Sampling adaptive block compressed sensing reconstruction algorithm for images based on edge detection

The Journal of China Universities of Posts and Telecommunications June 2013, 20(3): 97–103 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Sampling adaptive block compressed sensing reconstruction algorithm for images based on edge detection ZHENG Hai-bo (*), ZHU Xiu-chang School of Telecommunication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China

Abstract In this paper, a sampling adaptive for block compressed sensing with smooth projected Landweber based on edge detection (SA-BCS-SPL-ED) image reconstruction algorithm is presented. This algorithm takes full advantage of the characteristics of the block compressed sensing, which assigns a sampling rate depending on its texture complexity of each block. The block complexity is measured by the variance of its texture gradient, big variance with high sampling rates and small variance with low sampling rates. Meanwhile, in order to avoid over-sampling and sub-sampling, we set up the maximum sampling rate and the minimum sampling rate for each block. Through iterative algorithm, the actual sampling rate of the whole image approximately equals to the set up value. In aspects of the directional transforms, discrete cosine transform (DCT), dual-tree discrete wavelet transform (DDWT), discrete wavelet transform (DWT) and Contourlet (CT) are used in experiments. Experimental results show that compared to block compressed sensing with smooth projected Landweber (BCS-SPL), the proposed algorithm is much better with simple texture images and even complicated texture images at the same sampling rate. Besides, SA-BCS-SPL-ED-DDWT is quite good for the most of images while the SA-BCS-SPL-ED-CT is likely better only for more-complicated texture images. Keywords

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block compressed sensing, edge detection, sampling-adaptive, variance, directional transforms

Introduction

Compressed sensing (CS) [1–4] theory is a new signal processing theory, which is presented in recent years. CS can keep the original structure of the signal by using an adaptive linear projection. Under certain conditions, the original signal can be accurately reconstructed through numerical optimization method. Since CS achieves a sampling rate below the Nyquist criterion, it has wide application prospects. However, CS faces several challenges when applied to two-dimensional (2D) images. For example, the low reconstructed image qualities, a computationally expensive reconstruction process and huge memory required to store the random sampling operator. In order to solve those problems, several fast algorithms [5–7] have been developed recently. Gan and other researchers recently presented the reconstruction Received date: 12-07-2012 Corresponding author: ZHENG Hai-bo, E-mail: [email protected] DOI: 10.1016/S1005-8885(13)60056-4

algorithm based on block CS [8]. The advantage of this algorithm is its fast calculation speed. Mun et al. presented BCS-SPL [9] reconstruction algorithm based on Ref. [8]. The obvious advantages of Refs. [8–9] are that projection-based Landweber iterations were proposed to accomplish fast block CS reconstruction. Meanwhile, the imposing smoothing in the spatial domain using a Wiener filter with the goal of improving the reconstructed-image quality by eliminating blocking artifacts. Based on Ref. [9], Fowler et al. multiscale block compressed sensing with smoothed projected landweber (MS-BCS-SPL) [10] reconstruction algorithm. Additionally in Ref. [10], block-based compressed-sensing sampling is deployed independently within each subband of each decomposition level of a wavelet transform of an image. But in MS-BCS-SPL, the low frequency part of Wavelet decomposition results is not measured by the random measurement matrix. So, it is not a entire CS reconstruction algorithm. However, it still has an important reference value for the research of CS reconstruction

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algorithm. In this paper, an image sampling adaptive block compressed sensing reconstruction algorithm based on edge detection is presented based on BCS-SPL. Through the research of BCS-SPL, we find that BCS-SPL has a high reconstruction quality with simple texture images while a low reconstruction quality with complex texture images. Meanwhile, BCS-SPL can reconstruct a simple texture image with a well reconstruction quality even at a low sampling rate. For a given image, it often has some complex texture area and some simple texture area in the whole image. So, we can sample each block adaptively by its texture complexities. Blocks with complex texture measured by high sampling rate while those with simple texture measured by the low sampling rate. Thus, we can assign more samples to complex blocks and less to simple blocks, keeping up the same average sampling rate. Firstly, we do image edge detection for each block using Roberts method, calculate the variance of detected edge data, and assign the proper sampling rate for each block according to the variance value which denotes the block’s texture complexity. Secondly, in order to make the actual average sampling rate equals to the set up value, we calculate the multiples between the last assigned sampling rate of the whole image and the set up value, amplify or shrinking each block’s sampling rate once more according to the counted multiples. Finally, compare sampling rate with the set up value and iterate modification, until both rates are approximately equal. In the iteration process, we have set up the maximum sampling rate and the minimum sampling rate for each block. For example, when a block’s sampling rate is greater than 1, we set its sampling rate to 1. Meanwhile, when it is less than the set minimum value, we set its value to the set minimum value. The minimum sampling rate and the whole image sampling rate that ware set up before are related. In the process of smooth projected landweber (SPL), we also modify the iterative threshold value properly, to accelerate the convergence of SPL algorithm and improving the reconstruction quality. Experimental results show that our SA-BCS-SPL-ED algorithm is better than BCS-SPL in peak signal to noise ratio (PSNR) of reconstructed image.

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Background

In Ref. [8], the block-based CS (BCS) for the CS of 2D images was proposed. In this technique, an image with N

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pixels is divided into B×B blocks and sampled using an appropriately-sized measurement matrix. That is to say, suppose that xi is a vector representing, in raster-scan fashion, block i of input image x . The corresponding measurement yi is as follows: yi = Φa xi (1) where Φa

is an

m ´ B2

orthonormal measurement

matrix with m = êë MB / N úû . So the measurement matrix 2

Φ of the input image x is as follows: æ Φa ö ç ÷ Φa ÷ Φ=ç (2) ç ÷ O çç ÷ Φa ÷ø è BCS is memory efficient as we just need to store an m ´ B 2 orthonormal measurement matrix Φa , rather than a full matrix Φ . As an alternative to the pursuits class algorithms of CS reconstruction, techniques based on projection have been proposed in Ref. [7], they transformed the problem of CS s.t. y = Φx (3) to the problem as follow: xˆ = arg min (|| Ψx ||1 +l || y - Φx ||2 ) x

(4)

where Ψ is transformation matrix. Algorithms of this class form xˆ by successively projecting and thresholding; Initial approximation as x (0) = Φ T y (5) Forming the approximation at iteration i + 1 as 1 xˆˆ ( i ) = xˆ ( i ) + ΨΦ T ( y - ΦΨ -1 xˆ ( i ) ) (6) g ìï xˆˆ (i ) ; xˆˆ (i ) > t (i ) x (i +1) = í (7) ïî0; else where t (i ) is the thresholding value at iteration i. It was proposed in Ref. [9]. med xˆˆ (i ) (i ) t =l 2ln L (8) 0.674 5 where l is a constant convergence to control the convergent speed, and L is the number of the transform coefficients. In Ref. [8], Wiener filtering was incorporated into the basic projected Landweber (PL) framework described in Sect. 2 in order to remove blocking artifacts. In essence, this operation imposes smoothness in addition to the

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sparsity inherent to PL. A Wiener-filtering step was interleaved with the PL projection of Eqs. (6) and (7).

3 Sampling adaptive BCS with smoothed PL reconstruction In essence, adaptive block-based CS measurement adopt higher sampling rate in the complex texture area and with a low sampling rate in simple texture area according to the statistic characteristics of each block, making each block’s sampling rate match its complexities. In this paper, we calculate gradient variance of each block to roughly estimate the complexity of its texture. We have reason to believe that big variance with complex texture and small variance with simple texture. Assigning sampling rates strictly according to the variance may cause some blocks over-sampling (e.g. sampling rate over 1) and some of the other blocks sub-sampling. It is difficult to reconstruct the block data whose sampling rate is too low (sub-sampling) when sampled. So we should keep relative balance when assign sampling rates for each of the blocks. We designed the iterative approximation algorithm with upper limit and lower limit to meet the requirements as follows: 1) Each block’s sampling rate is proportional to its gradient variance with avoiding over-sampling and sub-sampling. 2) The actual sampling rate of the whole image approximately equals to the set up value. In the whole process, the flow of SA-BCS-SPL-ED is presented in Fig. 1.

Fig. 1

3.1

The flow diagram of SA-BCS-SPL-ED

Assigning sampling rates and approximation

Suppose that an image has N pixels, M samples, the whole image’s sampling rate is R = M / N . In BCS, the image is divided into B×B blocks. We detect the input image’s edges using Roberts gradient masks presented in Fig. 2. xi is a vector representing, in raster-scan fashion, block i of the gradient image x . xi is the mean of xi . xi =

1 B2

B2

åx j =1

ij

(9)

(a) Roberts masks

(b) Original image (c) Edge detection results Fig. 2 Roberts masks and an edge ditection example

d i 2 is the variance of xi . 1 di = 2 B 2

B2

å (x

ij

j =1

- xi )2

The mean of all blocks’ gradient ( d i 2 , i = 1, 2,3..., N / B 2 ) is as follows:

(10) variance

2

B2 N / B 2 å di N i =1 The assigned sampling rate for the block i is ri .

d2 =

ri = R

(11)

di2

K (12) d2 where K is a coefficient for expanding or reducing, with initial value 1. In order to avoid over-sampling and sub-sampling, we limit the value of ri as follows: ì1; ri ≥1 ï R ïR ri = í ; ri ≤ (13) g ïg ïîri ; else where g is a constant control to manage the minimum sampling rate. Its value is between 1.1 and 2.4. After the last assigned, updating the value of K. R K= (14) ri where ri is the mean of all blocks’ sampling rate ri after the last iterate. Executing Eqs. (12)~(14) once more until the value of K satisfying the inequality below: K - 1 < 0.001 (15) That is, the actual sampling rate of the whole image approximately equals to the set value. The process of iterative approximation algorithm is presented in Fig. 3. It iterates 3 or 4 times in general.

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x is as follows: æ Φ1 ö ç ÷ Φ2 ç ÷ ç ÷ O ÷ ΦS = ç (16) Φi ç ÷ ç ÷ O ç ÷ ç ΦN / B2 ÷ø è where, N / B 2 is the number of blocks. Suppose that Φb is an orthonormal measurement matrix. Its size is B 2 ´ B 2 . The relationship between Φi and Φb is as follows: Φi = Φb (1:1: ri B 2 ,:)

(17)

Eq. (17) is presented in the way of Matlab. That is, Φi Fig. 3

is composed by Φb ’s rows of 1 to ri B 2 . Where ri B 2 is

The processing of iterative approximation algorithm

Fig. 4 presents the sampling rates-distribution of each block for 512 ×512 Lenna image when the set up sampling rate is 0.4. Where the size of block is as well as 32×32 and the constant g in Eq. (13) is 2.4.

exactly the length of yi . In the program of Matlab, yi is stored by the way of cells. So SA-BCS is memory efficient as well as BCS as we just need to store an B 2 ´ B 2 orthonormal measurement matrix Φb , rather than a full measurement matrix ΦS . 3.3 The improvement of SPL In Ref. [9], SPL was presented detailed. In SA-BCS, as the length of yi is different with each other, we should reconstruct xi one by one. Meanwhile, in order to meet the characteristics of the SA-BCS, we reduce the value of l in Eq. (8) properly. In SA-BCS-SPL-ED, the function of SPL is as follows:

Fig. 4 The sampling rates distribution of each block when sampling rate is 0.4

Function x (i +1) = SPL( x ( i ) , y , Φb , l )

From Fig. 4, g =2.4, so the minimum sampling rate rmin = 0.4 / g = 0.166 7 , avoiding sub-sampling for some

xˆ ( i ) = Wiener( x ( i ) ) for each block j = 1 :1 : N / B 2 Φ j = Φb (1 :1 : length( yi ),:)

blocks whose gradient variance is too small.

xˆˆ j ( i ) = xˆ j ( i ) + Φ T j ( yi - Φi xˆ j ( i ) )

3.2 The storage of measurement matrix

end

In Ref. [8], for the sampling of BCS, the relationship between Φ and Φa is presented in Eq. (2). It is memory efficient as we just need to store an m ´ B orthonormal measurement matrix Φa , rather than a full 2

measurement matrix Φ . But in sampling adaptive for block compressed sensing (SA-BCS), as different block with different sampling rate ( ri ), the length of the measurement matrix Φi for the block i is different too. In SA-BCS, the measurement matrix ΦS of the input image

x%% ( i ) = Ψ xˆˆ ( i ) x% ( i ) = threshold( x%% ( i ) , l ) x ( i ) = Ψ -1 x% ( i ) for each block j = 1 :1 : N / B 2 Φ j = Φb (1 :1 : length( yi ),:) x j ( i +1) = x j ( i ) + Φ j T ( yi - Φ j x j ( i ) ) end

Here, Wiener (×) is presented in Eq. (7) and threshold (×) is presented in Eq. (8).

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Experimental results

In the Matlab platform, firstly, we compare SA-BCS-SPL-ED to BCS-SPL using the standard Lenna image (512×512, 8 bit). The block size of BCS is 32×32. In SA-BCS-SPL-ED, the constant g in Eq. (13) is 2.4 and we use hard threshold in Eq. (8) with l = 1.5 , in order to accelerate the convergence of SPL algorithm. As different measurement matrix will affect the reconstructed-image’s PSNR weakly, we use the same measurement matrix in all experiments in this paper. In order to compare SA-BCS-SPL-ED with BCS-SPL comprehensively, the directional transforms used in those experiments are DCT, DDWT, DWT and CT. The comparison of reconstruction qualities (PSNR) of the SA-BCS-SPL-ED and the BCS-SPL with the standard Lenna image are presented in Table 1. It shows that, SA-BCS-SPL-ED is much better than BCS-SPL no matter which directional transform was used and SA-BCS-SPL-ED-DDWT is the best for Lenna. Table 1 The comparison of reconstruction qualities (PSNR) of the SA-BCS-SPL-ED and the BCS-SPL with the standard Lenna image with different sampling rate Algorithm SA-BCS-SPL-ED-DCT BCS-SPL-DCT SA-BCS-SPL-ED-DDWT BCS-SPL-DDWT SA-BCS-SPL-ED-DWT BCS-SPL-DWT SA-BCS-SPL-ED-CT BCS-SPL-CT

PSNR/dB ri=0.1

ri=0.2

ri=0.3

ri=0.4

ri=0.5

28.86 27.74 29.54 28.03 29.15 27.71 28.40 28.09

32.25 30.52 33.23 31.40 32.81 30.90 32.32 30.99

34.68 32.55 35.54 33.54 35.08 32.98 34.83 33.00

36.78 34.28 37.38 35.24 37.18 34.65 36.88 34.70

38.77 35.84 39.07 36.78 39.02 36.17 38.82 36.23

Secondly, we compare SA-BCS-SPL-ED to BCS-SPL

using 5 simple texture gray images which presented in Fig. 5. The other experimental conditions are same as above. The comparison of reconstruction qualities (PSNR) of the SA-BCS-SPL-ED and the BCS-SPL with simple texture images are presented in Table 2.

(a) Airplane.pgm

(b) Scenery.pgm

(c) Castle.pgm

(d) Mondrian.pgm

(e) Sea.pgm Fig. 5

Simple texture gray images (sizes 512×512)

Table 2 The comparison of reconstruction qualities (PSNR) of the SA-BCS-SPL-ED and the BCS-SPL with simple texture images with different sampling rate Algorithm (image) SA-BCS-SPL-ED-DCT(Fig. 5(a)) BCS-SPL-DCT(Fig. 5(a)) SA-BCS-SPL-ED-DDWT(Fig. 5(a)) BCS-SPL-DDWT(Fig. 5(a)) SA-BCS-SPL-ED-DWT(Fig. 5(a)) BCS-SPL-DWT(Fig. 5(a)) SA-BCS-SPL-ED-CT(Fig. 5(a)) BCS-SPL-CT(Fig. 5(a)) SA-BCS-SPL-ED-DCT(Fig. 5(b)) BCS-SPL-DCT(Fig. 5(b)) SA-BCS-SPL-ED-DDWT(Fig. 5(b)) BCS-SPL-DDWT(Fig. 5(b)) SA-BCS-SPL-ED-DWT(Fig. 5(b)) BCS-SPL-DWT(Fig. 5(b)) SA-BCS-SPL-ED-CT(Fig. 5(b)) BCS-SPL-CT(Fig. 5(b)) SA-BCS-SPL-ED-DCT(Fig. 5(c)) BCS-SPL-DCT(Fig. 5(c)) SA-BCS-SPL-ED-DDWT(Fig. 5(c)) BCS-SPL-DDWT(Fig. 5(c)) SA-BCS-SPL-ED-DWT(Fig. 5(c)) BCS-SPL-DWT(Fig. 5(c)) SA-BCS-SPL-ED-CT(Fig. 5(c)) BCS-SPL-CT(Fig. 5(c)) SA-BCS-SPL-ED-DCT(Fig. 5(d)) BCS-SPL-DCT(Fig. 5(d)) SA-BCS-SPL-ED-DDWT(Fig. 5(d)) BCS-SPL-DDWT(Fig. 5(d)) SA-BCS-SPL-ED-DWT(Fig. 5(d)) BCS-SPL-DWT(Fig. 5(d)) SA-BCS-SPL-ED-CT(Fig. 5(d)) BCS-SPL-CT(Fig. 5(d)) SA-BCS-SPL-ED-DCT(Fig. 5(e)) BCS-SPL-DCT(Fig. 5(e)) SA-BCS-SPL-ED-DDWT(Fig. 5(e)) BCS-SPL-DDWT(Fig. 5(e)) SA-BCS-SPL-ED-DWT(Fig. 5(e)) BCS-SPL-DWT(Fig. 5(e)) SA-BCS-SPL-ED-CT(Fig. 5(e)) BCS-SPL-CT(Fig. 5(e))

ri=0.1 27.57 25.83 28.13 25.95 28.00 25.85 27.03 26.32 24.09 22.26 24.11 22.19 24.03 21.90 23.88 22.31 28.76 27.26 28.92 27.44 28.86 27.12 28.94 26.94 30.10 27.69 27.46 27.46 28.19 27.37 28.49 27.29 27.71 25.96 28.18 25.82 27.91 25.70 27.50 26.24

PSNR/dB ri=0.2 ri=0.3 ri=0.4 30.98 33.54 35.79 29.05 29.10 32.65 31.90 34.37 36.44 29.66 32.27 34.43 31.77 34.32 36.50 29.39 31.92 34.01 31.02 33.65 35.83 29.44 31.72 33.72 26.71 28.59 30.28 23.53 24.57 25.86 27.16 29.07 30.72 24.13 25.70 27.16 27.09 29.01 30.69 23.86 25.39 26.83 26.69 28.57 30.34 24.06 25.31 26.70 31.10 33.00 34.85 29.42 31.08 32.51 31.58 33.40 35.06 29.95 31.63 33.07 31.47 33.33 35.11 29.65 31.30 32.75 31.39 33.33 35.15 29.91 31.53 33.00 33.51 35.49 37.01 32.28 34.22 34.29 33.97 35.61 37.12 32.25 34.62 36.36 33.90 35.25 37.28 32.56 35.08 36.89 33.15 35.16 36.82 32.23 34.74 36.57 30.54 32.66 34.53 28.43 30.45 32.21 31.25 33.34 35.18 29.12 31.29 33.10 31.08 33.20 35.08 28.80 31.01 32.85 30.55 32.72 34.67 28.92 30.89 32.67

ri=0.5 37.91 32.44 38.37 36.41 38.61 35.97 38.01 35.57 32.03 27.15 32.36 28.55 32.39 28.19 32.11 27.98 36.92 33.87 36.91 34.47 37.13 34.09 37.17 34.40 38.42 35.54 38.42 37.92 38.66 38.50 38.31 38.16 36.54 33.88 37.06 34.76 37.07 34.53 36.71 34.38

Table 2 shows that, SA-BCS-SPL-ED’s advantage is

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more obvious for simple texture images and its PSNR of reconstructed-images are 1.00~5.47 dB higher than BCS-SPL. That is, SA-BCS-SPL-ED is much better than BCS-SPL at the same sampling rate. This is mainly because that there are a lot of flat backgrounds in these images, SA-BCS-SPL-ED can assign more sampling rates in the high frequency region of the image with less sampling rates in those flat backgrounds. The appropriate decrease of the sampling rates in background area does not affect the PSNR of these regions obviously, while the increase of the sampling rates in the high frequency region enhances the PSNR of these regions significantly. Particularly, when the sampling rate of an image is 0.5, the sampling rate can be up to 1 in parts of high frequency regions and those background areas can be reconstructed well. For example, compared to BCS-SPL-DCT, SA-BCS-SPL-ED-DCT’s PSNR of the reconstructed image is 5.47 dB higher for the image of Fig. 5(a) when subrate is 0.5. Besides, as we can see from Table 2, using the directional transform of DDWT is likely better than the other directional transforms for those simple texture images. In most cases, SA-BCS-SPL-ED-DDWT is better. Finally, we compare SA-BCS-SPL-ED to BCS-SPL using 5 complex texture gray images which presented in Fig. 6.

(a) Barbara.pgm

(c) mandrill.pgm

(b) bike.pgm

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The other experimental conditions are same as above. The comparison of reconstruction qualities (PSNR) of the SA-BCS-SPL-ED and the BCS-SPL with simple texture images are presented in Table 3. Table 3 The comparison of reconstruction qualities (PSNR) of the SA-BCS-SPL-ED and the BCS-SPL with complex texture images with different sampling rate Algorithm (image) SA-BCS-SPL-ED-DCT(Fig. 6(a)) BCS-SPL-DCT(Fig. 6(a)) SA-BCS-SPL-ED-DDWT(Fig. 6(a)) BCS-SPL-DDWT(Fig. 6(a)) SA-BCS-SPL-ED-DWT(Fig. 6(a)) BCS-SPL-DWT(Fig. 6(a)) SA-BCS-SPL-ED-CT(Fig. 6(a)) BCS-SPL-CT(Fig. 6(a)) SA-BCS-SPL-ED-DCT(Fig. 6(b)) BCS-SPL-DCT(Fig. 6(b)) SA-BCS-SPL-ED-DDWT(Fig. 6(b)) BCS-SPL-DDWT(Fig. 6(b)) SA-BCS-SPL-ED-DWT(Fig. 6(b)) BCS-SPL-DWT(Fig. 6(b)) SA-BCS-SPL-ED-CT(Fig. 6(b)) BCS-SPL-CT(Fig. 6(b)) SA-BCS-SPL-ED-DCT(Fig. 6(c)) BCS-SPL-DCT(Fig. 6(c)) SA-BCS-SPL-ED-DDWT(Fig. 6(c)) BCS-SPL-DDWT(Fig. 6(c)) SA-BCS-SPL-ED-DWT(Fig. 6(c)) BCS-SPL-DWT(Fig. 6(c)) SA-BCS-SPL-ED-CT(Fig. 6(c)) BCS-SPL-CT(Fig. 6(c)) SA-BCS-SPL-ED-DCT(Fig. 6(d)) BCS-SPL-DCT(Fig. 6(d)) SA-BCS-SPL-ED-DDWT(Fig. 6(d)) BCS-SPL-DDWT(Fig. 6(d)) SA-BCS-SPL-ED-DWT(Fig. 6(d)) BCS-SPL-DWT(Fig. 6(d)) SA-BCS-SPL-ED-CT(Fig. 6(d)) BCS-SPL-CT(Fig. 6(d)) SA-BCS-SPL-ED-DCT(Fig. 6(e)) BCS-SPL-DCT(Fig. 6(e)) SA-BCS-SPL-ED-DDWT(Fig. 6(e)) BCS-SPL-DDWT(Fig. 6(e)) SA-BCS-SPL-ED-DWT(Fig. 6(e)) BCS-SPL-DWT(Fig. 6(e)) SA-BCS-SPL-ED-CT(Fig. 6(e)) BCS-SPL-CT(Fig. 6(e))

ri=0.1 22.85 22.78 23.05 22.56 22.69 22.43 22.92 22.71 18.95 18.00 19.42 18.40 19.07 18.05 19.33 18.39 20.68 20.24 20.86 20.72 20.52 20.34 20.81 20.52 25.61 24.64 26.35 24.88 26.23 25.24 26.10 25.00 19.95 19.47 20.45 20.19 20.15 19.86 20.16 19.93

PSNR/dB ri=0.2 ri=0.3 ri=0.4 25.31 27.50 29.68 24.41 25.95 27.51 25.64 27.84 29.85 24.11 25.67 27.25 25.28 27.50 29.54 23.86 25.16 26.50 25.51 27.86 30.10 24.26 25.88 27.62 21.46 23.33 24.84 19.81 21.58 23.16 21.73 23.35 24.77 20.27 21.89 23.54 21.61 23.40 24.99 19.83 21.48 23.19 21.90 23.63 25.14 20.65 22.56 24.30 22.07 23.52 25.06 20.81 21.67 23.31 22.19 23.39 24.73 21.88 22.87 23.92 22.25 23.71 25.23 21.64 22.70 23.74 22.43 23.94 25.54 21.79 22.86 23.96 28.65 30.81 32.71 27.39 29.55 31.36 29.18 31.18 32.95 28.75 31.03 33.00 29.23 31.40 33.34 28.43 30.63 32.58 29.07 31.21 33.12 28.43 30.61 32.51 21.81 23.26 24.65 21.05 22.22 23.39 22.07 23.34 24.67 21.65 22.73 23.75 21.99 23.37 24.68 21.44 22.52 23.56 22.00 23.48 24.95 21.48 22.64 23.77

ri=0.5 32.08 29.22 31.96 28.86 31.79 28.06 32.49 29.38 26.31 24.61 26.25 25.15 26.59 24.80 26.65 25.84 26.92 24.45 26.31 25.06 27.02 24.86 27.35 25.14 34.50 33.15 34.69 34.92 35.33 34.44 35.03 34.33 26.22 24.56 26.21 24.83 26.25 24.66 26.56 24.98

(d) people.pgm

From Table 3, SA-BCS-SPL-ED still has some advantages for complex texture images. Although there are not a lot of flat backgrounds in these images, the sampling rate for each block is assigned still by its gradient variance adaptively. Using the directional transforms of DDWT and

Fig. 6

(e) Clothes.pgm Complex texture gray images (sizes 512×512)

CT are likely better than the other directional transforms for those simple texture images and CT is likely better than DDWT for more-complicated texture images particularly when the subrate is greater than 0.1.

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Conclusions

Based on BCS-SPL reconstruction algorithm, an image SA-BCS-SPL-ED is presented in this paper. This algorithm takes full advantage of the characteristics of BCS by matching each block’s sampling rate with its gradient variance. As we can see from the experimental results, compared to BCS-SPL, SA-BCS-SPL is much better with simple texture images and even complicated texture images at the same sampling rate. Besides, the directional transforms used in these experiments are DCT, DDWT, DWT and CT. DDWT is likely quite good both for the simple texture and the complex texture image while CT is likely better only for more-complicated texture images in the large subrate. However, because of assigning each block’s sampling rate by its gradient variance and reconstructing one by one in SPL, the time spent for sampling and reconstructing was increased inevitably in our algorithm. Acknowledgments This work was supported by the National Natural Science Foundation of China (61071091, 61071166), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institution-Information and Communication Engineering.

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