-based compression effective?

• Wavelet coefficient histogram

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Why is wavelet-based compression effective?

• Coefficient entropies

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Why is wavelet-based compression effective?

• Wavelet coefficient dependencies

X
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Why is wavelet-based compression effective?

• Lets define sets S (small) and L (large) wavelet
  coefficients
• The following two probabilities describe
  interscale dependancies

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Why is wavelet-based compression effective?

• Without interscale dependancies

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Why is wavelet-based compression effective?

• Measured dependancies from Lena

0.886 0.529 0.781 0.219
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Why is wavelet-based compression effective?

• Intra-scale dependencies

X1
X
X8
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Why is wavelet-based compression effective?

• Measured dependancies from Lena

0.912 0.623 0.781 0.219
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Why is wavelet-based compression effective?

• Have to use a causal neighbourhood for spatial
  prediction

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Example image compression algorithms

• We will look at 3 state of the art algorithms
• Set partitioning in hierarchical sets (SPIHT)
• Significance linked connected components analysis
  (SLCCA)
• Embedded block coding with optimal truncation
  (EBCOT) which is the basis of JPEG2000

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The SPIHT algorithm

• Coefficients transmitted in partial order

Coeff. number
1 2 3 4 5 6 7 8
9 10 11 12 13 14.
msb
5 4 3 2 1 0
0
lsb
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The SPIHT algorithm

• 2 components to the algorithm 
• Sorting pass
• Sorting information is transmitted on the basis
  of the most significant bit-plane
• Refinement pass
• Bits in bit-planes lower than the most
  significant bit plane are transmitted

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The SPIHT algorithm
N msb of (max(abs(wavelet coefficient))) for
(bit-plane-counter)N downto 1 transmit
significance/insignificance wrt bit-plane
counter transmit refinement bits of all
coefficients that are already significant
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The SPIHT algorithm

• Insignificant coefficients (with respect to
  current bitplane counter) organised into
  zerotrees

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The SPIHT algorithm

• Groups of coefficients made into zerotrees by set
  paritioning

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The SPIHT algorithm

• SPIHT produces an embedded bitstream

bitstream
.110010101110010110001101011100010111011011101
101.
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The SLCCA algorithm
Bit-plane encode significant coefficients
Wavelet transform
Quantise coefficients
Cluster and transmit significance map
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The SLCCA algorithm

• The significance map is grouped into clusters

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The SLCCA algorithm

• Clusters grown out from a seed

Seed
Significant coeff
Insignificant coeff
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The SLCCA algorithm

• Significance link symbol

Significance link
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Image compression results

• Evaluation 
• Mean squared error
• Human visual-based metrics
• Subjective evaluation

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Image compression results

• Mean-squared error 

Usually expressed as peak-signal-to-noise (in dB)
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Image compression results
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Image compression results
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Image compression results
SPIHT 0.2 bits/pixel
JPEG 0.2 bits/pixel
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Image compression results
SPIHT
JPEG
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EBCOT, JPEG2000

• JPEG2000, based on embedded block coding and
  optimal truncation is the state-of-the-art
  compression standard
• Wavelet-based
• It addresses the key issue of scalability
• SPIHT is distortion scalable as we have already
  seen
• JPEG2000 introduces both resolution and spatial
  scalability also
• An excellent reference to JPEG2000 and
  compression in general is JPEG2000 by D.Taubman
  and M. Marcellin

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EBCOT, JPEG2000

• Resolution scalability is the ability to extract
  from the bitstream the sub-bands representing any
  resolution level

.110010101110010110001101011100010111011011101
101.
bitstream
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EBCOT, JPEG2000

• Spatial scalability is the ability to extract
  from the bitstream the sub-bands representing
  specific regions in the image
• Very useful if we want to selectively decompress
  certain regions of massive images

.110010101110010110001101011100010111011011101
101.
bitstream
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Introduction to EBCOT

• JPEG2000 is able to implement this general
  scalability by implementing the EBCOT paradigm
• In EBCOT, the unit of compression is the
  codeblock which is a partition of a wavelet
  sub-band
• Typically, following the wavelet transform,each
  sub-band is partitioned into small blocks
  (typically 32x32)

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Introduction to EBCOT

• Codeblocks partitions of wavelet sub-bands

codeblock
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Introduction to EBCOT

• A simple bit stream organisation could comprise
  concatenated code block bit streams

Length of next code-block stream
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Introduction to EBCOT

• This simple bit stream structure is resolution
  and spatially scalable but not distortion
  scalable
• Complete scalability is obtained by introducing
  quality layers
• Each code block bitstream is individually
  (optimally) truncated in each quality layer
• Loss of parent-child redundancy more than
  compensated by ability to individually optimise
  separate code block bitstreams

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Introduction to EBCOT

• Each code block bit stream partitioned into a set
  of quality layers

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EBCOT advantages

• Multiple scalability
• Distortion, spatial and resolution scalability
• Efficient compression
• This results from independent optimal truncation
  of each code block bit stream
• Local processing
• Independent processing of each code block allows
  for efficient parallel implementations as well as
  hardware implementations

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EBCOT advantages

• Error resilience
• Again this results from independent code block
  processing which limits the influence of errors

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Performance comparison

• A performance comparison with other wavelet-based
  coders is not straightforward as it would depend
  on the target bit rates which the bit streams
  were truncated for
• With SPIHT, we simply truncate the bit stream
  when the target bit rate has been reached
• However, we only have distortion scalability with
  SPIHT
• Even so, we still get favourable PSNR (dB)
  results when comparing EBCOT (JPEG200) with SPIHT

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Performance comparison

• We can understand this more fully by looking at
  graphs of distortion (D) against rate (R)
  (bitstream length)

D
R-D curve for continuously modulated quantisation
step size
Truncation points
R
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Performance comparison

• Truncating the bit stream to some arbitrary rate
  will yield sub-optimal performance

D
R
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Performance comparison
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Performance comparison

• Comparable PSNR (dB) results between EBCOT and
  SPIHT even though
• Results for EBCOT are for 5 quality layers (5
  optimal bit rates)
• Intermediate bit rates sub-optimal
• We have resolution, spatial, distortion
  scalability in EBCOT but only distortion
  scalability in SPIHT