Binned Progressive Quantization for Compressive ... - Semantic Scholar
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Binned Progressive Quantization for Compressive Sensing Liangjun Wang, Xiaolin Wu, Fellow, IEEE, and Guangming Shi, Senior Member, IEEE
Abstract—Compressive sensing (CS) has been recently and enthusiastically promoted as a joint sampling and compression approach. The advantages of CS over conventional signal compression techniques are architectural: the CS encoder is made signal independent and computationally inexpensive by shifting the bulk of system complexity to the decoder. While these properties of CS allow signal acquisition and communication in some severely resource-deprived conditions that render conventional sampling and coding impossible, they are accompanied by rather disappointing rate–distortion performance. In this paper, we propose a novel coding technique that rectifies, to a certain extent, the problem of poor compression performance of CS and, at the same time, maintains the simplicity and universality of the current CS encoder design. The main innovation is a scheme of progressive fixed-rate scalar quantization with binning that enables the CS decoder to exploit hidden correlations between CS measurements, which was overlooked in the existing literature. Experimental results are presented to demonstrate the efficacy of the new CS coding technique. Encouragingly, on some test images, the new CS technique matches or even slightly outperforms JPEG. Index Terms—Compressive sensing (CS), convex optimization, integer programming, progressive refinement, quantization binning.
I. INTRODUCTION
A
TYPICAL digital image acquisition system consists of two cascade modules: 1) a dense sensor array that takes a large number of samples (pixels) of the light field and 2) a compressor that reduces the large amount of raw data created by the sensor array. According to the Nyquist–Shannon sampling theorem, the reconstruction of the original continuous image signal requires the sampling frequency to be at least twice as high as the highest frequency component of the image. In many Manuscript received June 17, 2011; revised November 27, 2011; accepted February 06, 2012. Date of publication February 24, 2012; date of current version May 11, 2012. This work was supported in part by the National Science Foundation of China under Grant 61070138, Grant 61072104, Grant 61003148, Grant 60902031, Grant 61033004, and Grant 61100155, in part by the Research Fund for the Doctoral Program of Higher Education of China under Grant 20090203110003, and in part by the Natural Sciences and Engineering Research Council of Canada. The associate editor coordinating the review of this paper and approving it for publication was Dr. James E. Fowler. L. Wang and G. Shi are with the Key Laboratory of Intelligent Perception and Image Understanding, Ministry of Education of China, School of Electronic Engineering, Xidian University, Xi’an 710071, China (e-mail: lj_wang@mail. xidian.edu.cn; [email protected]). X. Wu is with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4L8, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2012.2188810
high-end applications in sciences, medicine, and space, users need to capture very fine details and, hence, demand cameras of ultrahigh resolution. However, the cost and complexity of the camera system drastically increase in spatial resolution. In some application scenarios, such as on board of a satellite in outer space, the camera system is limited in both computing power and energy supply. In such a stringent system setting, the designer would naturally question why a large number of samples should be taken in the first place if sampled raw image data are highly compressible? The critique of conventional practice of “oversampling followed by massive dumping” has generated great enthusiasm in the new image acquisition strategy of compressive sensing (CS) [1], [2]. By unifying sampling and compression, the CS-based imager abandons the traditional sampling-then-processing cascade architecture and, hence, can be made simpler and more energy efficient. that is The CS theory states that a signal of length -sparse in space is recoverable from mearandom measurements , where is an surement matrix. As a joint sampling–compression scheme, CS streamlines the encoder of an image/video acquisition–communication system at the cost of high decoder complexity. This is because the decoder has to solve a nonlinear reconstruction problem, solved, for instance, with large-scale convex optimization programs. Shifting complexity from encoder to decoder is necessary whenever the encoder is severely resource deprived. However, up to now, CS-based image compression techniques suffer heavy losses in rate–distortion performance compared with conventional image coding techniques. The main cause of the inferior compression performance of CS is the wide use of a uniform scalar quantizer. Since CS measurements are real, they have to be quantized before can be transmitted or stored digitally. The inefficiency of uniform scalar quantization (SQ) of CS measurements was analyzed by Boufounos and Baraniuk [3] and by Goyal et al. [4]. To alleviate the coding loss of a uniform scalar quantizer, Dai et al. [5] and Jacques et al. [6] modified the constraints of the standard CS recovery algorithm. They proposed to confine each individual CS measurement within the quantization interval instead of applying an overall by the mean-squared error bound, that is, replacing quantization error bound. Although these authors improved the coding efficiency to some extent, they did not realize and exploit the statistical redundancy in CS measurements as we do in this paper. In [7], Pai studied how much the performance of CS can approach that of an adaptive encoding scheme by the quantization binning technique. Boufounos proposed a universal rate-efficient SQ that can reduce the quantization error with side information in [8]. However, these two papers
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were limited to theoretical analysis; neither of them developed a practical CS reconstruction algorithm. The decoding algorithm proposed in [7] has prohibitively high complexity and cannot be applied to real-world problems of even modest size. Very recently, Kamilov et al. have proposed an approximate message-passing estimation technique for recovering a signal from quantized linear measurements [9]. The rate–distortion performance of CS can be improved by increasing the efficiency of CS measurement quantization. The problem is that, however, rate–distortion optimized quantizers used in conventional signal compression systems (e.g., variable rate quantizers, entropy-constrained quantizers, etc.) have rather high complexity. The employment of these more sophisticated quantization techniques would make the CS encoder just as resource demanding as conventional image encoders, contradicting the CS design principle and neutralizing its most important strength. In this paper, we propose a progressive quantization (PQ) technique for lossy compression of CS measurements that can substantially improve the rate–distortion performance of the CS image acquisition system, while maintaining the low complexity of the encoder. The decoder is essentially the same as the traditional CS recovery algorithm. This new CS coding technique stems from the following observation. In a CS image acquisition–coding system, random measurements of an image are made; these measurements are quantized and sequentially transmitted to the decoder. However, instead of recovering after receiving all measurements, the decoder can recover the image upon receiving first measurements and obtain an estimated image . Since the first recovered image contains information about , it can be used to reduce the quantization precision of the subsequent CS measurements and consequently improve coding efficiency [10]. In spirit and functionality, very much resembles side information in distributed source coding [11], [12]. As in distributed source coding, the proposed quantization technique exploits correlation between and , or mutual information , only at the decoder. In this approach, we retain all the original advantages of the CS image acquisition methodology: reduced sampling rate, simple and energy-conserving encoder, and universality of the sampling scheme, while greatly improving the coding efficiency. In order to keep the encoder complexity at the minimum, in each stage of PQ, we stick to a fixed-length coded uniform scalar quantizer without any entropy coding. All coding gain is solely made by a novel cascade fixed-rate scalar quantizer of binning that can be 30% more efficient than the current CS quantization scheme without increasing complexity or energy consumption. Quite remarkably, on some test images, the proposed CS method matches or even slightly outperforms the JPEG compression standard in rate–distortion performance. This is the first instance, to our best knowledge, that CS is made competitive against a mainstream compression scheme in rate–distortion performance. However, this paper only qualifies as among initial attempts to lend CS credibility as a signal compressor in addition to being a sparse sampler. Our findings merely indicate the possibility of bringing CS on par with classical signal compression methods in coding
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Fig. 1. CS-based image acquisition–communication system.
performance with frugal source encoders. More studies on sophisticated quantizers of CS measurements are warranted. In particular, trellis coded quantization [13], with its low encoder complexity and high coding performance, holds the promise of further improving the reported results. The remainder of this paper is organized as follows. Section II introduces new system architecture of CS that differs from the current design in quantization operation: cascade fixed-rate SQ of CS measurements versus uniform SQ. It is explained how in the new CS architecture a simple binning technique can improve compression performance without increasing encoder complexity. Section III presents the process of estimated dequantization with binning to be performed by the CS decoder. In Section IV, we clarify how to determine some necessary parameters in our scheme. Section V discusses the optimal measurement allocation problem, and the real-time low-complexity implementation is introduced in Section VI. Experimental results are reported in Section VII, followed by conclusion in Section VIII. II. PROGRESSIVE CS QUANTIZATION The architecture of current CS-based image acquisition–communication systems is depicted in Fig. 1. At the encoder, an image is sampled into random measurements by a random projection matrix . The measurements are uniformly scalar quantized into . To keep the encoder simple, each is sequentially transmitted in fixed-length code of bits to the decoder. In the current CS architecture, the decoder collects all measurements and recovers in one step by solving an minimization problem. However, this CS recovery approach suffers from an intrinsic suboptimality in terms of rate–distortion performance. The deficiency of the all-at-once method is caused by its total disregard of the mutual information that can be significant. Although any two CS measurements and are independent of each other, two groups of CS measurements and are collectively correlated in the following hidden way. Let be the submatrix consisting of the first rows of , and let be the remainder submatrix of . If the input image is approximately recovered with the two subsets of random projections (1) separately, resulting two estimated images and the two signals and are correlated and so are
, then and
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Fig. 2. CS system with two-layer quantization of CS measurements. (a) Encoder. (b) Decoder.
. Based on this observation, we can improve the coding efficiency of CS by a progressive recovery strategy. The decoder first computes an initial or base estimate using the first measurements and then refines the base estimate by adding the remainder measurements in the CS recovery. At the refinement stage, the available base estimate greatly reduces the uncertainty of because
information provided by or . When encoding CS measurements , , in the refinement layer, bits can be saved by relating to the side information because
(2)
(4)
In other words, is a side information of , and it can be used to reduce the bit rate of the second patch of CS measurements without materially affecting the final CS recovery quality. Specifically, we propose a binning technique to reduce the quantization precision of . As argued in the introduction, the CS encoder should be kept as simple and energy efficient as possible. Hence, a fixed-rate uniform scalar quantizer is adopted to code both base and refinement patches of CS measurements and . Suppose in the final CS recovery the required quantization precision of each measurement is bits. For the first measurements , bits per measurement are transmitted. However, for each of the remaining measurements, only the least significant bits of the quantizer output index are transmitted. The most significant bits (MSBs) are dropped for , , and these dropped bits are instead estimated from by the decoder. The proposed CS architecture of two-layer quantization of CS measurements is schematically described in Fig. 2.
Given , the row random measurement vector mains Gaussian with each entry having the same variance but the conditional expectation of becomes
re,
(5) Therefore,
is a Gaussian random variable of the variance and the mean (6)
where . To simplify notations, we drop the subscript , , in and in the sequel. By Bayesian theorem, the posterior distribution can be computed from the posterior distribution and the prior distribution of . Thus
III. ESTIMATED DEQUANTIZATION WITH BINNING In this section, we discuss the dequantization process with binning for the CS measurements in the refinement layer, for which only the least significant bits of the quantization index are transmitted. Suppose the length of the signal is , for the CS measurement matrix whose entries are drawn from Gaussian distribution of zero mean and variance , all the measurements , where is the th row of the CS measurement matrix , , are Gaussian distributed as well, having zero mean and variance
(7)
where
(3) In other words, information available. However, for
when there was no extra , , there exists side
It follows from (7) that the conditional distribution Gaussian, and its variance is determined by A sufficiently large makes and
is still and . , hence,
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Fig. 3. Illustration of keeping the two least significant bits of a uniform scalar quantizer (where falls into one of the shaded bins.
and . In other words, the conditional variprimarily depends on the variance . By the benance efit of the mutual information between and , the variance of is reduced from to . Aided by the side information , the decoder can estimate a CS measurement in the refinement layer from the least significant bits of the quantized value. It can use either minimum mean square error (MSE) estimator or maximum a posteriori probability (MAP) estimator. Denote by the integer value of the least significant bits and by the quantization step size. The value of must fall into the following set of subintervals:
(8) as shown in Fig. 3. The MMSE estimator of
, and the two least significant bits are
). The value of
(base layer) signal (larger reduces ). How to optimally allocate the bit budget between the CS measurements in the base and refinement layers is the subject of Section V. IV. PROGRESSIVE QUANTIZER DESIGN PARAMETERS In this section, we derive the design parameters of the proposed PQ scheme for CS. Let the quantization precision of CS measurements be bits per sample. The step size of the uniform scalar quantizer is chosen such that the bins of size covers almost the entire dynamic range of CS measurements. Let be the probability mass to be covered by the quantizer bins, with a negligible overload error probability . Since the CS measurements obey a Gaussian distribution of variance , we have (11)
is
(12)
(9) where and are the probability density function and cumulative distribution function of the standard normal distribution, respectively. Alternatively, the decoder can use the MAP estimator of in dequantization, i.e., (10) Since the MAP estimate is given by the nearest value to the in the set , it is computationally much more efficient than the MSE estimate. The errors of the above two estimators mainly depend on the ratio . The greater the ratio, the smaller the estimation error. Therefore, the estimation precision of the refinement CS measurements improves as increases and, also, if more measurements are used to generate the initial estimated
where and are the error function and its inverse function of normal distribution, respectively. Now, we discuss how to determine , which is the number of the least significant bits to be transmitted in the refinement layer. The value of should be sufficiently large such that the MAP estimator can determine which bin measurement falls into with probability . As discussed in Section III, the MAP estimate is the point in the candidate set that has the minimum distance to . Thus, identifying the correct bin of requires the distance to be less than the half of the distance between two consecutive candidate bins (i.e., ) in , where is the quantized value of . Since , we have (13) that is (14) , which is the number of the The value of is related to measurements in the base layer. The larger is, the stronger the side information available to the refinement layer, and the smaller the value of can be. Specifically, using the approximation as discussed in Section III, we have (15)
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Fig. 4. Operational error functions of the (a) TV and (b) the MARX recovery methods.
where image
using the first
is the distortion of the recovered CS measurements. Therefore
performance of the proposed PQ scheme for CS. Let and be the number of base and refinement measurements, respectively. The base measurements are coded in bits, and the refinement measurements are coded in bits, where is as derived in the section above. Due to the quantization precision in two layer are nearly the same, in order to minimize the error in CS recovery of , we would like to maximize the total number of measurements subject to the bit budget constraint . Then, the optimal rate allocation problem can be stated as
subject to (20) Substituting (19) into it, we can get
(16) next. Generally speaking, the natural image We analyze signals are not strictly sparse, but they are still compressible due to the so-called near sparsity. Near sparsity means that the coefficients in a recovery space belong to an ball for . In other words, if the coefficient magnitude is in decreasing order, then (17) and the reconstruction error of bounded as [14]
base measurements can be (18)
where is a decay factor. Empirically, the above exponential behavior of CS reconstruction errors in the number of measurements is observed not only for the standard CS recovery method but also for other CS recovery methods. Fig. 4 shows the actual error functions of the popular total variation (TV) CS recovery method [15] and the method of model-based adaptive recovery of CS (MARX) [16]. The error curves are plotted for three representative images of different decay rates: “Pepper” (rapid), “Goldhill” (modest), and “Baboon” (slow). Substituting (18) into (16) yields
(19) where
. V. OPTIMAL MEASUREMENT ALLOCATION
Having discussed the determination of and , we now investigate the problem of bit allocation between the base and refinement layers, which plays a critical role in the rate–distortion
subject to (21) The above problem is one of integer programming because and are natural numbers, which is inherently difficult to solve. However, we can still get a good approximation solution in a relaxation approach: first allowing and to take on the real values in (21) and then rounding to integer the solution of the corresponding real-valued maximization problem. Now, we show how to solve the real-valued optimal bit allocation problem. It is proven in the Appendix that the solution space of (21) is convex. As such, the optimal solution is necessarily the point on the convex curve, i.e., (22) whose slope is
1, i.e., (23)
If the decay factor for the input image can be efficiently estimated, which is the subject of the next section, we can numerically solve the above nonlinear equation for optimal allocation for the base layer and then get optimal allocation for the refinement layer by subsisting into (22). Finally, we obtain the corresponding integer-valued allocation by rounding. VI. REAL-TIME LOW-COMPLEXITY IMPLEMENTATION We improve the rate–distortion performance of existing CS methods by making the measurement quantizer progressive and adaptive to the input image. The signal dependence of the proposed PQ scheme is shown in (19), (22), and (23). The values of , , and can be computed only if the decay factor and constant in the distortion function are known. A straightforward way of computing them is to recover the image signal for increasing and fit the distortion function in
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Fig. 5. Scatter plots and regressions of (d) MARX.
versus
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for (a) TV and (b) MARX. Scatter plots and regressions of
(18) to the resulting data points . However, this naive approach incurs a very high computational complexity of the encoder, and, as such, it nullifies the main advantage of CS: signal acquisition at low cost. The challenge is how to add signal adaptability to CS without materially increasing the computational complexity of image acquisition and communication. In order to meet the above objective, we develop a technique of estimating the decay factor of an input image . The new technique is based on the observation that higher decay rate of the distortion function corresponds to a greater degree of sparsity of , or increases as decreases. For most natural images, the level of energy concentration in the low-frequency subbands is highly correlated to the degree of sparsity. Therefore, we can empirically establish the relation of with respect to the allocation of signal energy in low frequencies. This allocation is defined to be the ratio , where is the total energy of the lowest frequency DCT coefficients of in percentage. Fig. 5(a) and (b) shows the scatter plots of versus for 100 natural images with the TV and MARX recovery methods, respectively, when setting . Such a functional relationship between and exhibited by Fig. 5(a) and (b), which can be satisfactorily fit by an exponential model, is learnt off line by using an appropriate training set. The training set can be made general to encompass all varieties of natural images or highly targeted if input images are of a particular type (e.g., faces). We can also empirically determine the constant (normalized by signal length ) in distortion function . It turns out that is nearly linear in the signal energy , as shown in Fig. 5(c) (TV) and (d) (MARX) that are the scatter plots of the same set of 100 natural images in Fig. 5(a) and (b). In real-time CS image acquisition and transmission, the encoder transmits to the decoder the signal energy and the aforementioned ratio (a proxy of the signal sparsity). Then, both the encoder and the decoder can obtain parameters and from the empirical functions discussed above and subsequently compute measurement allocation using (22) and (23). In addition, is needed to compute the quantization step size with (12). The encoder can readily compute as it is simply the energy of the CS measurements. In order to compute the ratio , the encoder measures in the of lowest frequency DCT bases and obtains . This can be done at very low computational complexity with the digital micromirror device technology [17], [18]. Granted, the empirically obtained real-time low-complexity solution of measurement allocation is suboptimal be-
versus
for (c) TV and
Fig. 6. Test images. (a) Lena. (b) Pepper. (c) Boats. (d) Monarch. (e) Cameraman. (f) Goldhill. (g) Birds. (h) Couple. (i) Paints. (j) Vessels.
cause it is based on estimated and . Nevertheless, the proposed approximate solution is found to be reasonably robust in practice. The difference in PSNR performance between our approximate solution and the truly optimal progressive quantizer (as though the function for the CS recovery algorithm on the input signal was known) is about 0.3 dB on average. VII. EXPERIMENTAL RESULTS We implemented the proposed technique of PQ of CS measurements and experimented it on natural images. In this section, we report our experimental results and compare them with those of the current CS methods. For the validity and fairness of our evaluations, we use a set of diverse images to test the proposed new CS quantization technique, some samples of which are shown in Fig. 6. These test images cover a wide range of image signals, ranging from relatively simple (e.g., Birds) to complex scenes (e.g., Goldhill). To achieve lower computation complexity, we use a random circular matrix to generate the random measurements. The first row of the matrix consists of Gaussian random variables, and the other rows are right-shift of the previous row. This matrix can speed up the computations at both the encoder and the decoder since the matrix multiplication can be efficiently implemented with fast Fourier transform [19]. In our experiments, we choose . The number of bits per CS measurement affects the rate–distortion performance of the CS-based signal compression system. The distortion of the recovered signal has two components: the approximation error due to insufficient number of CS measurements and the quantization error caused by insufficient quantization precision. Given a total bit budget , decreasing allows a larger number
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TABLE I PSNR (DB) RESULTS OF DIFFERENT IMAGES AT THE RATE OF 1.0 BPP
Fig. 7. Rate–distortion performance results of different CS recovery methods using simple scalar quantizer (SQ) and the proposed progressive scalar quantizer (PQ), in comparison with those of the JPEG image compression standard. (a) Lena. (b) Monarch. (c) Vessels. (d) Pepper.
of measurements but reduces the quantization precision per measurement, and vice versa. In other words, the approximation and quantization errors cannot be simultaneously minimized, and the optimal value of is the one that achieves the minimum distortion of the recovered image. Optimizing for minimum distortion seems to be an open problem, and no closed-form analytical solution or nontrivial algorithm (except exhaustive search) for optimal is known. Fortunately, in practice, the optimal value of was empirically found to be nearly constant for a given total bit budget over a wide range of natural images [20]. Fig. 7 presents the plots of PSNR versus the rate of uniform SQ without entropy coding (in the interest of inexpensive encoder) for four representative test images. The proposed PQ strategy makes significant coding gains over SQ when coupled with both the popular TV recovery method and the MARX method. In the comparison group in Fig. 7, we also include the results of the JPEG image compression standard, a classical case of dense-sampling, encoder-centric signal compression methodology. For two test images, namely, “Monarch” and “Vessels,” the PQ-based CS recovery method matches and even slightly outperforms JPEG in rate–distortion performance. This
is quite remarkable considering that, up to now, all existing CS methods are not competitive against traditional signal compression methods in rate–distortion performance. This paper suggests a promising direction to bring CS on par with traditional signal compression methods in coding performance while retaining the other advantages of CS. For more comprehensive comparisons, we add in Table I the PSNR results of ten test images at a bit rate of 1.0 bit per pixel (bpp). In this table, in addition to those of TV and MARX, the results of the minimization CS recovery in discrete wavelet transform (DWT) space are included as well. As clearly shown in Fig. 7 and Table I, the proposed PQ scheme consistently improves the reconstruction quality of all three recovery methods over the use of uniform SQ. It can be observed in Table I that the coding gain of PQ over SQ is greatest with MARX and smallest with DWT (1.96 dB versus 0.99 dB on average). This is because MARX is an adaptive recovery method, and it obtains a better initial estimate of the signal than the DWT recovery given the bit budget. The better the initial estimate, the higher precision the binned dequantization of the refinement measurements. In addition, Fig. 7 shows that the advantage of PQ over SQ becomes greater as the bit rate increases. The reason is that as the total bit budget increases, the correlation between the base and refinement layers of the CS measurements gets stronger, and SQ becomes even more inefficient. To evaluate the effectiveness of PQ in decoding the discarded refinement measurements, we report in Table II MSBs of the the probabilities that the binned decoder correctly estimates the first two MSBs for different coding parameters and different CS recovery methods. To see how the optimal progressive quantizer design process allocates measurements between the base and refinement layers in relation to the bit budget, we tabulate in Table III the optiand that are normalized by the signal mized values of length . We also include in the table the optimal values of of allocated bits to the base and refineand the ratio ment layers. For fixed and , the ratio generally increases as the bit budget increases, i.e., more measurements are allocated to the refinement layer. This is because only needs to be sufficient to make the recovered base layer good enough to facilitate the binned dequantization of the measurements in the refinement layer. For the same reason, when is fixed, the ratio with respect to a fixed bit rate is greater for a superior CS recovery algorithm (see MARX versus TV in Table III).
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TABLE II PROBABILITY OF THE DECODER ESTIMATING THE FIRST TWO MSBS CORRECTLY
TABLE III VALUES OF DESIGN PARAMETERS FOR DIFFERENT BIT RATES
TABLE IV COMPARISON OF SQ AND PQ SCHEMES IN THE NUMBER OF MEASUREMENTS ACCOMMODATED BY DIFFERENT BIT BUDGETS
By removing the statistical redundancy between two sets of CS measurements, the new PQ scheme can accommodate more CS measurements than SQ for a given bit budget. Let be the total number of CS measurements coded by PQ, and let be the number of CS measurements coded by SQ. To demonstrate the improvement of PQ over SQ in coding efficiency, we present the values of and normalized by and the ratio in Table IV. It can be observed in the table that the performance gap between PQ and SQ grows as the bit budget increases. With significant PSNR gains of PQ over SQ, one should expect visible improvement of visual quality by the new technique accordingly. Figs. 8–10 present and compare output images of the TV and MARX CS recovery methods using the two different measurement quantization schemes. In both TV and MARX cases, the images of PQ appear to be more pleasant and accurate recoveries of the originals than SQ, particularly in areas of edges and textures. VIII. CONCLUSION AND FUTURE WORK In CS, hidden correlations exist between different subsets of CS measurements but were overlooked in the existing literature. This was a cause of disappointing rate–distortion performance of present CS-based signal compression schemes. To overcome the drawback and improve the CS coding efficiency,
we developed a novel technique of progressive fixed-rate SQ with binning. The new technique can remove the redundancies between CS measurements without increasing the complexity and resource requirement of the CS encoder. Significant improvements over existing CS techniques were achieved. In some cases, the proposed CS quantization technique can match and even slightly outperform the traditional encoder-centric JPEG image compression standard. This paper demonstrates the importance of measurement quantization to the CS-based image acquisition and coding system. The specific technique developed and analyzed above is only one possible embodiment of the general principle of binned progressive CS quantization. Further improvement in rate–distortion performance of CS can be made by extending and fine tuning the proposed technique. For instance, the number of PQ levels can be increased to if the decoder has sufficient computation power to perform CS recovery times. The idea is to let a progressively refined reconstruction of the signal contribute to the estimation of subsequent measurements on the fly. In quest for higher coding efficiency, lattice vector quantizers can replace the uniform scalar quantizer in the system, and, accordingly, the binning ought to be performed in the vector code space. It will be also interesting to study the possibility and merits of “turbo binning”: initially dropping few least significant bits of the first batch of (base) measurements
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Fig. 8. Comparison of PQ and SQ on test image Boats at the rate of 1.0 bpp. dB . (c) TV (a) Original. (b) TV recovery with SQ scheme PSNR dB . (d) MARX recovery with SQ recovery with PQ scheme PSNR dB . (e) MARX recovery with PQ scheme PSNR scheme PSNR dB .
Fig. 10. Comparison of PQ and SQ on test image Lena at the rate of 1.0 bpp. dB . (c) TV (a) Original. (b) TV recovery with SQ scheme PSNR dB . (d) MARX recovery with SQ recovery with PQ scheme PSNR dB . (e) MARX recovery with PQ scheme PSNR scheme PSNR dB .
APPENDIX A In this Appendix, we show that the bit allocation problem (21) is one of convex optimization. The simple linear objective function is clearly convex and so are the inequalities and . The remaining task is to prove the convexity of the boundary curve corresponding to the bit budget constraint, i.e., (24) Fig. 9. Comparison of PQ and SQ on test image Vessels at the rate of 1.0 bpp. dB . (c) TV (a) Original. (b) TV recovery with SQ scheme PSNR dB . (d) MARX recovery with SQ recovery with PQ scheme PSNR dB . (e) MARX recovery with PQ scheme PSNR scheme PSNR dB .
and later estimating the dropped bits upon obtaining a refined reconstruction of the signal, and so forth.
The second derivative function of (24) is
(25) and, it is negative for all feasible values of . To see this, we note because for the proposed PQ scheme of CS measurements. Hence, the
WANG et al.: BINNED PROGRESSIVE QUANTIZATION FOR COMPRESSIVE SENSING
sign of of (25). Thus
only depends on the term in the numerator
(26) In the above inequality, we used the facts . Furthermore
and
(27) in which the first inequality is due to , and the second inequality follows from the fact that, for together with . Since is a monotonically nonincreasing function, for all
(28) Finally
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[8] P. Boufounos, “Universal rate-efficient scalar quantization,” ArXiv preprint arXiv:1009.3145 Sep. 2010. [9] U. Kamilov, V. K. Goyal, and S. Rangan, “Message-passing estimation from quantized samples,” ArXiv preprint arXiv:1105.6368 May 2011. [10] S. S. Pradhan and K. Ramchandran, “Generalized coset codes for distributed binning,” IEEE Trans. Inf. Theory, vol. 51, no. 10, pp. 3457–3474, Oct. 2005. [11] D. Slepian and J. Wolf, “Noiseless coding of correlated information sources,” IEEE Trans. Inf. Theory, vol. IT-19, no. 4, pp. 471–480, Jul. 1973. [12] A. Wyner and J. Ziv, “The rate–distortion function for source coding with side information at the decoder,” IEEE Trans. Inf. Theory, vol. IT-22, no. 1, pp. 1–10, Jan. 1976. [13] M. W. Marcellin and T. R. Fischer, “Trellis coded quantization of memoryless and Gauss_Markov sources,” IEEE Trans. Commun., vol. 38, no. 1, pp. 82–93, Jan. 1990. [14] E. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory, vol. 52, no. 12, pp. 5406–5425, Dec. 2006. [15] E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math., vol. 59, no. 8, pp. 1207–1223, Aug. 2006. [16] X. Wu and X. Zhang, “Model-guided adaptive recovery of compressive sensing,” in Proc. Data Compression Conf., Mar. 2009, pp. 123–132. [17] D. Dudley, W. Duncan, and J. Slaughter, Emerging Digital Micromirror Device (DMD) Applications. Dallas, TX: Texas Intrum., Feb. 2003, White paper. [18] D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in Proc. IS&T/SPIE Symp. Electron. Imag., 2006, pp. 43–52. [19] W. U. Bajwa, J. Haupt, G. Raz, S. J. Wright, and R. Nowak, “Toeplitzstructured compressed sensing matrices,” in Proc. 14th IEEE/SP Workshop SSP, Aug. 2007, pp. 294–298. [20] A. Schulz, L. Velho, and E. A. B. da Silva, “On the empirical rate–distortion performance of compressive sensing,” in Proc. Int. Conf. Image Process, Nov. 2009, pp. 3049–3052.
Liangjun Wang received the B.S. degree from Xidian University, Xi’an, China, in 2004, where he is currently working toward the Ph.D. degree with the School of Electronic Engineering. His research interests include compressive sensing, image processing, multimedia communication, and signal compression.
(29) and, the convexity of the solution space follows. REFERENCES [1] D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, Apr. 2006. [2] E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [3] P. Boufounos and R. Baraniuk, “Quantization of sparse representations,” in Proc. Data Compression Conf., Mar. 2007, pp. 378–387. [4] V. K. Goyal, A. K. Fletcher, and S. Rangan, “Compressive sampling and lossy compression,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 48–56, Mar. 2008. [5] W. Dai, H. Vinh Pham, and O. Milenkovic, “Quantized compressive sensing,” ArXiv preprint arXiv:0901.0749 Mar. 2009. [6] L. Jacques, D. K. Hammond, and J. M. Fadili, “Dequantizing compressed sensing: When oversampling and non-Gaussian constraints combine,” IEEE Trans. Inf. Theory, vol. 57, no. 1, pp. 559–571, Jan. 2011. [7] R. J. Pai, “Nonadaptive lossy encoding of sparse signals,” M.S. thesis, Massachusetts Inst. Technol., Cambridge, MA, 2006.
Xiaolin Wu (SM’96–F’11) received the B.S. degree in computer science from Wuhan University, Wuhan, China, in 1982 and the Ph.D. degree in computer science from the University of Calgary, Calgary, AB, Canada, in 1988. He started his academic career in 1988. He was with the faculty of the University of Western Ontario, London, ON, Canada, and New York Polytechnic University, New York. He is currently a Professor with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, and holds the NSERC senior industrial research chair in Digital Cinema. His research interests include multimedia signal compression, joint source-channel coding, multiple description coding, network-aware visual communication, and image processing. He has published over 230 research papers and holds two patents in these fields. Dr. Wu is an Associate Editor of the IEEE TRANSACTIONS ON IMAGE PROCESSING. He also served as an Associate Editor of the IEEE TRANSACTIONS ON MULTIMEDIA and on the technical committees of many IEEE international conferences/workshops on image processing, multimedia, data compression, and information theory.
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Guangming Shi (SM’10) received the B.S. degree in automatic control, the M.S. degree in computer control, and the Ph.D. degree in electronic information technology from Xidian University, Xi’an, China, in 1985, 1988, and 2002, respectively. He joined the School of Electronic Engineering, Xidian University, in 1988. From 1994 to 1996, he was a Research Assistant with the Department of Electronic Engineering, University of Hong Kong, Pokfulam, Hong Kong. Since 2003, he has been a Professor with the School of Electronic Engineering,
Xidian University. In 2004, he was the Head of the National Instruction Base of Electrician and Electronic. From June to December in 2004, he studied with the Department of Electronic Engineering, University of Illinois at Urbana-Champaign, Urbana. Currently, he is Deputy Director with the School of Electronic Engineering and the academic leader in the subject of Circuits and Systems. His research interests include compressed sensing, theory and design of multirate filter banks, image denoising, low-bit-rate image/video coding, and implementation of algorithms for intelligent signal processing. He has authored or coauthored over 60 research papers.
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 6, JUNE 2012
Binned Progressive Quantization for Compressive Sensing Liangjun Wang, Xiaolin Wu, Fellow, IEEE, and Guangming Shi, Senior Member, IEEE
Abstract—Compressive sensing (CS) has been recently and enthusiastically promoted as a joint sampling and compression approach. The advantages of CS over conventional signal compression techniques are architectural: the CS encoder is made signal independent and computationally inexpensive by shifting the bulk of system complexity to the decoder. While these properties of CS allow signal acquisition and communication in some severely resource-deprived conditions that render conventional sampling and coding impossible, they are accompanied by rather disappointing rate–distortion performance. In this paper, we propose a novel coding technique that rectifies, to a certain extent, the problem of poor compression performance of CS and, at the same time, maintains the simplicity and universality of the current CS encoder design. The main innovation is a scheme of progressive fixed-rate scalar quantization with binning that enables the CS decoder to exploit hidden correlations between CS measurements, which was overlooked in the existing literature. Experimental results are presented to demonstrate the efficacy of the new CS coding technique. Encouragingly, on some test images, the new CS technique matches or even slightly outperforms JPEG. Index Terms—Compressive sensing (CS), convex optimization, integer programming, progressive refinement, quantization binning.
I. INTRODUCTION
A
TYPICAL digital image acquisition system consists of two cascade modules: 1) a dense sensor array that takes a large number of samples (pixels) of the light field and 2) a compressor that reduces the large amount of raw data created by the sensor array. According to the Nyquist–Shannon sampling theorem, the reconstruction of the original continuous image signal requires the sampling frequency to be at least twice as high as the highest frequency component of the image. In many Manuscript received June 17, 2011; revised November 27, 2011; accepted February 06, 2012. Date of publication February 24, 2012; date of current version May 11, 2012. This work was supported in part by the National Science Foundation of China under Grant 61070138, Grant 61072104, Grant 61003148, Grant 60902031, Grant 61033004, and Grant 61100155, in part by the Research Fund for the Doctoral Program of Higher Education of China under Grant 20090203110003, and in part by the Natural Sciences and Engineering Research Council of Canada. The associate editor coordinating the review of this paper and approving it for publication was Dr. James E. Fowler. L. Wang and G. Shi are with the Key Laboratory of Intelligent Perception and Image Understanding, Ministry of Education of China, School of Electronic Engineering, Xidian University, Xi’an 710071, China (e-mail: lj_wang@mail. xidian.edu.cn; [email protected]). X. Wu is with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON L8S 4L8, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2012.2188810
high-end applications in sciences, medicine, and space, users need to capture very fine details and, hence, demand cameras of ultrahigh resolution. However, the cost and complexity of the camera system drastically increase in spatial resolution. In some application scenarios, such as on board of a satellite in outer space, the camera system is limited in both computing power and energy supply. In such a stringent system setting, the designer would naturally question why a large number of samples should be taken in the first place if sampled raw image data are highly compressible? The critique of conventional practice of “oversampling followed by massive dumping” has generated great enthusiasm in the new image acquisition strategy of compressive sensing (CS) [1], [2]. By unifying sampling and compression, the CS-based imager abandons the traditional sampling-then-processing cascade architecture and, hence, can be made simpler and more energy efficient. that is The CS theory states that a signal of length -sparse in space is recoverable from mearandom measurements , where is an surement matrix. As a joint sampling–compression scheme, CS streamlines the encoder of an image/video acquisition–communication system at the cost of high decoder complexity. This is because the decoder has to solve a nonlinear reconstruction problem, solved, for instance, with large-scale convex optimization programs. Shifting complexity from encoder to decoder is necessary whenever the encoder is severely resource deprived. However, up to now, CS-based image compression techniques suffer heavy losses in rate–distortion performance compared with conventional image coding techniques. The main cause of the inferior compression performance of CS is the wide use of a uniform scalar quantizer. Since CS measurements are real, they have to be quantized before can be transmitted or stored digitally. The inefficiency of uniform scalar quantization (SQ) of CS measurements was analyzed by Boufounos and Baraniuk [3] and by Goyal et al. [4]. To alleviate the coding loss of a uniform scalar quantizer, Dai et al. [5] and Jacques et al. [6] modified the constraints of the standard CS recovery algorithm. They proposed to confine each individual CS measurement within the quantization interval instead of applying an overall by the mean-squared error bound, that is, replacing quantization error bound. Although these authors improved the coding efficiency to some extent, they did not realize and exploit the statistical redundancy in CS measurements as we do in this paper. In [7], Pai studied how much the performance of CS can approach that of an adaptive encoding scheme by the quantization binning technique. Boufounos proposed a universal rate-efficient SQ that can reduce the quantization error with side information in [8]. However, these two papers
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were limited to theoretical analysis; neither of them developed a practical CS reconstruction algorithm. The decoding algorithm proposed in [7] has prohibitively high complexity and cannot be applied to real-world problems of even modest size. Very recently, Kamilov et al. have proposed an approximate message-passing estimation technique for recovering a signal from quantized linear measurements [9]. The rate–distortion performance of CS can be improved by increasing the efficiency of CS measurement quantization. The problem is that, however, rate–distortion optimized quantizers used in conventional signal compression systems (e.g., variable rate quantizers, entropy-constrained quantizers, etc.) have rather high complexity. The employment of these more sophisticated quantization techniques would make the CS encoder just as resource demanding as conventional image encoders, contradicting the CS design principle and neutralizing its most important strength. In this paper, we propose a progressive quantization (PQ) technique for lossy compression of CS measurements that can substantially improve the rate–distortion performance of the CS image acquisition system, while maintaining the low complexity of the encoder. The decoder is essentially the same as the traditional CS recovery algorithm. This new CS coding technique stems from the following observation. In a CS image acquisition–coding system, random measurements of an image are made; these measurements are quantized and sequentially transmitted to the decoder. However, instead of recovering after receiving all measurements, the decoder can recover the image upon receiving first measurements and obtain an estimated image . Since the first recovered image contains information about , it can be used to reduce the quantization precision of the subsequent CS measurements and consequently improve coding efficiency [10]. In spirit and functionality, very much resembles side information in distributed source coding [11], [12]. As in distributed source coding, the proposed quantization technique exploits correlation between and , or mutual information , only at the decoder. In this approach, we retain all the original advantages of the CS image acquisition methodology: reduced sampling rate, simple and energy-conserving encoder, and universality of the sampling scheme, while greatly improving the coding efficiency. In order to keep the encoder complexity at the minimum, in each stage of PQ, we stick to a fixed-length coded uniform scalar quantizer without any entropy coding. All coding gain is solely made by a novel cascade fixed-rate scalar quantizer of binning that can be 30% more efficient than the current CS quantization scheme without increasing complexity or energy consumption. Quite remarkably, on some test images, the proposed CS method matches or even slightly outperforms the JPEG compression standard in rate–distortion performance. This is the first instance, to our best knowledge, that CS is made competitive against a mainstream compression scheme in rate–distortion performance. However, this paper only qualifies as among initial attempts to lend CS credibility as a signal compressor in addition to being a sparse sampler. Our findings merely indicate the possibility of bringing CS on par with classical signal compression methods in coding
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Fig. 1. CS-based image acquisition–communication system.
performance with frugal source encoders. More studies on sophisticated quantizers of CS measurements are warranted. In particular, trellis coded quantization [13], with its low encoder complexity and high coding performance, holds the promise of further improving the reported results. The remainder of this paper is organized as follows. Section II introduces new system architecture of CS that differs from the current design in quantization operation: cascade fixed-rate SQ of CS measurements versus uniform SQ. It is explained how in the new CS architecture a simple binning technique can improve compression performance without increasing encoder complexity. Section III presents the process of estimated dequantization with binning to be performed by the CS decoder. In Section IV, we clarify how to determine some necessary parameters in our scheme. Section V discusses the optimal measurement allocation problem, and the real-time low-complexity implementation is introduced in Section VI. Experimental results are reported in Section VII, followed by conclusion in Section VIII. II. PROGRESSIVE CS QUANTIZATION The architecture of current CS-based image acquisition–communication systems is depicted in Fig. 1. At the encoder, an image is sampled into random measurements by a random projection matrix . The measurements are uniformly scalar quantized into . To keep the encoder simple, each is sequentially transmitted in fixed-length code of bits to the decoder. In the current CS architecture, the decoder collects all measurements and recovers in one step by solving an minimization problem. However, this CS recovery approach suffers from an intrinsic suboptimality in terms of rate–distortion performance. The deficiency of the all-at-once method is caused by its total disregard of the mutual information that can be significant. Although any two CS measurements and are independent of each other, two groups of CS measurements and are collectively correlated in the following hidden way. Let be the submatrix consisting of the first rows of , and let be the remainder submatrix of . If the input image is approximately recovered with the two subsets of random projections (1) separately, resulting two estimated images and the two signals and are correlated and so are
, then and
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Fig. 2. CS system with two-layer quantization of CS measurements. (a) Encoder. (b) Decoder.
. Based on this observation, we can improve the coding efficiency of CS by a progressive recovery strategy. The decoder first computes an initial or base estimate using the first measurements and then refines the base estimate by adding the remainder measurements in the CS recovery. At the refinement stage, the available base estimate greatly reduces the uncertainty of because
information provided by or . When encoding CS measurements , , in the refinement layer, bits can be saved by relating to the side information because
(2)
(4)
In other words, is a side information of , and it can be used to reduce the bit rate of the second patch of CS measurements without materially affecting the final CS recovery quality. Specifically, we propose a binning technique to reduce the quantization precision of . As argued in the introduction, the CS encoder should be kept as simple and energy efficient as possible. Hence, a fixed-rate uniform scalar quantizer is adopted to code both base and refinement patches of CS measurements and . Suppose in the final CS recovery the required quantization precision of each measurement is bits. For the first measurements , bits per measurement are transmitted. However, for each of the remaining measurements, only the least significant bits of the quantizer output index are transmitted. The most significant bits (MSBs) are dropped for , , and these dropped bits are instead estimated from by the decoder. The proposed CS architecture of two-layer quantization of CS measurements is schematically described in Fig. 2.
Given , the row random measurement vector mains Gaussian with each entry having the same variance but the conditional expectation of becomes
re,
(5) Therefore,
is a Gaussian random variable of the variance and the mean (6)
where . To simplify notations, we drop the subscript , , in and in the sequel. By Bayesian theorem, the posterior distribution can be computed from the posterior distribution and the prior distribution of . Thus
III. ESTIMATED DEQUANTIZATION WITH BINNING In this section, we discuss the dequantization process with binning for the CS measurements in the refinement layer, for which only the least significant bits of the quantization index are transmitted. Suppose the length of the signal is , for the CS measurement matrix whose entries are drawn from Gaussian distribution of zero mean and variance , all the measurements , where is the th row of the CS measurement matrix , , are Gaussian distributed as well, having zero mean and variance
(7)
where
(3) In other words, information available. However, for
when there was no extra , , there exists side
It follows from (7) that the conditional distribution Gaussian, and its variance is determined by A sufficiently large makes and
is still and . , hence,
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Fig. 3. Illustration of keeping the two least significant bits of a uniform scalar quantizer (where falls into one of the shaded bins.
and . In other words, the conditional variprimarily depends on the variance . By the benance efit of the mutual information between and , the variance of is reduced from to . Aided by the side information , the decoder can estimate a CS measurement in the refinement layer from the least significant bits of the quantized value. It can use either minimum mean square error (MSE) estimator or maximum a posteriori probability (MAP) estimator. Denote by the integer value of the least significant bits and by the quantization step size. The value of must fall into the following set of subintervals:
(8) as shown in Fig. 3. The MMSE estimator of
, and the two least significant bits are
). The value of
(base layer) signal (larger reduces ). How to optimally allocate the bit budget between the CS measurements in the base and refinement layers is the subject of Section V. IV. PROGRESSIVE QUANTIZER DESIGN PARAMETERS In this section, we derive the design parameters of the proposed PQ scheme for CS. Let the quantization precision of CS measurements be bits per sample. The step size of the uniform scalar quantizer is chosen such that the bins of size covers almost the entire dynamic range of CS measurements. Let be the probability mass to be covered by the quantizer bins, with a negligible overload error probability . Since the CS measurements obey a Gaussian distribution of variance , we have (11)
is
(12)
(9) where and are the probability density function and cumulative distribution function of the standard normal distribution, respectively. Alternatively, the decoder can use the MAP estimator of in dequantization, i.e., (10) Since the MAP estimate is given by the nearest value to the in the set , it is computationally much more efficient than the MSE estimate. The errors of the above two estimators mainly depend on the ratio . The greater the ratio, the smaller the estimation error. Therefore, the estimation precision of the refinement CS measurements improves as increases and, also, if more measurements are used to generate the initial estimated
where and are the error function and its inverse function of normal distribution, respectively. Now, we discuss how to determine , which is the number of the least significant bits to be transmitted in the refinement layer. The value of should be sufficiently large such that the MAP estimator can determine which bin measurement falls into with probability . As discussed in Section III, the MAP estimate is the point in the candidate set that has the minimum distance to . Thus, identifying the correct bin of requires the distance to be less than the half of the distance between two consecutive candidate bins (i.e., ) in , where is the quantized value of . Since , we have (13) that is (14) , which is the number of the The value of is related to measurements in the base layer. The larger is, the stronger the side information available to the refinement layer, and the smaller the value of can be. Specifically, using the approximation as discussed in Section III, we have (15)
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Fig. 4. Operational error functions of the (a) TV and (b) the MARX recovery methods.
where image
using the first
is the distortion of the recovered CS measurements. Therefore
performance of the proposed PQ scheme for CS. Let and be the number of base and refinement measurements, respectively. The base measurements are coded in bits, and the refinement measurements are coded in bits, where is as derived in the section above. Due to the quantization precision in two layer are nearly the same, in order to minimize the error in CS recovery of , we would like to maximize the total number of measurements subject to the bit budget constraint . Then, the optimal rate allocation problem can be stated as
subject to (20) Substituting (19) into it, we can get
(16) next. Generally speaking, the natural image We analyze signals are not strictly sparse, but they are still compressible due to the so-called near sparsity. Near sparsity means that the coefficients in a recovery space belong to an ball for . In other words, if the coefficient magnitude is in decreasing order, then (17) and the reconstruction error of bounded as [14]
base measurements can be (18)
where is a decay factor. Empirically, the above exponential behavior of CS reconstruction errors in the number of measurements is observed not only for the standard CS recovery method but also for other CS recovery methods. Fig. 4 shows the actual error functions of the popular total variation (TV) CS recovery method [15] and the method of model-based adaptive recovery of CS (MARX) [16]. The error curves are plotted for three representative images of different decay rates: “Pepper” (rapid), “Goldhill” (modest), and “Baboon” (slow). Substituting (18) into (16) yields
(19) where
. V. OPTIMAL MEASUREMENT ALLOCATION
Having discussed the determination of and , we now investigate the problem of bit allocation between the base and refinement layers, which plays a critical role in the rate–distortion
subject to (21) The above problem is one of integer programming because and are natural numbers, which is inherently difficult to solve. However, we can still get a good approximation solution in a relaxation approach: first allowing and to take on the real values in (21) and then rounding to integer the solution of the corresponding real-valued maximization problem. Now, we show how to solve the real-valued optimal bit allocation problem. It is proven in the Appendix that the solution space of (21) is convex. As such, the optimal solution is necessarily the point on the convex curve, i.e., (22) whose slope is
1, i.e., (23)
If the decay factor for the input image can be efficiently estimated, which is the subject of the next section, we can numerically solve the above nonlinear equation for optimal allocation for the base layer and then get optimal allocation for the refinement layer by subsisting into (22). Finally, we obtain the corresponding integer-valued allocation by rounding. VI. REAL-TIME LOW-COMPLEXITY IMPLEMENTATION We improve the rate–distortion performance of existing CS methods by making the measurement quantizer progressive and adaptive to the input image. The signal dependence of the proposed PQ scheme is shown in (19), (22), and (23). The values of , , and can be computed only if the decay factor and constant in the distortion function are known. A straightforward way of computing them is to recover the image signal for increasing and fit the distortion function in
WANG et al.: BINNED PROGRESSIVE QUANTIZATION FOR COMPRESSIVE SENSING
Fig. 5. Scatter plots and regressions of (d) MARX.
versus
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for (a) TV and (b) MARX. Scatter plots and regressions of
(18) to the resulting data points . However, this naive approach incurs a very high computational complexity of the encoder, and, as such, it nullifies the main advantage of CS: signal acquisition at low cost. The challenge is how to add signal adaptability to CS without materially increasing the computational complexity of image acquisition and communication. In order to meet the above objective, we develop a technique of estimating the decay factor of an input image . The new technique is based on the observation that higher decay rate of the distortion function corresponds to a greater degree of sparsity of , or increases as decreases. For most natural images, the level of energy concentration in the low-frequency subbands is highly correlated to the degree of sparsity. Therefore, we can empirically establish the relation of with respect to the allocation of signal energy in low frequencies. This allocation is defined to be the ratio , where is the total energy of the lowest frequency DCT coefficients of in percentage. Fig. 5(a) and (b) shows the scatter plots of versus for 100 natural images with the TV and MARX recovery methods, respectively, when setting . Such a functional relationship between and exhibited by Fig. 5(a) and (b), which can be satisfactorily fit by an exponential model, is learnt off line by using an appropriate training set. The training set can be made general to encompass all varieties of natural images or highly targeted if input images are of a particular type (e.g., faces). We can also empirically determine the constant (normalized by signal length ) in distortion function . It turns out that is nearly linear in the signal energy , as shown in Fig. 5(c) (TV) and (d) (MARX) that are the scatter plots of the same set of 100 natural images in Fig. 5(a) and (b). In real-time CS image acquisition and transmission, the encoder transmits to the decoder the signal energy and the aforementioned ratio (a proxy of the signal sparsity). Then, both the encoder and the decoder can obtain parameters and from the empirical functions discussed above and subsequently compute measurement allocation using (22) and (23). In addition, is needed to compute the quantization step size with (12). The encoder can readily compute as it is simply the energy of the CS measurements. In order to compute the ratio , the encoder measures in the of lowest frequency DCT bases and obtains . This can be done at very low computational complexity with the digital micromirror device technology [17], [18]. Granted, the empirically obtained real-time low-complexity solution of measurement allocation is suboptimal be-
versus
for (c) TV and
Fig. 6. Test images. (a) Lena. (b) Pepper. (c) Boats. (d) Monarch. (e) Cameraman. (f) Goldhill. (g) Birds. (h) Couple. (i) Paints. (j) Vessels.
cause it is based on estimated and . Nevertheless, the proposed approximate solution is found to be reasonably robust in practice. The difference in PSNR performance between our approximate solution and the truly optimal progressive quantizer (as though the function for the CS recovery algorithm on the input signal was known) is about 0.3 dB on average. VII. EXPERIMENTAL RESULTS We implemented the proposed technique of PQ of CS measurements and experimented it on natural images. In this section, we report our experimental results and compare them with those of the current CS methods. For the validity and fairness of our evaluations, we use a set of diverse images to test the proposed new CS quantization technique, some samples of which are shown in Fig. 6. These test images cover a wide range of image signals, ranging from relatively simple (e.g., Birds) to complex scenes (e.g., Goldhill). To achieve lower computation complexity, we use a random circular matrix to generate the random measurements. The first row of the matrix consists of Gaussian random variables, and the other rows are right-shift of the previous row. This matrix can speed up the computations at both the encoder and the decoder since the matrix multiplication can be efficiently implemented with fast Fourier transform [19]. In our experiments, we choose . The number of bits per CS measurement affects the rate–distortion performance of the CS-based signal compression system. The distortion of the recovered signal has two components: the approximation error due to insufficient number of CS measurements and the quantization error caused by insufficient quantization precision. Given a total bit budget , decreasing allows a larger number
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TABLE I PSNR (DB) RESULTS OF DIFFERENT IMAGES AT THE RATE OF 1.0 BPP
Fig. 7. Rate–distortion performance results of different CS recovery methods using simple scalar quantizer (SQ) and the proposed progressive scalar quantizer (PQ), in comparison with those of the JPEG image compression standard. (a) Lena. (b) Monarch. (c) Vessels. (d) Pepper.
of measurements but reduces the quantization precision per measurement, and vice versa. In other words, the approximation and quantization errors cannot be simultaneously minimized, and the optimal value of is the one that achieves the minimum distortion of the recovered image. Optimizing for minimum distortion seems to be an open problem, and no closed-form analytical solution or nontrivial algorithm (except exhaustive search) for optimal is known. Fortunately, in practice, the optimal value of was empirically found to be nearly constant for a given total bit budget over a wide range of natural images [20]. Fig. 7 presents the plots of PSNR versus the rate of uniform SQ without entropy coding (in the interest of inexpensive encoder) for four representative test images. The proposed PQ strategy makes significant coding gains over SQ when coupled with both the popular TV recovery method and the MARX method. In the comparison group in Fig. 7, we also include the results of the JPEG image compression standard, a classical case of dense-sampling, encoder-centric signal compression methodology. For two test images, namely, “Monarch” and “Vessels,” the PQ-based CS recovery method matches and even slightly outperforms JPEG in rate–distortion performance. This
is quite remarkable considering that, up to now, all existing CS methods are not competitive against traditional signal compression methods in rate–distortion performance. This paper suggests a promising direction to bring CS on par with traditional signal compression methods in coding performance while retaining the other advantages of CS. For more comprehensive comparisons, we add in Table I the PSNR results of ten test images at a bit rate of 1.0 bit per pixel (bpp). In this table, in addition to those of TV and MARX, the results of the minimization CS recovery in discrete wavelet transform (DWT) space are included as well. As clearly shown in Fig. 7 and Table I, the proposed PQ scheme consistently improves the reconstruction quality of all three recovery methods over the use of uniform SQ. It can be observed in Table I that the coding gain of PQ over SQ is greatest with MARX and smallest with DWT (1.96 dB versus 0.99 dB on average). This is because MARX is an adaptive recovery method, and it obtains a better initial estimate of the signal than the DWT recovery given the bit budget. The better the initial estimate, the higher precision the binned dequantization of the refinement measurements. In addition, Fig. 7 shows that the advantage of PQ over SQ becomes greater as the bit rate increases. The reason is that as the total bit budget increases, the correlation between the base and refinement layers of the CS measurements gets stronger, and SQ becomes even more inefficient. To evaluate the effectiveness of PQ in decoding the discarded refinement measurements, we report in Table II MSBs of the the probabilities that the binned decoder correctly estimates the first two MSBs for different coding parameters and different CS recovery methods. To see how the optimal progressive quantizer design process allocates measurements between the base and refinement layers in relation to the bit budget, we tabulate in Table III the optiand that are normalized by the signal mized values of length . We also include in the table the optimal values of of allocated bits to the base and refineand the ratio ment layers. For fixed and , the ratio generally increases as the bit budget increases, i.e., more measurements are allocated to the refinement layer. This is because only needs to be sufficient to make the recovered base layer good enough to facilitate the binned dequantization of the measurements in the refinement layer. For the same reason, when is fixed, the ratio with respect to a fixed bit rate is greater for a superior CS recovery algorithm (see MARX versus TV in Table III).
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TABLE II PROBABILITY OF THE DECODER ESTIMATING THE FIRST TWO MSBS CORRECTLY
TABLE III VALUES OF DESIGN PARAMETERS FOR DIFFERENT BIT RATES
TABLE IV COMPARISON OF SQ AND PQ SCHEMES IN THE NUMBER OF MEASUREMENTS ACCOMMODATED BY DIFFERENT BIT BUDGETS
By removing the statistical redundancy between two sets of CS measurements, the new PQ scheme can accommodate more CS measurements than SQ for a given bit budget. Let be the total number of CS measurements coded by PQ, and let be the number of CS measurements coded by SQ. To demonstrate the improvement of PQ over SQ in coding efficiency, we present the values of and normalized by and the ratio in Table IV. It can be observed in the table that the performance gap between PQ and SQ grows as the bit budget increases. With significant PSNR gains of PQ over SQ, one should expect visible improvement of visual quality by the new technique accordingly. Figs. 8–10 present and compare output images of the TV and MARX CS recovery methods using the two different measurement quantization schemes. In both TV and MARX cases, the images of PQ appear to be more pleasant and accurate recoveries of the originals than SQ, particularly in areas of edges and textures. VIII. CONCLUSION AND FUTURE WORK In CS, hidden correlations exist between different subsets of CS measurements but were overlooked in the existing literature. This was a cause of disappointing rate–distortion performance of present CS-based signal compression schemes. To overcome the drawback and improve the CS coding efficiency,
we developed a novel technique of progressive fixed-rate SQ with binning. The new technique can remove the redundancies between CS measurements without increasing the complexity and resource requirement of the CS encoder. Significant improvements over existing CS techniques were achieved. In some cases, the proposed CS quantization technique can match and even slightly outperform the traditional encoder-centric JPEG image compression standard. This paper demonstrates the importance of measurement quantization to the CS-based image acquisition and coding system. The specific technique developed and analyzed above is only one possible embodiment of the general principle of binned progressive CS quantization. Further improvement in rate–distortion performance of CS can be made by extending and fine tuning the proposed technique. For instance, the number of PQ levels can be increased to if the decoder has sufficient computation power to perform CS recovery times. The idea is to let a progressively refined reconstruction of the signal contribute to the estimation of subsequent measurements on the fly. In quest for higher coding efficiency, lattice vector quantizers can replace the uniform scalar quantizer in the system, and, accordingly, the binning ought to be performed in the vector code space. It will be also interesting to study the possibility and merits of “turbo binning”: initially dropping few least significant bits of the first batch of (base) measurements
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Fig. 8. Comparison of PQ and SQ on test image Boats at the rate of 1.0 bpp. dB . (c) TV (a) Original. (b) TV recovery with SQ scheme PSNR dB . (d) MARX recovery with SQ recovery with PQ scheme PSNR dB . (e) MARX recovery with PQ scheme PSNR scheme PSNR dB .
Fig. 10. Comparison of PQ and SQ on test image Lena at the rate of 1.0 bpp. dB . (c) TV (a) Original. (b) TV recovery with SQ scheme PSNR dB . (d) MARX recovery with SQ recovery with PQ scheme PSNR dB . (e) MARX recovery with PQ scheme PSNR scheme PSNR dB .
APPENDIX A In this Appendix, we show that the bit allocation problem (21) is one of convex optimization. The simple linear objective function is clearly convex and so are the inequalities and . The remaining task is to prove the convexity of the boundary curve corresponding to the bit budget constraint, i.e., (24) Fig. 9. Comparison of PQ and SQ on test image Vessels at the rate of 1.0 bpp. dB . (c) TV (a) Original. (b) TV recovery with SQ scheme PSNR dB . (d) MARX recovery with SQ recovery with PQ scheme PSNR dB . (e) MARX recovery with PQ scheme PSNR scheme PSNR dB .
and later estimating the dropped bits upon obtaining a refined reconstruction of the signal, and so forth.
The second derivative function of (24) is
(25) and, it is negative for all feasible values of . To see this, we note because for the proposed PQ scheme of CS measurements. Hence, the
WANG et al.: BINNED PROGRESSIVE QUANTIZATION FOR COMPRESSIVE SENSING
sign of of (25). Thus
only depends on the term in the numerator
(26) In the above inequality, we used the facts . Furthermore
and
(27) in which the first inequality is due to , and the second inequality follows from the fact that, for together with . Since is a monotonically nonincreasing function, for all
(28) Finally
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[8] P. Boufounos, “Universal rate-efficient scalar quantization,” ArXiv preprint arXiv:1009.3145 Sep. 2010. [9] U. Kamilov, V. K. Goyal, and S. Rangan, “Message-passing estimation from quantized samples,” ArXiv preprint arXiv:1105.6368 May 2011. [10] S. S. Pradhan and K. Ramchandran, “Generalized coset codes for distributed binning,” IEEE Trans. Inf. Theory, vol. 51, no. 10, pp. 3457–3474, Oct. 2005. [11] D. Slepian and J. Wolf, “Noiseless coding of correlated information sources,” IEEE Trans. Inf. Theory, vol. IT-19, no. 4, pp. 471–480, Jul. 1973. [12] A. Wyner and J. Ziv, “The rate–distortion function for source coding with side information at the decoder,” IEEE Trans. Inf. Theory, vol. IT-22, no. 1, pp. 1–10, Jan. 1976. [13] M. W. Marcellin and T. R. Fischer, “Trellis coded quantization of memoryless and Gauss_Markov sources,” IEEE Trans. Commun., vol. 38, no. 1, pp. 82–93, Jan. 1990. [14] E. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory, vol. 52, no. 12, pp. 5406–5425, Dec. 2006. [15] E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math., vol. 59, no. 8, pp. 1207–1223, Aug. 2006. [16] X. Wu and X. Zhang, “Model-guided adaptive recovery of compressive sensing,” in Proc. Data Compression Conf., Mar. 2009, pp. 123–132. [17] D. Dudley, W. Duncan, and J. Slaughter, Emerging Digital Micromirror Device (DMD) Applications. Dallas, TX: Texas Intrum., Feb. 2003, White paper. [18] D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in Proc. IS&T/SPIE Symp. Electron. Imag., 2006, pp. 43–52. [19] W. U. Bajwa, J. Haupt, G. Raz, S. J. Wright, and R. Nowak, “Toeplitzstructured compressed sensing matrices,” in Proc. 14th IEEE/SP Workshop SSP, Aug. 2007, pp. 294–298. [20] A. Schulz, L. Velho, and E. A. B. da Silva, “On the empirical rate–distortion performance of compressive sensing,” in Proc. Int. Conf. Image Process, Nov. 2009, pp. 3049–3052.
Liangjun Wang received the B.S. degree from Xidian University, Xi’an, China, in 2004, where he is currently working toward the Ph.D. degree with the School of Electronic Engineering. His research interests include compressive sensing, image processing, multimedia communication, and signal compression.
(29) and, the convexity of the solution space follows. REFERENCES [1] D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, Apr. 2006. [2] E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [3] P. Boufounos and R. Baraniuk, “Quantization of sparse representations,” in Proc. Data Compression Conf., Mar. 2007, pp. 378–387. [4] V. K. Goyal, A. K. Fletcher, and S. Rangan, “Compressive sampling and lossy compression,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 48–56, Mar. 2008. [5] W. Dai, H. Vinh Pham, and O. Milenkovic, “Quantized compressive sensing,” ArXiv preprint arXiv:0901.0749 Mar. 2009. [6] L. Jacques, D. K. Hammond, and J. M. Fadili, “Dequantizing compressed sensing: When oversampling and non-Gaussian constraints combine,” IEEE Trans. Inf. Theory, vol. 57, no. 1, pp. 559–571, Jan. 2011. [7] R. J. Pai, “Nonadaptive lossy encoding of sparse signals,” M.S. thesis, Massachusetts Inst. Technol., Cambridge, MA, 2006.
Xiaolin Wu (SM’96–F’11) received the B.S. degree in computer science from Wuhan University, Wuhan, China, in 1982 and the Ph.D. degree in computer science from the University of Calgary, Calgary, AB, Canada, in 1988. He started his academic career in 1988. He was with the faculty of the University of Western Ontario, London, ON, Canada, and New York Polytechnic University, New York. He is currently a Professor with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, and holds the NSERC senior industrial research chair in Digital Cinema. His research interests include multimedia signal compression, joint source-channel coding, multiple description coding, network-aware visual communication, and image processing. He has published over 230 research papers and holds two patents in these fields. Dr. Wu is an Associate Editor of the IEEE TRANSACTIONS ON IMAGE PROCESSING. He also served as an Associate Editor of the IEEE TRANSACTIONS ON MULTIMEDIA and on the technical committees of many IEEE international conferences/workshops on image processing, multimedia, data compression, and information theory.
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Guangming Shi (SM’10) received the B.S. degree in automatic control, the M.S. degree in computer control, and the Ph.D. degree in electronic information technology from Xidian University, Xi’an, China, in 1985, 1988, and 2002, respectively. He joined the School of Electronic Engineering, Xidian University, in 1988. From 1994 to 1996, he was a Research Assistant with the Department of Electronic Engineering, University of Hong Kong, Pokfulam, Hong Kong. Since 2003, he has been a Professor with the School of Electronic Engineering,
Xidian University. In 2004, he was the Head of the National Instruction Base of Electrician and Electronic. From June to December in 2004, he studied with the Department of Electronic Engineering, University of Illinois at Urbana-Champaign, Urbana. Currently, he is Deputy Director with the School of Electronic Engineering and the academic leader in the subject of Circuits and Systems. His research interests include compressed sensing, theory and design of multirate filter banks, image denoising, low-bit-rate image/video coding, and implementation of algorithms for intelligent signal processing. He has authored or coauthored over 60 research papers.
