1 Introduction
Current video coding techniques, such as HEVC SullivanOHW12 , are designed to have lowcomplexity decoders for broadcasting applications; this is based on the assumption that large amounts of resources are available at the encoder. However, many emerging realtime encoding applications, including lowpower sensor network applications or surveillance cameras, call for an opposite system design that can work with very limited computing and power resources at the encoder. Distributed video coding (DVC) hoangvan2012
is an alternate solution for a lowcomplexity encoder, in which the encoding complexity is substantially reduced by shifting the most computationallyintensive module of motion estimation/motion compensation to the decoder. Nonetheless, other than the encoding process, the processes of image/video acquisition also need to be considered to further reduce the complexity of the encoder
hoangvan2012 because current image/video applications capture large amounts of raw image/video data, most of which are thrown away in the encoding process for achieving highly compressed bitstream. In this context, compressive sensing (CS) has drawn interest since it provides a general signal acquisition framework at a subNyquist sampling rate while still enabling perfect or nearperfect signal reconstruction donoho2006 . More clearly, a sparse signal that has most entries equal to zero (or nearly zero) can be subsampled via linear projection onto sensing bases; this can be reconstructed later by a sophisticated recovery algorithm, which basically seeks its sparse approximation (i.e., the largest magnitude coefficients). Consequently, CS leads to simultaneous signal acquisition and compression to form an extremely simple encoder. Despite its simplicity, its recovery performance is heavily dependent on the recovery algorithm, in which some of the important factors are properly designing the sparsifying transforms and deploying appropriate denoising tools.Although many CS recovery algorithms have been developed, including NESTA (Nesterov’s algorithm) becker2011 , gradient projection for sparse reconstruction (GPSR) figueiredo2007 , Bayesian compressive sensing ji2008 ; he2009 ; he2010 , smooth projected Landweber (SPL) gan2007 , and total variation (TV)based algorithms li2013 ; zhang2010 ; xu2012 , their reconstructed quality has yet to be improved much, especially at a low subrate. For better CS recovery, Candes candes2008 proposed a weighted scheme based on the magnitude of signals to get closer to norm, while still using norm in the optimization problems. In a similar manner, Asif et al. asif2013 adaptively assigned weight values according to the homotopy of signals. As another approach, the authors in mun2009 ; van2013 ; van2014 utilized local smoothing filters, such as Wiener or Gaussian filters, to reduce blocking artifacts and enhance the quality of the recovered images. Despite these improvements, the performances of the aforementioned approaches are still far from satisfactory because much of the useful prior information of the image/video signals (e.g., the nonlocal statistics) was not taken into full account.
More recent investigations have sought to design a sparsifying transform to sparsify the image/video signal to the greatest degree because the CS recovery performance can be closer to that of sampling at the full Nyquist rate if the corresponding transform signal is sufficiently sparse donoho2006 . The direct usage of predetermined transform bases, such as the discrete wavelet transform (DWT) he2009 ; mun2009 , discrete cosine transform (DCT) he2010 ; mun2009 , or gradient transform li2013 ; zhang2010 ; xu2012 ; van2014 , is appealing due to their low complexity. However, predetermined transform bases cannot produce sufficient sparsity (i.e., the number of zero or closetozero coefficients is limited) for the signal of interest, thereby limiting their recovery performance. Because image and video signals are rich in nonlocal similarities (i.e., a pixel can be similar to other pixels that are not located close to it), usage of those nonlocal similarities buades2005 can generate a higher sparsity level to achieve better recovery performance; this is known as a patchbased sparse representation approach elad2006 . Note that this approach originally showed much success in image denoising elad2006 ; dabov2007 ; dabov2009bm3d ; chatterjee2012 and researchers have incorporated this idea into CS frameworks. Xu and Yin xu2014fast proposed a fast patch method for wholeimage sensing and recovery under a learned dictionary, while Zhang et al. zhang2012image took advantage of hybrid sparsifying bases by iteratively applying a gradient transform and a threedimensional (D) transform dabov2007 . By using the concept of decomposition, the authors in canh2016compressive also used a D transform for cartoon images to enhance the recovery quality. The D transform can be considered a global sparsifying transform because it is used for all patches of the recovered images. Dong et al. dong2014compressive
, motivated by the success of datadependent transforms for patches (referred to as local sparsifying transforms) such as principal component analysis (PCA) or singularvalue decomposition (SVD), proposed a method to enhance the sparsity level with the
logdet function to bring norm closer to norm, similar to the work of Candes candes2008 . Metzler et al. metzler2016denoising acquired a local sparsifying transform via block matching dabov2009bm3d and demonstrated the effectiveness of applying denoising tools to the CS recovery of the approximate message passing (AMP) method. However, because of the frame sensing that accesses the entire image at once, the work described in xu2014fast ; zhang2012image ; dong2014compressive ; metzler2016denoising requires extensive computation and huge amounts of memory for storing the sensing matrix fowler2012block ; thus, these approaches are not suitable as sensing schemes for realtime encoding applications or largescale images/video.Alternatively, block compressive sensing (BCS) has been developed to deal more efficiently with largesized natural images and video by sensing each block separately using a block sensing matrix with a much smaller size. The compressive sensor can instantly generate the measurement data of each block through its linear projection rather than waiting until the entire image is measured, as is done in frame sensing. The advantages of BCS are discussed in gan2007 ; fowler2012block ; dadkhah2014block ; dinh2016iterative . However, in BCS, the recovery performance has yet to be substantially improved in comparison to that of frame sensing. To address this problem on the sensing side, a Gaussian regression model between the coordinates of pixels and their gray levels can be used to achieve better performance compared to traditional Gaussian matrices han2015novel . Additionally, Fowler et al. fowler2011multiscale developed an adaptive subrate method (i.e., multiscale BCS) to exploit the different roles of wavelet bands. On the recovery side, for example, Dinh et al. dinh2014weighted designed overlapped recovery with a weighted scheme to reduce the blocking artifacts caused by block recovery. Chen et al. chen2011compressed used the Tikhonov regularization and residual image information to enhance the smooth projected Landweber gan2007 . Furthermore, to enrich the details of recovered images, the KSVD algorithm elad2006 was used in zhang2014image . By sharing the same idea in xu2014fast ; zhang2012image ; dong2014compressive ; metzler2016denoising ; zhang2014image where nonlocal similarities are exploited to design the local sparsifying transform, groupbased sparse representation (GSR) zhang2014group can achieve better recovery performance (in terms of the peak signaltonoiseratio (PSNR)) than other algorithms that were previously designed for BCS. However, its recovered images still contain many visual artifacts since the nonlocal searching and collecting patches based on the initial recovered images produced by chen2011compressed often have poor quality at low subrates. Consequently, this implies that more efforts are required for improving both the objective and subjective quality.
This paper attempts to improve the recovery performance of the BCS framework by using TV minimization, which is good at preserving edges li2013 , with multiple techniques consisting of reducing blocking artifacts in the gradient domain, denoising the Lagrangian multipliers, and enhancing the detailed information with patchbased sparse representation. Furthermore, the proposed recovery methods are easily extendible to compressive sensing and encoding problems of video do2009distributed ; mun2011residual ; tramel2011video ; van2014block ; kang2009distributed . Specifically, our main contributions are summarized as follows.

For BCS of images, we propose a method, referred to as multiblock gradient processing, that addresses the blocking artifacts caused by blockbyblock independent TV processing during recovery. Furthermore, based on our observation that both image information (e.g., edges and details) and highfrequency artifacts and staircase artifacts are still prevalent in the Lagrangian multiplier of the TV optimization, we propose a method to reduce such artifacts by denoising the Lagrangian multiplier directly with a nonlocal means (NLM) filter. Because the direct application of the NLM filter is not effective in preserving local details with low contrast buades2005 , we further propose enriching these lowcontrast details through an additional refinement process that uses patchbased sparse representation. We propose using both global and local sparsifying transforms because the single usage of either transform limits the effective sparse basis and achievement of a sufficient sparsity level for noisy data. The proposed recovery method demonstrates improvements for BCS of images compared to previous works he2009 ; he2010 ; mun2009 ; van2013 ; chen2011compressed ; zhang2014image ; zhang2014group .

For BCS of videos, we extend the proposed method to a compressive video sensing problem known as block distributed compressive video sensing (DCVS). An input video sequence is divided into groups of pictures (GOP), each of which consists of one key frame and several nonkey frames. These undergo block sensing by a Gaussian sensing matrix. The proposed method first recovers the key frame using the proposed recovery method. Then, for nonkey frames, side information is generated by exploiting measurements of the nonkey and previously recovered frames in the same GOP. Improved quality of the nonkey frames is sought by joint minimization of the sparsifying transforms and side information regularization. Our experimental results demonstrate that the proposed method performs better than existing recovery methods designed for block DCVS, including BCSSPL using motion compensation (MCBCSSPL) mun2011residual or BCSSPL using multihypothesis prediction (MHBCSSPL) tramel2011video .
The rest of this paper is organized as follows. Section 2 briefly presents works related to the BCS framework with some discussion. The proposed recovery method for BCS of images is described in Section 3, and its extension to the block DCVS model is addressed in Section 4. Section 5 evaluates the effectiveness of the proposed methods compared to other stateoftheart recovery methods. Finally, our conclusions are drawn in Section 6.
2 Block compressive sensing
In the BCS framework, a largesized image
is first divided into multiple nonoverlapping (small) blocks. Let a vector
of length denote the th block, which is vectorized by raster scanning. Its measurement vector is generated through the following linear projection by a sensing matrix(1) 
A ratio denotes the subrate (or subsampling rate, i.e., the measurement rate). BCS is memoryefficient as it only needs to store a small sensing matrix instead of a full one corresponding to the whole image size. In this sense, block sampling is more suitable for lowcomplexity applications.
The CS recovery performance heavily depends on the mutual coherence of the sensing matrix which is computed as eldar2012compressed :
(2) 
Here, and are any two arbitrary columns of ; denotes the inner product of two vectors. According to the Welch bounds eldar2012compressed , is limited in the range of . Additionally, at a low subrate (i.e., ), it can be approximated as
(3) 
Note that a low mutual coherence is preferred for CS, and that the lower bound of (3) is inversely proportional to . A low mutual coherence is harder to achieve with a small block size, although it is attractive in terms of memory requirements. This explains the limited recovery quality of BCS with a small block size, despite the great amount of research that has been conducted on this topic he2009 ; he2010 ; mun2009 ; dinh2014weighted ; chen2011compressed ; zhang2014image ; dinh2014measurement ; kulkarni2016reconnet .
sought to obtain structured sparsity of signals in the Bayesian framework with a Markovchain MonteCarlo
he2009 or variational Bayesian framework he2010 . However, for practical image sizes, these approaches demand impractical computational complexity; thus, the search must be terminated early, and the recovered images suffer from highfrequency oscillatory artifacts zhang2012image . The recoveries mun2009 ; chen2011compressed are much faster than those previous ones, but they are limited by their predetermined transforms. Specifically, noniterative recovery was demonstrated in kulkarni2016reconnetwith a convolutional neural network being used for the measurement data. The work in
zhang2014image ; zhang2014group uses a single patchbased sparse representation; therefore, its reconstructed quality is still not satisfactory. The initial images used for adaptive learning of sparsity contain a large amount of noise and artifacts; thus, using only one of them limits the definition of the effective sparse basis and makes it difficult to achieve a sufficient sparsity level for noisy data. These observations motivated us to design a new recovery method for high recovery quality, as discussed in the next section.3 Proposed recovery for block compressive sensing of image
In this section, we design a recovery method for BCS of images. For this, we modify the TVbased recovery method li2013 for BCS by introducing a multiblock gradient process to reduce blocking artifacts and directly denoise the nonlocal Lagrangian multiplier to mitigate the artifacts generated by the TVbased methods. Also, due to the limitations of the NLM filter in preserving the image texture, a patchbased sparse representation is also designed to enrich the local details of recovered images.
3.1 CS recovery for BCS framework
The first recovery method for BCS was proposed by L. Gan gan2007 , who incorporated a Wiener filter and Landweber iteration with the hard thresholding process. S. Mun et al. mun2009 further improved this idea by adding directional transforms of DCT, dualtree DWT (DDWT), or contourlet transform (CT). The visual quality of the best method mun2009 , namely the SPL with the dualtree wavelet transform (SPLDDWT), is shown in Figure (a)a for the image Leaves. The recovered image suffers from highfrequency oscillatory artifacts candes2006stable and has blurred edges. TV has been shown to be effective for frame CS li2013 in preserving edges and object boundaries in recovered images. As expected, the recovered image by TV, shown in Figure (b)b, looks much sharper than the image recovered via SPLDDWT mun2009 , although TV is applied to BCS for blockbyblock recovery. However, it shows significant blocking artifacts, due to the blockindependent TV processing. Motivated by this, we investigate a TVbased recovery scheme for BCS with an emphasis on improving the TV method that is applied to BCS such that it does not suffer from blocking artifacts. Our TVbased BCS recovery method has several implications, as follows.
3.2 Noise and artifacts reduction
As mentioned above, independent blockbyblock TV processing makes images suffer heavily from blocking artifacts as in Figure (b)b. When TV is computed separately for individual blocks as in xu2012 , a good deblocking filter should also be used to mitigate the blocking artifacts. When the BCS scheme gan2007 is in use, a block diagonal sensing matrix corresponding to a whole image of size (assuming it consists of blocks) and its measurement are given as
(4)  
(5) 
We design a method, referred to as multiblock total variation (MBTV), based on a multiblock gradient process, as depicted in Figure (a)a. This calculates the gradient for TV over multiple blocks such that the discontinuities at block boundaries can be reduced significantly by minimizing the gradient. Notice that, if this method is applied to all blocks in a recovered image, it is equivalent to a framebased gradient calculation. The visual quality of the recovered image Leaves, which is illustrated in Figure (c)c, demonstrates that many of the blocking artifacts are reduced compared to the blockbyblock TVbased recovery (Figure (b)b). Here, we use a small sensing matrix () to visualize the significant improvements of MBTV. Note that the recovered images usually suffer from more blocking artifacts with a small sensing operator when they are independently recovered blockbyblock, as shown in Figure (b)b.
The proposed MBTVbased recovery is described in detail below. The constrained problem of TVbased CS is expressed as
(6) 
where stands for the TV operator, which can be either isotropic or anisotropic. For isotropic TV, (6) is converted into an unconstrained problem by the augmented Lagrangian method li2013 ; van2014
(7) 
Here, , and , where and denote the horizontal and vertical gradient operators, respectively. and are Lagrangian multipliers, and and are positive penalty parameters. The key idea of the augmented Lagrangian method is to seek a saddle point of that is also the solution of (6). At the th iteration, by acquiring the splitting technique afonso2010fast , (7) is iteratively solved by two socalled subproblems, and , as shown below:
(8)  
(9) 
The solution of (8) is found by the shrinkagelike formula, where denotes the elementwise product:
(10) 
Because (9) is a quadratic function, its solution can be achieved by calculating the first derivative of the subproblem . However, to reduce the computational complexity of the MoorePenrose inverse li2013 , we also use gradient descent, as proposed in li2013 ; van2014 :
(11) 
The direction and the optimized step size in (11) are calculated by the BarzilaiBorwein method li2013 ; van2013 :
(12)  
(13) 
The two Lagrangian multipliers are then updated by li2013 :
(14)  
(15) 
Since TV basically assumes piecewise smoothness, it cannot avoid losing detailed information dong2013compressive ; this is a valid assumption for natural images in smooth regions but is less applicable in nonstationary regions near edges. Consequently, socalled staircase artifacts occur in the recovered image, as shown in Figure (c)c. Moreover, it is worth emphasizing that, even though the signal acquisition is assumed to be noisefree (i.e., ), image signals cannot be perfectly recovered because they cannot be described exactly by sparse approximation, as shown in ji2008 . Therefore, if a compressible signal of length in a selected transform domain is term approximated by its largest entries in magnitude , then the remaining
elements can be considered as recovery error or noise. By the central limit theorem, it is reasonable to assume that the noise is Gaussian if the sensing matrix is random
ji2008 ; in this scenario, the application of a filter will help smoothing the recovered images gan2007 ; mun2009 ; van2013 ; van2014 . Moreover, for image denoising, the idea of applying a denoising technique to a selected derived image (for example, the normal vectors of the level curves of the noisy image lysaker2004noise , the curvature image bertalmio2014denoising , or the combined spatialtransformed domain image knaus2013dual ) might be more effective than directly smoothing the corresponding original noisy image. Motivated by this idea, we suggest that smoothing the Lagrangian multiplier can effectively enhance the CS recovered image quality.Lagrangian multipliers are used to find the optimum solution for a multivariate function of CS recovery. The Lagrangian multipliers that represent the gradient image and the measurement vector ( and , respectively) should have their own roles in solving the illposed CS problems. Specially, is updated by the gradient image which naturally contains a rich image structure. Hence, in CS recovery, we note that in (14) can be seen as a noisy version of the gradient image. Indeed, the noise can actually be seen if Figure (a)a and (b)b are compared. With fullNyquist sampling (i.e., a subrate of ), there is no noise in ; however, there exists a large amount of noise if the subrate is lowered to . Also note that, according to the splitting technique, plays a role in estimating the solution . Therefore, a more exact will provide more accuracy to the subproblem and ultimately produce a superior recovered image. Consequently, this suggests the importance of improving the quality of in order to obtain better quality with the augmented Lagrangian TV recovery. A proper process should be designed to mitigate noise and artifacts in . Rather than utilizing Wiener or Gaussian filters, which might easily oversmooth the recovered image van2013 , we employ the nonlocal means (NLM) filter buades2005 , which is wellknown for its denoising ability while also preserving textures by employing an adaptive weighting scheme with a smoothing parameter that depends on the amount of noise in the signals. The new method to update the Lagrangian multiplier is designed as
(16) 
Here denotes the NLM filtering operator buades2005 . We first update by the traditional method li2013 in order to estimate a temporal version (denoted as in Step ). Next, the NLM filter is applied to reduce the noise in Step . Figure (c)c visualizes the efficiency of the proposed method. After NLM denoising, the Lagrangian multiplier is much cleaner and shows image structures better. Moreover, Figure (d)d shows the recovered image “Leaves” at a subrate of , indicating a reduction of the highfrequency artifacts. The proposed combination of MBTV and the nonlocal Lagrangian multiplier (NLLM) is referred to as MBTVNLLM and is summarized in Algorithm 1.
At this point, we stress that utilizing the NLM filter for the Lagrangian multiplier yields better recovered images in terms of both subjective and objective qualities. In addition, NLLM has lower computational complexity than nonlocal regularization methods that directly use an NLM filter for noisy images, as in zhang2010 ; zhang2013improved , since the Lagrangian multipliers are only updated if the recovered images are significantly changed. One can refer to Trinh et al. van2014 for both a theoretical analysis and a numerical comparison between NLLM and nonlocal regularizations zhang2010 ; zhang2013improved . In this paper, we focus on a valid explanation for the gain of the NLLM method, as discussed below.
Mathematically, the error bound of our method (MBTVNLLM) is smaller than that of traditional TV li2013 due to the following local convergence statement bertekas1982constrained . Assume that and are the solutions (see (11) and (16)) of the proposed method at the th iteration, where and are the solutions of TV li2013 . With a positive scalar , according to proposition bertekas1982constrained , the reconstructed errors of MBTVNLLM and TV li2013 are
(17)  
(18) 
Here, is the Lagrangian multiplier corresponding to the solution . Hence, we can set up error bounds:
(19)  
(20) 
Note that the solution prefers to be a clean version (see Figure (a)a). NLLM tries to reduce the noise in by acquiring the NLM filter (i.e., see (16)). This implies that is closer to than is (i.e., compare Figure (b)b with Figure (c)c), which can be represented as . Thus, we obtain
(21) 
The above error coincides with recent reports needell2013stable , confirming that the spatial error is bounded by the gradient error. Although the NLM filter provides nonlocal benefits, it still has difficulties in preserving many details in images with low contrast. This drawback is caused by the fact that two very similar pixels on opposite sides produce inaccurate weights for the NLM filter. As a result, some artifacts near edges cannot be sufficiently mitigated without losing detailed information maleki2012suboptimality . In this paper, additional processing to enrich the local details through patchbased sparse representation is designed to solve the problem, as proposed in the next subsection.
3.3 Refinement for recovered images with patchbased sparse representation
Classical predetermined transforms, such as DCT or wavelet transforms, cannot always attain a sparse representation of complex details. For example, sharp transitions and singularities in natural images are not expressed well by DCT. In the same way, D wavelets might perform poorly for textured or smooth regions dabov2007 . Recently, for image restoration applications such as denoising, inpainting, or deblurring, patchbased sparse representation has been actively investigated to deal with the complex variations of natural images. Suppose that (where is the total number of patches) denotes an column vector representing the th image patch extracted by a patchextracting operator through from an image of size represented by an column vector . In this scenario, the image is synthesized as
(22) 
where is the regular transpose. Moreover, assume to be sparse over a dictionary with its coefficient vector (that is, ). and denote the concatenation of dictionaries and , respectively. Then, in (22) is further expressed in a patchbased sparse representation as
(23) 
The operator makes the patchbased sparse representation more compact elad2006 . Briefly, utilizing a patchbased sparsifying transform to decorrelate the signal and noise in the transform domain is presented by the five following basic steps dabov2007 ; dabov2009bm3d ; chatterjee2012 :

Group similar patches: use a nonlocal search to find patches that are similar to the reference patch and stack them in a group.

Forward transform: apply a sparsifying transform (i.e., global or local sparsifying transforms) to each group for transformed coefficients.

Thresholding process: decollate signal and noise by keeping only the significant coefficients. The remaining coefficients are considered to be noise and are discarded. For the CS viewpoint, this step can be referred to as sparse approximation gan2007 .

Inverse transform: obtain the estimates for all grouped patches.

Weighting process: return the pixels of the patches to the original locations. The overlapping patches are appropriately weighted according to the number of times each pixel repeats.
For the sparse approximation (Steps , , and ), choosing a proper sparsifying basis will determine the recovered image quality. The authors in zhang2012image applied a global transform dabov2007 that combined D wavelet transform and D DCT transform for all grouped blocks. A predetermined global transform is advantageous in terms of its simplicity; however, it cannot reflect the various sparsity levels of all groups. Therefore, a local transform dabov2009bm3d ; chatterjee2012 was calculated for each individual group to adaptively support the various local sparsity levels. If the local sparsifying transform is poorly designed due to data heavily contaminated by noise, the recovered images will have serious visual artifacts.
Another important quality issue is determining how to collect the proper groups. Related to the nonlocal search discussed in buades2005 ; sutour2014adaptive , patchbased sparse representation still faces some explicit challenges with two outright problems sutour2014adaptive : for singular structures, it might fail to find similar patches, thereby producing poor results; and
due to noise, it may detect incorrect patches (i.e., selecting some patches that do not actually belong to the same underlying structure). This can eventually cause oversmoothing. Below, we show that the similarity between a recovered image and its original heavily depends on the variance of error.
Let us assume that the elements of the error vector
are independent and come from a normal distribution with zero mean and variance
Here, represents a restored version of an original image after performing patchbased sparse representation. Since are also independent and come from the halfnormal distribution with mean and variance, a new random variable
has mean and variance of(24)  
(25) 
Based on the Chebyshev inequality in probability, for a value
,(26) 
Substituting (24) and (25) into (26), the probability that expresses the similarity between and is
(27) 
With a sufficiently large image size (i.e., as becomes large), the probability of similarity between and in (27) approaches . The implication of this is twofold:

First, a good patchbased sparse representation should produce less estimation error (i.e., is small).

Second, as the noise becomes smaller, the patchbased sparse representation performs better.
The probability of the estimation error in (27) is important and suggests the idea of combining the local and global sparsifying transforms. At the th iteration, the recovered image is first updated by the five aforementioned steps with a local sparsifying transform. The output of this stage is referred to as , as shown in Figure 4. Thanks to the local sparsifying transform, has less noise and fewer artifacts than the input image . Additionally, this stage generates a more desirable input version for the second stage, in which we determine the sparsity levels of signals via a global sparsifying transform in order to produce the output image . Our suggested design is described below.
Generally, improving the sparsity level of a signal via a proper transform in CS can be carried out using xu2014fast ; zhang2012image ; dong2014compressive ; metzler2016denoising ; zhang2014image ; zhang2014group ; van2014block
(28) 
The unconstrained problem of (28), according to the patchbased sparse representation, is formulated to a more tractable optimization problem as
(29) 
where is a slack variable ensuring that (29) is equal to (28). We further note that (29) is a mixed optimization problem that aims at minimizing the cost between the sparsifying coefficients of all of the patches and compressive sensing. For a very simple encoder, the sparsifying transform is moved to the decoder gan2007 , which means that measurements are directly acquired in the spatial domain (i.e., ). According to the modified augmented Lagrangian approach in afonso2011augmented , which is a closed form of a sparsifying transform, (29) is changed to:
(30) 
Here, is a positive penalty parameter. The scaled vector is then updated as
(31) 
Using the splitting technique afonso2010fast , we minimize (30) by alternatively solving the and subproblems. More clearly, is solved by seeking sparsity levels with the five steps of patchbased sparse representation shown in Figure 4. The subproblem is solved by gradient descent with an optimized step , and the direction is calculated by the BarzilaiBorwein method li2013 ; van2013 ; van2014 :
(32)  
(33) 
Here,
is an identity matrix. The
combination of local and global sparsifying transforms (CST) and MBTVNLLM is referred to as MBTVNLLMCST. A summary of our proposed recovery method for the BCS framework is depicted in Algorithm 2. Figure 5 verifies the effectiveness of the suggested design. More clearly, MBTVNLLMCST (P) visualizes the reduction in error variance by the first stage in Figure 4 (i.e., using a local sparsifying transform), while MBTVNLLMCST (P) points out the noise reduction after the two stages in Figure 4. The gap between the two graphs of MBTVNLLMCST (P) and MBTVNLLMCST (P) indicates the gain from the additional global sparsifying transform. To better understand the nature of this gain, Figure 5 also shows two extra graphs corresponding to MBTVNLLM with a global sparsifying transform (MBTVNLLMGST) and the one with a local sparsifying transform (MBTVNLLMLST) only. We recall that the PSNR value is inversely proportional to the variation of error; thus, from Figure 5, we can confirm that our proposed algorithms always converge to a feasible solution. This is valuable because proving the convergence of a recovery algorithm that deploys a sparsifying transform is not trivial zhang2014image ; zhang2014group . More interestingly, the results also reveal that the proposed method is better than previous ones zhang2014image ; zhang2014group , which use either a global sparsifying transform or a local sparsifying transform.4 Block distributed compressive video sensing
In this section, we extend the proposed recovery method to block DCVS, as shown in Figure 6. The main advantage of this design over other existing ones, such as the design proposed in do2009distributed , is that it does not require full Nyquist sampling.
4.1 Key frame recovery
A key frame is recovered using the proposed recovery scheme in Algorithm 2, which was developed for still images. That is, an initial estimate is generated by MBTVNLLM, and then a sparsifying transform is applied to enrich the local details of the reconstructed key frame.
4.2 Side information generation
In distributed video decoding, side information (SI) plays an important role because inaccurate SI strongly degrades the recovery quality of nonkey frames. The frames that can be used for reconstructing a nonkey frame (denoted by ) as the side information are
(34) 
Here, denotes the th GOP in a video sequence, and the subscript indicates a nonkey frame. However, the definition in (34) is impractical when finding proper SI frames for DCVS simply because the only information available at the decoder is the measurement data of nonkey frames. According to the JohnsonLindenstrauss lemma baraniuk2008simple , the selection in (34) can be equivalently written as
(35) 
In a GOP, (35) allows all of the nonkey frames similar to the current nonkey frame in the measurement domain to be gathered. The selected SI frames are not much different from each other; thus, an initial nonkey frame is computed as their average. Otherwise, it is considered to be a frame with the minimum value of . Our goal is to find the best initial nonkey frame.
4.3 Recovery of nonkey frames
The nonkey frame is refined by its measurement vector, sparsifying transform, and the SI frame denoted by :
(36) 
Similar to (30), (36) can also be minimized using the augmented Lagrangian approach maleki2012suboptimality . This is converted to
(37) 
Here, and are positive penalty parameters. At each iteration, the side information is updated, and then the two subproblems and are solved. Additionally, the two vectors that represent the sparsifying transform and side information regularization (i.e., and , respectively) are updated as
(38) 
In detail, the side information is first initialized by (35) and then updated at each iteration by a multihypothesis (MH) prediction using Tikhonov regularization tramel2011video . Additionally, the subproblem is solved by the patchbased sparse representation with a combination of local and global sparsifying transforms, as explained previously. Finally, the subproblem is solved by gradient descent with the optimal step size and direction estimated by the BarzilaiBorwein method li2013 ; van2013 ; van2014 :
(39)  
(40) 
5 Experimental results
5.1 Test condition
The recovery performance of the proposed recovery schemes for the BCS framework is evaluated by extensive experiments using both natural images and video. The parameters in the proposed method are experimentally chosen to achieve the bestreconstructed quality. The positive penalty parameters and of MBTVNLLM are equal to and , respectively. The NLM filter has a patch size of and a search range of . The smoothing parameter is . The outer stopping criterion is defined as , while the inner stopping criterion is defined as . For patchbased sparse representations, the size of the groups is , and overlapping is used between patches with an overlapping step size of pixels. The training window for collecting groups is in size. The penalty parameters and of the refinement problems are set to , while is set to . Additionally, for video recovery, the scale vector is initially set to a zero vector, and the value of is set to . The testing conditions required for the other recovery methods are established based on their suggested recommendations he2009 ; he2010 ; mun2009 ; chen2011compressed ; zhang2014image ; zhang2014group . All of the experiments are performed by MatlabR2011a running on a desktop Intel Corei RAMG with the Microsoft Windows operating system. For objective analysis, we use the PSNR (in units of dB). Additionally, the Feature SIMilarity (FSIM) zhang2011fsim is used for visual quality evaluation. FSIM is in the range of , where a value of 1 indicates the best quality.
Image  Subrate  TSDWThe2009  TSDCThe2010  SPLCTmun2009  SPLDDWTmun2009  MHchen2011compressed  MBTVNLLM*  
PSNR  FSIM  PSNR  FSIM  PSNR  FSIM  PSNR  FSIM  PSNR  FSIM  PSNR  FSIM  
Lena  0.1  22.48  0.740  22.18  0.767  24.76  0.841  25.31  0.856  26.11  0.889  26.62  0.862 
0.2  25.18  0.839  26.57  0.879  27.48  0.894  28.11  0.906  29.71  0.933  29.31  0.912  
0.3  27.05  0.882  28.73  0.918  29.53  0.924  30.16  0.933  31.26  0.948  31.29  0.941  
0.4  28.50  0.908  30.35  0.940  31.35  0.945  31.98  0.952  33.76  0.966  33.30  0.958  
Leaves  0.1  15.98  0.589  17.06  0.620  18.56  0.680  18.66  0.685  20.68  0.761  21.11  0.825 
0.2  18.46  0.694  20.68  0.739  21.31  0.758  21.37  0.761  24.81  0.850  26.13  0.910  
0.3  20.51  0.772  23.42  0.807  23.31  0.805  23.30  0.805  27.65  0.898  29.07  0.941  
0.4  22.29  0.827  25.57  0.858  25.31  0.847  25.26  0.847  29.87  0.925  31.69  0.959  
Monarch  0.1  18.92  0.664  19.71  0.690  21.57  0.772  21.80  0.785  23.68  0.803  24.54  0.853 
0.2  21.50  0.759  23.33  0.782  24.71  0.832  25.26  0.850  27.23  0.869  28.72  0.915  
0.3  23.76  0.818  25.70  0.837  27.10  0.870  27.76  0.885  29.52  0.908  31.50  0.944  
0.4  25.90  0.868  27.68  0.875  29.05  0.898  29.81  0.912  31.28  0.928  33.47  0.959  
Cameraman  0.1  20.36  0.685  20.45  0.683  22.12  0.760  21.64  0.762  22.05  0.775  23.68  0.812 
0.2  22.22  0.763  23.05  0.778  24.83  0.825  24.79  0.838  25.41  0.843  26.68  0.875  
0.3  24.36  0.825  24.71  0.825  26.51  0.863  27.02  0.878  28.32  0.898  28.44  0.912  
0.4  25.97  0.863  26.60  0.869  28.41  0.894  28.99  0.909  29.81  0.918  30.38  0.937  
House  0.1  23.91  0.719  24.75  0.767  26.69  0.836  26.95  0.846  30.07  0.895  30.47  0.877 
0.2  27.20  0.837  29.23  0.871  29.95  0.894  30.56  0.902  33.73  0.938  33.51  0.920  
0.3  29.64  0.884  32.03  0.912  32.34  0.926  32.83  0.931  35.62  0.956  35.58  0.945  
0.4  32.30  0.920  34.09  0.938  34.18  0.947  34.67  0.949  37.20  0.967  36.98  0.958  
Parrot  0.1  21.93  0.755  22.58  0.830  23.26  0.865  23.32  0.876  24.29  0.887  25.15  0.880 
0.2  24.25  0.849  25.58  0.895  26.11  0.909  26.36  0.918  27.92  0.928  27.91  0.922  
0.3  25.54  0.890  27.48  0.926  28.29  0.935  28.63  0.943  30.79  0.952  30.33  0.942  
0.4  27.11  0.918  28.91  0.942  30.20  0.952  30.92  0.958  32.54  0.964  32.92  0.957  
Boat  0.1  21.78  0.679  19.82  0.722  24.22  0.799  24.52  0.802  26.12  0.852  26.76  0.844 
0.2  24.75  0.800  26.47  0.848  26.94  0.864  27.08  0.866  30.17  0.920  30.06  0.913  
0.3  26.57  0.854  28.84  0.900  29.05  0.903  28.97  0.900  32.42  0.944  32.54  0.943  
0.4  28.73  0.901  31.44  0.936  30.88  0.929  30.61  0.925  34.14  0.960  34.72  0.963  
Pepper  0.1  20.37  0.695  21.02  0.739  23.67  0.826  24.37  0.838  25.78  0.859  25.87  0.859 
0.2  22.50  0.785  25.00  0.836  27.03  0.883  27.63  0.890  29.32  0.910  30.28  0.921  
0.3  26.20  0.870  28.51  0.901  29.17  0.910  29.82  0.918  31.20  0.933  32.75  0.947  
0.4  29.49  0.912  30.98  0.931  30.98  0.932  31.72  0.939  32.91  0.950  34.62  0.962  
Average  24.24  0.805  25.70  0.836  26.84  0.866  27.19  0.874  29.23  0.904  29.89  0.915  
Gain by MBTVNLLM  5.65  0.110  4.19  0.079  3.05  0.049  2.70  0.041  0.66  0.011     
Image  Subrate  RALSzhang2014image  GSRzhang2014group 
MBTVNLLM GST 
MBTVNLLM LST 
MBTVNLLM CST 

PSNR  FSIM  PSNR  FSIM  PSNR  FSIM  PSNR  FSIM  PSNR  FSIM  
Lena  0.1  27.07  0.899  27.57  0.915  27.63  0.914  28.25  0.915  28.02  0.914 
0.2  30.49  0.943  30.88  0.952  30.98  0.951  31.57  0.952  31.66  0.954  
0.3  33.17  0.966  33.96  0.971  33.51  0.969  34.13  0.970  34.51  0.972  
0.4  35.50  0.978  36.46  0.981  35.90  0.979  36.21  0.980  36.71  0.981  
Leaves  0.1  21.55  0.801  23.22  0.876  23.69  0.890  25.68  0.914  26.55  0.926 
0.2  27.13  0.909  30.54  0.956  29.48  0.950  31.23  0.961  32.21  0.967  
0.3  31.20  0.951  34.40  0.976  33.32  0.972  34.81  0.978  36.13  0.983  
0.4  34.69  0.974  37.63  0.987  36.19  0.983  37.69  0.987  39.00  0.990  
Monarch  0.1  24.42  0.827  25.28  0.867  25.88  0.882  27.73  0.912  28.08  0.915 
0.2  28.36  0.892  30.77  0.941  30.27  0.931  31.98  0.953  32.80  0.956  
0.3  31.60  0.933  34.25  0.964  33.34  0.956  35.04  0.971  35.99  0.974  
0.4  34.39  0.957  36.86  0.976  35.58  0.969  37.35  0.980  38.32  0.982  
Cameraman  0.1  22.93  0.799  22.90  0.815  24.71  0.849  25.58  0.860  25.38  0.853 
0.2  26.59  0.875  26.76  0.889  28.17  0.912  28.74  0.915  28.78  0.915  
0.3  29.26  0.922  28.97  0.928  30.60  0.938  30.90  0.940  31.22  0.944  
0.4  31.19  0.945  31.25  0.953  32.39  0.956  32.42  0.956  32.85  0.961  
House  0.1  32.10  0.911  32.94  0.920  33.31  0.928  33.17  0.927  33.88  0.930 
0.2  36.08  0.955  37.37  0.964  36.43  0.959  36.37  0.958  37.19  0.963  
0.3  38.37  0.973  39.41  0.978  38.81  0.975  38.41  0.974  39.20  0.977  
0.4  40.17  0.982  40.90  0.984  40.32  0.982  40.03  0.982  40.88  0.985  
Parrot  0.1  25.33  0.908  25.98  0.923  26.28  0.919  27.41  0.921  27.47  0.924 
0.2  29.34  0.944  30.76  0.952  29.54  0.949  31.75  0.953  31.15  0.953  
0.3  32.49  0.963  34.17  0.968  32.39  0.963  34.15  0.966  34.15  0.967  
0.4  34.78  0.974  36.61  0.978  34.99  0.975  36.34  0.976  36.84  0.978  
Boat  0.1  28.01  0.890  28.28  0.901  28.69  0.901  29.11  0.908  29.18  0.907 
0.2  33.01  0.952  33.69  0.958  33.34  0.956  33.31  0.955  34.02  0.960  
0.3  36.29  0.973  36.67  0.975  36.14  0.973  36.00  0.972  36.90  0.978  
0.4  38.89  0.984  39.26  0.985  38.48  0.983  38.50  0.983  38.94  0.985  
Pepper  0.1  27.54  0.892  28.18  0.908  28.91  0.913  28.74  0.911  29.44  0.917 
0.2  31.64  0.941  32.59  0.951  32.73  0.953  32.56  0.952  33.28  0.955  
0.3  34.39  0.963  35.07  0.966  35.12  0.967  34.81  0.965  35.68  0.969  
0.4  36.52  0.974  37.00  0.976  37.03  0.978  36.56  0.975  37.42  0.978  
Average  31.39  0.930  32.52  0.945  32.32  0.946  33.02  0.951  33.56  0.954  
Gain by MBTVNLLMCST  2.17  0.024  1.04  0.009  1.24  0.008  0.54  0.003     
5.2 Test results with still images
Eight wellknown natural images are used, that is, Lena, Leaves, Monarch, Cameraman, House, Boat, and Pepper, as shown in Figure 7. For fair comparisons with previous works he2009 ; he2010 ; mun2009 ; chen2011compressed ; zhang2014image ; zhang2014group , the natural images are divided into nonoverlapping blocks ( in size). They are compressively sensed by an i.i.d. random Gaussian sensing matrix. Table 1 compares five wellknown existing CS recovery methods (i.e., the treestructured CS with variational Bayesian analysis using DWT (TSDWT) he2009 , the treestructured CS with variational Bayesian analysis using DCT (TSDCT) he2010 , SPLDDWT mun2009 , the SPL using the contourlet transform (SPLCT) mun2009 , and the multihypothesis CS method (MH) chen2011compressed ) with the proposed MBTVNLLM when patchbased sparse representation is not employed. It is worth emphasizing that, for this test case, MH is by far the most stateoftheart method. However, it turns out that the proposed MBTVNLLM is competitive with MH and much better than the others. In the best case, MBTVNLLM surpasses TSDWT, TSDCT, SPLCT, SPLDWT, and MH by dB, dB, dB, dB, and dB, respectively. Thus, it successfully demonstrates the effectiveness of the proposed schemes: MBTV and denoising of the Lagrange multiplier. The last rows of Tables 1 and 2 show the gains achieved by the proposed MBTVNLLM and MBTVNLLMCST, respectively, with respect to each individual method.
Further effectiveness of the proposed patchbased sparse representation is demonstrated in Table 2. In zhang2014image , the authors employed KSVD elad2006 to design a recovery method that used adaptivelylearned sparsifying (RALS), while the group sparse representation (GSR) zhang2014group acquired the local sparsifying transform. GSR is certainly better than RALS because of the local sparsifying basis for each group. In our three proposed methods in Table 2, MBTVNLLMGST attains better reconstructed quality than RALS, while MBTVNLLMLST and MBTVNLLMCST outperform GSR. This is because the better initial image created by MBTVNLLM has beneficial effects on grouping patches by facilitating a better nonlocal search and defining more appropriate sparsifying bases for each group (when using a local sparsifying transform). In particular, the PSNR of MBTVNLLMCST is as much as dB higher than GSR for the recovered image Leaves.
Furthermore, for a complex image with as much detail as is found in the image Lena, MBTVNLLMCST is not as successful as MBTVNLLMGST at a subrate of . The recovered image lacks spatial detail at a very lowsubrate such that the combination of local and global sparsifying transforms might make it slightly oversmoothed. The visual quality of the proposed schemes and previous work are compared in Figure 8 and Figure 9 using the image Monarch at subrate and Cameraman at subrate . This test shows that, while all conventional CS recovery schemes he2009 ; he2010 ; mun2009 ; chen2011compressed ; zhang2014image ; zhang2014group suffer from a large degree of highfrequency artifacts, including the stateofart method (GSR), the three proposed schemes seem to work much better. However, the recovered image of MBTVNLLMGST still has some artifacts at a very low subrate (e.g., see the image Monarch at subrate ). This indicates that a global sparsifying transform cannot adequately express the sparsifying levels for all groups.
Figure 10 quantifies the effectiveness of MBTVNLLMCST according to block sizes utilizing three images (i.e., Lena, Leaves, and Cameraman). Increasing the size of the sensing matrix yields better quality in the recovered images in terms of the PSNR. For example, at subrate with a block size of , the recovered image Lena has a PSNR of dB. However, this value can be increased up to dB with a block size of . These results coincide with our analysis in Section II based on the RIP property.
5.3 Test results with video for block DCVS
The effectiveness of the proposed CS recovery design is also evaluated with the first frames of three QCIF video sequences: News, Motherdaughter, and Salesman videotest . The GOP is set to . Input frames are split into nonoverlapping blocks ( in size), each of which is subject to BCS by an i.i.d. random Gaussian sensing matrix. To achieve better quality, key frames are sensed with a subrate of , while nonkey frames are sensed by a subrate ranging from to . Figure 11 shows the improvements of the reconstructed key frames of the proposed methods compared with MCBCSSPL and MHBCSSPL. All three proposed recovery schemes show far better visual quality than the previous block DCVS in mun2011residual ; tramel2011video . On average, over the three tested video sequences, MBTVNLLMCST shows dB and dB gains over MCBCSSPL and MHBCSSPL, respectively.
The improvements of nonkey frames for various block DCVS are shown in Figure 12. Due to the findings that TV can preserve edge objects, the nonlocal Lagrangian multiplier can reduce staircasing artifacts, and 3) the patchbased sparsifying transforms can enrich detail information, our proposed CS recovery schemes also produce far better PSNR values. Compared with MCBCSSPL, MBTVNLLMCST demonstrates gains between dB and dB depending on the subrate. In the best case, our recovery scheme is better by an average of dB compared to MCBCSSPL over nonkey frames.
Moreover, the visual quality of the first nonkey frame of the News sequence is illustrated in Figure 13. Because we utilized temporal redundancy over the frames, detailed information could be preserved for all block DCVS schemes. The high values of FSIM, even at a subrate of , demonstrate how crucial it is to exploit the correlation of frames in compressive video sensing. However, MCBCSSPL mun2011residual and MHBCSSPL tramel2011video still suffer from highfrequency oscillatory artifacts. Meanwhile, the proposed schemes no longer appear to have artifacts (i.e., the FSIM values are very close to ).
5.4 Computational complexity
Excluding patchbased sparse representation, the main computational complexity of the proposed MBTVNLLM comes from the high cost of the NLM filter. More specifically, if the search range and size of the similarity patches of the NLM filter are and , respectively, then, for an image of size , the computational complexity of this filter is . For natural images that are in size, with a subrate of , MBTVNLLM takes around min. to recover in our simulation. This is comparable to other methods that also do not use patchbased sparsifying transforms. That is, with the image Leaves at a subrate of 0.1, MBTVNLLM needs 63 s, MH takes s, and SPLDDWT consumes s. Alternatively, TSDCT requires much more decoding time than the others, requiring about min. to recover.
Patchbased sparse representation acts as a computational bottleneck. For a patch size of , search range of , group size of (where is the number of similar patches in a group), and two constant values and , the patchbased sparse representation using a local sparsifying transform demands a computational complexity of . In this way, MBTVNLLMCST is more complex due to the second stage containing the global sparsifying transform. Subsequently, for recovery of a QCIF video frame using MBTVNLLMCST, a key frame demands around min., and a nonkey frame requires s. Therefore, complexity optimization of patchbased sparse representation is an important task for future works. Specifically, to reduce complexity, we may be able to integrate our reconstructed algorithms with a robust sensing matrix such as Gaussian regressionbased han2015novel or multiscalebased sensing matrices fowler2011multiscale .
6 Conclusion
This paper proposed recovery schemes for BCS of still images and video that can recover pictures with highquality performance. For compressive imaging, the modified augmented Lagrangian total variation with a multiblock gradient process and nonlocal Lagrangian multiplier are used to generate an initial recovered image. Subsequently, the patchbased sparse representation enhances the local detailed information. Our design is also easily extendible to DCVS. More specifically, key frames are reconstructed to have improved quality and used to create initial versions of nonkey frames. Subsequently, nonkey frames are refined by patchbased sparsifying transformaided side information regularization. Our experimental results demonstrated the improvements made by the proposed recovery schemes compared to representative stateoftheart algorithms for both natural images and video.
Acknowledgements
This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No.), by the MSIP GITRC support program (IITP2016R6812160001) supervised by the IITP, and by the ERC via Grant EU FP 7  ERC Consolidator Grant 615216 LifeInverse.
References
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[link].
URL http://videocoders.com/yuv.html