This document discusses optimizing the rate allocation of hyperspectral images compressed using JPEG2000. It presents a mixed model for bit allocation that combines high and low bit rate models. This mixed model and an optimal rate allocation approach based on minimizing mean squared error under a rate constraint provide lower reconstruction errors than traditional approaches. Computational tests on hyperspectral data show the discrete wavelet transform allows for faster processing and less memory usage compared to the Karhunen-Loeve transform.

1. OPTIMIZED RATE ALLOCATION OF
HYPERSPECTRAL IMAGES IN COMPRESSED
DOMAIN USING JPEG2000 Part 2
Presented by
Vikram Jayaram
Authors: Silpa Attluri, Vikram Jayaram, Bryan Usevitch
Applied Signal Processing and Research Group
Division of Computing and Electrical Engineering
The University of Texas at El Paso
2006 IEEE Southwest Symposium on Image Analysis and Interpretation
 Denver, Colorado March 25th – March 28th

2. OUTLINE
 JPEG2000 Standard
 Description of Data
 DWT pre-processing
 Bit Rate Allocation
 2-D band-wise compression
 High bit rate quantizer model (Traditional model)
 Mixed Model
 Optimal Rate Allocation
 Results : MSE Comparison
 Future Work
 Conclusion

3. CASE STUDY: Hyperion
 Hyperion System on board of EO-1(developed at
Goddard NASA Center) platform
 Mt Fitton, Northern Flinders Ranges of South
Australia
- Semi-arid (<250 mm per year)
- 29 Deg. 55’ S, 139 Deg. 25’ E 700 Km NW of
Adelaide
- Region abundant in Talc
 Original 3-D set of data: 220 Spectral Bands, 6702 x
256 Dimension.
 Atmospheric Correction performed using Flaash.

4. Hyperspectral Remote
Sensing
Spectral
Dimension Each spatial
pixel has a
spectrum that
can be used to
analyze the
material
Spatial Dimension

6. DISCRETE WAVELET
TRANSFORM (DWT)
 Separates low and high frequencies, just as the
Fourier transform.
 Converts signal into a series of wavelets which
are easy for storage.
 Provides time-frequency information
simultaneously.

7. 1-D Wavelet Transform
h
g
2
g
h
g 2
2
2 2
2h
x[n]
y1h[n]
y1l[n]
y2h[n]
y2l[n]
YKh[n]
YKl[n]

8. 1-D DWT
 The 1-D sequence separates low-
frequency and high-frequency
coefficients.
 Low-pass and high-pass filters
together are called analysis filter-
banks.

9. Implementation
The 1-D wavelet can be implemented to get
similar results by using 2 methods, namely
 Convolution
 Lifting scheme

10. Forward Transform
oi
n = oi
n-1 + ∑ Pen(k) × ek
n-1 where n Є [1,2,3,….N]
ek
n = ei
n-1 + ∑ Udn(k) × ok
n-1 where n Є [1,2,3,….N]
C
+
++
+
Ud1(z) UdN(z)PeN(z)Pe1(z)
Sθ(z) S1(z)
R1(z)
C

11. Inverse DWT
 Each subband is interpolated by a factor of 2.
 Insert zeros between samples.
 Filter each resulting sequence with the
synthesis filter-bank.
 Filtered sequences when added gives an
approximation of the original signal.

12. Inverse Transform
+
+ +
+
+
-PeN(z) -UdN(z)
-Pe1(z)
-Ud1(z)
S1(z)
R1(z)
Sθ(z)
1/2C
1/2C

13. 3-Level 2-D DW Transform
1 HL
1 LH 1 HH
2 HL
2 LH 2 HH
3LL 3 HL
3 LH 3 HH

14. 2-D Band-wise Compression
DWT
Inverse
DWT
MSE
Computation
RAW to PGX
Conversion
J2K Compression
16 bits/pixel
Lossy
compression
Bit
Allocation
Decompression
At Multiple Bit
Rates

15. Need for Bit Allocation
BANDS
E
N
E
R
G
Y
E
N
E
R
G
Y
BANDS
Before Transformation After DWT

16. Optimization problem: Based on MSE
Minimize the overall mean squared error
under the constraint that the average bit rate is R

17. LaGrange Multiplier Technique
When subject to the rate constraint we use the
LaGrange Multiplier Technique which is
    0 xgxh 
  

N
n
nR
N
Rxg
1
1
    


N
n
N
n
n
R
rnr
n
xh
1 1
22222
2 

18. LaGrange Multiplier
Technique
After we differentiate with respect to Rn
and equate it to zero we have
From this we finally have
 
NN
j
j
n
n RR 1
1
2
2
2log
2
1




R
nMSE 22
2


19. Mixed Model (et al. Dr. Kosheleva)
From the Mallat and Falzon model we have
Here is the threshold bitrate
But this equation tends to infinity at low bit rates or zero. So we
modify it to









Rif2
Rif
1
~
2
RB
R
R
A
MSE
R
~










~
2
~
0
Rif2
Rif
)(
1
)(
RB
R
RR
A
RMSE
R

~
R

20. Mixed Model (contd…)
1222
2212
x
loglog
)(log)(log
RR
RMSERMSE



)(log)((log)(log))((loglog 12222212
2
RRMSERRMSEA xx 
oxR
x
RMSE
A
RR
x
x
ox

1
)( 






Here we consider R1 =0.5, R2 =0.75 and R=2 to find the constants
A, , and B using the below
B = MSE(R) 2R
x

21. Mixed Model Example
Low Bit Rate Model
Original RD Curve
High Bit Rate Model
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
2000
4000
6000
8000
10000
12000
14000
Bit rates (bpppb)
M
S
E

22. Results based on
MSE

23. 10
-2
10
-1
10
0
10
1
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
4
Compression Rate (bpppb) on LOG Scale
MeanSquaredError(MSE)
DWT Mixed Model RDO
DWT LOG of Variances

24. 10
-2
10
-1
10
0
10
1
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
4
Compression Rate (bpppb) on LOG Scale
MeanSquaredError(MSE)
DWT Mixed Model &RDO
DWT & Traditional LOG of variances
KLT & RDO
KLT & Traditional LOG of variances

26. Computational Time
Comparison
3440.531
475378
0
500
1000
1500
2000
2500
3000
3500
4000
PCA DWT (5/3) DWT(9/7)
T
I
M
E
(sec)

27. Conclusions
 The rate distortion curves obtained from the Mixed
Model are very close to the R-D curves obtained
experimentally.
 Mixed Model and RDO approach gives lower MSE
over the traditional high bit rate quantizer model.
 In the case of KLT the computation time is very high
when compared to DWT and also the memory
requirement is very high in case of KLT. (Reference
Master’s Thesis Vikram Jayaram, 2004)
 The DWT allows for us to divide the huge data set into
parts and pre-process each of these independent of
the other. This enables parallel processing which
cannot be done with KLT.