REWEIGHTED COMPRESSIVE SAMPLING FOR IMAGE ... - Lu's Vision
REWEIGHTED COMPRESSIVE SAMPLING FOR IMAGE COMPRESSION Yi Yang, Oscar C. Au, Lu Fang, Xing Wen and Weiran Tang Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong, China email:{yyang, eeau, fanglu, wxxab, tangweir}@ust.hk ABSTRACT Compressive Sampling (CS), is an emerging theory which points us a promising direction of designing novel efficient data compression techniques. However, the conventional CS adopts a non-discriminated sampling scheme which usually gives poor performance on realistic complex signals. In this paper we propose a reweighted Compressive Sampling for image compression. It introduces a weighting scheme into the conventional CS framework whose coefficients are determined in encoding side according to the statistics of image signals. Experimental results demonstrate that our proposed method notably outperforms the conventional Compressive Sampling framework in coding performance in the sense that the reconstruction quality is greatly enhanced with same number of measurements and computational complexity. Index Terms— Compressive Sampling, natural image statistics, reweighted sampling, sparsity, 1 minimization 1. INTRODUCTION Shannon sampling theory tells us that if we want to reconstruct a band limited signal without distortion, then sampling the original signal at a rate which is at least twice of its highest frequency is necessary. However, a newly emerged theory called Compressive Sampling (CS), also known as Compressive Sensing proves that this is not always true. It tells that as long as the signal is sparse or compressible, then we can exactly recover the original signals through only a few measurements[1][2], namely, we can achieve data sampling and compression at the same time. Compressive Sampling theory points us a promising direction for designing highly efficient data compression techniques while the performance of existing ones seem to reach the bottleneck. Some initial exploration about applying CS theory in data compression has been made[3], however, the performance is limited. The main problem that prevents CS framework from being applied in real data compression system is the efficiency decrease for complex signals which are usually hard to be perfectly sparsely represented. One major reason causing this efficiency droping is that conventional CS
framework samples data randomly without showing discrimination to different components, in other words, it does not consider about the characteristics of the signals. However, this may not be the best way to sample when it comes to the complex signals, like images. A lot of efforts have been made trying to improve the performance of the CS signal recovery algorithms by adjusting the behavior of the decoder adaptively to the signal characteristics[4][5] (see http://www.dsp.ece.rice. edu/cs/ for a comprehensive list of related work). However, usually their methods are based on the blind estimation of the signal magnitudes and iterative refining process in the decoding side. For instance, in [4], the authors propose an iterative reweighting scheme to enhance the sparsity of the reconstruction results. In their work, the solution of an unweighted 1 minimization reconstruction is used as the initial guess, then the weighting coefficients are refined iteratively, until the convergence is reached. However, the complexity for these iterative processing schemes is too high for the methods to be practical, and they are often guaranteed only to converge to a local (not necessarily) optimum. In this paper, we propose a reweighted Compressive Sampling for CS based image compression by introducing a weighting scheme into the encoding procedure. This method works on the block-based processing mechanism. First the images will be divided into small blocks, and then the weighting coefficients are determined by taking advantage of the statistical characteristics of natural images and subsequently sent together with the CS measurements to help the decoder to achieve better reconstruction results. The main advantage of our proposed method is that it can sufficiently utilize image characteristics to achieve considerable performance gain without notably increasing the complexity of the system structure or the computation procedures. The rest of the paper is organized as follows, Section 2 gives a brief overview of the conventional Compressive Sampling framework, then we describe our proposed reweighted Compressive Sampling scheme in detail in Section 3. The corresponding experimental results are demonstrated in Section 4, and finally, we conclude this paper in Section 5.
2. COMPRESSIVE SAMPLING FRAMEWORK OVERVIEW Consider a real-value, finite length, discrete time signal x with dimension N , the compressive sampling process just takes linear non-adaptive measurements of x through dimension reduction projection[2], which can be described as, y = Φx
(1)
where y is the sampled measurements with dimension n(n N )) , and Φ is an n × N sampling matrix. Usually, the real world signals could not be perfectly sparse in time/space domain (e.g. images, audio signals), but they may be sparsely represented in a certain transform domain Ψ (e.g. DCT or Wavelet). The above sampling problem could also be described in a more general form[2]: y = ΦΨs = Θs
(2)
where Θ = ΦΨ is an n × N matrix. Since n N , the problem of reconstructing x from y is ill conditioned. However, the compressive sensing theory says that as long as x is sparse in some domain, we can reconstruct the original signal from the condensed measurements exactly by solving a combinational optimization problem: (3) min : s 0 s.t. : y = Θs n where s 0 = #{i : si = 0} = i=1 |si |0 and s is the estimation of s. However, this problem cannot be solved efficiently since it is non-convex. A common alternative is to solve the following linear programming problem[2], (4) min : s 1 s.t. : y = Θs n where s 1 = i=1 |si |. It is proved in [1][2] that if Φ and Ψ are incoherent and their product Θ satisfies the Restricted Isometry Property (RIP) [1], then x can be well reconstructed through (4) from the n = O(c log N ) measurements[1], where c is a constant related to the sparsity of original signal. A commonly used sampling matrix is i.i.d. Gaussian matrix with entries being outcomes of i.i.d. Gaussian variables [1][2]. It is universal in the sense that the product of an i.i.d Gaussian matrix and the transform matrix Θ = ΦΨ is also i.i.d. Gaussian thus will have the RIP with high probability regardless the choice of transform basis Ψ. In this paper, we call the standard non-weighted Compressive Sampling framework described above as conventional CS framework. 3. PROPOSED REWEIGHTED COMPRESSIVE SAMPLING SCHEME 3.1. Weighted Sampling Scheme In some previous work, it has been shown that the performance of the conventional CS framework could be notably
enhanced by introducing weighting schemes[4][5]. Our proposed work follows the same insight. In stead of using the conventional sampling scheme in (1) and (2), we consider the following weighted sampling: y = ΦΨW s = ΘW s
(5)
where W is a diagonal weighting matrix with weighting coefficients {w1 , w2 , ..., wn } on the diagonal and zeros elsewhere. Then the corresponding reconstruction problem becomes: min : s 1 s.t. : y = ΘW s
(6)
It is easy to show that (6) is equivalent to the following problem: (7) min : Λs 1 s.t. : y = Θs with Λ = W −1 , in the sense that for a given W or Λ, the optimal solution of (7) is also the optimal solution for (6). Note that (7) is the same problem stated in [4]. Let us define Λ = diag{α1 , α2 , ..., αn } which is also a diagonal weighting matrix, then we have αi = wi−1 . Usually, people use 1 norm as the alternative of 0 norm to solve the reconstruction problem. The key difference between 1 and 0 norm is the dependence on the signal magnitude, which means large coefficients are penalized more heavily in the 1 norm than small ones, while 0 does not take the coefficient magnitudes into count[4]. Therefore a proper principle of determining the value of weighting coefficients is to counteract the influence of the signal magnitude on the 1 penalty function. A reasonable way is to set the weights in (7) to be inverse proportional to the signal magnitude[4], i.e., ⎧ ⎨ 1 , si = 0, (8) |si | αi ∝ ⎩ ∞, si = 0. accordingly, weighting coefficients in (6) should be: wi ∝ si
(9)
For (7), if the input signal s is sparse, then the large (infinity) entries of αi will force the corresponding small value in the solution to approach zero, contrarily the large entries in the solution will be precisely reconstructed since the corresponding weights are very small. The same effect will also take place in (6), the large value of wi will favor the signal entries with large magnitudes to be more precisely reconstructed, and as a result, the quality of reconstructed image will be enhanced. The proposed weighting sampling scheme can also be interpreted in an geometric way. Consider a N dimensional signal space RN , a random sampling matrix forms a subspace H orienting to a random direction. A CS measurement sequence y is a projection of the original signal x from the high dimensional space onto the subspace H. Typically, all the components of x will contribute the same amount of energy with
the same probability. However, by introducing the weighting matrix with the coefficients set as defined in (9), then the subspace is forced to approach the directions where the signal components have large magnitudes, therefore more energy of them will be captured, which improves recovery precision.
mean of DC coefficients all the blocks, which is an unbiased estimation for Gaussian random variables, which is: m n i=1 j=1 Idc (i, j) (13) μ ˆdc = mn
3.2. Determination of Weighting Coefficients
N where m = M B , n = B , i, j are block indexes, and Idc (i, j) is the value of DC coefficient in each block. For the variance, the maximum likelihood estimator is adopted and the estimation is expressed as:
3.2.1. Natural Image Statistics It is well known that the DC component of natural images DCT coefficients can be modeled to satisfy a Gaussian distribution with the Probability Density Function (PDF) of fdc (x) =
2 2 1 √ e−(x−μdc ) /2σdc σdc 2π
(10)
2 where μdc and and σdc are the mean value and variance of DC coefficient, respectively. However, the behavior of AC components are more difficult to model. A proper choice balancing the simplicity of model and fidelity to the empirical data is zero-mean Laplacian distribution [6]:
fac (x) =
1 −|x|/λX e 2λX
(11)
where λX is the rate parameter. Therefore the mean value 2 = and variance for AC coefficients are: μac = 0 and σac 2λ2X . In block processing scheme, we vectorize the DCT coefficients of each block into a 1-D sequence and assign {w1 , w2 , ..., wn } to be the corresponding weighting coefficients. According to the criteria set in (9), we define the weighting coefficient for each frequency component to be the linear combination of the absolute value of its mean and the square root of its variance: wi = η|μi | + (1 − η)σi
(12)
where η(0 ≤ η ≤ 1) is a small constant. Typically, for DCT coefficients of natural images, the frequency components with large magnitudes concentrate at low frequency part which also have large mean and variance, while the high frequency components often have small magnitudes with small mean and variance. Therefore, in most cases, our definition of weighting coefficients is a well approximation that is proportional to the real signal magnitude. 3.2.2. Parameter Estimation In this paper, we employ a block based parameter estimation scheme, namely, the candidate M × N image to be encoded will be divided into B × B blocks and processed. The frequency components at same locations in every block are considered as the outcomes of the same random variable. For DC component, the mean value is estimated by the sample
2 = σ ˆdc
1 (Idc (i, j) − μdc )2 mn i=1 j=1 m
n
(14)
At last, maximum likelihood estimator is again employed to get the estimation of the variance for AC coefficients with zero mean: m n 1 2 σ ˆac = Iac (i, j)2 (15) mn i=1 j=1 similarly Iac (i, j) is the value of the AC coefficient at a certain frequency location in each block. After all the parameters are obtained, the same weighting coefficients set will be used in every block and they will be sent to the decoder as overheads together with CS measurements. In total, we only need to send additional B×B floating point numbers. Though, the compression efficiency will drop a little bit, comparing with the great performance gain the decoder will achieve, the increase in data size is neglectable. 4. EXPERIMENTAL RESULTS We compared our proposed sampling scheme with the the conventional CS framework as well as the reweighted 1 minimization which is the state-of-art reconstruction algorithm proposed recently. The test data are on 512 × 512 8-bit grey level images with different features. Block size of 8 × 8 and 16 × 16 are tested in our simulation. This is because for block size smaller than 8 × 8, the signal length is too short for CS framework to work properly, while for block size larger than 16 × 16, the computational complexity is too high for our test bed. We choose the sparse binary random matrix[7] as the sampling matrix for (5). It has the similar performance as the i.i.d. Gaussian random matrix but simplifies the computation since it is sparse and all the entries are just 0 and 1. According to our experiment, the optimal value for η in (12) is between 0.5 ∼ 0.7 in most cases. It is expected that choosing η adaptively will give better results, however, for the ease of simulation, we choose η = 0.65 uniformly over all experiments. For reweighted 1 minimization, we set the maximum iteration number L = 4 and the constant = 0.1, the same value as stated in [4]. Due to the limitation of space, we only demonstrate the visual quality comparison at sample rate 1 0.4 as shown in Fig.1 1 Sample rate is defined as the ratio of measurements set length to the original signal length.
(a) Original
(b) Conventional
(c) Reweighted 1
(d) Proposed
Fig. 1. Visual quality comparison at sample rate 0.4 with block size 16 × 16. PSNR for (b) ∼ (d) are 27.1dB, 33.0dB and 33.8dB (for more results, please visit: http://ihome.ust.hk/ ˜yyang/). We can see that our proposed method achieves much better reconstruction quality than the conventional CS framework. A more detailed comparison in PSNR performance is tabulated in Table 1. For block size of 8×8, our proposed method achieve an average gain of 7.38dB for all the images listed over all sample rates. And for block size of 16 × 16 the gain is 6.51dB.
to the utilization of the statistics information of natural images, the weighting matrix we introduced helps the sampling procedure to capture more information from significant components in DCT sequence which leads to the enhancement of recovery precision. The simulation results reveal that our proposed algorithm greatly enhances the performance of Compressive Sampling framework without notably increasing the system complexity comparing with the existing conventional Compressive Sampling framework.
Table 1. PSNR Performance (in dB)
6. ACKNOWLEDGEMENT
Sample Rate Proposed Lenna Reweight 1 8×8 Conventional Proposed Peper Reweight 1 8×8 Conventional Proposed Boat Reweight 1 8×8 Conventional Proposed Lenna Reweight 1 16 × 16 Conventional Proposed Peper Reweight 1 16 × 16 Conventional Proposed Boat Reweight 1 16 × 16 Conventional
0.2 31.7 30.6 23.3 32.7 31.7 24.0 28.4 27.5 21.5 32.7 31.7 25.5 33.7 32.4 26.6 29.2 28.0 22.9
0.3 34.5 33.1 26.4 34.9 33.7 27.2 31.0 30.1 24.0 35.4 34.1 28.2 35.6 34.6 29.2 31.7 30.4 25.0
0.4 37.0 35.6 29.0 36.7 35.9 30.1 33.2 32.5 26.2 37.5 36.2 30.5 37.1 36.4 31.6 33.8 33.0 27.1
0.5 38.7 37.4 31.1 37.9 37.5 32.2 34.9 34.7 28.2 39.4 38.1 32.5 38.5 38.1 33.5 35.4 35.3 29.1
One thing we would like to mention is that, comparing with reweighted 1 reconstruction, our proposed method is also notably better, though their performance is supposed to be similar since they are mathematically equivalent. This is because the convergence of reweighted 1 is hard to be reached in the affordable simulation time for image signals, and it is guaranteed to only converge to a local optimum. This phenomenon demonstrates the efficiency of our method from another angle. 5. CONCLUSION In this paper, a novel weighted sampling scheme is proposed for Compressive Sampling based image compression. Due
This work has been supported in part by the Innovation and Technology Commission of the Hong Kong Special Administrative Region, China (project no GHP/048/08). 7. REFERENCES [1] E.J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Information Theory, IEEE Transactions on, vol. 52, no. 2, pp. 489–509, 2006. [2] E.J. Candes, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006, vol. 3, pp. 1433–1452. [3] Y. Zhang, S. Mei, Q. Chen, and Z. Chen, “A Novel Image/Video Coding Method Based on Compressive Sensing Theory,” in Proc. Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), 2008, pp. 1361–1364. [4] E.J. Candes, M.B. Wakin, and S.P. Boyd, “Enhancing Sparsity by Reweighted L1 Minimization,” Arxiv preprint arXiv:0711.1612, 2007. [5] R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” in Proc. Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), 2008, pp. 3869–3872. [6] E.Y. Lam and J.W. Goodman, “A mathematical analysis of the DCT coefficient distributions forimages,” Image Processing, IEEE Transactions on, vol. 9, no. 10, pp. 1661–1666, 2000. [7] R. Berinde and P. Indyk, “Sparse recovery using sparse random matrices,” Preprint, 2008.
framework samples data randomly without showing discrimination to different components, in other words, it does not consider about the characteristics of the signals. However, this may not be the best way to sample when it comes to the complex signals, like images. A lot of efforts have been made trying to improve the performance of the CS signal recovery algorithms by adjusting the behavior of the decoder adaptively to the signal characteristics[4][5] (see http://www.dsp.ece.rice. edu/cs/ for a comprehensive list of related work). However, usually their methods are based on the blind estimation of the signal magnitudes and iterative refining process in the decoding side. For instance, in [4], the authors propose an iterative reweighting scheme to enhance the sparsity of the reconstruction results. In their work, the solution of an unweighted 1 minimization reconstruction is used as the initial guess, then the weighting coefficients are refined iteratively, until the convergence is reached. However, the complexity for these iterative processing schemes is too high for the methods to be practical, and they are often guaranteed only to converge to a local (not necessarily) optimum. In this paper, we propose a reweighted Compressive Sampling for CS based image compression by introducing a weighting scheme into the encoding procedure. This method works on the block-based processing mechanism. First the images will be divided into small blocks, and then the weighting coefficients are determined by taking advantage of the statistical characteristics of natural images and subsequently sent together with the CS measurements to help the decoder to achieve better reconstruction results. The main advantage of our proposed method is that it can sufficiently utilize image characteristics to achieve considerable performance gain without notably increasing the complexity of the system structure or the computation procedures. The rest of the paper is organized as follows, Section 2 gives a brief overview of the conventional Compressive Sampling framework, then we describe our proposed reweighted Compressive Sampling scheme in detail in Section 3. The corresponding experimental results are demonstrated in Section 4, and finally, we conclude this paper in Section 5.
2. COMPRESSIVE SAMPLING FRAMEWORK OVERVIEW Consider a real-value, finite length, discrete time signal x with dimension N , the compressive sampling process just takes linear non-adaptive measurements of x through dimension reduction projection[2], which can be described as, y = Φx
(1)
where y is the sampled measurements with dimension n(n N )) , and Φ is an n × N sampling matrix. Usually, the real world signals could not be perfectly sparse in time/space domain (e.g. images, audio signals), but they may be sparsely represented in a certain transform domain Ψ (e.g. DCT or Wavelet). The above sampling problem could also be described in a more general form[2]: y = ΦΨs = Θs
(2)
where Θ = ΦΨ is an n × N matrix. Since n N , the problem of reconstructing x from y is ill conditioned. However, the compressive sensing theory says that as long as x is sparse in some domain, we can reconstruct the original signal from the condensed measurements exactly by solving a combinational optimization problem: (3) min : s 0 s.t. : y = Θs n where s 0 = #{i : si = 0} = i=1 |si |0 and s is the estimation of s. However, this problem cannot be solved efficiently since it is non-convex. A common alternative is to solve the following linear programming problem[2], (4) min : s 1 s.t. : y = Θs n where s 1 = i=1 |si |. It is proved in [1][2] that if Φ and Ψ are incoherent and their product Θ satisfies the Restricted Isometry Property (RIP) [1], then x can be well reconstructed through (4) from the n = O(c log N ) measurements[1], where c is a constant related to the sparsity of original signal. A commonly used sampling matrix is i.i.d. Gaussian matrix with entries being outcomes of i.i.d. Gaussian variables [1][2]. It is universal in the sense that the product of an i.i.d Gaussian matrix and the transform matrix Θ = ΦΨ is also i.i.d. Gaussian thus will have the RIP with high probability regardless the choice of transform basis Ψ. In this paper, we call the standard non-weighted Compressive Sampling framework described above as conventional CS framework. 3. PROPOSED REWEIGHTED COMPRESSIVE SAMPLING SCHEME 3.1. Weighted Sampling Scheme In some previous work, it has been shown that the performance of the conventional CS framework could be notably
enhanced by introducing weighting schemes[4][5]. Our proposed work follows the same insight. In stead of using the conventional sampling scheme in (1) and (2), we consider the following weighted sampling: y = ΦΨW s = ΘW s
(5)
where W is a diagonal weighting matrix with weighting coefficients {w1 , w2 , ..., wn } on the diagonal and zeros elsewhere. Then the corresponding reconstruction problem becomes: min : s 1 s.t. : y = ΘW s
(6)
It is easy to show that (6) is equivalent to the following problem: (7) min : Λs 1 s.t. : y = Θs with Λ = W −1 , in the sense that for a given W or Λ, the optimal solution of (7) is also the optimal solution for (6). Note that (7) is the same problem stated in [4]. Let us define Λ = diag{α1 , α2 , ..., αn } which is also a diagonal weighting matrix, then we have αi = wi−1 . Usually, people use 1 norm as the alternative of 0 norm to solve the reconstruction problem. The key difference between 1 and 0 norm is the dependence on the signal magnitude, which means large coefficients are penalized more heavily in the 1 norm than small ones, while 0 does not take the coefficient magnitudes into count[4]. Therefore a proper principle of determining the value of weighting coefficients is to counteract the influence of the signal magnitude on the 1 penalty function. A reasonable way is to set the weights in (7) to be inverse proportional to the signal magnitude[4], i.e., ⎧ ⎨ 1 , si = 0, (8) |si | αi ∝ ⎩ ∞, si = 0. accordingly, weighting coefficients in (6) should be: wi ∝ si
(9)
For (7), if the input signal s is sparse, then the large (infinity) entries of αi will force the corresponding small value in the solution to approach zero, contrarily the large entries in the solution will be precisely reconstructed since the corresponding weights are very small. The same effect will also take place in (6), the large value of wi will favor the signal entries with large magnitudes to be more precisely reconstructed, and as a result, the quality of reconstructed image will be enhanced. The proposed weighting sampling scheme can also be interpreted in an geometric way. Consider a N dimensional signal space RN , a random sampling matrix forms a subspace H orienting to a random direction. A CS measurement sequence y is a projection of the original signal x from the high dimensional space onto the subspace H. Typically, all the components of x will contribute the same amount of energy with
the same probability. However, by introducing the weighting matrix with the coefficients set as defined in (9), then the subspace is forced to approach the directions where the signal components have large magnitudes, therefore more energy of them will be captured, which improves recovery precision.
mean of DC coefficients all the blocks, which is an unbiased estimation for Gaussian random variables, which is: m n i=1 j=1 Idc (i, j) (13) μ ˆdc = mn
3.2. Determination of Weighting Coefficients
N where m = M B , n = B , i, j are block indexes, and Idc (i, j) is the value of DC coefficient in each block. For the variance, the maximum likelihood estimator is adopted and the estimation is expressed as:
3.2.1. Natural Image Statistics It is well known that the DC component of natural images DCT coefficients can be modeled to satisfy a Gaussian distribution with the Probability Density Function (PDF) of fdc (x) =
2 2 1 √ e−(x−μdc ) /2σdc σdc 2π
(10)
2 where μdc and and σdc are the mean value and variance of DC coefficient, respectively. However, the behavior of AC components are more difficult to model. A proper choice balancing the simplicity of model and fidelity to the empirical data is zero-mean Laplacian distribution [6]:
fac (x) =
1 −|x|/λX e 2λX
(11)
where λX is the rate parameter. Therefore the mean value 2 = and variance for AC coefficients are: μac = 0 and σac 2λ2X . In block processing scheme, we vectorize the DCT coefficients of each block into a 1-D sequence and assign {w1 , w2 , ..., wn } to be the corresponding weighting coefficients. According to the criteria set in (9), we define the weighting coefficient for each frequency component to be the linear combination of the absolute value of its mean and the square root of its variance: wi = η|μi | + (1 − η)σi
(12)
where η(0 ≤ η ≤ 1) is a small constant. Typically, for DCT coefficients of natural images, the frequency components with large magnitudes concentrate at low frequency part which also have large mean and variance, while the high frequency components often have small magnitudes with small mean and variance. Therefore, in most cases, our definition of weighting coefficients is a well approximation that is proportional to the real signal magnitude. 3.2.2. Parameter Estimation In this paper, we employ a block based parameter estimation scheme, namely, the candidate M × N image to be encoded will be divided into B × B blocks and processed. The frequency components at same locations in every block are considered as the outcomes of the same random variable. For DC component, the mean value is estimated by the sample
2 = σ ˆdc
1 (Idc (i, j) − μdc )2 mn i=1 j=1 m
n
(14)
At last, maximum likelihood estimator is again employed to get the estimation of the variance for AC coefficients with zero mean: m n 1 2 σ ˆac = Iac (i, j)2 (15) mn i=1 j=1 similarly Iac (i, j) is the value of the AC coefficient at a certain frequency location in each block. After all the parameters are obtained, the same weighting coefficients set will be used in every block and they will be sent to the decoder as overheads together with CS measurements. In total, we only need to send additional B×B floating point numbers. Though, the compression efficiency will drop a little bit, comparing with the great performance gain the decoder will achieve, the increase in data size is neglectable. 4. EXPERIMENTAL RESULTS We compared our proposed sampling scheme with the the conventional CS framework as well as the reweighted 1 minimization which is the state-of-art reconstruction algorithm proposed recently. The test data are on 512 × 512 8-bit grey level images with different features. Block size of 8 × 8 and 16 × 16 are tested in our simulation. This is because for block size smaller than 8 × 8, the signal length is too short for CS framework to work properly, while for block size larger than 16 × 16, the computational complexity is too high for our test bed. We choose the sparse binary random matrix[7] as the sampling matrix for (5). It has the similar performance as the i.i.d. Gaussian random matrix but simplifies the computation since it is sparse and all the entries are just 0 and 1. According to our experiment, the optimal value for η in (12) is between 0.5 ∼ 0.7 in most cases. It is expected that choosing η adaptively will give better results, however, for the ease of simulation, we choose η = 0.65 uniformly over all experiments. For reweighted 1 minimization, we set the maximum iteration number L = 4 and the constant = 0.1, the same value as stated in [4]. Due to the limitation of space, we only demonstrate the visual quality comparison at sample rate 1 0.4 as shown in Fig.1 1 Sample rate is defined as the ratio of measurements set length to the original signal length.
(a) Original
(b) Conventional
(c) Reweighted 1
(d) Proposed
Fig. 1. Visual quality comparison at sample rate 0.4 with block size 16 × 16. PSNR for (b) ∼ (d) are 27.1dB, 33.0dB and 33.8dB (for more results, please visit: http://ihome.ust.hk/ ˜yyang/). We can see that our proposed method achieves much better reconstruction quality than the conventional CS framework. A more detailed comparison in PSNR performance is tabulated in Table 1. For block size of 8×8, our proposed method achieve an average gain of 7.38dB for all the images listed over all sample rates. And for block size of 16 × 16 the gain is 6.51dB.
to the utilization of the statistics information of natural images, the weighting matrix we introduced helps the sampling procedure to capture more information from significant components in DCT sequence which leads to the enhancement of recovery precision. The simulation results reveal that our proposed algorithm greatly enhances the performance of Compressive Sampling framework without notably increasing the system complexity comparing with the existing conventional Compressive Sampling framework.
Table 1. PSNR Performance (in dB)
6. ACKNOWLEDGEMENT
Sample Rate Proposed Lenna Reweight 1 8×8 Conventional Proposed Peper Reweight 1 8×8 Conventional Proposed Boat Reweight 1 8×8 Conventional Proposed Lenna Reweight 1 16 × 16 Conventional Proposed Peper Reweight 1 16 × 16 Conventional Proposed Boat Reweight 1 16 × 16 Conventional
0.2 31.7 30.6 23.3 32.7 31.7 24.0 28.4 27.5 21.5 32.7 31.7 25.5 33.7 32.4 26.6 29.2 28.0 22.9
0.3 34.5 33.1 26.4 34.9 33.7 27.2 31.0 30.1 24.0 35.4 34.1 28.2 35.6 34.6 29.2 31.7 30.4 25.0
0.4 37.0 35.6 29.0 36.7 35.9 30.1 33.2 32.5 26.2 37.5 36.2 30.5 37.1 36.4 31.6 33.8 33.0 27.1
0.5 38.7 37.4 31.1 37.9 37.5 32.2 34.9 34.7 28.2 39.4 38.1 32.5 38.5 38.1 33.5 35.4 35.3 29.1
One thing we would like to mention is that, comparing with reweighted 1 reconstruction, our proposed method is also notably better, though their performance is supposed to be similar since they are mathematically equivalent. This is because the convergence of reweighted 1 is hard to be reached in the affordable simulation time for image signals, and it is guaranteed to only converge to a local optimum. This phenomenon demonstrates the efficiency of our method from another angle. 5. CONCLUSION In this paper, a novel weighted sampling scheme is proposed for Compressive Sampling based image compression. Due
This work has been supported in part by the Innovation and Technology Commission of the Hong Kong Special Administrative Region, China (project no GHP/048/08). 7. REFERENCES [1] E.J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” Information Theory, IEEE Transactions on, vol. 52, no. 2, pp. 489–509, 2006. [2] E.J. Candes, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006, vol. 3, pp. 1433–1452. [3] Y. Zhang, S. Mei, Q. Chen, and Z. Chen, “A Novel Image/Video Coding Method Based on Compressive Sensing Theory,” in Proc. Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), 2008, pp. 1361–1364. [4] E.J. Candes, M.B. Wakin, and S.P. Boyd, “Enhancing Sparsity by Reweighted L1 Minimization,” Arxiv preprint arXiv:0711.1612, 2007. [5] R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” in Proc. Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), 2008, pp. 3869–3872. [6] E.Y. Lam and J.W. Goodman, “A mathematical analysis of the DCT coefficient distributions forimages,” Image Processing, IEEE Transactions on, vol. 9, no. 10, pp. 1661–1666, 2000. [7] R. Berinde and P. Indyk, “Sparse recovery using sparse random matrices,” Preprint, 2008.
