2006 ssiai
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The document discusses optimizing rate allocation for compressing hyperspectral images using JPEG2000. It proposes using the discrete wavelet transform instead of the Karhunen-Loeve transform for decorrelation due to lower computational complexity. A mixed model is used for rate distortion optimal bit allocation instead of experimentally obtained rate distortion curves. Comparisons show the mixed model approach results in lower mean squared error than traditional bit allocation schemes, while having lower implementation complexity than prior methods.
![OPTIMIZED RATE ALLOCATION OF HYPERSPECTRAL IMAGES IN
COMPRESSED DOMAIN UNDER JPEG2000 PART 2
Silpa Attluri, Vikram Jayaram, Bryan Usevitch
Division of Computing and Electrical Engineering
The University of Texas at El Paso
500 W. University Ave, El Paso, Texas 79968-0523
{silpa, jayaram, usevitch }@ece.utep.edu ∗
Abstract
This paper studies the application of bit allocation using
JPEG2000 for compressing multi-dimensional remote sensing data. Past experiments have shown that the KarhunenLo`ve transform (KLT) along with rate distortion optimal
e
(RDO) bit allocation produces good compression performance. However, this model has the unavoidable disadvantage of paying a price in terms of implementation complexity. In this research we address this complexity problem by
using the discrete wavelet transform (DWT) instead of the
KLT as the decorrelator. Further, we have incorporated a
mixed model (MM) to find the rate distortion curves instead
of the prior method of using experimental rate distortion
curves for RDO bit allocation. We compared our results to
the traditional high bit rate quantizer bit allocation model
based on the logarithm of variances among the bands. Our
comparisons show that by using the MM-RDO bit rate allocation method result in lower mean squared error (MSE)
compared to the traditional bit allocation scheme. Our approach also has an additional advantage of using DWT as
a computationally efficient decorrelator when compared to
the KLT.
1
Introduction
The Hyperion (Hyperspectral Imager) provides a high
resolution hyperspectral image capable of resolving 220
spectral bands (0.4 - 2.5 µm) with a 30 meters pixel
resolution. The data used in these experiments is the Mt.
Fitton scene in southern Australia. The layers or bands
of this Hyperion data are highly correlated to each other
in the spectral direction. Decorrelation of these bands
∗ This work was supported by National Geospatial Intelligence Agency
NMA401-02-2017 and Texas Instruments Foundation.
1-4244-0069-4/06/$20.00/©2006 IEEE.
allows a more compact representation of an image by
packing the energy into fewer numbers of bands. Part 2
of the JPEG2000 standard recommends the use of either
the KLT or the DWT for decorrelation. Past results have
shown that the KLT RDO combination works very well
for bit allocation [2]. Due to the good accuracy of the
results achieved in this method has become a very useful
alternative to the lossless compression for this capacious
data set. Nevertheless, due to the high computational complexity KLT is not a very attractive option as a decorrelator.
Therefore, we use DWT decorrelation instead of the KLT.
Our research refers to past experiments [1] which have
implemented the mixed model for the bit allocation using
the KLT and no decorrelation at different rates.
The high bit rate quantizer model performs bit allocation
using the log of variances (of the bands). The major problems of this high bit rate quantizer model are non integer rates, negative allocations and when these are forced to
non-negative integer rates it leads to suboptimal solutions.
Moreover the present day challenges lies in looking at low
and very low bit rates where this model fails. To overcome
this our past research [2] implemented the approach of experimentally obtained rate distortion curves, which allows
optimal bit allocation with minimized MSE. However, this
method has an unavoidable complexity of generating rate
distortion curves for each band. This problem is addressed
in our current research by using the mixed model which
mathematically models each band’s rate distortion curves
using two distinct regions to get a distortion model that suits
both high and low bit rates.
2
Description of Data
The Hyperion instrument can image a 7.5 km by 100 km
land area per image and provide detailed spectral mapping
228](https://image.slidesharecdn.com/2006ssiai-131120175747-phpapp02/85/2006-ssiai-1-320.jpg)
![across all 220 channels with high radiometric accuracy.
The Hyperion instrument, in conjunction with Advanced
Land Imager (ALI) instrument which provides accurate
radiometric calibration on orbit using precisely controlled
amount of Solar irradiance, forms the standalone camera
system on the Earth Observatory-1 (EO-1) satellite [15].
The analysis of Hyperspectral Imagery by means of such
spectrometers (Hyperion) exploits the fact that each material radiates different amount of electromagnetic energy
throughout all the spectra. This unique characteristic of the
material is commonly known as spectral signature which
can be read using such airborne or space borne-based
detectors.
The data used for experiments were collected by the Hyperion over the semiarid Mount Fitton area which is located
in the Northern Flinders Ranges of South Australia. Centered at −29◦ 55 S and 139◦ 25 E, about 700 kms NNW of
Adelaide. This Hyperion image consists of 194 atmospherically corrected contiguous spectral bands. Each band has
dimensions of 6702 x 256 [12].
each code-block’s embedded stream should be truncated in
order to achieve a pre-defined bit-rate or distortion bound
for the entire data. This bit-stream rescheduling module is
referred to as Tier 2. It establishes a multi-layered representation of the final bit-stream, guaranteeing an optimal performance at several bit-rates or resolutions. Figure 2 illustrates the general flowchart of the JPEG2000 architecture.
The Tier 2 component optimizes the truncation process, and
tries to reach the desired bit-rate while minimizing the introduced distortion (MSE), utilizing Lagrangian rate allocation principles. This following procedure is known as PostCompression Rate-Distortion (PCRD) optimization [4] and
the basic principles behind it are extensively discussed in
[7].
4
Description of our approach
Due to the computationally inefficient nature of the KLT
we employed DWT for decorrelating the bands. The DWT
can be implemented by either a convolution or a lifting
scheme. We have used the lifting scheme implementation
of DWT due to its lesser computations and memory requirements. Here the 9/7 Daubechies filter is used as we are
performing the lossy compression.
Figure 1. Hyperion color composite image of
Mount Fitton test site.
Figure 3. 2-D Wavelet Decomposition
3
The JPEG2000 Standard
The JPEG2000 standard has many advantages compared
to the earlier compression standards. Such advantages include superior low bit rate performance, bit rate scalability
and progressive transmission by quality or resolution. The
coder and decoder for JPEG2000 use a wavelet based compression method which implements the Embedded Block
Coding with Optimized Truncation (EBCOT) [4] scheme.
Quality scalability is achieved by dividing the wavelet transformed image into code-blocks. After each code-block
is encoded, a post-processing operation determines where
Once the data is decorrelated by the use of the transform, all
of the energy or information is packed into just a few bands
instead of all the bands. Figure 4 is a hypothetical portrayal
of energy distributions: before and after decorrelation. The
right plot clearly shows that we cannot just allocate the same
number of bits for each band. Assigning the same bit rate
for every band neglects the information importance of each
band. To avoid this we allocate bits optimally to the separate slices or image bands. The individual bands were compressed and reconstructed depending on the different rate
allocation strategies. We performed experiments using several different approaches to bit rate allocation for individual
229](https://image.slidesharecdn.com/2006ssiai-131120175747-phpapp02/85/2006-ssiai-2-320.jpg)
![ORIGINAL
IMAGE
Forward
DC Level
Shift
Forward
Component
Transform
Forward
Wavelet
Transform
Tier−1
Encoder
Quantization
DECODED
IMAGE
Inverse
DC Level
Shift
Tier−2
Encoder
COMPRESSED
IMAGE
Inverse
Component
Transform
Inverse
Wavelet
Transform
Inverse
Quantization
Tier−1
Decoder
Tier−2
Decoder
Figure 2. JPEG2000 architectural flowchart
bands. One of them is the traditional bit rate allocation approach based on logarithm of variances of individual bands
(see [9] for more details). In this model the MSE’s in each
band can be approximated as function of rate R and variance
σ 2 . Assumptions of a high bit rate model lead to a quantizer
distortion given by:
bit rates [1]. The relation between the MSE and the bit rate
corresponding to the information about the data can still be
described by a modified Mallat-Falzon model,i.e.,
M SE(R) ∼ σ 2 · 2−2R
=
where A, Ro , α come from the low rate model and B
which is nothing but the variance of each band is the parameter from the high rate model (same as 1). To find Ro ,
we use the fact that the MSE of each band is nothing but
˜
its entropy when Ro = 0. R is the cross over bit rate between the low and the high rate models. The determination
of these mixed model parameters are derived in [1].
(1)
where σ 2 is the variance (of the bands) and R being the
average rate for each band.
M SE(R) =
1
A (R+Ro )α
B · 2−2R
˜
if R ≤ R,
˜
if R ≥ R
(2)
Figure 4. Original band energy Vs.
formed band energy
Trans-
The traditional or the log of variances bit allocation scheme
is not fully compatible, since this high bit rate quantizer
distortion model breaks down at low bit rates. Instead we
use the mixed model and the RDO [8], which is based on
post compression rate distortion (PCRD) extensively used
in JPEG2000, for bit allocation. Our research implemented
the RDO by using RD curves that have been obtained by
using the mixed model. The mixed model is essentially a
hybrid splicing of the low and the high bit rate models. The
low bit rate of the model follows the results of Mallat and
Falzon [5] . Past research [1] tried modifying this so that
we can make it work for our problem incorporating lower
mean squared error (MSE)
14000
12000
Low bit rate model
High bit rate model
Experimental RD curve
10000
8000
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Bit rates
Figure 5. Comparison of experimental RD
curve with Mixed model. Notice that the hybrid splicing of mixed model shown as 2 dotted curves is close to the RD curves (solid
lines) obtained experimentally.
Advantages of the mixed model approach are that it can give
230](https://image.slidesharecdn.com/2006ssiai-131120175747-phpapp02/85/2006-ssiai-3-320.jpg)
![advantage of an optimum compaction transform, the DWT
manages to match up with its MSE performance and also
does slightly better compared to the KLT traditional bit allocation in the low bit rate region as seen in Figure 8.
4
4
mean squared error (MSE)
performance nearly identical to the RDO approach with a
much lower implementation complexity. Here we have generated this approach for the data at bit rates in the range
0.0375 bpppb to 4 bpppb (where bpppb stands for bits per
pixel per band) and further compute the corresponding distortions (MSE) at each average bit rate using JPEG2000
coder and decoder. Results show that the mixed model approach gives distortion results that are nearly identical to
the RDO approach on this data, while significantly reducing complexity of implementation. We notice from Figure
x 10
3.5
DWT & MM RDO
DWT & MM Traditional
3
2.5
2
1.5
1
0.5
0 −2
10
−1
10
0
10
1
10
Compression rate (bpppb) on Log Scale
Figure 7. MSE comparison of mixed model
RDO and the high rate quantizer model.
4
Figure 6. 2-D band-wise compression.
5 that the mixed model approach gives distortion results that
are nearly identical to the experimental RDO approach on
this data, while significantly reducing the complexity. Further, our experiments are performed as shown in Figure 6.
Each band of the Hyperspectral cube were compressed at
16 bpppb and decompressed at bit-rates obtained from the
two bit allocation schemes. Next we computed the MSE’s
by comparing the original set of bands to the compressed
decompressed bands for the given set of average bit rates.
5
mean squared error (MSE)
4
3.5
DWT & Mixed Model−RDO
DWT & Traditional Model
3
KLT & RDO
KLT & Traditional
2.5
2
1.5
1
0.5
0 −2
10
−1
10
0
10
1
10
Compression Rate on (bpppb) on Log Scale
Figure 8. MSE comparisons of DWT mixed
model, DWT traditional model, KLT RDO and
KLT traditional model.
Results
Figure 7 shows a comparison of MSE’s obtained for each
average bit rates corresponding to traditional and the MMRDO bit allocation schemes using DWT as a decorrelator. Our results confirm that MM-RDO has lower MSE’s
compared to the traditional model. Moreover, MM-RDO
when used in conjunction with DWT as decorrelator further produces lower MSE’s compared to KLT decorrelation
with traditional bit allocation for lower bit rates. Figure 8
shows a comparison our past experimental results [2] that
employed KLT docorrelation followed by RDO bit allocation with the present DWT mixed model. In spite of KLT’s
x 10
6
Conclusions
In the case of KLT the computation time and memory
requirements is very high (about 8 times higher) when
compared to DWT. One other main advantage of DWT, that
it allows to divide a huge data set into parts and pre-process
each of these independent of the other. This enables parallel
231](https://image.slidesharecdn.com/2006ssiai-131120175747-phpapp02/85/2006-ssiai-4-320.jpg)
![[6] ITU-T Recommendation T.801 | ISO/IEC 154441:2001 Information technology – JPEG2000 image coding system – Part 2: Extensions, 2000.
processing which cannot be done with KLT. At higher bit
rates we observe that both rate allocation strategies produce
similar MSE results. The rate distortion curves obtained
from the Mixed Model are very close to the R-D curves
obtained experimentally. The Mixed Model and RDO
approach gives lower MSE than the traditional model and
also slightly out performs our past KLT traditional method
at lower rates.
[7] H. E. Everett III, Generalized lagrange Multiplier
method for solving problems of optimum allocation of resources, Operations Research, vol. 11,
pp. 399–417, 1963.
[8] Olga M. Kosheleva, Task-Specific Metrics and
Optimized Rate Allocation Applied to Part 2 of
JPEG2000 and 3-D Meteorological Data. Ph.D.
Dissertation, University of Texas at El Paso, August 2003.
However, we do see that KLT-RDO provides lower MSE’s
in most cases compared to the present DWT approach. This
is because of the fact that the KLT is a better decorrelator than a DWT. The KLT is theoretically the ideal transform and is expected to give better results for both classification performance and compression. Nevertheless, its
biggest disadvantage is that it is computationally expensive.
7
[9] R. Jayant and P. Noll, Digital Coding of Waveforms. Upper Saddle River: Prentice Hall, 1984.
[10] A. Gersho and R. M. Gray, Vector Quantization
and Signal Compression. Boston: Kluwer, 1992.
Acknowledgment
We would like to thank Dr. Olga Kosheleva and Alberto
Aguirre for their assistance with this research. We also
would like to thank Mary Williams (EOIR) for performing atmospheric corrections on our data and Raed Aldouri
(PACES) for providing us ENVI software package.
[11] Bryan Usevitch, “A Tutorial on Modern Lossy
Wavelet Image Compression: Foundations of
JPEG2000”, IEEE Signal Processing Magazine,
Vol. 18, Issue 5, September 2001.
[12] T. J. Cudahy, R. Hewson, J. F. Huntington, M. A.
Quigley and P. S. Barry, “The performance of the
Satellite-borne Hyperion Hyperspectral VNIR–
SWIR imaging system for mineral mapping at
Mount Fitton, South Australia,” in Proceedings of
IEEE International Geoscience and Remote Sensing Symposium IGARSS01, Sydney, July 2001.
References
[1] Olga Kosheleva, Bryan Usevitch, Sergio Cabrera,
and Edward Vidal Jr., “Rate distortion optimal
bit allocation methods for volumetric data using
JPEG2000,” submitted to IEEE Transactions on
Image Processing, January 2004.
[13] Shaw Gary and Manolakis Dimitris, “Signal Processing for Hyperspectral Image Exploitation”,
IEEE Signal Processing Magazine, Vol. 19, Issue
1, January 2002, ISSN 1053-5888.
[2] Vikram Jayaram, Bryan Usevitch and Olga
Kosheleva, “Detection from Hyperspectral images
compressed using rate distortion and optimization techniques under JPEG2000 Part 2,” in Proc.
IEEE 11th DSP Workshop & 3rd SPE Workshop
(cdrom), August 2004.
[14] Manolakis Dimitris and Shaw Gary, ”Detection
Algorithms for Hyperspectral Imaging Applications”, IEEE Signal Processing Magazine, Vol. 19,
Issue 1, January 2002, ISSN 1053-5888.
[3] M. D. Pal, C. M. Brislawn, and S. P. Brumby,
“Feature Extraction from Hyperspectral Images
Compressed using the JPEG-2000 Standard,” in
Proceedings of Fifth IEEE Southwest Symposium
on Image Analysis and Interpretation (SSIAI’02),
Santa Fe, NM, pp. 168–172, April 2002.
[15] http://eo1.gsfc.nasa.gov/
[4] D. Taubman and M. Marcellin, JPEG2000 Image
Compression Fundamentals, Standards and Practice. Boston, Dordrecht, London: Kluwer, 2002.
[5] F. Falzon and S. Mallat, “Low bit rate image coding over bases,” in proceedings of 14th International Conference on Pattern Recognition, vol 2,
pp. 692-695, August 1998.
232](https://image.slidesharecdn.com/2006ssiai-131120175747-phpapp02/85/2006-ssiai-5-320.jpg)
2006 ssiai
- 1. OPTIMIZED RATE ALLOCATION OF HYPERSPECTRAL IMAGES IN COMPRESSED DOMAIN UNDER JPEG2000 PART 2 Silpa Attluri, Vikram Jayaram, Bryan Usevitch Division of Computing and Electrical Engineering The University of Texas at El Paso 500 W. University Ave, El Paso, Texas 79968-0523 {silpa, jayaram, usevitch }@ece.utep.edu ∗ Abstract This paper studies the application of bit allocation using JPEG2000 for compressing multi-dimensional remote sensing data. Past experiments have shown that the KarhunenLo`ve transform (KLT) along with rate distortion optimal e (RDO) bit allocation produces good compression performance. However, this model has the unavoidable disadvantage of paying a price in terms of implementation complexity. In this research we address this complexity problem by using the discrete wavelet transform (DWT) instead of the KLT as the decorrelator. Further, we have incorporated a mixed model (MM) to find the rate distortion curves instead of the prior method of using experimental rate distortion curves for RDO bit allocation. We compared our results to the traditional high bit rate quantizer bit allocation model based on the logarithm of variances among the bands. Our comparisons show that by using the MM-RDO bit rate allocation method result in lower mean squared error (MSE) compared to the traditional bit allocation scheme. Our approach also has an additional advantage of using DWT as a computationally efficient decorrelator when compared to the KLT. 1 Introduction The Hyperion (Hyperspectral Imager) provides a high resolution hyperspectral image capable of resolving 220 spectral bands (0.4 - 2.5 µm) with a 30 meters pixel resolution. The data used in these experiments is the Mt. Fitton scene in southern Australia. The layers or bands of this Hyperion data are highly correlated to each other in the spectral direction. Decorrelation of these bands ∗ This work was supported by National Geospatial Intelligence Agency NMA401-02-2017 and Texas Instruments Foundation. 1-4244-0069-4/06/$20.00/©2006 IEEE. allows a more compact representation of an image by packing the energy into fewer numbers of bands. Part 2 of the JPEG2000 standard recommends the use of either the KLT or the DWT for decorrelation. Past results have shown that the KLT RDO combination works very well for bit allocation [2]. Due to the good accuracy of the results achieved in this method has become a very useful alternative to the lossless compression for this capacious data set. Nevertheless, due to the high computational complexity KLT is not a very attractive option as a decorrelator. Therefore, we use DWT decorrelation instead of the KLT. Our research refers to past experiments [1] which have implemented the mixed model for the bit allocation using the KLT and no decorrelation at different rates. The high bit rate quantizer model performs bit allocation using the log of variances (of the bands). The major problems of this high bit rate quantizer model are non integer rates, negative allocations and when these are forced to non-negative integer rates it leads to suboptimal solutions. Moreover the present day challenges lies in looking at low and very low bit rates where this model fails. To overcome this our past research [2] implemented the approach of experimentally obtained rate distortion curves, which allows optimal bit allocation with minimized MSE. However, this method has an unavoidable complexity of generating rate distortion curves for each band. This problem is addressed in our current research by using the mixed model which mathematically models each band’s rate distortion curves using two distinct regions to get a distortion model that suits both high and low bit rates. 2 Description of Data The Hyperion instrument can image a 7.5 km by 100 km land area per image and provide detailed spectral mapping 228
- 2. across all 220 channels with high radiometric accuracy. The Hyperion instrument, in conjunction with Advanced Land Imager (ALI) instrument which provides accurate radiometric calibration on orbit using precisely controlled amount of Solar irradiance, forms the standalone camera system on the Earth Observatory-1 (EO-1) satellite [15]. The analysis of Hyperspectral Imagery by means of such spectrometers (Hyperion) exploits the fact that each material radiates different amount of electromagnetic energy throughout all the spectra. This unique characteristic of the material is commonly known as spectral signature which can be read using such airborne or space borne-based detectors. The data used for experiments were collected by the Hyperion over the semiarid Mount Fitton area which is located in the Northern Flinders Ranges of South Australia. Centered at −29◦ 55 S and 139◦ 25 E, about 700 kms NNW of Adelaide. This Hyperion image consists of 194 atmospherically corrected contiguous spectral bands. Each band has dimensions of 6702 x 256 [12]. each code-block’s embedded stream should be truncated in order to achieve a pre-defined bit-rate or distortion bound for the entire data. This bit-stream rescheduling module is referred to as Tier 2. It establishes a multi-layered representation of the final bit-stream, guaranteeing an optimal performance at several bit-rates or resolutions. Figure 2 illustrates the general flowchart of the JPEG2000 architecture. The Tier 2 component optimizes the truncation process, and tries to reach the desired bit-rate while minimizing the introduced distortion (MSE), utilizing Lagrangian rate allocation principles. This following procedure is known as PostCompression Rate-Distortion (PCRD) optimization [4] and the basic principles behind it are extensively discussed in [7]. 4 Description of our approach Due to the computationally inefficient nature of the KLT we employed DWT for decorrelating the bands. The DWT can be implemented by either a convolution or a lifting scheme. We have used the lifting scheme implementation of DWT due to its lesser computations and memory requirements. Here the 9/7 Daubechies filter is used as we are performing the lossy compression. Figure 1. Hyperion color composite image of Mount Fitton test site. Figure 3. 2-D Wavelet Decomposition 3 The JPEG2000 Standard The JPEG2000 standard has many advantages compared to the earlier compression standards. Such advantages include superior low bit rate performance, bit rate scalability and progressive transmission by quality or resolution. The coder and decoder for JPEG2000 use a wavelet based compression method which implements the Embedded Block Coding with Optimized Truncation (EBCOT) [4] scheme. Quality scalability is achieved by dividing the wavelet transformed image into code-blocks. After each code-block is encoded, a post-processing operation determines where Once the data is decorrelated by the use of the transform, all of the energy or information is packed into just a few bands instead of all the bands. Figure 4 is a hypothetical portrayal of energy distributions: before and after decorrelation. The right plot clearly shows that we cannot just allocate the same number of bits for each band. Assigning the same bit rate for every band neglects the information importance of each band. To avoid this we allocate bits optimally to the separate slices or image bands. The individual bands were compressed and reconstructed depending on the different rate allocation strategies. We performed experiments using several different approaches to bit rate allocation for individual 229
- 3. ORIGINAL IMAGE Forward DC Level Shift Forward Component Transform Forward Wavelet Transform Tier−1 Encoder Quantization DECODED IMAGE Inverse DC Level Shift Tier−2 Encoder COMPRESSED IMAGE Inverse Component Transform Inverse Wavelet Transform Inverse Quantization Tier−1 Decoder Tier−2 Decoder Figure 2. JPEG2000 architectural flowchart bands. One of them is the traditional bit rate allocation approach based on logarithm of variances of individual bands (see [9] for more details). In this model the MSE’s in each band can be approximated as function of rate R and variance σ 2 . Assumptions of a high bit rate model lead to a quantizer distortion given by: bit rates [1]. The relation between the MSE and the bit rate corresponding to the information about the data can still be described by a modified Mallat-Falzon model,i.e., M SE(R) ∼ σ 2 · 2−2R = where A, Ro , α come from the low rate model and B which is nothing but the variance of each band is the parameter from the high rate model (same as 1). To find Ro , we use the fact that the MSE of each band is nothing but ˜ its entropy when Ro = 0. R is the cross over bit rate between the low and the high rate models. The determination of these mixed model parameters are derived in [1]. (1) where σ 2 is the variance (of the bands) and R being the average rate for each band. M SE(R) = 1 A (R+Ro )α B · 2−2R ˜ if R ≤ R, ˜ if R ≥ R (2) Figure 4. Original band energy Vs. formed band energy Trans- The traditional or the log of variances bit allocation scheme is not fully compatible, since this high bit rate quantizer distortion model breaks down at low bit rates. Instead we use the mixed model and the RDO [8], which is based on post compression rate distortion (PCRD) extensively used in JPEG2000, for bit allocation. Our research implemented the RDO by using RD curves that have been obtained by using the mixed model. The mixed model is essentially a hybrid splicing of the low and the high bit rate models. The low bit rate of the model follows the results of Mallat and Falzon [5] . Past research [1] tried modifying this so that we can make it work for our problem incorporating lower mean squared error (MSE) 14000 12000 Low bit rate model High bit rate model Experimental RD curve 10000 8000 6000 4000 2000 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Bit rates Figure 5. Comparison of experimental RD curve with Mixed model. Notice that the hybrid splicing of mixed model shown as 2 dotted curves is close to the RD curves (solid lines) obtained experimentally. Advantages of the mixed model approach are that it can give 230
- 4. advantage of an optimum compaction transform, the DWT manages to match up with its MSE performance and also does slightly better compared to the KLT traditional bit allocation in the low bit rate region as seen in Figure 8. 4 4 mean squared error (MSE) performance nearly identical to the RDO approach with a much lower implementation complexity. Here we have generated this approach for the data at bit rates in the range 0.0375 bpppb to 4 bpppb (where bpppb stands for bits per pixel per band) and further compute the corresponding distortions (MSE) at each average bit rate using JPEG2000 coder and decoder. Results show that the mixed model approach gives distortion results that are nearly identical to the RDO approach on this data, while significantly reducing complexity of implementation. We notice from Figure x 10 3.5 DWT & MM RDO DWT & MM Traditional 3 2.5 2 1.5 1 0.5 0 −2 10 −1 10 0 10 1 10 Compression rate (bpppb) on Log Scale Figure 7. MSE comparison of mixed model RDO and the high rate quantizer model. 4 Figure 6. 2-D band-wise compression. 5 that the mixed model approach gives distortion results that are nearly identical to the experimental RDO approach on this data, while significantly reducing the complexity. Further, our experiments are performed as shown in Figure 6. Each band of the Hyperspectral cube were compressed at 16 bpppb and decompressed at bit-rates obtained from the two bit allocation schemes. Next we computed the MSE’s by comparing the original set of bands to the compressed decompressed bands for the given set of average bit rates. 5 mean squared error (MSE) 4 3.5 DWT & Mixed Model−RDO DWT & Traditional Model 3 KLT & RDO KLT & Traditional 2.5 2 1.5 1 0.5 0 −2 10 −1 10 0 10 1 10 Compression Rate on (bpppb) on Log Scale Figure 8. MSE comparisons of DWT mixed model, DWT traditional model, KLT RDO and KLT traditional model. Results Figure 7 shows a comparison of MSE’s obtained for each average bit rates corresponding to traditional and the MMRDO bit allocation schemes using DWT as a decorrelator. Our results confirm that MM-RDO has lower MSE’s compared to the traditional model. Moreover, MM-RDO when used in conjunction with DWT as decorrelator further produces lower MSE’s compared to KLT decorrelation with traditional bit allocation for lower bit rates. Figure 8 shows a comparison our past experimental results [2] that employed KLT docorrelation followed by RDO bit allocation with the present DWT mixed model. In spite of KLT’s x 10 6 Conclusions In the case of KLT the computation time and memory requirements is very high (about 8 times higher) when compared to DWT. One other main advantage of DWT, that it allows to divide a huge data set into parts and pre-process each of these independent of the other. This enables parallel 231
- 5. [6] ITU-T Recommendation T.801 | ISO/IEC 154441:2001 Information technology – JPEG2000 image coding system – Part 2: Extensions, 2000. processing which cannot be done with KLT. At higher bit rates we observe that both rate allocation strategies produce similar MSE results. The rate distortion curves obtained from the Mixed Model are very close to the R-D curves obtained experimentally. The Mixed Model and RDO approach gives lower MSE than the traditional model and also slightly out performs our past KLT traditional method at lower rates. [7] H. E. Everett III, Generalized lagrange Multiplier method for solving problems of optimum allocation of resources, Operations Research, vol. 11, pp. 399–417, 1963. [8] Olga M. Kosheleva, Task-Specific Metrics and Optimized Rate Allocation Applied to Part 2 of JPEG2000 and 3-D Meteorological Data. Ph.D. Dissertation, University of Texas at El Paso, August 2003. However, we do see that KLT-RDO provides lower MSE’s in most cases compared to the present DWT approach. This is because of the fact that the KLT is a better decorrelator than a DWT. The KLT is theoretically the ideal transform and is expected to give better results for both classification performance and compression. Nevertheless, its biggest disadvantage is that it is computationally expensive. 7 [9] R. Jayant and P. Noll, Digital Coding of Waveforms. Upper Saddle River: Prentice Hall, 1984. [10] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston: Kluwer, 1992. Acknowledgment We would like to thank Dr. Olga Kosheleva and Alberto Aguirre for their assistance with this research. We also would like to thank Mary Williams (EOIR) for performing atmospheric corrections on our data and Raed Aldouri (PACES) for providing us ENVI software package. [11] Bryan Usevitch, “A Tutorial on Modern Lossy Wavelet Image Compression: Foundations of JPEG2000”, IEEE Signal Processing Magazine, Vol. 18, Issue 5, September 2001. [12] T. J. Cudahy, R. Hewson, J. F. Huntington, M. A. Quigley and P. S. Barry, “The performance of the Satellite-borne Hyperion Hyperspectral VNIR– SWIR imaging system for mineral mapping at Mount Fitton, South Australia,” in Proceedings of IEEE International Geoscience and Remote Sensing Symposium IGARSS01, Sydney, July 2001. References [1] Olga Kosheleva, Bryan Usevitch, Sergio Cabrera, and Edward Vidal Jr., “Rate distortion optimal bit allocation methods for volumetric data using JPEG2000,” submitted to IEEE Transactions on Image Processing, January 2004. [13] Shaw Gary and Manolakis Dimitris, “Signal Processing for Hyperspectral Image Exploitation”, IEEE Signal Processing Magazine, Vol. 19, Issue 1, January 2002, ISSN 1053-5888. [2] Vikram Jayaram, Bryan Usevitch and Olga Kosheleva, “Detection from Hyperspectral images compressed using rate distortion and optimization techniques under JPEG2000 Part 2,” in Proc. IEEE 11th DSP Workshop & 3rd SPE Workshop (cdrom), August 2004. [14] Manolakis Dimitris and Shaw Gary, ”Detection Algorithms for Hyperspectral Imaging Applications”, IEEE Signal Processing Magazine, Vol. 19, Issue 1, January 2002, ISSN 1053-5888. [3] M. D. Pal, C. M. Brislawn, and S. P. Brumby, “Feature Extraction from Hyperspectral Images Compressed using the JPEG-2000 Standard,” in Proceedings of Fifth IEEE Southwest Symposium on Image Analysis and Interpretation (SSIAI’02), Santa Fe, NM, pp. 168–172, April 2002. [15] http://eo1.gsfc.nasa.gov/ [4] D. Taubman and M. Marcellin, JPEG2000 Image Compression Fundamentals, Standards and Practice. Boston, Dordrecht, London: Kluwer, 2002. [5] F. Falzon and S. Mallat, “Low bit rate image coding over bases,” in proceedings of 14th International Conference on Pattern Recognition, vol 2, pp. 692-695, August 1998. 232