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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
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
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.
Hyperspectral Remote
Sensing Spectral
Dimension Each spatial
pixel has a
spectrum that
can be used to
analyze the
material
Spatial Dimension
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.
1-D Wavelet Transform
h
g
2
g
h
g 2
2
2 2
2 h
x[n]
y1h[n]
y1l[n]
y2h[n]
y2l[n]
YKh[n]
YKl[n]
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.
Implementation
The 1-D wavelet can be implemented to get
similar results by using 2 methods, namely
Convolution
Lifting scheme
Forward Transform
oi
n = oin-1 + ∑ Pen(k) × ek
n-1 where n Є [1,2,3,….N]
ekn = ei
n-1 + ∑ Udn(k) × okn-1 where n Є [1,2,3,….N]
C +
+ +
+
Ud1(z)
UdN(z)
PeN(z)
Pe1(z)
Sθ(z) S1(z)
R1(z)
C
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.
Inverse Transform
+
+ +
+
+
-PeN(z) -UdN(z)
-Pe1(z) -Ud1(z)
S1(z)
R1(z)
Sθ(z)
1/2C
1/2C
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
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
Need for Bit Allocation
BANDS
E
N
E
R
G
Y
E
N
E
R
G
Y
BANDS
Before Transformation After DWT
Optimization problem: Based on MSE
Minimize the overall mean squared error
under the constraint that the average bit rate is R
LaGrange Multiplier Technique
When subject to the rate constraint we use the LaGrange Multiplier Technique which is
0 xgxh
N
n
nRN
Rxg1
1
N
n
N
n
n
R
rnrnxh
1 1
22222 2
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
nn RR
1
1
2
2
2log2
1
R
nMSE 222
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
R if 2
R if 1
~2 RB
RR
AMSE
R
~
~
2
~
0
R if 2
R if )(
1
)(
RB
RRR
ARMSE
R
~
R
Mixed Model (contd…)
1222
2212x
loglog
)(log)(log
RR
RMSERMSE
)(log)((log)(log))((loglog 122222122
RRMSERRMSEA xx
oxR
x
RMSE
ARR
x
xox
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
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
Results based on
MSE
10-2
10-1
100
101
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
4
Compression Rate (bpppb) on LOG Scale
Mean S
quare
d E
rror
(MS
E)
DWT Mixed Model RDO
DWT LOG of Variances
10-2
10-1
100
101
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
4
Compression Rate (bpppb) on LOG Scale
Mean S
quare
d E
rror
(MS
E) DWT Mixed Model &RDO
DWT & Traditional LOG of variances
KLT & RDO
KLT & Traditional LOG of variances
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)
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.