CN108053455B - Vector quantization hyperspectral image compression method based on linear prediction decorrelation - Google Patents

Vector quantization hyperspectral image compression method based on linear prediction decorrelation Download PDF

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CN108053455B
CN108053455B CN201711421608.1A CN201711421608A CN108053455B CN 108053455 B CN108053455 B CN 108053455B CN 201711421608 A CN201711421608 A CN 201711421608A CN 108053455 B CN108053455 B CN 108053455B
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潘志斌
李�瑞
王洋
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Xian Jiaotong University
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Abstract

The invention discloses a vector quantization hyperspectral image compression method based on linear prediction decorrelation, which comprises the following steps: the method comprises the following steps: collecting a hyperspectral image to be compressed, dividing all spectral bands of the hyperspectral image to be compressed into a plurality of classes by using a clustering method, and taking the clustering center of each class as a generated reference spectral band; step two: predicting all spectral bands in the hyperspectral image to be compressed by using reference spectral bands, regarding the reference spectral bands as a group of substrates, projecting all spectral bands in the hyperspectral image to be compressed onto the group of substrates, and predicting the spectral bands through projected coefficients; the prediction residual is the result after redundancy removal; step three: carrying out VQ coding on the prediction residual error to complete the compression of the hyperspectral image to be compressed; the final image is compressed into two parts, the reference spectral band and the projection coefficients of the spectral band prediction algorithm, and the codebook and index values in the VQ algorithm. The invention provides an effective method for removing redundancy of a spectrum band, and the effectiveness of the method is verified through experiments.

Description

Vector quantization hyperspectral image compression method based on linear prediction decorrelation
Technical Field
The invention belongs to the field of hyperspectral image processing, and particularly relates to a vector quantization hyperspectral image compression method based on linear prediction decorrelation.
Background
Hyperspectral images are typically the result of imaging the same scene in hundreds of spectral bands. The hyperspectral images have obvious advantages in reflecting material components of targets in the images, and nowadays, due to the fact that hyperspectral data are paid more and more attention in the aspects of national defense, agriculture, geology and the like. However, the transmission and processing of the hyperspectral images are restricted by a large amount of redundancy existing in the hyperspectral images, and therefore the compression of the hyperspectral images is also a popular research subject. Since there are a lot of redundancies in hyperspectral images, these redundancies are divided into spatial and inter-spectral redundancies. The compression method for hyperspectral images can be summarized into three categories: predictive coding methods, transform-based methods and vector quantization-based methods. Vector quantization is a theoretically optimal block coding algorithm, and for a compression method of a hyperspectral image, VQ is mainly applied to removing spectral band redundancy. At present, point-by-point prediction schemes are mostly used for spectral band redundancy removal modes of hyperspectral images, the method has large calculation amount, and the effect of the method using spectral band direct prediction is not ideal.
Disclosure of Invention
The invention aims to provide a vector quantization hyperspectral image compression method based on linear prediction decorrelation, and the method is used for solving the problem that in the prior art, due to the fact that a large amount of spatial redundancy and inter-spectral redundancy exist in a hyperspectral image, transmission and processing of the hyperspectral image are severely limited. According to the invention, the hyperspectral images are researched, and it is concluded that the inter-spectral redundancy in the hyperspectral images is obviously stronger than the spatial redundancy. The present invention therefore focuses on eliminating inter-spectral redundancy. Meanwhile, the strong correlation among the spectrums is considered, and the method performs integral prediction on the spectrum band to achieve a better effect of eliminating the redundancy among the spectrums.
In order to achieve the purpose, the invention adopts the following technical scheme to realize the purpose:
a vector quantization hyperspectral image compression method based on linear prediction decorrelation comprises the following steps:
the method comprises the following steps: collecting a hyperspectral image to be compressed, dividing all spectral bands of the hyperspectral image to be compressed into a plurality of classes by using a clustering method, and taking the clustering center of each class as a generated reference spectral band;
step two: predicting all spectral bands in the hyperspectral image to be compressed by using reference spectral bands, regarding the reference spectral bands as a group of substrates, projecting all spectral bands in the hyperspectral image to be compressed onto the group of substrates, and predicting the spectral bands through projected coefficients; the prediction residual is the result after redundancy removal;
step three: carrying out VQ coding on the prediction residual error to complete the compression of the hyperspectral image to be compressed; the final image is compressed into two parts, the reference spectral band and the projection coefficients of the spectral band prediction algorithm, and the codebook and index values in the VQ algorithm.
Further, the first step specifically includes:
reference band generation procedure:
1.1) calculating a cross-correlation coefficient matrix among all the spectral bands;
1.2) taking each row in the cross-correlation coefficient matrix as a characteristic vector to carry out K-means clustering;
1.3) averaging the bands of the same class to be used as reference bands;
in the clustering algorithm, cross-correlation coefficient vectors among all spectral bands are calculated to serve as input features of the clustering algorithm, and the cross-correlation coefficient calculation method among any spectral bands is as follows:
Figure BDA0001523145300000021
in the formula (1), i and j respectively represent the spatial position of any pixel point, M and N represent the length and width of a hyperspectral image to be compressed, and p and q represent two different spectral bands; r isq(p)＝rp(q)，rp(q) represents the cross-correlation coefficient of spectral bands p and q, I (I, j, q) represents any point in the hyperspectral image,
Figure BDA0001523145300000022
represents the mean of the bands q;
obtaining a cross-correlation coefficient matrix R by calculating cross-correlation coefficients among all the bands, wherein R is formed by vecq(q ═ 1,2,. cndot, P);
vecq＝{rq(1),rq(2),...,rq(p),...,rq(P)},p＝1,2,...,P (2)
the cross correlation coefficient vector vec of any spectral band is obtainedqThen, clustering all the cross-correlation coefficient vectors by using a K-means algorithm;
the clustering algorithm proceeds as follows:
inputting: p cross correlation coefficient vectors vec for all spectral bandsqAnd the cluster number K, q ═ 1, 2.., P;
2.1) randomly generating K clustering centers
Figure BDA0001523145300000031
2.2) calculating the distance from all the cross-correlation coefficient vectors to K clustering centers, and dividing each cross-correlation coefficient vector into a class to which a clustering center with the nearest distance belongs;
2.3) calculating the mean value of all the cross-correlation coefficient vectors of each class as a new clustering center of the class;
Figure BDA0001523145300000032
2.4) when
Figure BDA0001523145300000033
When the algorithm is stopped, the output is output
Figure BDA0001523145300000034
The final clustering center is obtained, the class to which each cross correlation coefficient vector belongs is the final classification result, and otherwise, the step 2.2) is repeated;
and (3) outputting: classification results and clustering centers for all input cross-correlation coefficient vectors
Figure BDA0001523145300000035
Finally, obtaining a clustering result through a clustering algorithm as a final classification result of all spectral bands;
Figure BDA0001523145300000036
ref is obtained by the formula (4)kK is the final reference band obtained.
Further, the second step specifically comprises:
in the case of band prediction:
Figure BDA0001523145300000037
in the formula y denotes the band to be predicted, refkDenotes the reference band, and wkRepresenting the projection of the spectral band y onto this reference spectral band. n iskFor noise, by solving for wkA prediction coefficient is obtained.
Further, solving the optimization problem constrained by the formulas (6) and (7) to obtain a final prediction coefficient;
Figure BDA0001523145300000041
Figure BDA0001523145300000042
wherein X denotes the reference band refkA matrix of X ═ ref1,ref2,...,refK}；
w is a prediction coefficient, w ═ w1,w2,...,wK)，wlsIs a prediction coefficient obtained by a least square method; i ispIs a spectral band to be predicted and is,
Figure BDA0001523145300000043
is a prediction of the band; by solving this optimization problem, the final result is expressed as formula (8) and formula (9);
wls＝(XTX)-1XTIp(8)
Figure BDA0001523145300000044
Figure BDA0001523145300000045
is a downward rounding function; the final prediction result is the result of removing the inter-spectral correlation of the spectral bands, and the linear prediction process linearly represents all the spectral bands by taking the reference spectral band as a base through the inter-spectral overall prediction so as to achieve the effect of removing the inter-spectral correlation.
Further, the VQ encoding process is specifically as follows:
3.1) carrying out block processing on the image;
3.2) generating a code book by using an LBG algorithm;
3.3) for each block, finding the index idx corresponding to the closest block in the codebook,
Figure BDA0001523145300000046
wherein c isdIs any code word in the code book, and the code book contains sbA code word;
the final codebook and the index value idx are the final result of the coding;
the specific process of the LBG algorithm is as follows:
inputting: the number s of code words contained in all input image blocks v and codebooksb；
4.1) random Generation of sbAn initial code word
Figure BDA0001523145300000051
s＝1,2,…,sb；
4.2) calculating all input image blocks to sbThe distance of each code word divides each input image block into the class to which the code word with the closest distance belongs;
4.3) calculating the average value of all input image blocks of each class as a new code word of the class;
Figure BDA0001523145300000052
wherein a(s) represents
Figure BDA0001523145300000053
The number of input image blocks contained in the class to which the input image block belongs;
4.4) when
Figure BDA0001523145300000054
When the algorithm is stopped, the output is output
Figure BDA0001523145300000055
Is the final code word, otherwise repeats 4.2);
and (3) outputting: all code words
Figure BDA0001523145300000056
Further, the method also comprises the following steps:
step four: when decoding the compressed image, the method is divided into two parts: decoding by using a VQ algorithm to obtain a prediction residual error, predicting all spectral bands by using reference spectral bands, and finally obtaining a reconstructed hyperspectral image.
Further, the number K of reference spectral bands generated by carrying out K-means clustering is 4.
Further, the code word subjected to VQ encoding in step three is a 4 × 4 block.
Compared with the prior art, the invention has the following advantages: the redundancy among the spectrums can be eliminated through the scheme of overall prediction of the spectrum bands, and the method has the performance of simpler and more convenient calculation; when the whole spectral band prediction is carried out, the correlation coefficient is used as a characteristic vector, so that the calculation amount of clustering is reduced; the mode of acquiring the reference spectral band through clustering is simpler and more effective than the mode of conversion; finally, the hyperspectral image can be well compressed at a lower code rate.
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FIG. 1 is a flow chart of the present invention.
FIG. 2 is a graphical illustration of the results of the statistical band-to-space correlation on Indian _ pings; wherein FIG. 2(a) is the mean inter-spectral correlation; FIG. 2(b) is the average correlation listed in band 50.
Fig. 3 is a matrix of all cross-correlation coefficient vectors.
FIG. 4 is band IpProjection results on the reference spectral band.
Fig. 5 is a composition of the method compression-encoded stream.
FIG. 6 is a comparison of a compressed reconstructed image and an original image; wherein FIG. 6(a) is the 50 th spectral band of the original drawing; FIG. 6(b) shows the 50 th spectral band of the reconstructed image after compression;
fig. 7 is a graph of the rate-distortion curves for the compression method of the present invention using different size codebook sizes sb.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings:
referring to fig. 1 to 7, a vector quantization hyperspectral image compression method based on linear prediction decorrelation according to the present invention includes the following steps:
the method comprises the following steps: collecting the hyperspectral images to be compressed, and dividing all spectral bands in the hyperspectral images to be compressed into a plurality of classes by using a clustering method through the fact that the spectral bands have strong correlation, as shown in figure 2, so as to generate reference spectral bands.
The reference band generation procedure was as follows:
(1) calculating a cross correlation coefficient matrix among all the spectral bands;
(2) taking each row in the cross-correlation coefficient matrix as a characteristic vector to perform K-means clustering;
(3) averaging the bands of the same kind to be used as reference bands.
In the clustering algorithm, cross-correlation coefficient vectors among all spectral bands are calculated to serve as input features of the clustering algorithm, and the cross-correlation coefficient calculation method among any spectral bands is as follows:
Figure BDA0001523145300000061
in the formula (1), i and j respectively represent the spatial position of any pixel point, M and N represent the length and width of a hyperspectral image to be compressed, and p and q represent two different spectral bands. r isq(p)＝rp(q)，rp(q) represents the cross-correlation coefficient of spectral bands p and q, I (I, j, q) represents any point in the hyperspectral image,
Figure BDA0001523145300000071
represents the mean of the bands q.
Obtaining a cross-correlation coefficient matrix R by calculating cross-correlation coefficients among all the bands, wherein R is formed by vecq(q ═ 1,2,.., P). As shown in fig. 3.
vecq＝{rq(1),rq(2),...,rq(p),...,rq(P)},p＝1,2,...,P (2)
The cross correlation coefficient vector vec of any spectral band is obtainedqAnd then, clustering all the cross-correlation coefficient vectors by using a K-means algorithm.
The clustering algorithm proceeds as follows:
inputting: p cross correlation coefficient vectors vec for all spectral bandsq(q ═ 1,2,.., P), and the number of clusters K.
1) Randomly generating K cluster centers
Figure BDA0001523145300000072
2) And calculating the distances from all the cross-correlation coefficient vectors to the K clustering centers, and dividing each cross-correlation coefficient vector into the class to which the nearest clustering center belongs.
3) And calculating the mean value of all the cross-correlation coefficient vectors of each class as a new clustering center of the class.
Figure BDA0001523145300000073
ε represents a minimum value near 0 to limit the iterative convergence of the algorithm.
4) When in use
Figure BDA0001523145300000074
When the algorithm is stopped, the output is output
Figure BDA0001523145300000075
And (4) the final clustering center is obtained, the class to which each cross-correlation coefficient vector belongs is the final classification result, and otherwise, the step 2) is repeated.
And (3) outputting: classification results and clustering centers for all input cross-correlation coefficient vectors
Figure BDA0001523145300000076
And finally, obtaining a clustering result through a clustering algorithm as a final classification result of all spectral bands.
Figure BDA0001523145300000077
Ref is obtained by the formula (4)kK is the final reference band obtained.
Step two: prediction is performed using reference spectral bands, which are considered as a set of bases, onto which all spectral bands in the hyperspectral image are projected, which are predicted by the coefficients of the projection, as shown in fig. 4. The prediction residual is the result of redundancy removal. In the case of band prediction:
Figure BDA0001523145300000081
in formula (5), y represents the band to be predicted, refkDenotes the reference band, and wkRepresenting the projection of the spectral band y onto this reference spectral band. n iskIs noise;
the final prediction coefficients can be obtained by solving the optimization problem constrained by equations (6) and (7).
Figure BDA0001523145300000082
Figure BDA0001523145300000083
Wherein X denotes the reference band refkA matrix of X ═ ref1,ref2,...,refK}。
w is a prediction coefficient, w ═ w1,w2,...,wK)，wlsIs a prediction coefficient obtained by the least square method. I ispIs a spectral band to be predicted and is,
Figure BDA0001523145300000084
is a prediction of the band; by solving this optimization problem, the final result is expressed as formula (8) and formula (9).
wls＝(XTX)-1XTIp(8)
Figure BDA0001523145300000085
Figure BDA0001523145300000086
Is a downward rounding function. The final prediction result is the result of removing the inter-spectral correlation of the spectral bands, and the linear prediction process linearly represents all the spectral bands by taking the reference spectral band as a base through the inter-spectral overall prediction so as to achieve the effect of removing the inter-spectral correlation.
Step three: and carrying out VQ coding on the prediction residual error with a large amount of correlation eliminated, thereby achieving the purpose of improving the coding performance. The final image is compressed into two parts, the reference spectral band and the projection coefficients of the spectral band prediction algorithm, and the codebook and index values in the VQ algorithm. Since a large amount of correlation is eliminated by the prediction residual, the coding performance can be improved. In the third step, the code word for VQ coding is a 4 × 4 block, and the size of the codebook is selected according to the size of the spectral band in the hyperspectral image to be compressed. The encoding process of the VQ algorithm is specifically as follows:
(1) the image is processed in blocks, and overlap between blocks can occur.
(2) The codebook is generated using the LBG algorithm.
(3) For each block formed vector v, the index idx corresponding to the closest block is searched in the codebook,
Figure BDA0001523145300000091
wherein c isdIs any code word in the code book, and the code book contains sbA code word.
The final codebook and index value idx are the final result of the encoding.
The specific process of the LBG algorithm is as follows:
inputting: all input image blocks v and codebooks contain the number K of codewords.
1) Random generation of sbAn initial code word
Figure BDA0001523145300000092
2) Calculating all input image blocks to sbDistance of each codeword, dividing each input image block into the class to which the closest codeword belongs.
3) All input image blocks of each class are averaged as a new codeword of that class.
Figure BDA0001523145300000093
Where a(s) represents the number of input image blocks comprised by the class.
4) When in use
Figure BDA0001523145300000094
When the algorithm is stopped, the output is output
Figure BDA0001523145300000095
Is the final codeword, otherwise repeat 2).
And (3) outputting: all code words
Figure BDA0001523145300000096
The encoded stream obtained by this step is as shown in fig. 5.
Step four: the compressed image is decoded and reconstructed. The steps can be divided into the following modes:
(1) decoding by using a VQ algorithm to obtain a prediction residual error, wherein the decoding process is the inverse process of encoding, and the reconstruction process of the prediction residual error is completed by searching a code word corresponding to the index value in a code book.
(2) All bands are predicted using a reference band, and the process of generating the predicted bands is calculated using the reference band and the prediction vector (prediction coefficient) for each band. Each dimension of the prediction vector is the projection value of the original spectral band to be predicted on the corresponding base (reference spectral band). Predicting the spectral band is therefore the process of combining these projections. Equation (12) shows that this process finally yields a reconstructed hyperspectral image.
Figure BDA0001523145300000102
The code rate of the final method is calculated as follows
Figure BDA0001523145300000101
In the formula, K is the number of reference bands, and the parameter is determined when K-means are clustered, and s isbIs the size of the codebook, swIs the dimension of the codeword. The final output coded stream consists of three parts, namely a reference spectral band, coefficients of a predictor and a codebook and an index in VQ.
It can be seen from fig. 6 and 7 that the hyperspectral image compression method can well complete the compression of the hyperspectral image. FIG. 6(a) shows a spectral band in a hyperspectral image, and (b) shows an image recovered after being compressed by the method, so that the image can be compressed well without greatly influencing the image quality. Fig. 7 shows the comparison effect of the method of the present invention in selecting codebooks of different sizes, and it can be seen that the rate-distortion performance of the method of the present invention can already have better performance on a codebook of 256 codewords.

Claims (8)

1. A vector quantization hyperspectral image compression method based on linear prediction decorrelation is characterized by comprising the following steps:
the method comprises the following steps: collecting a hyperspectral image to be compressed, dividing all spectral bands of the hyperspectral image to be compressed into a plurality of classes by using a clustering method, and taking the clustering center of each class as a generated reference spectral band;
step two: predicting all spectral bands in the hyperspectral image to be compressed by using reference spectral bands, regarding the reference spectral bands as a group of substrates, projecting all spectral bands in the hyperspectral image to be compressed onto the group of substrates, and predicting the spectral bands through projected coefficients; the prediction residual is the result after redundancy removal;
step three: carrying out VQ coding on the prediction residual error to complete the compression of the hyperspectral image to be compressed; the final image is compressed into two parts, the reference spectral band and the projection coefficients of the spectral band prediction algorithm, and the codebook and index values in the VQ algorithm.
2. The linear prediction decorrelation-based vector quantization hyperspectral image compression method according to claim 1, wherein the first step specifically comprises:
reference band generation procedure:
1.1) calculating a cross-correlation coefficient matrix among all the spectral bands;
1.2) taking each row in the cross-correlation coefficient matrix as a characteristic vector to carry out K-means clustering;
1.3) averaging the bands of the same class to be used as reference bands;
in the clustering algorithm, cross-correlation coefficient vectors among all spectral bands are calculated to serve as input features of the clustering algorithm, and the cross-correlation coefficient calculation method among any spectral bands is as follows:
Figure FDA0002381811560000011
in the formula (1), i and j respectively represent the spatial position of any pixel point, M and N represent the length and width of a hyperspectral image to be compressed, and p and q represent two different spectral bands; r isq(p)＝rp(q)，rp(q) represents the cross-correlation coefficient of spectral bands p and q, I (I, j, q) represents any point in the hyperspectral image,
Figure FDA0002381811560000012
represents the mean of the bands q;
obtaining a cross-correlation coefficient matrix R by calculating cross-correlation coefficients among all the bands, wherein R is formed by vecq(q ═ 1,2,. cndot, P);
vecq＝{rq(1),rq(2),...,rq(p),...,rq(P)},p＝1,2,...,P (2)
p represents the number of spectral bands;
the cross correlation coefficient vector vec of any spectral band is obtainedqThen, clustering all the cross-correlation coefficient vectors by using a K-means algorithm;
the clustering algorithm proceeds as follows:
inputting: p cross correlation coefficient vectors vec for all spectral bandsqAnd the cluster number K, q ═ 1, 2.., P;
2.1) randomly generating K clustering centers
Figure FDA0002381811560000021
2.2) calculating the distance from all the cross-correlation coefficient vectors to K clustering centers, and dividing each cross-correlation coefficient vector into a class to which a clustering center with the nearest distance belongs;
2.3) calculating the mean value of all the cross-correlation coefficient vectors of each class as a new clustering center of the class;
Figure FDA0002381811560000022
2.4) when
Figure FDA0002381811560000023
When the algorithm is stopped, the output is output
Figure FDA0002381811560000024
The final clustering center is obtained, the class to which each cross correlation coefficient vector belongs is the final classification result, and otherwise, the step 2.2) is repeated;
and (3) outputting: classification results and clustering centers for all input cross-correlation coefficient vectors
Figure FDA0002381811560000025
Finally, obtaining a clustering result through a clustering algorithm as a final classification result of all spectral bands;
Figure FDA0002381811560000026
ref is obtained by the formula (4)kK is the final reference band obtained.
3. The linear prediction decorrelation-based vector quantization hyperspectral image compression method according to claim 2, wherein the second step specifically comprises:
in the case of band prediction:
Figure FDA0002381811560000031
in the formula y denotes the band to be predicted, refkDenotes the reference band, and wkRepresents the projection of the spectral band y onto the reference spectral band; n iskFor noise, by solving for wkA prediction coefficient is obtained.
4. The linear prediction decorrelation-based vector quantization hyperspectral image compression method according to claim 3, wherein a final prediction coefficient is obtained by solving an optimization problem constrained by formulas (6) and (7);
Figure FDA0002381811560000032
Figure FDA0002381811560000033
wherein X denotes the reference band refkA matrix of X ═ ref1,ref2,...,refK}；
w is a prediction coefficient, w ═ w1,w2,...,wK)，wlsIs a prediction coefficient obtained by a least square method; i ispIs a spectral band to be predicted and is,
Figure FDA0002381811560000034
is a prediction of the band; by solving this optimization problem, the final result is expressed as formula (8) and formula (9);
wls＝(XTX)-1XTIp(8)
Figure FDA0002381811560000035
Figure FDA0002381811560000036
is a floor function; the final prediction result is the result of removing the inter-spectral correlation of the spectral bands, and the linear prediction process linearly represents all the spectral bands by taking the reference spectral band as a base through the inter-spectral overall prediction so as to achieve the effect of removing the inter-spectral correlation.
5. The linear prediction decorrelation-based vector quantization hyperspectral image compression method according to claim 2, wherein the VQ encoding process is specifically as follows:
3.1) carrying out block processing on the image;
3.2) generating a code book by using an LBG algorithm;
3.3) for each block, finding the index idx corresponding to the closest block in the codebook,
Figure FDA0002381811560000041
wherein c isdIs any code word in the code book, and the code book contains sbA code word;
the final codebook and the index value idx are the final result of the coding;
the specific process of the LBG algorithm is as follows:
inputting: the number s of code words contained in all input image blocks v and codebooksb；
4.1) random Generation of sbAn initial code word
Figure FDA0002381811560000042
4.2) calculating all input image blocks to sbThe distance of each code word divides each input image block into the class to which the code word with the closest distance belongs;
4.3) calculating the average value of all input image blocks of each class as a new code word of the class;
Figure FDA0002381811560000043
wherein a(s) represents the number of input image blocks included in the class;
4.4) when
Figure FDA0002381811560000044
When the algorithm is stopped, the output is output
Figure FDA0002381811560000045
Is the final code word, otherwise repeats 4.2);
and (3) outputting: all code words
Figure FDA0002381811560000046
6. The vector quantization hyperspectral image compression method based on linear prediction decorrelation according to claim 1, characterized by further comprising the following steps:
step four: when decoding the compressed image, the method is divided into two parts: decoding by using a VQ algorithm to obtain a prediction residual error, predicting all spectral bands by using reference spectral bands, and finally obtaining a reconstructed hyperspectral image.
7. The vector quantization hyperspectral image compression method based on linear prediction decorrelation according to claim 2, wherein the number K of reference spectral bands generated by K-means clustering is 4.
8. The linear prediction decorrelation-based vector quantization hyperspectral image compression method according to claim 5, wherein code words subjected to VQ coding in the third step are 4 x 4 blocks.
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