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.