LCT-wavelet based algorithms for data compression

LCT-wavelet based algorithms for data compression
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International Journal of Wavelets, Multiresolution
and Information Processing
Vol. 11, No. 5 (2013) 1350032 (25 pages)
c World Scientific Publishing Company
DOI: 10.1142/S021969131350032X
LCT-WAVELET BASED ALGORITHMS
FOR DATA COMPRESSION
AMIR Z. AVERBUCH∗ , VALERY A. ZHELUDEV†
and MOSHE GUTTMANN‡
School of Computer Science
Tel Aviv University, Tel Aviv 69978, Israel
∗[email protected][email protected][email protected]
DAN D. KOSLOFF
Department of Earth and Planetary Sciences
Tel Aviv University, Tel Aviv 69978, Israel
[email protected]
Received 16 August 2012
Revised 31 May 2013
Accepted 1 June 2013
Published 29 August 2013
We present an algorithm that compresses two-dimensional data, which are piece-wise
smooth in one direction and have oscillatory events in the other direction. Fine texture,
seismic, hyper-spectral and fingerprints have this mixed structure. The transform part
of the compression process is an algorithm that combines the application of the wavelet
transform in one direction with the local cosine transform (LCT) in the other direction.
This is why it is called hybrid compression. The quantization and the entropy coding
parts in the compression process were taken from SPIHT codec but it can also be taken
from any multiresolution based codec such as EZW. To efficiently apply the SPIHT
codec to a mixed coefficients array, reordering of the LCT coefficients takes place. When
oscillating events are present in different directions as in fingerprints or when the image
comprises of a fine texture, a 2D LCT with coefficients reordering is applied. These algorithms outperform algorithms that are solely based on the the application of 2D wavelet
transforms to each direction with either SPIHT or EZW coding including JPEG2000
compression standard. The proposed algorithms retain fine oscillating events including
texture even at a low bitrate. Its compression capabilities are also demonstrated on multimedia images that have a fine texture. The wavelet part in the mixed transform of the
hybrid algorithm utilizes the Butterworth wavelet transforms library that outperforms
the 9/7 biorthogonal wavelet transform.
Keywords: Wavelet transform; local cosine transform (LCT); compression; hybrid compression; SPIHT; hyper-spectral; seismic; fingerprints.
AMS Subject Classification: 42C40, 68P30, 94A08
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1. Introduction
3D seismic data and hyper-spectral images are considered to be very large datasets.
Efficient algorithms are needed to compress these datasets in order to store them or
transmit them from planes, boats and satellites to base stations for further analysis.
The compression ratio should be as high as possible without damaging the analytical capabilities that take place after decompression. The compression of these
datasets should preserve fine details. Since wavelet transforms have a successful
history in achieving high compression ratios for still images, these techniques were
ported to handle seismic compression.14,19,32 However, the outcomes were less satisfactory due to the great variability of seismic data, the inherent noisy background
and their oscillatory nature. Moreover, some researchers argued that seismic signals are not wavelet-friendly2,23 due to the oscillatory patterns that are present in
seismic data.
These oscillatory patterns can be properly handled by cosine transforms. The
local cosine transform (LCT),2,23,33 which uses the lapped DCTa -IV transform
(LCT)13 with adaptive partition, was applied to compress 2D seismic sections. The
LCT catches well oscillatory patterns and sparsifies it as the wavelet transform sparsifies smooth data — see Refs. 6–8. Although these methods provided an excellent
energy compaction, the bottleneck was the absence of an efficient quantization —
coding scheme comparable with the schemes designed for the wavelet transforms
such as EZW,30 SPIHT28 and JPEG2000 standard.18
It was observed in Ref. 34 that the cosine transforms coefficients can be rearranged in a way where their structure becomes similar to the structure of wavelet
transform coefficients. This observation paved the way to combine the cosine transforms with the wavelet-oriented coding schemes (WOCS).17,20 The DCT-II–based
algorithm coupled with the SPIHT coder for compressing the segmented seismic cube is presented in Ref. 35. An embedded tarp filtering combined with the
classification of the reordered DCT-II coefficients (TCE) is applied to the image
compression in Ref. 29. Evaluation of the image compression quality is given
in Ref. 9.
Seismic data has a different structure in its vertical and horizontal directions.
While the horizontal structure is piece-wise smooth, the vertical traces comprise of
oscillatory patterns. To some extent, the same can be said on hyper-spectral data
where each spatial pixel is represented by a vector (also called multipixel) of the
intensities in all the available wavebands (typically ≈ 200). Compression of hyperspectral data should retain the spectral characteristic features of the multipixels.
In seismic compression, the oscillating events, which bear the information on the
subsurface layers, must be retained.
Seismic and hyper-spectral images have different structures in different directions. This triggered the idea to apply a different transform to a different direction.
a Discrete
Cosine Transform.
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One realization of this approach is presented in Ref. 16 where 3D hyper-spectral
data cubes were compressed by the application of a 2D wavelet transform in the
spatial direction and the Karhunen–Loéve transform in the spectral direction. The
transforms coefficients were coded by JPEG2000. Another scheme for a 3D hyperspectral data compression was presented in Ref. 25 where different types of wavelet
transforms were applied to the spatial and spectral directions followed by JPEG2000
coding.
As was mentioned above, compression of seismic and hyper-spectral data pursues two goals:
(1) Transmission of the data from the capturing devices to the analysis station
(so-called onboard compression).
(2) Storage of these huge data volumes (on-the-ground compression).
In this paper, we focus on the compression of 2D rectangles, which comprise one
spatial direction and one spectral (or in-depth) direction. Such data rectangles result
from data acquisition schemes in hyper-spectral imaging and in seismic exploration.
Thus, the proposed algorithms can be directly used for the onboard compression.
For on-the-ground compression, 3D methods produce better results but our scheme
can be easily extended to 3D setting.
We exploit the fact that wavelets provide a sparse representation for such a
mixed-structure data in the horizontal direction while the LCT handles well oscillatory patterns. Thus, we propose to apply the wavelet transform to the horizontal
direction and the LCT to the vertical direction.
The proposed transform produces a mix of LCT and wavelet coefficients. In
order to be processed properly by WOCS, the mixed coefficients array should be
reshaped to mimic the structure of a 2D array of wavelet coefficients. To achieve it,
we supply the LCT coefficients with a spatial meaning by partitioning the data in
the vertical direction. The joint coefficients array is organized in a tree-like form by
reordering the LCT coefficients. Therefore, this scheme is called an hybrid compression algorithm (HCA). If there are oscillations in both directions, then the LCT
is applied to both directions followed by reordering the transform coefficients and
then the WOCS is applied. This method is abbreviated by RLCTA.
We compare between the performances of HCA and RLCTA where both use the
SPIHT codec, and the performance of JPEG2000. In addition, we also compare with
the performance of 2D wavelet-based algorithms that use SPIHT codec. In most
of the experiments, which operate on hyper-spectral and seismic data, HCA and
RLCTA that uses SPIHT outperforms JPEG2000 and wavelet-based algorithms
that use SPIHT. In addition, our experiments demonstrated that the biorthogonal
wavelet transforms, which are based on infinite impulse response (IIR) Butterworth
filters,3,4 provide better performance in comparison with the transforms that are
based on the popular 9/7 transforms,1 which are also utilized in the JPEG2000
standard. This is true for the 2D wavelet transforms as well as for the hybrid
transforms.
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Compression of fingerprints images is another successful application of HCA
and RLCTA. The FBI uses the compression standard11 that is based on the 9/7
2D wavelet transform. This transform, which uses finite impulse response (FIR)
filters of lengths 7 and 9, is included also in JPEG2000 compression standard. The
HCA and RLCTA produce higher PSNR results and retain better the structure of
the fingerprints in comparison to the outputs from the application of 2D wavelet
transforms and JPEG2000 to fingerpints.
The HCA performance on multimedia type images, which are relatively smooth
such as “Lena” and “Fabrics”, was close (sometimes even inferior) to the 2D waveletbased algorithms performance. On the other hand, HCA and RLCTA outperform
the 2D wavelet based algorithms on images that have fine texture or oscillating
patterns such as “Barbara”, “Elaine”, “Wood fence” etc. HCA and RLCTA retain
the texture and the oscillations even in a very low bit rate.
To summarize, the contributions of the paper are threefold:
(1) We exploit the fact that wavelets provide a sparse representation for such
a mixed-structure data in the horizontal direction while the LCT handles
well oscillatory patterns. This provides a new way to enhance wavelet based
compression.
(2) We demonstrate that the biorthogonal wavelet transforms, which are based
on infinite impulse response (IIR) Butterworth filters, provide better performance in comparison with the transforms that are based on the popular 9/7
transforms, which are also utilized in the JPEG2000 standard.
(3) The proposed methodology outperforms the existing state-of-the art compression schemes including JPEG2000 applied to fine texture, seismic, hyperspectral and fingerprints images that are characterized by having sharp
oscillations. The proposed method does not have special advantages when it
is applied to regular rich multimedia images.
The paper is organized as follows. Section 2 presents some known facts about
LCT, wavelet transforms and SPIHT coding. Section 3 describes the hybrid algorithm. Section 4 presents the experimental compression results of seismic sections,
hyper-spectral data, fingerprints and multimedia images.
2. Preliminaries
2.1. Local cosine transform (LCT)
The discrete cosine transform of type IV (DCT-IV)26 of the signal f and its inverse
−1
N −1 ˆIV
π
1
1
2
are defined as fˆIV (k) = N
(k) ×
n=0 fn cos[ N (k + 2 )(n + 2 )], fn = N
k=0 f
π
π
(k + 12 )(n + 12 )]. The basis signals {cos[ N
(k + 12 )(n + 12 )]}, n = 0, . . . , N − 1,
cos[ N
of DCT-IV are even on the left side with respect to − 21 and odd on the right side
with respect to N − 12 . Therefore, direct application of the DCT-IV to a partitioned
data leads to severe boundary discrepancies. However, this transform serves as a
base for the so called local cosine bases,13 which are the windowed lapped DCT-IV
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transforms. These bases were successfully exploited in Refs. 2, 5, 23, 22, 33 for
image compression in general and seismic data compression in particular.
−1
Assume we have a signal S {sk }N
k=0 and some partition P of the interval
0 : N −1. The idea behind the lapped DCT-IV transform, also called the local cosine
transform (LCT) or the local Fourier basis of a signal, is to apply overlapped bells to
adjacent sub-intervals. Then, the overlapped parts are folded back to sub-intervals
across the endpoints of the sub-intervals and each sub-interval is expanded by the
application of the DCT-IV transform. In the reconstruction phase, the transform
coefficients are unfolded. For details, see Refs. 2, 5, 13, 21. It is important that
LCT produces no discrepancies between adjacent intervals. The choice of a bell
is discussed in Refs. 5, 22. The bell we chose is given in Sec. 4. There are fast
algorithms to implement the LCT.
To achieve a sparse representation of the oscillatory kernel a multiscale adaptive
local cosine transform rather than a one-level local cosine transform is used as
described in Ref. 6. Multiscale adaptive use of the local cosine transform enables
us to achieve impressive compact coding description of oscillatory data. It can be
used in sparsification of operator kernels.
2.2. Wavelet transforms
Currently, wavelet transforms constitute a recognized tool for image processing
applications. In particular, they have gained a proven success in image compression.
We summarize here some other well known facts that are needed later.
The multiscale wavelet transform of a signal is implemented via iterated multirate filtering by a pair of L (low-pass) and H (high-pass) filters. Once implemented,
the wavelet transform of a signal produces a partition of the Nyquist frequency
band in a logarithmic mode. The diagram of a three-scale wavelet transform and
the layout of the transform coefficients are displayed in Fig. 1.
On the other hand, the wavelet coefficients have a spatial meaning. A transform
coefficient from a certain decomposition scale is a correlation coefficient of the signal
with a translation of the waveform (wavelet) related to this scale. The wavelets
coefficients from the first decomposition scale are translated by two-sample steps,
the steps for the second scale are four samples and so on.
The wavelets ψ j (k), j = 1, . . . , 4, Φ4 (k) and their frequency responses are displayed in Fig. 2.
2.2.1. Tree structure of the wavelet transform coefficients
Assume that the number of scales is J = 3. The high-frequency coefficient h3m from
3
, which is centered around the
the coarsest scale is related to the waveform ψ8m+4
2
sample 8m + 4. The high-frequency coefficients h2m and h22m+1 from the second
2
2
and ψ8m+6
, which are centered around the
scale are related to the waveforms ψ8m+2
samples 8m + 2 and 8m + 6, respectively. These waveforms occupy, approximately,
the same area as the waveform ψ 3 (k)8m+4 . In that sense, we claim that the finer
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Fig. 1.
Three-scale wavelet transform and the layout of the transform coefficients (color online).
1
ψ
ψ2
ψ3
4
ψ
4
φ
Fig. 2. Left: The high-frequency wavelets ψj (k), j = 1, . . . , 4, and the low-frequency wavelet
Φ4 (k). Right: Their frequency responses (color online).
scale coefficients h22m and h22m+1 are the “offsprings” of the coarsest scale coefficient
h3m . In turn, their “offsprings” are the fine scale coefficients h14m , h14m+1 , h14m+2 and
h12m+3 . Thus, the coarsest scale coefficient h3m is the root of the tree
h3m h14m
h24m+1
.
h14m+2
h22m+1 2
h4m+3
h22m (2.1)
The wavelet transform of a two-dimensional array T = {tn,m } of size N × M
is implemented as a tensor product of 1D wavelet transform. First, a pair of filters
L and H is applied to the columns of T and the results are downsampled. The
coefficients arrays W1L and W1H of size N/2 × M are produced. Then, the filters
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L and H are applied to the rows of W1L and W1H . This filtering is followed by
downsampling which results in four sub-arrays coefficients W1LL , W1LH , W1HL and
W1HH of size N/2 × M/2. The 2D Nyquist frequency domain is split accordingly.
Then, the above procedure is applied to the coefficient array W1LL to decompose
it into the sub-arrays W2LL , W2LH , W2HL and W2HH of size N/4 × M/4. Then,
this procedure is iterated using W2LL instead of W1LL and so on. The layout of the
transform coefficients, which corresponds to the Nyquist frequency partition, for
the three-scale wavelet transform, is displayed in Fig. 3.
J
∈ WJHL and
Similarly to the 1D case, each coefficient lhJn,m ∈ WJLH , hln,m
J
J
hhn,m ∈ WHH from the coarse scale has four “offsprings” from the finer scale
J − 1.
lhJ−1
2n,2m
J−1
lh
2n,2m+1
,
lhJn,m J−1
lh2n+1,2m
lhJ−1
2n+1,2m+1
J−1
hl2n,2m
J−1
J hl2n,2m+1
hln,m
,
J−1
hl2n+1,2m
J−1
hl2n+1,2m+1
hhJ−1
2n,2m ,
J−1
hh
2n,2m+1
hhJn,m .
hhJ−1
2n+1,2m
hhJ−1
2n+1,2m+1
(2.2)
Similar relations exist between the coefficients from scales J − 1 and J − 2, and so
on. These relations for J = 3 are illustrated in Fig. 4.
J
∈ WJHL or hhJn,m ∈ WJHH from the
Thus, each coefficient lhJn,m ∈ WJLH , hln,m
coarse scale can be treate as the root of a quad-tree of coefficients. This relationship
between wavelet coefficients in different scales is exploited in the embedded zerotree
wavelet (EZW) codec.30 This codec takes advantage of the self-similarity between
wavelet coefficients across the decomposed scales and their decay toward high frequency scales. One of the most efficient algorithms, which is based on the zerotree
concept, is the SPIHT algorithm.28 This algorithm combines adaptive quantization
of wavelet coefficients with coding. The produced bitstream is further compressed
(LL) LH
LH
(LL) HL
(LL) HH
HL
Fig. 3.
HH
Three-scale layout coefficients of a 2D wavelet transform.
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Fig. 4.
Relationship among wavelet coefficients from different scales.
losslessly by the application of an adaptive arithmetic coding.27 The SPIHT coding
procedure is fast and its decoding procedure is even faster.
3. The Hybrid Algorithm
We propose to apply the wavelet and the LCT transforms in the horizontal and
vertical directions, respectively. The mixed wavelet and LCT transforms coefficients
are encoded by the SPIHT algorithm or by JPEG2000 coder. To fit these codecs,
the array of the mixed transform coefficients is organized in a wavelet-like way. It
requires to reorder the LCT coefficients.
The array of DCT coefficients of a signal of length 2k produces a natural logarithmic split of the frequency band once being separated into k + 1 blocks:
(c0 | c1 | c2 c3 |c4 c5 c6 c7 |c8 c9 c10 c11 c12 c13 c14 c15 | . . .)T .
(3.1)
The partition in Eq. (3.1) appears automatically when the coefficients indices are
given in a binary representation:
(c0 |c1 |c10 c11 |c100 c101 c110 c111 |c1000 c1001 c1010 c1011 c1100 c1101 c1110 c1111 | . . .)T .
(3.2)
Thus, the array is partitioned according to the number of bits in the coefficients
indices. We call this a bit-wise partitioning.
Assume we are given an N × M data array T, where N = 2k Q, M = 2J R. We
define the partition P of the interval I [0, 1, . . . , N − 1] by splitting it into Q
i
k
subintervals I = Q
i=1 I of length 2 each. We apply the P -based LCT transform
to each column of T. Thus, the array T is transformed into the array C of LCT
coefficients.
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For each column, we get the array c of N LCT coefficients, which consists
Q
k
−1
of Q blocks c = i=1 ci , where ci {cin }2n=0
. Each of them can be a bit-wise
k
i
i
partitioning c = β=0 bβ as in Eq. (3.2) where bi0 = ci0 , bi1 = ci1 and biβ is the set
of coefficients cin , whose indices can be represented by β bits.
In order to obtain a wavelet-like structure of the array c of N LCT coefficients,
we rearrange it in a bit-wise mode:
c→b
k
bβ ,
Q
bβ biβ .
(3.3)
i=1
β=0
Q T
T
1 2
1 1 2 2
Thus, we get b0 (c10 , c20 , . . . , cQ
0 ) , b1 (c1 , c1 , . . . , c1 ) , b2 (c2 , c3 ; c2 , c3 ; . . . ;
Q Q T
c2 , c3 ) , and so on. This rearrangement is illustrated in Fig. 5. The structure of
the array b is similar to the structure of the coefficients array w of the wavelet
transform, where the signal was decomposed into the k − 1 scale. The similarity
relations are
k−1
k−1
k−2
, b1 ∼ wH
, b2 ∼ wH
, . . . , bk−1 ∼ w1H .
b0 ∼ wL
(3.4)
The ancestor-descendant relationships in the array b are similar to that in w.
We perform this LCT coefficients reordering for all the columns of the array C.
Thus, we get
C→B=
k
Bβ ,
Bβ β=0
Q
Biβ .
(3.5)
i=1
Then, each row in the array B is decomposed into scale J by the application of
the wavelet transform. This produces the hybrid LCT-wavelet coefficients array
denoted as CW. The structure of CW is similar to the the structure of a 2D
wavelet coefficients array, where the transform on the columns decomposed to scale
k − 1, while the transform on the rows was decomposed till scale J. A three scales
decomposition structure of the CW array is illustrated in Fig. 6.
C1
C
2
C
B
+
+
+
+
+
+
+
+
+
*
+
*
+
+
*
*
+
+
+
+
*
*
*
*
*
*
*
*
*
*
*
*
Fig. 5.
B0
B1
B2
B3
Reordering scheme of the LCT coefficients.
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WL3 WH3 WH2
WH1
B0
B1
B2
B3
Fig. 6.
Three-scales of the LCT-wavelet coefficients layout.
The array CW is the input to SPIHT or JPEG2000 coders that implement the
quantization followed by the implementation of the entropy coding.
If the data arrays have oscillating structures in both vertical and horizontal
directions then the LCT application to both directions followed by reordering of
the transform coefficients in both directions is recommended. We call this transform
the reordered LCT denoted by (RLCTA).
4. Experimental Results
The hybrid and the RLCTA algorithms were applied to compress different data
types: seismic, hyper-spectral, fingerprints and multimedia images.
We compared between the performances of these algorithms where the quantization and entropy coding was borrowed from the SPIHT codec. They were compared
with the performance of JPEG2000 compression standard. For the latter, we used
the MATLAB script imwrite. We also compared between the compression results
from HCA and RLCTA algorithms and the results from the application of the 2D
wavelet transform using the same SPIHT codec in all the transforms. An additional
experimental goal was to compare between the performance of the Butterworth
wavelet transforms3,4 and that of the 9/7 biorthogonal wavelet transform.12,1
The choice of a bell in the LCT construction (see Sec. 2.1) has some effect on
the performance of the LCT and, consequently, on the hybrid algorithm. A library
of bells was introduced in Ref. 22. A comparative study of their effects on the
performance of an LCT-based image compression algorithm was given in Ref. 24.
However, such a comparison is beyond the scope of this paper and we did not check
the effects of different bells on the performance of the algorithm. Our goal was to
demonstrate the capabilities of the new method. For this purpose, it was sufficient
to implement LCT with the simple “sine” bell b(x) = sin π2 (x + 1/2).5
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All the transforms in the paper have the following notation:
Jpeg2000 standard is denoted by JP2k.
2D wavelet transforms using 9/7 filters is denoted by W9/7.
2D wavelet transforms of Butterworth wavelet transform with M vanishing moments is denoted by WButt/M.
Hybrid transforms are composed from the 1D wavelet transforms in the horizontal direction and from the LCT in the vertical direction, which partitions the vertical
section into Q horizontal rectangles. The transform coefficients were reordered to fit
SPIHT encoding as was explained in Sec. 3. H9/7/Q denotes the 9/7 wavelet transform in the horizontal direction; HButt/M/Q denotes the Butterworth wavelet
transform with M vanishing moments.
2D RLCTA: RLCTA/Q/P denotes the 2D LCT transform, which partitions
the image into Q × P rectangles where Q is the partition in one direction and P
is the partition in the other direction. The transform coefficients were reordered to
fit the SPIHT encoding.
All the images, which were used in the experiments, are 8-bit grayscale
images. Assume that the image X, whose pixels array is x {xk }N
k=1 , was subjected to lossy compression by some method M and, then, reconstructed to the
image X̃, whose pixels array is x̃ {x̃k }N
k=1 . We compare between the quality
of different compression methods by the peak signal to noise ratio (PSNR) in
decibels


N 2552
 N

(4.1)
PSNR 10 log10   dB,
(xk − x̃k )2
k=1
in addition to a visual inspection. The SPIHT codec was enhanced by incorporating
into it an adaptive arithmetic coder,27 which is a popular entropy coder. The bold
numbers in the following tables indicate the best achieved results.
4.1. Seismic sections compression
Compression of seismic data, which is used for the reconstruction of subsurface
layers structure, should retain all the seismic events in the traces.
4.1.1. Stacked CMP data section
In the experiments, we used a stacked common mid point (CMP) data section. We
display the original section of size 512 × 512 and a fragment of size 200 × 200 from
it in Fig. 7.
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Fig. 7. The original stacked CMP section. Left: The whole section. Right: A fragment from the
left image. The X-axis corresponds to the horizontal direction along the earth surface while the
Y -axis corresponds to the vertical (depth) direction.
Table 1.
The PSNR values after decompression of the stacked CMP seismic section.
bit/pixel
JP2k
W9/7
WButt/10
H9/7/8
1/8
1/4
1/2
1
2
31.03
33.12
35.91
40.61
48.66
30.86
32.79
35.39
39.83
46.91
31.01
32.99
35.84
40.42
48.82
31.19
33.39
36.68
41.8
51.26
HButt/10/8
31.39
33.79
37
42.26
51.6
RLCTA/8/8
31.46
33.83
37
42.31
51.63
Five types of transforms were used: W9/7, WButt/10, H9/7/8, HButt/
10/8 and RLCTA/8/8, where the transform coefficients were coded by SPIHT.
The results were compared with the results from the application of JP2k. The
achieved PSNR values are presented in Table 1.
We observe that the PSNR values after the application of the hybrid transforms
and the RLCTA are significantly higher than those after the application of the
2D wavelet transforms whether JP2k or SPIHT is used. The best PSNR values
appear after the application of RLCTA/8/8, which only slightly exceeds the values
produced by HButt/10/8. The reason for that is in the presence of oscillations
in the vertical and in the horizontal directions. Note that the 2D wavelet and
hybrid transforms, which used the Butterworth wavelets, produced higher PSNR
in comparison to the 9/7 based transform. The hybrid transforms and the RLCTA
retain seismic events much better than the 2D wavelet transforms. It is illustrated
in Fig. 8.
Figure 8 displays fragments from the reconstruction of the CMP section in Fig. 7
(right) after the application of the JP2k compression algorithm where the compression ratio is 1/4 bit per pixel (left image in Fig. 8) and from the application of
the RLCTA/8/8–SPIHT compression (right image in Fig. 8). The RLCTA/8/8–
SPIHT compression retains the main structure of the data unlike the outcome from
the application of the JP2k algorithm.
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Fig. 8. A fragment from the reconstructed CMP section in Fig. 7 (right). Left: Output from the
application of the JP2k algorithm. Right: The output from the application of the RLCTA/8/8–
SPIHT algorithm. The transforms coefficients were encoded by 1/4 bit per pixel. The X-axis
corresponds to the horizontal direction along the earth surface while the Y -axis corresponds to
the vertical (depth) direction.
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120
110
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100
90
90
80
150
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250
300
350
400
450
500
80
150
200
250
300
350
400
450
500
Fig. 9. The differences between the original line #110 from the vertical trace in the CMP section
(Fig. 7 (right)) and the reconstructed line after the applications of different compression algorithms. Dotted line: Original. Solid line: Restored. Left: Restored after the application of JP2k
algorithm. Right: Restored after the application of the HButt/10/8–SPIHT algorithm. Both
were compressed to 1/2 bit per pixel (color online).
Figure 9 displays line #110 from the vertical trace from the original section versus the same line from the reconstructed section from 1/2 bit per
pixel compression by the applications of JP2k and HButt/10/8–SPIHT based
algorithms.
We observe that the reconstructed trace in the right image in Fig. 9 is very close
to the original trace. This is not the case for the trace in the left image.
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Fig. 10. The original marine shot gather (MSG) section. Left: The whole section. Right: A
fragment from the section on the left. The X-axis corresponds to the horizontal direction along
the earth surface while the Y -axis corresponds to the vertical (depth) direction.
4.1.2. Marine shot gather data section
For another seismic experiments, we used a marine shot gather (MSG) data section
of size 512×512. As before, we compared between the performance of the 2D wavelet
transforms and the hybrid transforms that use different wavelets. Each data pixel
is quantized to 8 bits. We display the original section of size 512 × 512 and fragment
of size 200 × 200 from it in Fig. 10.
Five types of transforms were applied: W9/7, WButt/10, H9/7/8, HButt/
10/8 and RLCTA/8/8, where the transform coefficients were coded by SPIHT.
The results were compared with the results from the application of JP2k. The
achieved PSNR values are presented in Table 2.
The compression results from the applications of W9/7, WButt/10 and JP2k
wavelet transforms are close to each other where JP2k has a small advantage over
the other two. The HButt/10/8 transform outperforms H9/7/10 and JP2k.
Figure 11 displays the fragments of the reconstructed MSG section after the application of JP2k (left image) and Hybrid HButt/10/10 with SPIHT coding (right
image). The compression ratio for both is 1 bit per pixel. The HButt/10/8 transform retains the structure of the data much better than JP2k.
Figure 12 displays the line #270 from the vertical trace from the original section
taken from Fig. 10 (right) versus the same line taken from the reconstructed section
after the application of JP2k and HButt/10/8+SPIHT algorithms.
Table 2.
bit/pixel
1/4
1/2
1
2
The PSNR values after decompression of the MSG seismic section (Fig. 10 (right)).
JP2k
32.82
34.65
38
44
W9/7
WButt/10
H9/7/8
HButt/10/8
RLCTA/8/8
32.76
34.42
37.69
43.28
32.69
34.31
37.64
43.25
33.07
35.11
39.10
45.55
33.22
35.23
39.17
45.67
33.11
35.13
39.16
45.62
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Fig. 11. A fragment from the reconstructed MSG section (Fig. 10 (right)). Left: Reconstruction
from the application of JP2k, PSNR = 34.63 dB. Right: The reconstruction from the application of
the Hybrid HButt/10/8+SPIHT, PSNR = 35.23 dB. The compression ratio is 1 bit per pixel. The
X-axis corresponds to the horizontal direction along the earth surface while the Y -axis corresponds
to the vertical (depth) direction.
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300
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400
450
500
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150
200
250
300
350
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500
Fig. 12. Line #270 from the vertical trace in the MSG section. Dotted lines: Original. Solid line:
Restored. Left: After the application the JP2k algorithm. Right: The same after the application
of the HButt/10/8+SPIHT algorithm. Both are compressed to 1 bit per pixel (color online).
We see that the trace, which was reconstructed from HButt/10/8+SPIHT
encoding/decoding algorithm, is much closer to the original trace in comparison to
the trace reconstructed after the application of the JP2k algorithm that missed
many details.
Conclusion: For seismic data, the Hybrid compression algorithm significantly outperforms compression algorithms that are based on the 2D wavelet transforms
including JPEG2000 coding scheme.
4.2. Compression of hyper-spectral images
The hybrid compression was applied to hyper-spectral data cubes. These cubes
were captured from a plane that took simultaneously pictures in many (≈ 200)
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wavebands from the ground surface. This type of camera can also be placed in a
satellite. We used the SPECIM camera.31 In this camera, the spectrum of intensities for all the wavebands is assigned at once to each spatial pixel that is called
multipixel. The proposed compression methodology fits this capturing type and no
buffering is needed in order to compress the hyper-spectral data. An hyper-spectral
image is treated as a 3D cube where X, Y are the spatial axes and Z is the wavelength axis. The camera captures parallel lines of multipixels that are placed in
the X/Z planes. The structure of the Z direction is very different from that of the
X direction. This justifies the application of the hybrid transform. This processing
enables a real-time transmission of hyper-spectral data without the need to buffer
the data before the application of the hybrid compression algorithm. The results are
compared with the results from the application of JPEG2000 compression standard.
When an hyper-spectral data is compressed, it is important to preserve the spectral characteristic features of the multipixels. We applied the compression algorithm
to different hyper-spectral data cubes scenarios. Here, we present the results from
the application of the compression algorithm to an urban scenery that has many
details and thus contains many oscillations.
Figure 13 displays an urban ground scene at one of the X/Z multipixel planes.
This image was captured by the Specim31 camera placed on a airplane flying 10,000
feet above sea level. The resolution is 1.5 meter/pixel, 3000 × 300 pixels per waveband with 180 wavebands.
Five transforms types were applied: W9/7, WButt/4, H9/7/8, HButt/4/8
and RLCTA/8/8, where the transform coefficients were coded by the SPIHT
mechanism. The results were compared with the results from the application of
Fig. 13. An hyper-spectral image. Left: One horizontal plane from the data cube. Right: A
multipixel plane. Collection of all multipixels that belong to one line from the horizontal plane.
The X-axis corresponds to the direction along a horizontal line of pixels in the left image while
the Y -axis indicates the wavebands.
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Table 3. PSNR values for the hyper-spectral compression of X/Z plane #200 (plane
200 from the 3000 planes of the whole hyper-spectral cube) from the urban scene
Fig. 13 (right).
bit/pixel
1/4
1/2
1
2
0
20
JP2k
W9/7
36.79
40.9
46
51.65
40
60
35.39
39
43.62
49.97
80
100
WButt/4
H9/7/8
HButt/4/8
RLCTA/8/8
35.32
38.83
43.58
50.01
37.73
42.03
46.64
52.1
38.18
42.25
46.51
52.24
38.03
41.95
46.21
51.69
120
0
20
40
60
80
100
120
Fig. 14. One multipixel from X/Z plane #200. Dotted lines: Original. Solid line: Restored. Left:
After the application of JP2k. Right: After the application of HButt/10/8+SPIHT encoding/decoding. The compression ratio for both is 0.25 bits per pixel (color online).
JP2k. Table 3 presents the comparison results on the X/Z plane #200. The performances are given in PSNR values.
We observe that the hybrid transforms with SPIHT produce PSNR values, which
are higher than the results from the applications of the JP2k algorithm, which,
in turn, outperforms the wavelet transforms. Figure 14 displays a multipixel from
plane #200 from 3000 planes in Fig. 13 (left), which was reconstructed after the
application of JP2k (left image) and HButt/4/8 (right image) transforms followed by SPIHT encoding/decoding where the compression ratio is 0.25 bit per
pixel.
Almost all the “events” in the original data are present in the reconstructed
signal after the application of the hybrid transform with even high compression
rate. This is not the case following the application of the JP2k algorithm.
4.3. Fingerprints
The 2D wavelet, hybrid and LCT transforms followed by SPIHT coding were applied
to compress a fingerprint image of size 512 × 512, shown in Fig. 15. It was downloaded from C. Brislawn’s web page http://www.c3.lanl.gov/∼brislawn/index.html.
It has an oscillating structure in each direction. Each pixel has 8 bits.
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50
50
100
100
150
200
150
250
300
200
350
400
250
450
500
50
Fig. 15.
image.
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300
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450
500
300
260
280
300
320
340
360
380
400
420
440
460
The original fingerprint image. Left: The whole image. Right: A fragment from the left
Table 4.
PSNR values for fingerprint compression.
bit/pixel
JP2k
W9/7
WButt/10
H9/7/16
HButt/10/16
RLCTA/16/16
1/8
1/4
1/2
1
23.54
26.23
29.84
33.94
23.05
25.51
29.33
33.3
24.03
26.42
29.91
33.83
23.87
26.53
30.06
33.97
24.38
26.77
29.99
34.27
24.55
26.99
30.23
34.83
The performances of the SPIHT based algorithms were compared with the performance of JPEG2000 compression standard. The LCT transforms partition the
image into 16 horizontal and vertical rectangles. The achieved PSNR values are
given in Table 4.
We can see from Table 4 that RLCTA/16/16+SPIHT outperforms all the
other algorithms in all bitrates. This is due to better handling of the oscillatory
events in comparison to wavelet transforms based algorithms.
Figure 16 displays fragments from the reconstructed fingerprint image after the
applications of JP2k (left) and RLCTA/16/16+SPIHT (right). The compression
ratio is 1/8 bit per pixel. The right fragment retains much better the structure of
the data than that in the left fragment.
Figure 17 displays the column line #300 from the original section versus the
same column line from the reconstructed sections after the applications of JP2k
(left) and RLCTA/16/16+SPIHT (right). The compression ratio is 1/8 bit per
pixel.
We observe that, unlike JP2k, RLCTA/16/16+SPIHT restores the curve to
be very close to its original form even when the compression ratio is very high.
4.4. Compression of multimedia images
The hybrid compression algorithms also proved to be efficient for multimedia
images. Their superior performances are more evident for images that have
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50
50
100
100
150
150
200
200
250
250
300
250
300
350
400
300
450
260
280
300
320
340
360
380
400
420
440
460
Fig. 16. A fragment from the reconstructed fingerprint image. Left: After the application of
JP2k, PSNR = 23.54. Right: After the application of RLCTA/16/16+SPIHT, PSNR = 24.55.
The compression ratio is 1/8 bit per pixel.
0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
Fig. 17. Column line #300 from the fingerprint image. Dotted line: Original. Solid line: Restored.
Left: The two lines after the application of JP2k. Right: The two lines after the application of
RLCTA/16/16+SPIHT (right). The compression ratio is 1/8 bit per pixel (color online).
oscillating texture. The algorithm restores the texture even at a very low bitrate.
On the other hand, it sometimes produces artifacts on the boundaries between
smooth and textured areas. Barbara image is of size 512 × 512 and its fragment
is of size 280 × 280. They are displayed in Fig. 18. Each pixel in this image has 8
bits.
This image comprises areas with oscillatory textures. We compared between the
performances of 6 transforms where the hybrid and RLCTA transforms partition
the vertical direction into Q = 16 horizontal rectangles each of height 32. The
achieved PSNR values are given in Table 5.
RLCTA/16/16+SPIHT, which produced the best results except the result for
1 bit per pixel, retains the fine texture of the image much better than JP2k. This
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Fig. 18.
image.
The original “Barbara” image. Left: The whole image. Right: A fragment from the left
Table 5.
The PSNR values after the compression/decompression of “Barbara”.
bit/pixel
JP2k
W9/7
WButt/10
1/8
1/4
1/2
1
25.78
28.68
32.71
37.95
25.18
27.85
31.57
36.82
25.14
28.29
32.32
37.9
H9/7/16
25.71
28.63
32.72
38
HButt/10/16
RLCTA/16/16
26.08
29.18
33.28
38.38
26.78
29.54
33.28
38.15
fact is illustrated in Fig. 19, which displays fragments of the reconstructed “Barbara” image after the applications of JP2k (left) and RLCTA/16/16+SPIHT
(right) algorithms. The compression ratio is 1/8 bit per pixel.
Figure 20 displays the column line #246 from the original section versus this
column line from the reconstructed sections after the application of JP2k (left)
and RLCTA/16/16+SPIHT (right). The compression ratio is 1/4 bit per pixel.
We observe that RLCTA/16/16+SPIHT algorithm retains most of the oscillatory events that were missed by JP2k.
Elaine image is of size 512 × 512 and its fragment of size 236 × 256 are displayed
in Fig. 21. Each pixel has 8 bits.
We compared between the performances of 6 transforms where the hybrid and
the RLCTA transforms partition the vertical direction into Q = 8 horizontal rectangles each of height 64. The achieved PSNR values are given in Table 6.
The highest PSNR values in all bitrates except for 1/8 bit per pixel were produced by RLCTA/8/8+SPIHT, while the 2D wavelet transform with Butterworth
wavelet WButt/4+SPIHT was better with 1/8 bit per pixel. At bitrates 1/8 and
1/4 bit per pixel, reconstruction from RLCTA/8/8+SPIHT and hybrid algorithms
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Fig. 19. A fragment from the reconstructed “Barbara” image after the applications of: Left:
JP2k. Right: RLCTA/16/16+SPIHT. The compression ratio is 1/8 bit per pixel.
250
200
150
100
50
0
50
100
150
200
250
300
350
400
450
500
0
0
100
200
300
400
500
Fig. 20. Column line #246 from Barbara image. Dotted line: Original. Solid line: Restored.
Left: Restored after the application of JP2k. Right: Restored after the application of
RLCTA/16/16+SPIHT (right). The compression ratio is 1/4 bit per pixel (color online).
produced a better visual quality of the images in comparison to JP2k, which over
smoothed the images. This fact is illustrated in Fig. 22, which displays fragments
of “Elaine” image that were reconstructed after the application of JP2k and of
RLCTA/8/8+SPIHT. The compression ratio is 1/8 bit per pixel.
Over smoothing by JP2k is clearly seen in Fig. 23, which displays the column
line #280 from the original section versus this column line from the reconstructed
sections after the applications of JP2k (left) and RLCTA/8/8+SPIHT (right).
The compression ratio is 1/4 bit per pixel.
We observe that the RLCTA/8/8+SPIHT algorithm retains most of the fine
texture that was blurred by JP2k.
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Fig. 21.
image.
The original “Elaine” image. Left: The whole image. Right: A fragment from the right
Table 6.
PSNR values after compression/decompression of “Elaine”.
bit/pixel
JP2k
W9/7
WButt/4
H9/7/16
HButt/10/16
RLCTA/8/8
1/8
1/4
1/2
1
31.11
32.29
33.48
36.06
31.1
32.33
33.48
35.75
31.12
32.33
33.36
35.46
30.41
31.96
33.94
36.41
30.65
32.22
34.15
36.79
30.77
32.8
34.9
37.72
Fig. 22. A fragment of the reconstructed “Elaine” image after the applications of: Left: JP2k.
Right: RLCTA/8/8+SPIHT. The compression ratio is 1/4 bit per pixel.
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0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
Fig. 23. Column line #280 from the Elaine image. Dotted line: Original. Solid line: Restored.
Left: Restored after the application of JP2k. Right: Restored after the application of
RLCTA/8/8+SPIHT (right). The compression ratio is 1/4 bit per pixel (color online).
5. Conclusions
The experimental results strongly support our assumption that the hybrid
wavelet — LCT–SPIHT and 2D LCT algorithms with reordering of the transform
coefficients is a new powerful tool to compress seismic data, hyper-spectral images,
fingerprints and other data that comprise oscillatory structures and (or) a fine
texture. Although all the components of the algorithm are well known, their combined operation via reordering of the transform coefficients outperforms compression schemes that are based on multidimensional wavelet transforms with SPIHT
coding and JPEG2000. Additional flexibility of the method stems from the use
of the library of Butterworth wavelet transforms of different orders. In almost all
the experiments, the hybrid transforms with Butterworth wavelets outperform the
transforms with the popular 7/9 wavelets based filters. The 2D LCT algorithm with
reordering of the transform coefficients demonstrates an excellent performance when
oscillating events are present in different directions as in fingerprints or when the
image comprises a fine texture as in the “Elaine” image.
The algorithm retains fine oscillating events even at a low bitrate. This is important for seismic, hyper-spectral and fingerprint images processing and less critical
for multimedia type images. The extension of the algorithm to compress 3D seismic
or hyper-spectral cubes is straightforward. In the horizontal planes, the 2D wavelet
transform is applied, while the LCT is applied to vertical directions followed by
coefficients reordering. The joint array of coefficients is the input to SPIHT encoding. The algorithm is completely automatic and the encoding/decoding operations
can be implemented in a fast way.
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