Vector quantization for saturated SAR raw data compression

Available online at www.sciencedirect.com

Advances in Space Research 45 (2010) 1330–1337 www.elsevier.com/locate/asr

Vector quantization for saturated SAR raw data compression Bin Hua a,b,c,*, Haiming Qi a,c,*, Ping Zhang a, Xin Li a,b a

Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China b Graduate School of Chinese Academy of Sciences, Beijing 100039, China c National Key Laboratory of Microwave Imaging Technology, Beijing 100190, China Received 3 July 2009; received in revised form 27 December 2009; accepted 7 January 2010

Abstract Spaceborne SAR involves the storage and transmission of large-size sampling data. Block adaptive quantization (BAQ) is now the most widely used onboard data compression algorithm due to its good tradeoff between system performance and complexity. However, when spaceborne SAR raw data is saturated, the performance of conventional BAQ deteriorates dramatically because its precondition of Gaussian distribution of raw data no longer holds. In order to solve this problem, an improved vector quantization (VQ) algorithm is proposed. This algorithm firstly introduces saturation modification to a conventional vector quantizer, obtains the saturation codebook based on Gaussian density function, and then obtains the new vector quantizer for the whole set of Saturation Degree (SD). This algorithm makes the vector quantizer match statistical model of data for the whole set of SD, so the performance of the compression is improved. The case of the 2D signal is explicitly computed. The performance of the proposed algorithm is verified by simulated and real data experiments. Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Synthetic aperture radar (SAR); Data compression; Raw data; Saturation; Block adaptive quantization (BAQ); Vector quantization (VQ)

1. Introduction A spaceborne SAR (Synthetic Aperture Radar) system faces the challenge of storage and transmission of mass data. A SAR system may collect data at a high rate that easily exceeds the capacity of the downlink channel. In addition, when no ground-station can be reached, the data must be recorded on some mass storage medium, the storage capacity of which can easily be exhausted. The situation has become even more severe in the last few years with the increased requirements of modern SAR systems, including high resolution, multi-polarization, multi-beam, three-dimensional mapping, wide swath,

*

Corresponding authors. Address: Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China. Tel.: +86 13811405265 (B. Hua); tel.: +86 13581727216 (H. Qi). E-mail addresses: [email protected], [email protected] (B. Hua), [email protected] (H. Qi).

multi-frequency, and multi-operation mode. As a result, effective data compression becomes necessary. Various raw data compression algorithms have been proposed in the past 30 years; these method can generally be divided into three categories: (1) scalar compression algorithms, (2) vector compression algorithms and (3) transform domain compression algorithms. Scalar compression algorithms include block adaptive quantization(BAQ) (Kwok and Johnson, 1989), block floating point quantization (BFPQ) (Boustani et al., 2001), fuzzy BAQ (FBAQ) (Benz et al., 1995; Boustani et al., 2001), entropy-constrained BAQ (ECBAQ) (Benz et al., 1995; Boustani et al., 2001), and flexible block adaptive quantization (FBAQ) (Mcleod et al., 1998). Vector compression algorithms involve vector quantization (VQ) (Gresho and Gray, 1992), block adaptive vector quantization (BAVQ) (Benz et al., 1995; Boustani et al., 2001), block gain adaptive vector quantization (BGAVQ) (Lebedeff et al., 1995), and trellis coded vector quantization (TCVQ) (Boustani et al., 2001).

0273-1177/$36.00 Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2010.01.007

B. Hua et al. / Advances in Space Research 45 (2010) 1330–1337

Transform domain compression algorithms include FFT-BAQ (Benz et al., 1995; Fischer et al., 1999), WHTBAQ (Fischer et al., 1999), DCT-BAQ (Fischer et al., 1999), wavelet transform (WT) (Boustani et al., 2001) and compressed sensing (Baraniuk and Steeghs, 2007; Bhattacharya et al., 2007). Although transform domain algorithms and vector quantization algorithms have better performance than that of scalar compression algorithm, their complexity has prevented their onboard implementation. Furthermore, BAQ is the most widely used onboard data compression technique due to its good tradeoff between performance and complexity. An important problem in SAR raw data compression is that the large dynamic range of SAR echo may cause the analog-to-digital converter (ADC) to saturate (Shimada, 1996, 1999a,b). Once the saturation occurs, the output of ADC becomes a truncated Gaussian random signal instead of a Gaussian one, which does not satisfy the precondition of a Lloyd-Max quantizer (Lloyd, 1982; Max, 1960). This will result in the degradation of the conventional BAQ algorithm. This disadvantage also exists for the VQ algorithms. This paper firstly reviews the relationships between standard deviation of the input signal (SDIS), average signal magnitude (ASM), and SD (Qi and Yu, 2009). Because the density matching property of VQ is very powerful, this paper studies the principle of VQ and obtains a universal codebook for unsaturated SAR data based on Gaussian density function. In the case of saturation, standard VQ performs rather poorly. This paper therefore modifies the VQ and proposes an anti-saturation vector quantizer in the whole set of SD. Section 2 reviews the relationships between ASM, SDIS and SD. Section 3 introduces the principle of VQ as well as the design of a codebook based on Gaussian density function. Section 4 makes the modification to the vector quantizer which results in the optimal vector quantizer for the whole set of SD. In Section 5, experiments are carried out based on simulated and real data. Conclusions are drawn in the last section. In this paper, boldface variables represent vectors, while non-boldface variables represent scalars or functions in a continuous domain.

2. Mutual relationships between ASM, SDIS and SD 2.1. Statistical properties of SAR raw data after ADC It is important to know the statistics of the sampled echoes during SAR raw data compression. SAR echo can be thought of as the superposition of the responses of many small scatterers in all the azimuth-range resolution cells. Based on central limit theorem, the in-phase (I) and quadrature (Q) components are both Gaussian distributed. In other words, the amplitude is in Rayleigh distribution, while the phase is uniformly distributed in interval ½p; p. Since the received signals in both I and Q channels are zero-mean Gaussian, the distribution can be specified with a single parameter. For simplicity, the parameter ASM is selected in the engineering implementation. For an 8-bit ADC, Kwok and Johnson (1989) have given the relationship between ASM and SDIS, and Qi et al. (2008) revised it as Eq. (1)   126 X nþ1 erf pffiffiffi ð1Þ jIj ¼ jQj ¼ 127:5  2r n¼0 Rx where erf ðxÞ ¼ p2ffiffip 0 expðt2 Þdt, and r is the SDID of the ADC. When saturation occurs, the SD can be measured from the raw data as follows (Qi and Yu, 2009; Shimada, 1999a,b), SD ¼

N satu  100% N total

SD ¼ 2

Z

1

f ðxÞdx

100 80 60 40

ð3Þ

M

where f ðxÞ is the probability density function (PDF) of normalized Gaussian distribution, M is the normalization saturation threshold. Fig. 1 shows the relationships between ASM, SDIS and SD, respectively.

(c) 100 Saturation degree (%)

120

ð2Þ

where N total is the total number of sampled data, and N satu denotes the number of saturated ones. Eq. (2) can be generalized for continuously random variables (Qi and Yu, 2009)

(b) 100 Saturation degree (%)

Average signal magnitude

(a) 140

1331

80 60 40 20

80 60 40 20

20 0

0

200

400

600

800

1000

Standard deviation of input signal

0

0

200

400

600

800

1000

Standard deviation of input signal

0

0

20

40

60

80

100

120

Average signal magnitude

Fig. 1. (a) Relationship between ASM and SDIS; (b) relationship between ASM and SDIS; (c) relationship between SD and ASM.

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In practice, M corresponds to the peak-to-peak value (PPV) of ADC. For standard Gaussian signal, the relationships between SD, the PPV and SDIS (Qi and Yu, 2009) are shown in Fig. 2, while the relationships among SD, PPV and ASM are shown in Fig. 2(b). As a result, all the statistical properties of SAR raw data after ADC can be obtained if the ASM of a specific ADC is known. 2.2. Statistical model of saturated data in multivariate Suppose that f ðxÞ is multivariate Gaussian distributed as:

  1 1 T 1 f ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  ðx  lx Þ Cx ðx  lx Þ 2 k ð2pÞ jCx j

Fig. 4. Bi-variants histogram of ERS-1 saturated raw data.

ð4Þ

ital number range. Fig. 3 shows the bi-variant standard Gaussian distribution and bi-variant saturated standard Gaussian distribution respectively. Fig. 4 is the statistical bi-variant histograms of ERS-1 saturated raw data, with 5-bit ADC.

where lx and Cx are, respectively, the mean and covariance matrix of x, and k is the dimension of vector x. When the input data exceed the ADC conversion range, its output is no longer scattered in the tails of the distribution, but forced into the minimum and maximum values of the dig-

(a)

(b)

100

Saturation degree (%)

Saturation degree (%)

100 80 60 40 20 0 0

C

60 40 20 0 0

20

AD

80

40

pe

ak

to

AD 20 C 40 pe ak 60 to pe 80 ak

0

pe

60

50 100

ak 80 100

n of in

eviatio

150

200

ard d Stand

nal put sig

100

60

80

100

de

agnitu

nal m

e sig verag

0

20

40

A

Fig. 2. (a) Relationship between SD, PPV and SDIS; (b) relationship between SD, PPV and ASM.

(a)

(b) 0.14

0.14 0.15

0.15

0.1

0.1

0.08

0.12

density

density

0.12 0.1

0.1 0.08

0.05

0.05 0.06 0 3

0.04 2 1 0

y

−1 −2 −3

−3

−2

−1

0

x

1

2

3

0.06 0 3

0.04 2

0.02

1 0

y

−1 −2

Fig. 3. Bi-variants standard Gaussian distribution.

−3

−3

−2

−1

0

x

1

2

3 0.02

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3. Principle of vector quantization

Step 1 can guarantee optimal partition (Linde et al., 1980).

Vector quantization (VQ) is a well known technique for signal compression, and it is also the generalization of the scalar quantization. It fully takes advantages of (Lebedeff et al., 1995): (1) the space filling properties and (2) the probability density function shape of the source and (3) both linear and non linear dependencies between vector components. VQ can be defined as a mapping of k dimensional Euclidean space R into the codebook Y, which can be represented as :

Step 2 (CC): Assuming that R is given, a necessary condition for D to be at a stationary value is (Chen, 1977) :

ð5Þ yi ¼ QðyÞ for i ¼ 1; 2; . . . ; N  T T  T T where Y ¼ y1 ; y2 ; . . . ; yN ; x ¼ ½x1 ; x2 ; . . . ; xk  ; yi ¼ ½y i1 ; y i2 ; . . . ; y ik T , and N is the length of the codebook. This paper uses mean-square error (MSE) as a distortion measure, which is defined as 2

D ¼ Eðkx  QðxÞk Þ ¼

N X

2

Eðkx  yi k Þ

i¼1

¼

N X i¼1

Z

2

kx  yi k pðxÞdx

ð6Þ

Ri

where pðxÞ is joint probability density function. The main purpose of VQ design is to find a codebook that minimizes the value of D. In the following, two main steps of VQ, called the Nearest Neighbor Condition (NNC) and the Centroid Condition (CC) (Patane´ and Russo, 2001), are introduced: Step 1 (NNC): with preassigned Y, the best partition of Ri can be written as: 2

2

Ri ¼ fx : kx  yi k 6 kx  yj k ; for all j – ig and

N [

Ri ¼ R

i¼1

ð7Þ

rY D ¼ 0N k

ð8Þ

where r is the gradient operator. Eq. (8) can be rewritten as: 2 @D 3    @y@D @y 11 1k 7 @D 6 6 .. .. 7 ¼ 6 ... ð9Þ rY D ¼ 7 . . 5 @Y 4 @D    @y@D @y N1

Substituting Eq. (6) into Eq. (9), we get Z @D ¼ 2 ðxj  y ij ÞpðxÞdx ¼ 0 @y ij Ri

raw data sub-block

Vector Formation

By iteratively applying the two conditions, we can obtain a VQ design that is at least locally optimal. Lebedeff et al. (1995) proposed an adaptive vector quantization scheme for space-borne raw SAR data compression. It operates as a set of optimum vector quantizers – based on the LBG (Linde et al., 1980) algorithm – with different gain settings. Its block diagram is shown in Fig. 5. The block gain adaptive vector quantization (BGAVQ) utilizes the statistical characteristics of echo and divides them into sub-blocks of the same size, and then calculates the ASM of each sub-block and obtains the SDIS for the ADC. After that, the sampled data is normalized using Downlink

Vector Search Nearest Neighbor Rule 1

SDIS N

ASM

ð10Þ

for all i ¼ 1; 2; . . . ; N ; j ¼ 1; 2; . . . ; K. It can be further written as: R xj pðxÞ dx ð11Þ y ij ¼ RRi pðxÞ dx Ri

Data Forming system onboard

Raw Data (I/Qchannel)

Nk

codebook VQ Fig. 5. Block diagram of BGAVQ.

Index Ground Data Processing System

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the SDIS, and assigned a corresponding codeword. Finally, the index of the codeword is transmitted to the ground data processing system for subsequent processing.

Start

Initial

4. Anti-saturation optimal vector quantizer When saturation occurs, the normalized signal is not joint Gaussian distribution random signal any more, so the vector quantizer design in Section 3 is not optimal for the saturated output signal, and needs to be modified. However, the partitioned regions within the ADC conversion range remain in the original distribution; therefore, in those regions Eq. (11) can still be used. For simplicity, Ris and Rins are adopted to denote the regions with and without S saturation data respectively, and they satisfy Ri ¼ Ris Rins . So Eq. (11) may be rewritten as: R R x pðxÞ dx þ Ris xj pðxÞ dx Rins j R ð12Þ y ij ¼ pðxÞ dx Ri

Partition region

No

Judge Saturation

Normal centroid

Consider the truncation: 8 if xj > M; > : M if xj 6 M:

Yes

Saturated centroid

Distortion calculation Yes

(Dm-1 - Dm )/Dm <

No

ð13Þ Eq. (12) may be modified as: R R x pðxÞ dx þ Ris M 1 pðxÞ dx Rins j R y ij ¼ pðxÞ dx Ri

Final codebook ð14Þ

Fig. 6 shows the partition of two-dimensional cell. From Fig. 6 we may see that saturation threshold M partitions the region into nine segments. The saturation data exist VQ Cell 3 2.5

IV

V

III M

2 R7

m=m+1

R8

R9

1.5 1

End Fig. 7. Flow chart of the codebook design.

in all segments except IX. ADC saturation differs in each segment ðI–VIIIÞ. Using the centroid equation Eq. (11), we may get the centroid of partition region R1 -R6 , and from Eq. (14), we may get the centroid of partition region R7 –R16 . The flow chart of the codebook design is shown in Fig. 7. The difference between conventional VQ and our

R2 R16

Y

0.5

R10

R1

0

R3

R5 VI

II

−0.5

R11

R15

R6

R4

−1 IX

−1.5

R14

R12

R13

−2 −2.5 −3 −3

VII

−2.5

I

VIII

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

X

Fig. 6. Practical example of partition of two-dimensional cell.

Table 1 Comparison of performances of BAQ and new algorithm on simulated data in data domain. File

SD (%)

BAQ (SNR/dB)

New algorithm (SNR/dB)

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10

0 4.55 19.36 31.73 39.53 50.93 61.71 76.42 84.15 92.03

14.63 15.13 15.36 11.99 14.40 12.42 5.89 10.87 7.98 3.81

15.24 16.52 18.88 20.57 21.69 23.17 25.09 27.69 30.09 33.05

B. Hua et al. / Advances in Space Research 45 (2010) 1330–1337

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Fig. 8. Scheme of SNR calculation.

algorithm is the partition region blocks (see Appendix A), and saturated centroid block. 5. Numerical experiments 5.1. Simulation of raw data for the whole set of saturated degree In order to verify the effectiveness of the proposed modification to the vector quantizer, we present a series of experiments based on the simulated and real raw data. To compare the performance of BAQ and the new algorithm, take the output signal of ADC as reference, simulation results in data domain are listed in Table 1. The compression ratio is 8:3. Signal to noise ratio (SNR) is defined here as: " # PK a PK r 2 i¼0 j¼0 zij ð15Þ SNR ¼ 10 lg PK a PK r 2 i¼0 j¼0 ðzij  zij Þ where K a ; K r are azimuth and range sample numbers, respectively. In the data domain, zij is the original SAR raw data (zdij in Fig. 8) and zij is the reconstructed raw data (zdij in Fig. 8), while in the image domain zij is the gray value of the original image (ziij in Fig. 8) and zij is the gray value of the reconstructed image (ziij in Fig. 8). In order to explain the performance more clearly, in the data domain we rewrite Eq. (15) as:

" SNR ¼ 10 lg ¼ 10 lg

1 KaKr 1 KaKr

#

PK a PK r i¼0

PK a PK r i¼0

d 2 j¼0 ðzij Þ

d j¼0 ðzij

 zdij Þ

2

r2r MSEr

ð16Þ

As shown in Fig. 8 zdij is the raw data obtained from ADC, zdij is its reconstructed raw data, r2r is the variance of the reference signal, MSEr is mean square error between zdij and zdij , and other parameters’ meaning is the same as Eq. (15). When saturation occurs, r2r becomes larger with the increase of SD. For BAQ algorithm, with the SD increasing, Lloyd-Max quantizer is unmatched with the signal from ADC, so the larger the SD the larger MSEr and the performances of conventional BAQ deteriorate. For new algorithm, with the SD increasing, the modified vector quantizer still matches statistics characteristics of the signal from ADC, and therefore MSEr is smaller. So the new algorithm performs better. From Table 1 we may conclude that the performance of the new algorithm in this case is better than that of BAQ over the whole set of SD. 5.2. Experiment using real SAR data Fig. 9(a) shows the image of the original SAR raw data with size 3500  6016 ðrange  azimuthÞ. The image is

Fig. 9. (a) Original image; (b) reconstructed image with BAQ; (c) reconstructed image with new algorithm.

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(b)

(a)

70

Saturation degree (%)

60

50

40

30

20

10 0

0.376

0.752

1.128

1.504

1.88

2.256

2.632

3.008 4

x 10

The number of sub−images

Fig. 10. (a) Sub-block partition of original image; (b) SD curves of original image.

divided into 30,080 sub-blocks as shown in Fig. 10(a). Fig. 10(b) shows the saturation degree of each sub-block in raw data domain. Fig. 9(b) is obtained from the raw data compressed by BAQ while Fig. 9(c) is obtained from the raw data compressed by new algorithm. From Table 1 we may conclude that in the data domain the performance of the new algorithm is better than that of BAQ over the whole set of SD. SAR imaging is a linear process that correlates the SAR raw data with the two-dimensional (2D) response function. So in the image domain the performance of the new algorithm also better than that of BAQ over the whole set of SD. Close visual examination indicates that Fig. 9(a) and (c) have a greater resemblance than Fig. 9(a) and (b). Histograms of the three images in Fig. 9 are shown in Fig. 11, and the SNR performances are compared in Table 2. From Fig. 11 we may see that the histograms of original image and new algorithm images are practically overlying. Results in Table 2 show that the new algorithm achieves a good performance on both data domain and image domain. All results show that when the SAR raw data are saturated, the performance of the new algorithm is better than that of BAQ.

(a)

x 10

BAQ (SNR/dB)

New algorithm (SNR/dB)

10.9058 14.8494

22.8728 31.6911

Automatic gain control (AGC) is a system to maintain the proper voltage range at the input to the ADC. Therefore, automatic gain control (AGC) usually is used to control the dynamic range of input signal under the peak to peak value of ADC in practice. But sometimes losing control of AGC also cause data saturation. When saturation occurs, the proposed algorithm could be used on the resulting raw SAR data. The saturation of the ADC will also cause power losses for the input signal, if the power loss is not compensated at all, an effect on the image quality will be present, so the new algorithm should be combined with the power loss compensation method on the ground. 6. Conclusions A new algorithm is proposed for space-borne SAR raw data compression corresponding to the whole set of SD.

Original Image BAQ Image New−algorithm Image

6

1.5

x 10

1.45 1.4

Frequency Count

2

Frequency Count

Data domain Image domain

(b)

6

2.5

Table 2 SNR performance for BAQ and new algorithm, both on data domain and image domain.

1.5

1

0.5

1.35 1.3 1.25 1.2 1.15 1.1

Original Image BAQ Image New−algorithm Image

1.05 0

0

0.5

1

1.5

2

2.5

Gray Value

3

3.5

4 4

x 10

1 7500

8000

8500

9000

9500

Gray Value

Fig. 11. Histograms for three images: original image, reconstructed images with BAQ and new algorithm.

10000

B. Hua et al. / Advances in Space Research 45 (2010) 1330–1337

1337

3

and let Lij denote the hyperplane:

boundary R2

2

Lij ¼ fx : yij  x þ bij ¼ 0g for

y2

y3

R3

ðA:4Þ

ThenTthe Ri may be described as the intersection of halfspaces j – i H ij and each face of the polytope Ri must lie in Lij for some j. Fig. A.1 depicts a demo of partition region. The optimal partition is obtained by drawing perpendicular bisectors between each pair of codewords, i.e., the blue line segments.

1

0

j ¼ 1; 2; . . . ; N

R1 y1 y6

−1

References R4

R6 y4

−2

R5 y5

−3 −3

−2

−1

0

1

2

3

Fig. A.1. Demo of partitioning a region.

Results based on simulated and real SAR data show that the performance of the new algorithm is better than that of BAQ over the whole set of SD . The advantage of this algorithm is that the existing data compression system onboard can be largest used. Compared with the conventional VQ algorithm, the new algorithm only needs update the codebook, compared with the conventional BAQ algorithm, the new algorithm needs more read only memories (ROM’s) and random access memories (RAM’S) (Benz et al., 1995). For spaceborne processing consideration, future work will focus on investigate the reduction of the encoding complexity. Acknowledgement This work was supported by Special Fund to the winner of CAS Excellent Doctoral Dissertation President Reward (Project No. 0813260042), and National Key Laboratory of Microwave Imaging Technology Fund (Project No. 9140C1903041003). We also appreciate the help from Dr. Wang Shuo on the modification of this paper. Appendix A. Partitioning of regions For Y preassigned, the best partition of Ri is given as : 2

2

Ri ¼ fx : kx  yi k 6 kx  yj k ; for all j – ig

ðA:1Þ

rewrite Eq. (A.1) in the form: Ri ¼ fx : yij  x þ bij P 0; for all j – ig 1 ðkyi k2 2

ðA:2Þ 2

where yij ¼ yi  yj and bij ¼  kyj k Þ , “”denotes the inner product operation. Let H ij denote the half-space defined by the relation (Gersho, 1982) H ij ¼ fx : yij  x þ bij P 0g

ðA:3Þ

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