RMF Based EZW Algorithm

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RMF Based EZW Algorithm

• School of Computer Science,
• University of Central Florida,
• VLSI and M-5 Research Group

June 1, 2000
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Organization

• The EZW algorithm
• Basic concept
• Introduction of the algorithm
• An example
• A brief introduction of RMF
• 1-D RMF example
• 2-D RMF example
• The RMF based EZW algorithm
• Basic Idea
• Band_max construction algorithm
• An example
• Experimental results
• Conclusions

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EZW basic concepts(1)

• E The EZW encoder is based on progressive
  encoding. Progressive encoding is also known as
  embedded encoding
• Z A data structure called zero-tree is used in
  EZW algorithm to encode the data
• W The EZW encoder is specially designed to use
  with wavelet transform. It was originally
  designed to operate on images (2-D signals)

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EZW basic concepts(2)
Lower octave has higher resolution and contains
higher frequency information
A Multi-resolution Analysis Example
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EZW basic concepts(3)

• The EZW algorithm is based on two observations
• Natural images in general have a low pass
  spectrum. When an image is wavelet transformed,
  the energy in the sub-bands decreases with the
  scale goes lower (low scale means high
  resolution), so the wavelet coefficient will, on
  average, be smaller in the lower levels than in
  the higher levels.
• Large wavelet coefficients are more important
  than small wavelet coefficients.

631 544 86 10 -7 29 55 -54 730 655 -13
30 -12 44 41 32 19 23 37 17 -4 13
-13 39 25 -49 32 -4 9 -23 -17 -35
32 -10 56 -22 -7 -25 40 -10 6 34 -44
4 13 -12 21 24 -12 -2 -8 -24 -42
9 -21 45 13 -3 -16 -15 31 -11 -10 -17
typical wavelet coefficients for a 88 block in
a real image
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EZW basic concepts(4)

• The observations give rise to the basic
  progressive coding idea
• We can set a threshold T, if the wavelet
  coefficient is larger than T, then encode it as
  1, otherwise we code it as 0.
• 1 will be reconstructed as T (or a number
  larger than T) and 0 will be reconstructed as
  0.
• We then decrease T to a lower value, repeat 1 and
  2. So we get finer and finer reconstructed data.
• The actual implementation of EZA algorithm should
  consider
• What should we do to the sign of the
  coefficients. (positive or negative) ? answer
  use POS and NEG
• Can we code the 0s more efficiently? --
  answer zero-tree
• How to decide the threshold T and how to
  reconstruct? answer see the algorithm

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EZW basic concepts(5)
coefficients that are in the same spatial
location consist of a quad-tree.
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EZW basic concepts(6)

• The definition of the zero-tree
• There are coefficients in different subbands
  that represent the same spatial location in the
  image and this spatial relation can be depicted
  by a quad tree except for the root node at top
  left corner representing the DC coeeficient which
  only has three children nodes.
• Zero-tree Hypothesis
• If a wavelet coefficient c at a coarse scale is
  insignificant with respect to a given threshold
  T, i.e. cltT then all wavelet coefficients of
  the same orientation at finer scales are also
  likely to be insignificant with respect to T.

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EZW the algorithm(1)

• First step The DWT of the entire 2-D image will
  be computed by FWT
• Second step Progressively EZW encodes the
  coefficients by decreasing the threshold
• Third step Arithmetic coding is used to entropy
  code the symbols

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EZW the algorithm(2)
What is inside the second step?
threshold initial_threshold do
dominant_pass(image) subordinate_pass(image)
threshold threshold/2 while
(threshold gt minimum_threshold)
The main loop ends when the threshold reaches a
minimum value, which could be specified to
control the encoding performance, a 0 minimum
value gives the lossless reconstruction of the
image
The initial threshold t0 is decided as
Here MAX() means the maximum coefficient value in
the image and y(x,y) denotes the coefficient.
With this threshold we enter the main coding loop
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EZW the algorithm(3)

• In the dominant_pass
• All the coefficients are scanned in a special
  order
• If the coefficient is a zero tree root, it will
  be encoded as ZTR. All its descendants dont need
  to be encoded they will be reconstructed as
  zero at this threshold level
• If the coefficient itself is insignificant but
  one of its descendants is significant, it is
  encoded as IZ (isolated zero).
• If the coefficient is significant then it is
  encoded as POS (positive) or NEG (negative)
  depends on its sign.

This encoding of the zero tree produces
significant compression because gray level images
resulting from natural sources typically result
in DWTs with many ZTR symbols. Each ZTR indicates
that no more bits are needed for encoding the
descendants of the corresponding coefficient
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EZW the algorithm(5)

• At the end of dominant_pass
• all the coefficients that are in absolute value
  larger than the current threshold are extracted
  and placed without their sign on the subordinate
  list and their positions in the image are filled
  with zeroes. This will prevent them from being
  coded again.
• In the subordinate_pass
• All the values in the subordinate list are
  refined. this gives rise to some juggling with
  uncertainty intervals and it outputs next most
  significant bit of all the coefficients in the
  subordinate list.

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EZW An example(1)
Wavelet coefficients for a 88 block
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EZW An example(2)
The initial threshold is 32 and the result from
the dominant_pass is shown in the figure
Data without any symbol is a node in the
zero-tree.
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EZW An example(3)
The result from the dominant_pass is output as
the following
POS, NEG, IZ, ZTR, POS, ZTR, ZTR, ZTR, ZTR, IZ,
ZTR, ZTR, IZ, IZ, IZ, IZ, IZ, POS, IZ, IZ
POS01, NEG11, ZTR00, IZ--10
The significant coefficients are put in a
subordinate list and are refined. A one-bit
symbol is output to the decoder.
Original data 63 34 49 47
Output symbol 1 0 1 0
Reconstructed data 56 40 56 40
For example, the output for 63 is sign 32
16 8 4 2 1 0 1 1
? ? ? ? If T.5T is less than data item
take the average of 2T and 1.5T. So 63 will be
reconstructed as the average of 48 and 64 which
is 56. If it is more, put a 0 in the code and
encode this as t.5T.25T. Thus, 34 is
reconstructed as 40.
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EZW An example(4)
After dominant_pass, the significant coefficients
will be replaced by or 0 Then the threshold is
divided by 2, so we have 16 as current threshold
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EZW An example(5)
The result from the second dominant_pass is
output as the following
IZ, ZTR, NEG, POS, IZ,IZ, IZ, IZ, IZ, IZ, IZ, IZ
The significant coefficients are put in the
subordinate list and all data in this list will
be refined as
Original data 63 34 49 47 31 23
Output symbol 1 0 0 1 1 0
Reconstructed data 60 36 52 44 28 20
For example, the output for 63 is sign 32
16 8 4 2 1 0 1 1
1 ? ? ? The computatin is now extended
with respect to the next significant bit. So 63
will be reconstructed as the average of 56 and 64
- 60!
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EZW An example(6)
The process is going on until threshold 1, the
final output as
D1 pnztpttttztttttttptt S1 1010 D2
ztnptttttttt S2 100110 D3 zzzzzppnppnttnnptptt
nttttttttptttptttttttttptttttttttttt S3
10011101111011011000 D4 zzzzzzztztznzzzzpttptppt
pnptntttttptpnpppptttttptptttpnp S4
11011111011001000001110110100010010101100 D5
zzzzztzzzzztpzzzttpttttnptppttptttnppnttttpnnpttpt
tppttt S5 10111100110100010111110101101100100000
000110110110011000111 D6 zzzttztttztttttnnttt He
re ppos, nneg, ziz, tztr
For example, the output for 63 is sign 32
16 8 4 2 1 0 1 1
1 1 1 1 So 63 will be reconstructed as
3216842163! Note, how progressive
transmission can be done.
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The Limitations of EZW algorithm
bad

• It is not possible to encode sub-images because
  the entire image must be transformed before the
  encoding can start.
• EZW algorithm is computational expensive

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1-D RMF Example
Complete wavelet coefficients for (x1, x2)
Complete wavelet coefficients for (x1, x2, x3, x4)
S1 D1 d1 d2 S2 D2 d3 d4
filter
input
Wavelet coefficients
The RMF computation maintains the spatial
locality coherence
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2-D RMF Example
The new HH band is the concatenation of the four
smaller HH bands The new HL band is the
concatenation of the four smaller LH bands
followed by the column-wise 1-D RMF The new LH
band is the concatenation of the four smaller HL
bands followed by the row-wise 1-D RMF The new LL
band is the concatenation of the four smaller LL
bands followed by the 2-D RMF
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RMF-EZW basic idea (1)

• The sub-image level coding is possible because
• we have the complete wavelet coefficients for
  each sub-image.
• For each of them, we apply the EZW algorithm.

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RMF-EZW basic idea (2)

• In the original EZW algorithm, each time when we
  check one coefficient is a ZTR or not, we compare
  all its descendants with the threshold This is
  very time-consuming!
• However, if we know the maximum absolute value
  among these descendants, we only need to compare
  this maximum absolute value with the threshold.
  If the maximum absolute value is smaller than the
  threshold, we can claim that there is no
  descendant of the current coefficient has an
  absolute value that is larger than the threshold
  and the current node is a zero-tree root
• Obviously, the maximum value for each sub-band
  should be maintained, we call this maximum value
  band_max value.

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RMF-EZW basic idea (3)

• There are two kinds of band_max value.
• The current_band_max value of each band is the
  maximum absolute value of wavelet coefficients
  for that band and its descendant sub-bands.
• The previous_band_max value is the value that is
  propagated to the next step of the RMF
  computation in order to efficiently generate the
  current_band_max values for the next step
  band_max construction.
• The band_max could be constructed as a
  by-product of the RMF computation

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RMF-EZW band_max construction (1)
The construction of the band_max in HH band

• current_band_max_HHoctave
• max(maxWT1WT4(previous_band_max_HHoctave,
  current_band_max_HHoctave-1
• previous_band_max_HHoctave
• current_band_max_HHoctave

Because in 2-D RMF computation, the new HH band
is just the concatenation of the four smaller HH
bands, the maximum in the new HH band is simply
the maximum among the the four maximums in the
smaller HH bands.
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RMF-EZW band_max construction (2)
The construction of the band_max in HL band

• current_band_max_HLoctave
• max(maxWT1WT4(previous_band_max_HLoctave),
  max(the first two rows in HL octave)
  current_band_max_HLoctave-1)
• In the new HL band, only the coefficients in the
  first two rows are new.
• previous_band_max_HLoctave
• max(maxWT1WT4(previous_band_max_HLoctave),
• max(second row in HL octave))
• The coefficients in the first row will be
  replaced by new data so they shouldnt be
  considered.

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RMF-EZW band_max construction (3)
The construction of the band_max in LH band

• current_band_max_LHoctave
• max(maxWT1WT4(previous_band_max_LHoctave)
  max(the first two columns in the LH octave),
  current_band_max_LHoctave-1)
• In the new LH band, only the coefficients in the
  first two columns are new.
• previous_band_max_LHoctave
• max(maxWT1WT4(previous_band_max_LHoctave,
• max(the second column in LH octave)
• The coefficients in the first column will be
  replaced by new data so they shouldnt be
  considered.

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RMF-EZW an example (1)
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RMF-EZW an example (2)
ZTR
When we encode 23, instead of comparing all its
descendants with T, we only need to compare
current_band_max_HH2 with T. We save 19
comparisons! Suppose we have a ZTR in the
highest octave of a 512512 image, In the
original EZW algorithm, we need 87380 comparisons
but now we need only ONE!
Current_band_max_HH16
Current_band_max_HH214
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RMF-EZW The experimental results (1)
The sub-image level coding
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RMF-EZW The experimental results (2)
Comparison of the Execution Time by EZW algorithm
and RMF-EZW algorithm
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RMF-EZW The experimental results (3)
Comparison of the Execution Time by EZW algorithm
and RMF-EZW algorithm
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RMF-EZW conclusions

• The RMF algorithm makes the sub-image level EZW
  coding possible.
• The band_max information, which is generated as
  the by-product of RMF computation, could improve
  the speed of the EZW coding by 11 (for 3232
  image) to 50 (for 512512 image) or even higher
  depends on the image size.