Hyperspectral Images

Abstract—In this work, we propose a compression algorithm based on Partitioned Vector Quantization for 3D scientific data, and in particular those generated by remote sensors. One of the dimensions of the input data is treated as a single vector that is partitioned into subvectors of (possibly) different lengths, each quantized separately. Quantization indices and residuals are then entropy coded. Partitioning serves different purposes: vector quantization looses efficiency in high dimension and partitioning is a way of improving it; it allows a better decorrelation of the data in the given dimension, improving compression; the concatenation of the quantization indices can be seen as a projection to a lower dimensional subspace that preserves the original statistical properties of the data; partition indices can be used for fast (remote) browsing and classification. In fact, experimental results with NASA AVIRIS hyperspectral images show that the proposed algorithm achieves state of the art lossless compression on such data, provides near-lossless and lossy capabilities (that could be used for fast browsing and broadcasting on limited bandwidth communication channels), is scalable to arbitrary data size, allows fast classification and target detection in the compressed domain.

[1] Index Terms—About four key words or phrases in alphabetical order, separated by commas. For a list of suggested keywords, send a blank e-mail to keywords@ieee.org or visit the IEEE web site at http://www.ieee.org/web/developers/webthes/index.htm.

I. INTRODUCTIONN increasing number of scientific instruments produce high volume

of data in the form of two-dimensional matrices of vectors, also

called data cubes. Typical examples are biomedical images, multi, hyper and

ultra spectral images, measurements of volumetric parameters like

atmospheric temperature and pressure by way of microwaves sounders.

Compressing these data presents a challenge for the currently standardized

A

2 CHAPTER xx

compressors. Beside the three dimensional nature of the measurements,

which exhibit correlation along each dimension, the single measurements

have values of 16 bits or more. Current state of the art compressors do not

perform well on data sources with large alphabets. Furthermore, it is

desirable to have a single compressor that can perform well in all of lossless,

near lossless, and lossy modes and can scale to data cubes of arbitrary sizes.

Lossless compression is often required for data collection and archiving due

to the cost of the data collection and that it may be destined to analysis and

further elaboration. However, when data is downloaded by users, a lossy

mode can be employed to speed up the transmission, since the final users

may not need all measurements at their full precision, and in any case, can

request further precision only when needed.

In [1] we propose a compression algorithm based on Partitioned Vector

Quantization, where one dimension of the cube is treated as a single vector,

partitioned and quantized. Partitioning the vector into subvectors of

(possibly) different lengths is necessary because the possibly large number of

components. Subvectors are individually quantized with appropriate

codebooks. The adaptive partitioning uses a novel locally optimal algorithm

and provides a tool to reduce the size of the source alphabet. Correlation

among the other two dimensions is exploited with methods derived from the

image coding domain. Lossless or near-lossless entropy coding can be

CHAPTER xx 3

applied to the residual error. An additional feature of this compression

method is the possibility to tightly bound error on a pixel by pixel basis. Due

to the hierarchical structure of the data compressed with the proposed

algorithm, it is possible to perform classification and target detection directly

in compressed domain, with considerable speed and memory savings. The

high speed classification and target detection algorithm that we describe here

can process 90 percent of the compressed pixels with a simple table lookup,

full decompression is only required if exact classification is necessary and it

is limited to the 10 percent remaining pixels. Algorithms have been verified

experimentally on the lossless and near lossless compression of NASA

AVIRIS images.

Section 2 briefly reviews the algorithm in order to introduce the necessary

notation. Besides good compression performance, LPVQ presents a number

of features that are highly desirable for these kinds of images. A

communication paradigm that shows these advantages is outlined and

discussed in Section 3.

The problem of classification and the necessary notation is introduced in

Section 4. Section 5 presents experimental results on the classification error

introduced by LPVQ lossy and near lossless compression modes. Section 6

builds on these experimental observations with an algorithm that speeds up

the classification by accepting or rejecting vectors based only on bounds of

4 CHAPTER xx

the VQ error. These error bounds are available at design time or can be

extracted from the compressed image. In the same section, the asymptotic

behavior is derived. Given a compressed image and a classification threshold,

the speed up achieved on the naïve algorithm depends on the acceptance and

rejection rates. Experiments on typical AVIRIS images are performed in the

same section in order to assess these rates for a number of classification

thresholds. The acceptance and rejection rates only depend on the bounds on

the quantization error and can be easily derived during the design of the

LPVQ. So, these considerations also provide a tool useful to select the

quantization parameters (codebook entries and number of partitions) in order

to match a desired range of classification speeds.

II. The Communication ModelAn application for which our algorithms were designed is the encoding of

information collected by a remote sensing device having limited

computational power (satellite, airplane, biomedical imager) and losslessly

transmitted to a powerful central server (see Figure 1). The central server

decompresses and elaborates the data, performing all necessary adjustments

and corrections. Then, re-codes the data in a more compact representation

and distributes the result among a number of users that may subscribe the

service at different quality levels. A common example is weather information

collected by geostationary satellites. The satellites observe of events evolving

CHAPTER xx 5

in a specific geographic area, such as hurricanes, storms, flooding and

volcanic eruptions. The collected data have to be faithfully transmitted to a

land based central station that analyzes the data, applies some form of

enhancement, compressed the data further and sends the result back to the

satellite in order to broadcast it to the final users. In this setting, the same

satellite is used for both acquisition and broadcasting. More common is the

case of the central server performing the transmission. While the

compression hardware used in the satellite must be simple and require very

low power to function, there is typically little or no limitation to the

computational power of the base station.

III. DefinitionsWe model a three dimensional data cube as a discrete time, discrete values,

bi-dimensional random source that emits pixels that are -

dimensional vectors . Each vector component , , is

drawn from a finite alphabet and is distributed according to a space

variant probability distribution that may depend on other components. We

assume that the alphabet has the canonical form .

Our encoder is based on a Partitioned Vector Quantizer (or PVQ), which

consists of independent, -levels, -dimensional exhaustive search

vector quantizers , such that and:

6 CHAPTER xx

is a finite indexed subset of called codebook.

Its elements are the code vectors.

is a partition of and its equivalence classes (or

cells) satisfy:

and for .

is a function defining the relation between the codebook

and the partition as if and only if .

The index of the codeword , result of the quantization of the -

dimensional subvector , is the information that is sent to the decoder.

With reference to the previously defined vector quantizers

, a Partitioned Vector Quantizer can be formally described by

a triple where:

is a codebook in ;

is a partition of ;

is computed on an input vector as the

concatenation of the independent quantization of the subvectors

of . Similarly, the index vector sent to the decoder is obtained as a

concatenation of the indices.

The design of a Partitioned VQ aims at the joint determination of the

partition boundaries and at the design of the

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independent vector quantizers having dimension , .

In the following, given a source vector , we indicate the

subvector of boundaries and with the symbol (for simplicity,

the and spatial coordinates are omitted when clear from the context).

The mean squared quantization error between the vector and its quantized

representation , is given by

where is the codeword of the codebook

minimizing the reconstruction error on , and:

IV. A Locally Optimal PVQ DesignThe complexity of building a quantizer that scales to vectors of arbitrary

dimension is known to be computationally prohibitive. Furthermore, the

efficiency of a VQ decreases with its dimensionality: the fraction of the

volume of a hypersphere inscribed in a hypercube of the same dimension d

goes to zero when d goes to infinity ([2]-[3]). Partitioning the vectors in

consecutive, non-overlapping subvectors is a common solution to this

problem (see the book of Gersho and Gray [4]). While a partitioned VQ leads

to a sub-optimal solution in terms of Mean Squared Error (MSE) because it

8 CHAPTER xx

does not exploit correlation among subvectors, the resulting design is

practical and coding and decoding present a number of advantages in terms

of speed, memory requirements and exploitable parallelism.

In a Partitioned VQ, we divide the input vectors into subvectors and

quantize each of them with an -levels exhaustive search VQ. Having

subvectors of the same dimension (except perhaps one, if does not divide

) is not a possibility here, since we assume that the components of

may be drawn from different alphabets. In this case their distributions may be

significantly different and partitioning the components uniformly into

blocks may not be optimal. We wish to determine the size of the sub

vectors (of possibly different size) adaptively, while minimizing the

quantization error, measured for example in terms of MSE. Once the

codebooks are designed, input vectors are encoded by partitioning them into

subvectors of appropriate length, each of which is quantized

independently with the corresponding VQ. The index of the partitioned

vector is given by the concatenation of the indices of the subvectors.

Given the number of partitions N and the number of levels per codebook L,

it is possible to find the partition boundaries achieving minimum distortion

with a brute-force approach. We first determine, for every , the

distortion that an -levels vector quantizer achieves on the input

subvectors of boundaries and . Then, with a dynamic program, we traverse

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the matrix and find the costs corresponding to the input partition

of boundaries and whose sum is minimal. A more

sophisticated approach to partitioned vector quantization is discussed in

Matsuyama [5]. It uses dynamic programming at each step to decide the

current optimal partition, instead of designing one vector quantizer for each

possible partition configuration first and then applying dynamic

programming. Nevertheless, even this approach is computational intensive.

The locally optimal partitioning algorithm that we propose here (LPVQ)

provides an efficient alternative to dynamic programming, while performing

comparably in most applications. Our partitioning algorithm is based on a

variation of the Generalized Lloyd Algorithm (or GLA, Linde, Buzo and

Gray [6]).

Unconstrained vector quantization can be seen as the joint optimization of

an encoder (the function described before) and a decoder (the

determination of the codewords for the equivalence classes of the partition

). GLA is an iterative algorithm that, starting from the

source sample vectors chooses a set of codewords (also called centroids) and

optimizes in turns encoder and decoder until improvements on the predefined

distortion measure are negligible. In order to define our PVQ, the boundaries

of the vector partition have to be determined as

well. Our design follows the same spirit of the GLA. The key observation is

10 CHAPTER xx

that, once the partition boundaries are kept fixed, the Mean Square Error (the

distortion we use in our application) is minimized independently for each

partition by applying the well-known optimality conditions on the centroids

and on the cells. Similarly, when the centroids and the cells are held fixed,

the (locally optimal) partitions boundaries can be determined in a greedy

fashion. The GLA step is independently applied to each partition. The

equivalence classes are determined as usual, but as shown in Figure 2, the

computation keeps a record of the contribution to the quantization error of

the leftmost and rightmost components of each partition:

and

Except for the leftmost and rightmost partition, two extra components are

also computed:

and

The reconstruction values used in the expressions for and are

determined by the classification performed on the components . The

boundary between the partitions and is changed according to the

criteria

= min( , , )if ( = )

else if ( = )

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V.Entropy CodingThe partitioned vector quantizer is used here as a tool to reduce the

dimensionality of the source vectors. Dependence between the other two

dimensions has to be exploited by means of an entropy coder. Furthermore,

when a lossless coding is required, quantization residuals have to be encoded

as well, entropy coded with a conditioning on the subvector indices.

The index vectors and the codewords , with

and , are sufficient to a lossy reconstruction of the data.

When higher accuracy is needed, the compressed data can be augmented with

the quantization error:

where, for each :

The unconstrained quantizers work independently from each other and

independently on each source vector and an entropy encoder must be used to

exploit this residual redundancy. In particular, each VQ index is

encoded conditioning its probability with respect to a set of causal indices

spatially and spectrally adjacent. The components of the residual vector

are entropy coded with their probability conditioned on the VQ

index . In near-lossless applications, a small, controlled error can be

introduced at this stage. Figure 3 depicts a diagram of the LPVQ encoder.

12 CHAPTER xx

VI. Computational ComplexityVector Quantization is known to be computationally asymmetrical, in the

sense that encoding has a computational complexity which is orders of

magnitude higher than decoding. Even more time consuming is the design of

the dictionary which is also not guaranteed to converge into an easily

predictable number of iterations. Numerous methods have been designed to

speed up the convergence, for example, by constraining the search of the

closest codeword like in [7], or by seeding the dictionary with vectors having

special properties [8]. Adding structure to the VQ can also simplify encoding

at the price of a limited degradation of the output quality [4]. All these

improvements clearly apply to our method as well; furthermore, practical

scenarios arise in which our algorithm can be applied as is without the

computational complexity being an obstacle.

A possible way of mapping the LPVQ algorithm to the communication

model described in Section II is to equip platform used for the remote

acquisition with an encoder able to download a dictionary transmitted from

the central server (see Figure 1). This dictionary can be, for the very first

scene, bootstrapped as described in [8], with a number of vectors randomly

selected from the image itself, with a dictionary designed for another image

or with any dictionary that believed to be statistically significant. Due to the

nature of our algorithm, the choice of the first dictionary will only impact the

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compression ratio of the very first transmission.

Once the central server receives a scene, let’s say at time instant t, a few

iterations of LBG may be necessary to refine the dictionary, adjust the

partition boundaries and improve compression. The refined version of the

dictionary is transmitted back to the remote platform and used for the

encoding of the scene acquired at time t+1. In Section VIII we report on

experiments intended to assess the performance degradation derived from a

mismatched dictionary.

It has been observed in [1] that if the codewords in each codebook are

sorted according to their energy, the planes constituted by the indices

retain important information on the structure of the original image. These

planes alone can be used to browse and select areas of interest and, as we

show in section XX, to extract important information on the nature of the

original pixel vectors.

VII. Experimental resultsLPVQ has been tested on a set of five AVIRIS images ([9]). AVIRIS

images are obtained by flying a spectrometer over the target area. They are

614 pixels wide and typically have a height on the order of 2,000 pixels,

depending on the duration of the flight. Each pixel represents the light

reflected by a 20m x 20m area (high altitude) or 4m x 4m area (low altitude).

The spectral response of the reflected light is decomposed into 224

14 CHAPTER xx

contiguous bands (or channels), approximately 10nm wide and spanning

from visible to near infrared light (400nm to 2500nm). Each of these vectors

provides a spectral signature of the area. Spectral components are acquired

in floating point 12-bit precision and then scaled and packed into signed 16

bit integers. After acquisition, AVIRIS images are processed to correct for

various physical effects (geometrical distortion due to the flight trajectory,

time of day, etc.) and stored in scenes of 614 by 512 pixels each (when the

image height is not a multiple of 512, the last scene will be smaller). All files

for each of the 5 test images were downloaded from the NASA web site (JPL

[9]) and the scenes belonging to the same flight merged together to form a

complete image.

Several experiments have been performed for various numbers of partitions

and for different codebook sizes. The results that we describe here were

obtained for partitions and codebook levels. The choice of

the number of levels makes also practical the use of off-the-shelf image

compression tools that are fine-tuned for 8 bit data. The LPVQ algorithm

trains on each image independently and the codebooks are sent to the decoder

as side information. The size of the codebook is negligible with respect the

size of the compressed data (256 x 224 x 2 bytes = 112 KB) and its cost is

included in the reported results.

The partition boundaries for each of the five images are depicted in Figure

CHAPTER xx 15

4. While similarities exist, the algorithm converges to different optimal

boundaries on different input images. This is evidence that LPVQ adapts the

partitions to input statistics. Starting with a codebook obtained by random

image pixels, we have found that adaptation is fairly quick and boundaries

converge to their definitive values in less than one hundred iterations.

If LPVQ is bootstrapped with the method described in [8], which populates

the dictionary with the vector with highest norm and then adds the vector

whose minimum distance from the current dictionary is maximum, the

algorithms shows a much faster convergence. Furthermore, the constrained

search for the closest codeword is generally faster by a factor of 2.

LPVQ performances are analyzed in terms of Compression Ratio (defined

as the size of the original file divided by the size of the compressed one),

Signal to Quantization Noise Ratio, Maximum Absolute Error and

Percentage Maximum Absolute Error.

The Signal to Quantization Noise Ratio (SQNR) is defined here as:

The correction factor is introduced to take into account the error

introduced by the 12 bit analog-to-digital converter used by the AVIRIS

spectrometer ([9]). This solution also avoids unbounded values in the case of

a band perfectly reconstructed.

The Maximum Absolute Error (MAE) is defined in terms of the MAE for

16 CHAPTER xx

the band as , where is:

The average Percentage Maximum Absolute Error (PMAE) for the band

having canonical alphabet is defined as:

Table I shows that LPVQ achieves on the five images we have considered

an average compression of 3.14:1, 45% better than the 1-D lossless

compressor bzip2 when applied on the plane–interleaved images (worse

results are achieved by bzip2 on the original pixel–interleaved image format)

and 55% better than the standard lossless image compressor JPEG-LS. We

also compared LPVQ with the method published in [11], even though it

reports experiments on a subset of our data set. The algorithm in [11] uses

LBG to cluster data. An optimized predictor is computed for each cluster and

for each band. Prediction error is computed and entropy coded, along with

side information regarding the optimized predictors. The resulting method

outperforms LPVQ, while sharing time complexity similar to the design

stage of LPVQ and lacking any extra feature provided by the latter, like lossy

and near-lossless mode, fast browsing and classification, et cetera. The last

column of Table I reports, as a reference, the compression and the SQNR

when only the indices are encoded and the quantization error is fully

discarded. As we can see from the table, on average we achieve 55:1

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compression with 25.27dB of SQNR.

More interesting and practical are the results obtained with the

near-lossless settings, shown in Table II. At first, the introduction of a small

and constant quantization error across each dimension is considered; that is,

before entropy coding, each residual value x is quantized by dividing x

adjusted to the center of the range by the size of the range; i.e., q(x) =

(x+)/(2+1). This is the classical approach to the near-lossless

compression of image data and results into a constant MAE across all bands.

With this setting, it is possible to reach an average compression ratio ranging

from 4:1 with the introduction of an error and a to 10:1

with and error of and . While the performance in this

setting seem to be acceptable for most applications and the SQNR is

relatively high even at high compression, the analysis of the contribution to

the PMAE of the individual bands shows artifacts that might be

unacceptable. In particular, while the average PMAE measured across the

224 bands of the AVIRIS cube is low, the percentage error peaks well over

50% on several bands (see Figure 5). Since the PMAE is relevant in

predicting the performance of many classification schemes, we have

investigated two different approaches aimed at overcoming this problem. In

both approaches we select a quantization parameter that is different for each

band and it is inversely proportional to the alphabet size (or dynamic). In

18 CHAPTER xx

general, high frequencies, having in AVIRIS images higher dynamic, will be

quantized more coarsely than low frequencies. We want this process to be

governed by a global integer parameter .

The first method, aiming at a quasi–constant PMAE across all bands,

introduces on the band a distortion such that:

Since the band has alphabet we must have:

The alternative approach, aims at a quasi–constant SQNR across the bands.

If we allow a maximum absolute error on the band, it is reasonable to

assume that the average absolute error on that band will be . If we indicate

with the average energy of that band and with the target average

maximum absolute error, then the absolute quantization error allowed on

each band is obtained by rounding to the nearest integer the solution of this

system of equations:

As can be seen from Table II, the three methods for near-lossless coding of

AVIRIS data are equivalent in terms of average SQNR at the same

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compression. However, the quasi-constant PMAE method is indeed able to

stabilize the PMAE across each band (Figure 6). The small variations are due

to the lossless compression of some bands and the rounding used in the

equations. The average SQNR is not compromised as well. Similar results

are observed for the quasi-constant SQNR approach. The SQNR is almost

flat (Figure 7), except for those bands that are losslessly encoded and those

with small dynamic. The PMAE is also more stable than the constant MAE

method (Figure 8).

VIII. LPVQ and Random DictionariesIn order to evaluate the proposed approach in the framework described in

Section II, we performed the following experiments for each scene of the

Moffett Field image:

1) we compute compression ratio and RMSE at each step of the LPVQ

algorithm, starting with a random dictionary;

2) same except that the starting dictionary is the locally optimal one

for the previous scene.

As we can see from Figure 9, compression performance is quite stable

already after just 50 iterations. The compression ratio is slightly lower when

LPVQ is fed with the dictionary from the previous scene, while RMSE, on

the other hand, is much lower. This suggests that the entropy coder needs to

be tuned-up for this case and that the lossy compression is slightly better

20 CHAPTER xx

when only few iterations are allowed.

A different experimental setting uses six different schemes to generate

random dictionaries. One hundred dictionaries for each scheme are used with

PVQ (i.e., the dictionary is not refined with the proposed LBG-like scheme

to obtain a locally optimal one) and with both uniform and optimal partitions.

Compression results for each scheme are box-plotted in Figure 10. The

results suggest that taking a small sample of the input data is enough to

obtain compression performances reasonably close to the locally optimal

solution produced by LPVQ, and that the non-uniform partitioning is much

more effective at high compression.

IX. The Problem of ClassificationLet each image pixel be a -dimensional vectors

divided into disjoint subvectors as

, the -th partition having dimension ,

for . For simplicity the boundaries are omitted here.

Upon quantization each vector is represented by an -dimensional vector

of indices and by

, a -dimensional vector representing

the quantization error. Each component indexes one of the

codebook centroids for the VQ corresponding to the partition , .

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If is the quantized value for , LPVQ decomposes each component of an

input vector into as , where .

Given a target vector , a distortion measure

and a classification threshold , we wish to determine, for every

image pixel , whether . When this condition is

satisfied, we say that the pixel is a member of the class .

The Euclidean Distance is distortion measures

commonly used in classification.

The classification threshold , plays an important role since it determines

whether the classification is used to detect a rare pixel instance, eventually

present in the image (problem commonly referred to as target detection) or

whether the objective of the classification is the determination of a thematic

map, where almost every pixel belongs to one of several classes.

X.Effect of Coding Error on the ClassificationAs we did in the case of the compression we used AVIRIS hyperspectral

images to assess the effect of the coding error on classification. Classification

and target detection are algorithms frequently applied to hyperspectral

images; when the images are lossily encoded, it is licit to ask whether and in

which measure the coding error affects the results of these algorithms.

In order to assess the classification errors introduced by LPVQ lossy

22 CHAPTER xx

modes, we have experimented with the standard images for various errors

and threshold parameters. It is important to notice that, while the lossy results

are peculiar of our algorithm, the near lossless apply to any compression

scheme that bounds the maximum absolute error on each vector component.

The results for three images, Cuprite, Moffet Field and Low Altitude, are

presented here. Results on the other two images were very similar.

In our experiments, the absolute error introduced by LPVQ in near lossless

mode is respectively bounded to a maximum of 1, 2, 4 and 8 units. The

classification uses Euclidean Distance with an average threshold per band

; typically, values in the range are of some interest

in practical applications. The figures show the percentage of errors in the

classification of 8 targets selected from each image. The targets are hand

picked from the reduced resolution image constituted by the index planes and

are meant to represent targets of common interest (buildings, roads, water,

vegetation, etc...). The lossless encoding is used as a reference for the

classification and the error percentages reflect the sum of all false positive

(classifications that were not classified in the lossless), false negative (vectors

classified in the lossless and not classified in the lossy) and misclassifications

(different clusters in lossless and lossy).

As it is possible to see from Figures 11-13, in near lossless, the

classification is fundamentally unaffected; even for an error of , the

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percentage of classification errors is, in the worst case, below 0.1%. This is

sufficient for most applications. Smaller coding errors, achieve even lower

percentages. For the compression parameters used in Section VII (i.e., 256

code vectors and 16 partitions) the indices and the code vectors alone (Figure

14, lossy case) achieve a classification error always lower than 3%. While

this error rate may be unsuitable for some applications, as we show in the

next section, it is possible to take advantage of this characteristic to design a

faster classifier.

XI. Speeding up the Classification Since the LPVQ indices provide most of the information necessary to

classify the pixels, we designed a simple algorithm that, bounding the

minimum and maximum quantization error for each component and for each

cluster, can decide the classification of most pixels without inspecting their

error vector . Bounds on the quantization error are available at design

time or can be inferred from the encoded image. The advantages of this

algorithm are twofold. Firstly, the number of partitions and the number of

centroids are typically small, so pre-computed values can be arranged into a

lookup table. If most vectors can be classified without the need of the error,

this will result into a substantial speed up. Secondly, we can imagine a user

on the field, browsing and trying different classifications on the LPVQ

indices instead of the whole image. Then, when a parameterization of interest

24 CHAPTER xx

is found, the user can query the server for the error vectors necessary to

resolve any classification ambiguity.

Given a threshold , a target vector and a distortion measure

, we wish to determine, from the quantization indices of a pixel

, two quantities and , respectively lower

and upper bounding the distortion .

Since belongs to the class only if , we can

observe that:

If , the pixel clearly does not belong to

the class ;

If , the pixel trivially belongs to the class

;

If then the quantization indices

and the bounds on the error are not sufficient to decide the

classification. In this case, and only in this case, it is necessary to

access the error vector and compute .

Figure 15 shows the pseudocode that implements this scheme. The main

assumption is that the quantities and can be

computed by summing the individual contributions of each vector component

, upper and lower bounded by the minimum and maximum of the

quantization residuals and .

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This is clearly the case of Euclidean Distance, here defined as:

Changing the threshold into , we can focus on the contribution of

and to the individual terms of the sum. For a given

component , , in the partition , we have:

.

If has been quantized with the centroid , :

and

.

Summing these bounds over all components of the partition we obtain

and . These quantities are precomputed in the

algorithm in Figure 15 by the two nested “for” loops. Since there are only

possible values and and are typically small ( and

in Section VII), and can be stored in a

lookup table indexed by and . The computation of and

requires a total of lookups and sums. If one

of the two comparisons and succeeds,

then the pixel is classified from its indices only, with a number of

operations proportional to .

Otherwise, the error vector must be retrieved, exactly

26 CHAPTER xx

reconstructed as and finally

computed. This requires a sum a subtraction and a product for each

component and sums to accumulate the result, i.e. a total of

subtractions, sums and products, a number of operations

proportional to (for AVIRIS images, ).

The speed of the new classification algorithm depends on the probability

that the two tests fail and that we have to use the error vector in order to

classify. Let this probability be:

If , and are respectively the times to perform a subtraction, a

sum and a multiplication on a given computing platform, the total running

time of the new classification is:

Figures 16-18 show an experimental assessment of the quantity

for three AVIRIS images. They have

been classified using 100 randomly selected targets. Average results are

reported for thresholds and quantizers having 256, 512 and 1024

levels. It is possible to see how the percentage of vectors on which the

simplified test fails to classify the points reaches a maximum between 7.5%

and 11% for a 1024 levels quantizer and 12% to 18% for a 256 level

quantizer. Practical figures are typically better. Problems like target detection

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are solved by applying a low threshold, while classification requires a high

one; in these regions of the curve, the percentages may be much lower than

the maxima. These results can be used in two ways. Given an image encoded

with LPVQ, we can improve the speed the naïve classification with the use of

a lookup table. Alternatively, we can use this as a tool to select the number of

quantization levels and match a desired average classification speed

XII. ConclusionsWe have presented an extension of the GLA algorithm to the locally

optimal design of a partitioned vector quantizer (LPVQ) for the encoding of

source vectors drawn from a high dimensional source on . It breaks down

the input space into independent subspaces and for each subspace designs a

minimal distortion vector quantizer. The partition is adaptively determined

while building the quantizers in order to minimize the total distortion.

Experimental results on lossless and near-lossless compression of

hyperspectral imagery have been presented, and different paradigms of

near-lossless compression are compared. Aside from competitive

compression and progressive decoding, LPVQ has a natural parallel

implementation and it can also be used to implement search, analysis and

classification in the compressed data stream. High speed implementation of

our approach is made possible by the use of small independent VQ

codebooks (of size 256 for the experiments reported), which are included as

28 CHAPTER xx

part of the compressed image (the total size of all codebooks is negligible as

compared to the size of the compressed image and our experiments include

this cost). Decoding is no more than fast table look-up on these small

independent tables. Although encoding requires an initial codebook training,

this training may only be necessary periodically (e.g., for successive images

of the same location), and not in real time. The encoding itself involves

independent searches of these codebooks (which could be done in parallel

and with specialized hardware).

Abstract—We present a new low complexity algorithm for hyperspectral image compression based on linear prediction targeted at spectral correlation. We describe a simple heuristic to detect contexts in which linear prediction is likely to perform poorly and a context modeling mechanism with one band look-ahead capability, that improve the overall compression with only marginal extra storage space. Therefore the proposed method is suitable to spacecraft on-board implementation, where limited hardware and low power consumption are key requirements. Finally we present a least squares optimized linear prediction method which, to the best of our knowledge, achieves better compression than any other method targeted at AVIRIS images published so far. We tested the proposed algorithms on data cubes acquired with the NASA JPL’s Airborne Visible/Infrared Imaging Spectrometer (AVIRIS).

[2] Index Terms—Linear predictive coding, data compression, remote sensing, 3D data.[3] EDICS(TSPL)—1.FAST, 2.IMMD[4] EDICS(TSP)—3.LOSS,

Compression of Hyperspectral Imagery via Linear Prediction

Francesco Rizzo, Bruno Carpentieri, Giovanni Motta, and James A. Storer

CHAPTER xx 29

XIII. INTRODUCTION

EMOTE acquisition of high definition electro-optic images is

becoming central to military and civilian applications, such as

surveillance, geology, environmental monitoring, and meteorology. In fact, it

is possible to recognize materials and their physical state from the spectrum

of the electromagnetic energy they reflect or emit. Hyperspectral sensors

span visible and invisible light spectra by taking regularly spaced

measurements in contiguous bands. A typical hyperspectral detector is the

NASA JPL’s Airborne Visible/Infrared Imaging Spectrometer (AVIRIS [1]).

The operational spectral range is the visible and near infrared region (from

0.4 to 2.4 μm). Spectral information is quantized into 224 contiguous bands,

of approximately 10 nm each, with a spatial resolution of 20 m2 at

operational altitude. Spectral components are acquired with a 12 bit DAC and

then represented with 16 bit precision after calibration and geometric

corrections. The unit size of the recorded image is the scene, a data cube of

512 lines by 614 columns by 224 bands, for a total of 140MB; the final

image (flight) typically consists of three or more consecutive scenes.

R

With the rapid development of analytic tools, the demand for higher spatial

and spectral resolution will increase dramatically in the near future. On the

other hand, transmission and distribution bandwidth is growing at a much

30 CHAPTER xx

slower rate than the data volume, hence efficient data compression is

required. If the scope of the remote acquisition is target detection and/or

classification, a compression algorithm that provides lossless or near lossless

quality may be required. In addition, it may be desirable to have low

complexity algorithms suitable to on board implementation with limited

hardware and power consumption. Traditional approaches to the compression

of hyperspectral imagery are based on differential PCM ([2]-[4]), direct

vector quantization ([5]-[8]) or dimensionality reduction through principal

component analysis ([9],[10]). An inter-band linear prediction approach

based on least squares optimization is presented in [11]; this compression

method optimizes, for each sample, the prediction coefficients with spatial

and spectral support.

In previous work ([12]) we presented a locally optimal partitioned vector

quantizer (LPVQ) for encoding high dimensional data, and applied it to

lossless, near-lossless, and lossy compression of AVIRIS hyperspectral

images. The compression achieved by LPVQ is aligned with the current state

of the art, and its decoding complexity is extremely low (entropy decoding

followed by table lookup). However, encoding is relatively complex, and still

more complex than decoding even when codebooks are supplied.

Here we introduce a new low complexity algorithm targeted at the

compression of hyperspectral images. Section 2 begins by presenting a low

CHAPTER xx 31

complexity hyperspectral compression algorithm based on inter-band

prediction (LP), and then introduces SLSQ, a more aggressive method that

optimizes the predictor for each pixel and each band. Section 3 describes

experiments with these predictive compression methods applied to AVIRIS

images, improvements to the baseline methods and complexity analysis. The

latter shows, surprisingly, that SLSQ is roughly equivalent to LP in terms of

time and leaner in terms of space complexity. As expected, SLSQ is also

superior to LP and its compression improves the current state of the art.

Section 4 discusses future research.

XIV. Inter-Band Linear Prediction

The standard algorithm for lossless image coding, JPEG-LS [13], is based

on the modeling/coding paradigm: an image is traversed in a specific order

(usually raster); a prediction value , based on a finite subset of past data, is

generated for the current sample ; the prediction context, also based on the

past, is also determined; a probability model, for the prediction residual

is generated, conditioned on the prediction context and the residual

is finally entropy coded using that model. In order to reduce complexity,

JPEG-LS makes stringent assumptions about the input data, resulting in a

low cost, very efficient compression algorithm for photographic 8 bit images.

Unfortunately the same assumptions do not hold when compressing

32 CHAPTER xx

hyperspectral imagery.

Remote sensed images, like AVIRIS, show two forms of correlation:

spatial (adjacent pixels are likely to share common materials) and spectral

(one band can be fully or partly predicted from other bands). Spectral

correlation is generally, but not always, stronger than spatial correlation.

Furthermore, dynamic range and noise level (instrument noise, reflection

interference, aircraft movements, etc.) of AVIRIS data are higher than those

in photographic images. For these reasons a spatial predictor like the median

predictor in JPEG-LS tends to fail on this kind of data. For example, Figure

1 shows the performance (in bits per sample) of the median predictor versus

LPVQ: the predictor is inefficient almost everywhere and especially in the

visible part of the spectrum, characterized by large dynamic range and

accounting for almost half of the data. Nevertheless, JPEG-LS fast and

efficient compression would be highly desirable to an on-board, hardware

implementation.

Motivated by these considerations, we propose a new approach (here

named LP) that uses a median predictor for bands marked for intra-band

coding ( set) and a new inter-band linear predictor for the rest. Both inter

and intra band predictors rely on a small causal data subset of the pixel

in the -th row, -th column of the -th band to compute the prediction

. The inter-band prediction is formed by simply adding the average

CHAPTER xx 33

difference between the current band the previous one to the value of . If

the deviation between the maximum and the minimum difference is larger

than a given threshold , LP assumes that the inter-band prediction is likely

to fail and corrects it with the average prediction error taken on the previous

two bands. Figure 2 shows a block diagram of the proposed method

(inclusive of the context modeling mechanism mentioned later). LP

prediction is formally written as

(1)where , , .

The prediction error is encoded with an arithmetic coder.

A. Least Squares Optimization

To assess the performance of LP when in inter-band ,mode, we use a

Spectral oriented Least SQuares (SLSQ) optimized linear predictor. SLSQ

determines for each sample the coefficients of a linear predictor that is

optimal, in the lest square sense, with respect a 3D subset of past data. We

employ a prediction structure and notation similar to [14]: two context

34 CHAPTER xx

enumeration shown in Figure 3, are defined by the distance functions

(2)

(3)

In the following, denotes the -th pixel of the 2D context in the above

enumeration. Moreover, denotes the -th pixel in the 3D context of

. The -th order prediction of the current pixel ( , we drop

the subscript and the parenthesis when referring to the current pixel) is

computed as:

(4)

The coefficients minimizing the energy of the prediction

error

(5)

are calculated by using the well-known theory on optimal linear prediction.

Notice that the data used in the prediction are a causal, finite sub–set of the

past data and no side information needs to be sent to the decoder.

Using matrix notation, we write where

(6)

CHAPTER xx 35

By taking the derivative with respect to and setting it to zero, the

optimal predictor coefficients are the solution of the linear system

(7)Once the optimal predictor coefficients have been determined for the

current sample, the prediction error is entropy coded as in the

previous method.

XV. Experimental results

Table I reports the compression ratio obtained by LP and SLSQ on five

publicly available AVIRIS images. We compare these results to the baseline

JPEG-LS, JPEG2000 ([15]), and LPVQ. An average compression of 3.06 to

1 is also reported in [11], although experiments reported there refer to a

subset of our data set and to images having slightly smaller dimension. As we

can see, LP is on average only 2% worse than LPVQ, while SLSQ is 4%

better. The baseline LP algorithm is applied with and no

prediction threshold ( ). SLSQ uses the same set, with and

.

A. Time complexity

In order to compare the time efficiency of the proposed methods, we

assume a model in which the cost is determined by the number of additions

and multiplications, with multiplications being computationally more

36 CHAPTER xx

expensive than additions. From (1) we can see that LP requires at each step

in the worst case. This count assumes the buffering of and

, and ignores the cost of context modeling used in LP-CTX (which is

not constant). ***** !?!?!? FIRST TIME YOU MENTION LP_CTX !?!?!?!

*****

Least squares optimization can be solved with the method of normal

equations with floating point operations (additions,

multiplications, and subscripting; [16]). In the case of SLSQ, we

experimented with different values of and . Setting and

achieves an excellent tradeoff between computational burden and

compression performance. Larger contexts provide slight compression

improvements (if any), at much higher computational cost. With and

, we can write (7) as

(8).

Using limited buffering, we need to compute and only.

It follows that each prediction step of SLSQ could be implemented with just

, a computational cost roughly double with respect to LP (with no

context modeling), but with much lower memory requirements.

CHAPTER xx 37

B. Improvements of baseline LP and SLSQ.

To improve the baseline LP and SLSQ methods, we introduce the

following mechanism:

LP-CTX uses a simplified version of the context modeling

described in [12]. The prediction threshold T is here set to .

In the HEU option the IB set, common to all input images, has been

determined by an off-line heuristic: for each scene of the data set

we check which band is better compressed spatially (median

predictor) rather than spectrally (LP or SLSQ). Bands that are inter

encoded at least 60% of the time form the IB set.

OPT assumes that the encoder checks for the best method first. This

requires virtually no side information (1 bit/band) and a one band

look-ahead

C. Results

Results of improved LP and SLSQ are reported in Table II. LP-CTX with

one band look-ahead closes the gap with LPVQ at the cost of a small increase

of storage requirements over baseline LP. Regarding the SLSQ approach,

compression efficiency is about 6% higher than LPVQ, at a fraction of the

high computational cost of its design stage. ***** !?!?!? BEFORE YOU

SAID IT IS 4% BETTER !?!?!?! *****

38 CHAPTER xx

XVI. Conclusion

We propose two novel approaches to lossless coding of AVIRIS data. One

is based on an inter-band, linear predictor the other on an inter-band, least

squares optimized predictor. Coupled with a simple entropy coder, they

respectively compete with and outperform the current state of the art. The

low complexity of the proposed methods and their raster scan nature, make

them amenable for on-board implementations: AVIRIS and other space

borne sensors acquire data in a spectral interleaved fashion and have reduced

storage capacity. ***** !?!?!? AVIRIS IS NOT SPACE BORNE !?!?!?!

***** Parallel implementations are also possible. Finally, the proposed

methods depend loosely on the entropy coder; experiments are in progress to

assess the effect of replacing the arithmetic coder with the CCSDS standard

algorithm ([17]) for lossless data compression for space applications, whose

hardware implementation is available and widely used in aerospace

applications.

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[5] NASA, “AVIRIS home page,” http://popo.jpl.nasa.gov, February 2003

[6] G. P. Abousleman, “Compression of hyperspectral imagery using hybrid

DPCM/DCT and entropy constrained trellis coded quantization,” in

CHAPTER xx 39

Proceedings Data Compression Conference, J.A. Storer and M. Cohn,

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lossless compression of AVIRIS images,” in Proceedings of the XI

40 CHAPTER xx

European Signal Processing Conference, EUSIPCO-2002, Toulouse,

France, September 2002, EURASIP.

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Hyperspectral Data Compression Using Principal Component

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SPIE Optical Engineering, 41:9, 1-10, 2002.

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4725, no. 8, pp. 600-608, 2002.

[16] G. Motta, F. Rizzo, and J. A. Storer, “Compression of Hyperspectral

Imagery,” in Proceedings Data Compression Conference, J.A. Storer and

M. Cohn, Eds. 2003, pp. 333-342, IEEE Computer Society Press.

[17] M. J. Weinberger, G. Seroussi, and G. Sapiro, “The LOCO-I lossless

image compression algorithm: Principles and standardization into

JPEG-LS,” IEEE Transactions Image Processing, vol. 9, pp. 1309-1324,

August 2000.

[18] D. Brunello, G. Calvagno, G. A. Mian, and R. Rinaldo. “Lossless

video coding using optimal 3D prediction,” in Proceedings of the 9th

IEEE International Conference on Image Processing (ICIP 2002),

CHAPTER xx 41

Volume 1, Rochester, NY, pp. 89–92. IEEE Signal Processing Society,

September 2002.

[19] D. S. Taubman and M. W. Marcellin, Jpeg2000: Image Compression

Fundamentals, Standards, and Practice. Kluwer Academic Publishers,

2001

[20] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd Edition.

The Johns Hopkins University Press, 1996.

[21] Consulting Committee for Space Data Systems, “Recommendation

for space data system standards: Lossless Data Compression”, CCSDS

121.0-B-1, Blue Book, May 1997.

42 CHAPTER xx

Fig. 1. Empirical entropy per band of the Median predictor.

Fig. 2. LP block diagram.

CHAPTER xx 43

Fig. 3. Two and three-dimensional pixel enumerations based on (3)-(4).

TABLE ICOMPARISON OF COMPRESSION ACHIEVED

JPEG-LS JPEG2000 LPVQ LP SLSQCuprite 2.09 1.91 3.13 3.03 3.15Jaspder Ridge 2.00 1.80 2.82 2.94 3.15Low Altitude 2.14 1.96 2.89 2.76 2.98Lunar Lake 1.99 1.82 3.23 3.05 3.15Moffett Field 1.91 1.78 2.94 2.88 3.14AVERAGE 2.03 1.85 3.00 2.93 3.12

44 CHAPTER xx

TABLE IICOMPARISON OF COMPRESSION ACHIEVED

LP-CTX SLSQBASE HEU OPT HEU OPT

Cuprite 3.04 3.07 3.09 3.23 3.24Jasper Ridge 2.96 2.98 3.00 3.22 3.23Low Altitude 2.79 2.79 2.83 3.02 3.04Lunar Lake 3.06 3.08 3.10 3.23 3.23Moffett Field 2.93 2.94 2.96 3.20 3.21AVERAGE 2.96 2.97 3.00 3.18 3.19

Figure 1: Communication model.

Figure 2: Error contributions for two adjacent partitions.

Figure 3: LPVQ lossless encoder.

Figure 4: Partition sizes and alignment.

CHAPTER xx 45

Figure 5: PMAE for near-lossless coding with constant Maximum Absolute Error.

Figure 6: PMAE for near-lossless coding with quasi-constant PMAE.

Figure 7: SQNR for near-lossless coding with quasi-constant SQNR.

Figure 8: PMAE for near-lossless coding with quasi-constant SQNR.0

10

20

30

40

50

60

70

80

1 11 21 31 41 51 61 71 81 91

LPVQ Iterations

2.84

2.86

2.88

2.90

2.92

2.94

2.96

0

10

20

30

40

50

60

70

1 11 21 31 41 51 61 71 81 91

LPVQ Iterations

2.86

2.88

2.90

2.92

2.94

2.96

2.98

3.00

3.02

0

10

20

30

40

50

60

1 11 21 31 41 51 61 71 81 91

LPVQ Iterations

3.06

3.08

3.10

3.12

3.14

3.16

3.18

3.20

3.22

0

5

10

15

20

25

30

35

40

45

1 11 21 31 41 51 61 71 81 91

LPVQ Iterations

2.88

2.90

2.92

2.94

2.96

2.98

3.00

Figure 9: Moffett Field RMS/C.R. vs LPVQ iterations. RMSE(p) and C.R.(p) values indicates that the starting dictionary was the one of the previous scene, rather than random vectors.

46 CHAPTER xx

Figure 10: performances of Partitioned Vector Quantization with Random Dictionaries on a scene 2 of Moffett Field. xR1 = 25600 random vectors from the input data are grouped in blocks of 100, the centroid of each block will be one of the 256 vectors used by PVQ. xR2 = similar to xR1 with the random vectors sorted by position in the input before clustering and averaging. xR3 = 256 random vectors in the first lines of input are used as dictionary. xR4 = like xR2 but with sorting by energy instead. xR5 = like xR4 but with just 2560 input vectors. LR4.10 = 256 random vectors (drawn from the entire image, not just the first lines, like in xR3).

0

0.02

0.04

0.06

0.08

0.1

0.12

Threshold

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Threshold

Figure 11: Cuprite, classification errors in near-lossless compressed images.

Figure 12: LowAltitude, classification errors in near-lossless compressed images.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Threshold

0

0.5

1

1.5

2

2.5

3

Threshold

CHAPTER xx 47

Figure 15: MoffetField, classification errors in near-lossless compressed images.

Figure 14: Classification error in LPVQ lossy compressed images.

Input:; // Target vector

; // Quantized image and // Min/Max error

for // component k and

// centroid lOutput:

// Classification

For each centroid with // PrecomputationFor each partition with

Compute from Compute from

For each pixel

If = 0; // Not in the class

Else if = 1; // In the class

Else = ; // Use the error

Figure 15: Fast classification algorithm.

48 CHAPTER xx

Cuprite (Avg on 100 Random Targets)

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

18.00%

20.00%

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

Threshold

2565121024

Figure 16: Image Cuprite. Percentage of vectors that are not classified by the

simplified test. Results averaged for 100 random targets. Quantizers have

256, 512 and 1024 levels.

LowAltitude (Avg on 100 Random Targets)

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

Threshold

2565121024

Figure 17: Image Low Altitude. Percentage of vectors that are not classified by the simplified test. Results averaged for 100 random targets. Quantizers have 256, 512 and 1024 levels.

CHAPTER xx 49

MoffetField (Avg on 100 Random Targets)

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

14.00%

16.00%

18.00%

5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

Threshold

2565121024

Figure 18: Image Moffet Field. Percentage of vectors that are not classified by the simplified test. Results averaged for 100 random targets. Quantizers have 256, 512 and 1024 levels.

AVIRIS Lossless Indices Onlygzip bzip2 JPEG-LS JPEG 2K [11] LPVQ CR SQNR

Cuprite 1.35 2.25 2.09 1.91 3.42 3,27 53,44 24,15Jasper Ridge 1.39 2.05 1.91 1.78 3.46 3,12 51,08 24,44Low Altitude 1.38 2.13 2.00 1.80 N.A. 2,97 54,32 25,50Lunar Lake 1.36 2.30 2.14 1.96 3.37 3,31 59,17 26,91Moffett Field 1.41 2.10 1.99 1.82 3.46 3,01 58,03 25,32Average 1.38 2.17 2.03 1.85 3.43 3,14 55,21 25,27

Table I: Compression ratio for lossless and lossy mode.

Constant MAE Quasi-Constant PMAE Quasi-Constant SQNRCR RMSE SQNR MAE PMAE CR RMSESQNR MAE PMAE CR RMSESQNR MAE PMAE

1 4.03 0.82 46.90 1.00 0.19 3.41 0.73 52.15 0.57 0.00 3.45 0.78 51.81 0.62 0.012 4.80 1.41 42.49 2.00 0.39 3.97 1.50 47.34 1.54 0.02 3.86 1.56 48.11 1.54 0.023 5.49 1.98 39.64 3.00 0.58 4.46 2.27 44.35 2.53 0.03 4.33 2.36 45.06 2.54 0.034 6.16 2.52 37.60 4.00 0.77 4.95 3.03 42.04 3.56 0.04 4.75 3.15 42.91 3.53 0.045 6.83 3.04 36.05 5.00 0.96 5.36 3.80 40.39 4.54 0.06 5.15 3.95 41.18 4.54 0.066 7.50 3.51 34.85 6.00 1.16 5.77 4.56 38.97 5.55 0.07 5.51 4.74 39.85 5.52 0.07

50 CHAPTER xx

7 8.16 3.97 33.88 7.00 1.35 6.16 5.29 37.83 6.52 0.08 5.88 5.54 38.61 6.55 0.088 8.80 4.39 33.08 8.00 1.54 6.55 6.04 36.83 7.53 0.10 6.23 6.31 37.57 7.54 0.109 9.42 4.80 32.41 9.00 1.72 6.95 6.76 35.88 8.53 0.11 6.55 7.06 36.73 8.52 0.11

10 10.01 5.19 31.84 9.98 1.85 7.32 7.48 35.14 9.53 0.13 6.90 7.82 35.91 9.53 0.1311 - - - - - 7.68 8.15 34.48 10.51 0.14 7.25 8.53 35.19 10.54 0.1412 - - - - - 8.07 8.80 33.87 11.53 0.16 7.57 9.21 34.61 11.51 0.1613 - - - - - 8.45 9.42 33.32 12.54 0.17 7.94 9.84 34.01 12.51 0.1714 - - - - - 8.83 10.01 32.84 13.53 0.19 8.29 10.47 33.50 13.53 0.1915 - - - - - 9.17 10.56 32.44 14.51 0.20 8.63 11.03 33.08 14.52 0.2016 - - - - - 9.55 11.08 32.05 15.51 0.22 8.97 11.57 32.67 15.53 0.2117 - - - - - 9.91 11.58 31.69 16.51 0.23 9.31 12.08 32.32 16.52 0.2318 - - - - - 10.32 12.04 31.33 17.53 0.25 9.66 12.55 31.97 17.52 0.2419 - - - - - 10.67 12.47 31.04 18.51 0.26 10.00 12.99 31.66 18.52 0.2620 - - - - - 11.02 12.89 30.78 19.52 0.28 10.35 13.40 31.35 19.52 0.27

Table II: Average Compression Ratio (CR), Root Mean Squared Error (RMSE), Signal to Quantization Noise Ratio (SQNR), Maximum Absolute Error (MAE) and Percentage Maximum Absolute Error (PMAE) achieved by the near-lossless LPVQ on the test set for different .

REFERENCES

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CHAPTER xx 51

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design”, IEEE Transactions on Communications 28, 84-95.[7] M. Manohar and J. C. Tilton [2000]. “Browse Level Compression of AVIRIS

Data Using Vector Quantization on Massively Parallel Machine,” Proceedings AVIRIS Airborne Geoscience Workshop.

52 CHAPTER xx

[8] Y. Matsuyama [1987]. “Image Compression Via Vector Quantization with Variable Dimension”, TENCON 87: IEEE Region 10 Conference Computers and Communications Technology Toward 2000, Aug. 25-28, Seoul, South Korea.

[9] R. E. Roger, J. F. Arnold, M. C. Cavenor and J. A. Richards, [1991]. “Lossless Compression of AVIRIS Data: Comparison of Methods and Instrument Constraints”, Proceedings AVIRIS Airborne Geoscience Workshop.

[10] S. R. Tate [1997], “Band ordering in lossless compression of multispectral images”, IEEE Transactions on Computers, April, 477-483.

[11] G. Shaw and D. Manolakis [2002], “Signal Processing for Hyperspectral Image Exploitation”, IEEE Signal Processing Magazine 19:1

[12] S. Subramanian, N. Gat, A. Ratcliff and M. Eismann [1992]. “Real-Time Hyperspectral Data Compression Using Principal Component Transform”, Proceedings AVIRIS Airborne Geoscience Workshop.

[13] G. O. Young, “Synthetic structure of industrial plastics (Book style with paper title and editor),” in Plastics, 2nd ed. vol. 3, J. Peters, Ed. New York: McGraw-Hill, 1964, pp. 15–64.

[1] [2] NASA, “AVIRIS home page,” http://popo.jpl.nasa.gov, February 2003[3].[4]. [5] M. Manohar and J.C. Tilton, “Browse level compression of AVIRIS data using vector quantization on massively parallel machine,” in Proceedings AVIRIS Airborne Geosciences Workshop, 2000.SPURGARE LE REFERENCES ED AGGIUNGERLE

In the 1993 the ITU-T (International Telecommunication Union / Telecommunication Standardization Sector) developed H.261, a video coding standard for audiovisual services at bit rates multiple of 64 Kbit/s. Bit rate was chosen because of the availability of ISDN (Integrated Services Digital Network) transmission lines that could be allocated in multiples of 64Kbit/s. As a consequence of that choice, in older drafts, H.261 is also referred as Kbit/s.

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