Nonlocaly Multi-Morphological Representation for ... - IEEE Xplore

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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 26, NO. 12, DECEMBER 2017

Nonlocaly Multi-Morphological Representation for Image Reconstruction From Compressive Measurements Jiao Wu, Member, IEEE, Feilong Cao, and Juncheng Yin Abstract— A novel multi-morphological representation model for solving the nonlocal similarity-based image reconstruction from compressed measurements is introduced in this paper. Under the probabilistic framework, the proposed approach provides the nonlocal similarity clustering for image patches by using the Gaussian mixture models, and endows a multimorphological representation for image patches in each cluster by using the Gaussians that represent the different features to model the morphological components. Using the simple alternating iteration, the developed piecewise morphological diversity estimation (PMDE) algorithm can effectively estimate the MAP of morphological components, thus resulting in the nonlinear estimation for image patches. We extend the PMDE to a piecewise morphological diversity sparse estimation by using the constrained Gaussians with the low-rank covariance matrices, to gain the performance improvements. We report the experimental results on image compressed sensing in the case of sensing nonoverlapping patches with Gaussian random matrices. The results demonstrate that our algorithms can suppress undesirable block artifacts efficiently, and delivers reconstructed images with higher qualities than other state-of-the-art methods. Index Terms— Compressed sensing, Gaussian model, morphological diversity, latent variable model.

I. I NTRODUCTION

T

HE theory of compressed sensing (CS) [1]–[3] provides a possible way of exactly recovering signals from their compressed measurement, i.e., the projection onto a small number of random vectors, by means of their intrinsic sparsity. Under the framework of CS, y ∈ R Q is an observed measurement of an unknown true signal x ∈ R D (such as an image): y = x + e, where ∈ R Q×D is a sensing matrix with Q D, and e is a white Gaussian noise with variance σ02 . In the noiseless case, if x is sparsely represented under a dictionary U: x = Uα, that is α is a sparse coefficient vector with few non-zero elements, and and U satisfy certain conditions (e.g. Restrictive Isometry Property (RIP)) [3]–[5], α and hence x can be correctly recovered from y by solving the under-determined inverse problem with the regularization Manuscript received October 8, 2016; revised June 29, 2017; accepted August 7, 2017. Date of publication August 16, 2017; date of current version September 15, 2017. This work was supported by the National Natural Science Foundation of China under Grant 61302190, Grant 61272023, Grant 91330118, Grant 61571410, Grant 61572449, and Grant 61273018. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ivana Tosic. (Corresponding author: Jiao Wu.) The authors are with the College of Sciences, China Jiliang University, Hangzhou 310018, China (e-mail: [email protected]; [email protected]; [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.2017.2740566

technique, such as the classical 1 −minimization [6], [7], matching pursuit [8], [9], Bayesian method [10], and so on. Instead of the sparse prior used in conventional CS, many structured prior models (e.g., multiple measurement vector model, tree-structured supports model, block sparse model) have been introduced to provide the structured sparse representation of images and improve the reconstruction performances, thus leading to a flurry of structured CS algorithms [11]–[16]. Another popular catalog, local-patch based sparse representation learning methods, has been developed and obtained the impressive results for various image restoration problems. On the one hand, the learned dictionaries from the sample patches have the advantage of being better adapted to the images, thereby enhancing the sparsity (e.g., K-SVD [17], [18]). On the other hand, exploiting the selfsimilarity of overlapping image patches has led to a number of state-of-the-art nonlocal algorithms, e.g., nonlocal mean [19], BM3D [20]. Furthermore, there is quite a bit of nonlocal algorithms that incorporate the dictionary learning and the clustering of self-similarity patterns (associated with the local and nonlocal sparsity constraints respectively) into a unified framework, e.g., locally learned dictionaries K-LLD [21], clustering-based sparse representation (CSR) [22], nonlocally centralized sparse representation (NCSR) [23], piecewise linear estimation (PLE) [24]. Among them, PLE, that is based on the Gaussian mixture models, is used for image reconstruction from the compressed measurements under the probabilistic framework in [25], and developing a new framework of CS, namely statistical compressed sensing (SCS). PLE is based on a Max-Max algorithm that alternates between the synchronous estimation and clustering (or model selection) of image patches, and the parameter estimation of Gaussian models. Each cluster corresponds to a piecewise Gaussian model locally, the best fit Gaussian model is selected for each image patch and the MAP is calculated with a linear estimation therein. The mixture of Gaussian models (GMM), in a global view, implies a union of PCA bases (where the PCA basis is given by the eigendecomposition of Gaussian covariance matrix), which leads to a nonlinear block sparse representation for image patch. The simple model and efficient piecewise linear estimation lead to comparable or even better results than other state-of-theart algorithms in image restorations based on overlapping image patches. A related work, the expected patch log likelihood (EPLL) based on a Gaussian mixture prior, is independently given by [26]. The most difference of EPLL from

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WU et al.: NONLOCALY MULTI-MORPHOLOGICAL REPRESENTATION

PLE is that, the GMM in EPLL is learned from a train set of natural image patches, while PLE learns the GMM by using the self-trained approach and therefore incorporates the dictionary learning naturally. For the model selection and estimation of image patches, EPLL uses the similar mechanism as PLE. In an overlapping way, EPLL performs well on various problems of image restoration. However, compressed sensing only allows sensing nonoverlapping patches, e.g. if the sensing operators are the Gaussian random matrices. Although the reconstruction experiments for individual image patches have shown that SCS outperforms conventional CS based on the sparse model, aggregating nonoverlapping patches to a whole images produces significant block artifacts [25]. For improving the reconstructed performance of nonoverlapping patches from the compressed measurements, a novel image reconstruction algorithm, called piecewise morphological diversity estimation (PMDE), via a nonlocally multimorphological representation of image patches is developed in this paper. This algorithm is based on the Gaussian models and designed under the probabilistic framework, includes three steps: the patch clustering, the patch estimation, and the model parameter estimation, similar to the nonlocal schemes mentioned above. The global GMM is used in the first step for clustering the image patches, and in the third step, the parameters of each Gaussian model are updated with the maximum-likelihood (ML) estimates by using all the estimated patches (obtained in the second step). PMDE differs in the patch estimation step from PLE used in SCS. As mentioned earlier, PLE uses a single Gaussian model to depict the image patches in each cluster, and produces a linear estimation for image patch. However, an image, or a patch, contains the various features that represent the different morphological aspects, and therefore it seems that a single Gaussian is not enough to well describe the real image (patch). The question now is whether we can recover the morphological components of image patch from the compressed measurements by determining the reasonable model and the reconstruction method to achieve better final results? It is well known that the morphological component analysis (MCA) [27], [28] is an effective tool to decompose an image into its building blocks (e.g., texture and piecewise-smooth parts), and has been applied in various image processing problems including blind source separation [29], denoising [27], inpainting [30], and supersesolution [31], etc. Recently, a morphological regularization method is successfully used for CS image reconstruction [32]. This motivates us to incorporate the multi-morphological representation of image patch into the nonlocal method. In our work, the Gaussian models that represent the different directional features are used to model the morphological components of image patch, and an alternating iterative algorithm is developed to estimate the MAP of Gaussian morphological components, resulting in the nonlinear estimation for patches in each cluster. Considering the inherent sparsity of natural images, the morphological components, which depict the structural features of images, also have sparsity. In addition, the Gaussian model with the low-rank covariance matrix can provide the better

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describe for sparse patches [33], [34]. However, the Gaussian model learned from the compressed measurements usually does not have an exact low-rank covariance. This inspire us to learn the low-rank Gaussian models from the measurements, and use them as the morphological models to depict the image patches. This model is known as the constrained Gaussian model in probabilistic framework. In this paper, we extend the PMDE to a piecewise morphological diversity sparse estimation (PMDSE) algorithm by introducing a latent variable model for image patch. The developed PMDSE algorithm updates the parameters of each constrained Gaussian by using the technique of probabilistic PCA (PPCA) [35]. Given the rank r of the covariance matrix of constrained Gaussian (i.e., r is also called as the dimension of latent variable space), PPCA iteratively estimates the r principal axes of the corresponding PCA basis, therefore the sparse estimations of morphological components can be obtained in the patch estimation step of PMDSE by using the developed alternating iterative algorithm. In experimental section, we firstly discuss the impact of the number of morphological models and the rank of constrained Gaussian of our algorithms on the reconstructed qualities. We observe that the different number of morphological models should be used for the image patches with different features. To this end, we develop a method to adaptively determine the number of morphological models according to the statistical properties of image patches. The proposed algorithms, PMDE and PMDSE, are applied to the compressed sensing of natural images under the nonoverlapped sampling mode, and the performances of PMDE and PMDSE are compared to other state-of-the-art algorithms. The remainder of this paper is organized as follows. In Section II, we introduce the multi-morphological representation for image patches, and elaborate on the details of formulating the PMDE problem and how to solve PMDE. In Section III, we extent PMDE to PMDSE algorithm by introducing the latent variable model for image patches. Section IV reports the experimental results, with comparisons to other state-of-the art algorithms. Conclusions and future work are discussed in section V. II. P IECEWISE M ORPHOLOGICAL D IVERSITY E STIMATION In this√paper,√an image is decomposed into the nonoverN , lapping D × D (typically 8 × 8) local patches {xn }n=1 D the vector xn ∈ R is the nth patch, and N is the total number of patches. Each patch is sampled with a row orthogonal Gaussian sensing matrix1 ∈ R Q×D ( i.e. T = I Q ), N , resulting in a collection of compressed measurements {yn }n=1 in which

yn = xn + en

(1)

where yn ∈ R Q , and en ∈ R Q is the additive Gaussian noise with en ∼ N (0, σ02 I Q ). Thus we can write p(yn | xn ) = N (yn | xn , σ02 I Q )

(2)

1 The orthogonal Gaussian random matrix is generated by randomly selecting Q rows from an D × D orthogonal matrix with i.i.d. draws of a Gaussian distribution.

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This section describes how we reconstruct the image from the compressed measurements by introducing the nonlocally multi-morphological representation for image patches. The developed PMDE includes the following three steps: • Structural clustering and morphological model selection: We first cluster the image patches by calculating the posterior class probabilities p(k|yn ) of the nth patch xn for K Gaussian models and assigning xn to the Gaussian cluster corresponding the maximum posterior probability. This clustering process is similar to the model selection of PLE and EPLL. Another cluster methods (e.g., K-means, KNN) can also be used to cluster image patches, however the clustering based on maximizing posterior probability connects the clustering process with the Gaussian models (or underlying PCA bases), and makes it possible for us to select the morphological models simultaneously. Here the morphological model selection is based on the fact that the Gaussian model with the larger the posterior probability is more descriptive for the structure of image patch. Thus the first few Gaussian distributions with the largest posterior probabilities are chosen as the morphological component models. • MAP estimate based on morphological diversity: In each cluster, instead of calculating the linear MAP estimates for image patches by PLE, we introduce a morphological representation for image patches and develop an alternating iterative algorithm, referred to as MAPMD, to calculate the MAP estimates of the morphological components, thus producing the nonlinear estimates for image patches. The linear MAP estimates of PLE can be viewed as the initial estimations of the proposed MAPMD algorithm. • Re-estimate of Gaussian models: Using the recovered partial image patches in each cluster, we re-estimate the parameters of Gaussian models. In some sense, this model updating implies a self-learning of dictionary. The use of the self-trained approach has the advantages of not requiring a lot of training samples in advance and making the updated models (or dictionaries) more suitable for signals to be recovered. In this section, the mean and variance parameters of each Gaussian model are updated by the ML solutions as PLE has done. In fact, the models used in PMDE are some of unconstrained Gaussians. In next section, we will consider to use some of constrained Gaussian models for image patches, therefore the model parameters will be updated by using the technique of PPCA [35]. In order to make the notations simpler, and concisely describe the multi-morphological probability representation of image patches and the developed algorithms, the subscript n is omitted in the following context. We denote x as a certain image patch, y is the corresponding measurement, and x j is the j th morphological component of x. A. Structural Clustering and Morphological Model Selection Given the K unconstrained Gaussian distributions {N (μk , k )}1≤k≤K , we cluster the image patches into

K clusters by maximizing the log posterior class probabilities. Assume that the patch x is taken to be a realization from one of the K Gaussians, e.g., the kth Gaussian p(x | k) = N (x | μk , k )

(3)

which is parameterized by the mean μk and the covariance k . Then using the probability distribution (2), the conditional distribution of y given the kth cluster is calculated as p(y | k) = p(y | x) p(x | k)dx = N (y | μk , Ck ) (4) by integrating over the variable x. Where the covariance matrix Ck ∈ R Q×Q is given by Ck = k T + σ02 I Q

(5)

Using Bayes’ theorem, the posterior class probability conditioned on y is given by p(k | y) ∝ p(y | k) p(k)

(6)

Thus maximizing log p(k | y) is equivalent to maximizing log p(y | k) when all K Gaussians are equally likely, i.e., the prior probabilities p(k) = 1/K , for all k ∈ [1, . . . , K ]. Let ρk = log p(y | k)

(7)

Then the cluster index of x is given by k˜ = arg min {ρk } k

(8)

˜ Gaussian cluster. i.e., the patch x is assigned to the kth The morphological component models can be chosen simultaneously by using the scores calculated in (7). To this end, sorting these scores in ascending order, ρk1 ≤ ρk2 ≤ · · · ≤ ρk K , then the smallest one is used for clustering x. The smaller the value of ρk , the better the corresponding Gaussian model represents x. Thus we simply select those Gaussians corresponding to the first few small scores as morphological models, e.g. the first J Gaussians. B. MAP Estimate Based on Morphological Diversity 1) Nonlocal Multi-Representation of Image Patch: As mentioned earlier, we introduce a multi-morphological representation to image patch after the clustering and the morphological model selection. For a patch x that belongs to the kth cluster, its multi-morphological representation is given by x = μk +

J

xj

(9)

j =1

where μk is the mean vector of the kth Gaussian model, x j is the j th morphological component, and J is the number of morphological components. The different morphological components depict the different morphological (or directional) features of x. Here x is recovered by estimating the morphological components x j from the measurement y.


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2) MAPMD Algorithm: Assume that the morphological components are independent of each other, and each of them is a realization of a zero mean Gaussian, i.e., p(x j | k j ) = N (x j | 0, k j ), for k j ∈ [1, . . . , K ]. Then we obtain a factorizable prior of (x1 , . . . , x J ) p(x1 , . . . , x J | k1 , . . . , k J ) =

J

p(x j | k j )

Algorithm 1 MAPMD

(10)

j =1

From (2), the probability distribution of measurement vector y conditioned on the morphological components x1 , . . . , x J has the form p(y | x1 , . . . , x J ) = N (y | μk +

J

x j , σ02 I Q ) (11)

j =1

Then the MAP of x1 , . . . , x J can be estimated by maximizing the log posteriori probability 3) Convergence: The optimization problem (13) in fact is a special example of the regularized block multi-convex optimization [36], which has the form as

(˜x1 , . . . , x˜ J ) = arg max log p(x , . . . , x | y) 1

J

x1 ,...,x J

= arg max {log p(y | x1 , . . . , x J ) + x1 ,...,x J

J

log p(x j | k j )}

j =1

(12) or equivalently minimizing the following optimization problem ⎧ J ⎨ 1 J (˜x , . . . , x˜ ) = arg min ¯y − x j 2 x1 ,...,x J ⎩ j =1 ⎫ J ⎬ j + σ02 (x j )T −1 (x ) (13) kj ⎭ j =1

where y¯ = y − μk , i.e., we center the measurement y with respect to the mean μk of the kth Gaussian. The solution of (13) can be calculated via the block coordinate descent (BCD) method alternatively solving a sequence of subproblems. In (i + 1)th iteration, the estimate of x j is obtained by solving the following subproblem (˜x j )(i+1) = arg min (r j )(i+1) − x j 2 xj j + σ02 (x j )T k−1 (x ) (14) j

j −1

where (r j )(i+1) = y¯ − t =1 (˜xt )(i+1) − tJ= j +1 (˜xt )(i) . A closed linear solution of (14) is given by j (i+1) (˜x j )(i+1) = k j T C−1 k j (r )

(15)

where Ck j is given by (5). (15) can be viewed as a linear filter. This iterative MAP algorithm is based on the BCD method, referred as MAPMD, is shown in Algorithm 1. Starting with the initial residual r¯ (0) = y¯ , MAPMD iteratively estimates the J morphological components by calculating a set of linear filters (15), such that the residual decreases continuously. The algorithm is terminated until the difference of residuals is small enough. Then the patch x is recovered by using the estimates of J morphological components.

min{ f (x1 , . . . , x J ) +

x∈X

J

g j (x j )}

(16)

j =1

where variable x is decomposed into J blocks x1 , . . . , x J , the set X of feasible points is a closed and block multi-convex set, f is a differentiable and block multi-convex function (i.e., f is a convex function of x j while all the other blocks are fixed), and g j , j = 1, . . . , J , are extended-value convex functions. Using the BCD method of the Gauss-Seidel type, the objective function in (16) can be cyclically minimized over each of x1 , . . . , x J while fixing the remaining blocks at their last updated values. In the (i + 1)th iteration, x j can be estimated by solving the following subproblem (i+1)

(x j )(i+1) = arg min{ f j xj

(x j ) + g j (x j )}

(17)

where (x j )(i+1) is the value of x j after its (i + 1)th update, and f j(i+1) (x j ) f (x1 )(i+1) , . . . , (x j −1 )(i+1) , x j , (18) (x j +1 )(i) , . . . , (x J )(i)

In (13), f (x1 , . . . , x J ) = ¯y − Jj=1 x j 2 is differentiable j and block multi-convex, g j (x j ) = σ02 (x j )T −1 k j (x ) is a convex function, and in the subproblem (14), f j(i+1) (x j ) = (r j )(i+1) − x j 2 is a quadratic function of x j with the strongly convex. The theoretical analysis in [36] shows that the BCD algorithm for solving the above regularized block multiconvex optimization converges. See [36] for more details. C. Re-Estimate the Unconstrained Gaussian Models In each cluster, using the recovered partial image patches {˜xn }n∈Ck , ∀k, the mean and covariance parameters of the

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unconstrained Gaussian model are re-estimated by calculating the ML estimates 1 μk = xn (19) Nk n∈Ck

1 k = ( xn − μk )( xn − μk )T Nk

(20)

n∈Ck

where Ck is the set of indices corresponding to the patches assigned to the kth Gaussian model, Nk = |Ck | is the number of patches included in the kth cluster.

B. PMDSE 1) Structural Clustering and Morphological Model Selection: Given the K constrained Gaussians N (μk , k ) with the low-rank rk < D − 1, and k is given by (25), we can use (3)-(8) given in Section II-A to cluster image patches and select the morphological models. 2) MAPMD in Latent Space: A multi-morphological representation based on the latent variables for patch x is introduced as x = μk +

III. P IECEWISE M ORPHOLOGICAL D IVERSITY S PARSE E STIMATION

A. Latent Variable Model The image patch x ∈ R D assigned to the kth cluster can be viewed as being generated from a linear latent variable model (21)

Rrk

where zk ∈ is a Gaussian latent variable with the zeromean and unit-covariance, i.e., p(zk ) ∼ N (zk | 0, Irk )

(23)

in which the mean of x is a linear function of zk governed by the matrix Wk ∈ R D×rk . And the marginal distribution for x is again Gaussian p(x | k) = p(x | zk ) p(zk )dzk = N (x | μk , k ) (24) by using the product rule of probability and integrating out the latent variable, where the covariance matrix k with the rank rk is given by k =

Wk WkT

+

σk2 I D

(25)

The model (24) represents a zero-mean Gaussian distribution governed by the parameters μk , Wk and σk2 . For the case of the latent-space dimension (or the rank of covariance matrix) rk = D − 1, (24) is equivalent to a full-covariance Gaussian distribution, while for rk < D − 1 it represents a constrained Gaussian. We can see that the reduction of the latent-space dimension gives the counterpart of rk -sparsity under the above linear transform for Gaussian signals. In other words, the low-rank constrained Gaussian model can provide the sparse representation for image patch or its Gaussian morphological components, thus the image patch or its morphological components can be recovered by calculating their corresponding sparse estimates in the latent space.

J

(Wk j zk j + ek j )

(26)

j =1

The j th morphological component is assumed to be generated from the linear latent variable model corresponding to the k j th Gaussian distribution x j = Wk j zk j + ek j

(27)

r

where the latent variable zk j ∈ R k j with rk j < D − 1, and the noise ek j ∼ N (0, σk2j I D ). From (22) and (23), and the independence among the morphological components and the corresponding latent variables, we have p(x1 , . . . , x J | zk1 , . . . , zk J ) =

J

p(x j | zk j ) =

j =1

(22)

and ek is an additive Gaussian noise with ek ∼ N (0, σk2 I D ). Then the zk -conditional probability distribution over x is given by p(x | zk , k) = N (x | μk + Wk zk , σk2 I D )

x j = μk +

j =1

In this section, we introduce the latent variable model for depicting the image patches, from which a collection of constrained Gaussian models with the low-rank covariance matrices can be obtained. The developed PMDSE calculates the sparse estimates of morphological components of patches on the latent space, and uses the technique of PPCA [35] to re-estimate the model parameters in model updating step.

x = μk + Wk zk + ek

J

J j =1

N (x j | Wk j zk j , σk2j I D )

(28)

and p(zk1 , . . . , zk J ) =

J

p(zk j ) =

j =1

J j =1

N (zk j | 0, Irk j )

(29)

The probability distribution of y conditional on zk1 , . . . , zk Jn is calculated as p(y | zk1 , . . . , zk J ) = p(y | x1 , . . . , x J ) p(x1 , . . . , x J | zk1 , . . . , zk J ) d(x1 , . . . , x J ) ⎛ = N ⎝y | μk +

J j =1

Wk j zk j , (σ02 +

J j =1

⎞ σk2j )I Q ⎠ (30)

by using (11) and (28), and integrating over the morphological components x1 , . . . , x J . Then the MAP of zk1 , . . . , zk J is estimated by maximizing the log posteriori probability (˜zk1 , . . . , z˜ k J ) = arg max log p(zk1 , . . . , zk J | y) zk1 ,...,zk J ⎫ ⎧ J ⎬ ⎨ = arg max log p(zk j ) log p(y | zk1 , . . . , zk J ) + zk1 ,...,zk J ⎩ ⎭ j =1

(31)


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Gaussian parameters μk , Wk and σk for the kth cluster are derived by [35], in which the mean is also updated by the empirical estimate μ˜ k shown in (19), and the estimates of Wk and σk are given by

Algorithm 2 MAPMD∗

k = Sk Wk (σk2 Irk + M−1 WkT Sk Wk )−1 (36) W k 1 kT (37) Tr Sk − Sk Wk Mk−1 W σ˜ k2 = Nk where Sk is the sample covariance matrix of the kth cluster given by 1 ( xn − μk )( xn − μ k )T (38) Sk = Nk n∈Ck

and Mk = WkT Wk + σk2 Irk

or equivalently minimizing the following optimization problem ⎧ J ⎨ (˜zk1 , . . . , z˜ k J ) = arg min Wk j zk j 2 ¯y − zk1 ,...,zk J ⎩ j =1 ⎫ J J ⎬ + (σ02 + σk2j ) (zk j )T (zk j ) (32) ⎭ j =1

j =1

where y¯ is a centered measurement with respect to the mean μk as mentioned above. Here we also use the block coordinate descent method to solve this optimization problem. In (i + 1)th iteration, the subproblem for estimating zk j is given by (zk j )(i+1) = arg min (r j )(i+1) − Wk j zk j 2 zk j ⎫ J ⎬ + (σ02 + σk2j )(zk j )T (zk j ) (33) ⎭ j =1

j −1 (r j )(i+1) = y¯ − zkt )(i+1) − t =1 Wkt (˜ (i) zkt ) . The linear solution is given by t = j +1 Wkt (˜

where

J

T T j (i+1) (˜zk j )(i+1) = D−1 k j Wk j (r )

(34)

where Dk j =

WkTj T Wk j

+

(σ02

+

J j =1

σk2j )Irk j

(35)

Under the current values of parameters, zk1 , . . . , zk J are iteratively estimated by using the set of linear filter (34) until the difference of residuals is small enough. The final estimate of x is given by x˜ = μk + Jj=1 Wk j z˜ k j . The derived algorithm, referred as MAPMD∗ , is shown in Algorithm 2. 3) Re-Estimate the Constrained Gaussian Models: From the reconstructed image patches, the model parameters of each constrained Gaussian can be updated by using the technique of PPCA when the dimension of latent space is given. Given the recovered patches {˜xn }n∈Ck , the re-estimate equations of

(39)

[35] has proved that the standard ML estimate (20) for the unconstrained Gaussian parameter will be obtained from (36) when the latent-space dimension rk = D. The latent-space dimension can also be determined automatically by using the technique of Bayesian PCA. In [37], a following hierarchical prior over Wk is introduced p(Wk |αk ) =

rk

−1 N (wki |0, αki ID )

i=1

where αki is the inverse variance parameter of wki , so that if the value for αki is driven to infinity, the corresponding wki will tend to zero, and the corresponding direction in latent space will be effectively switched off. For more details about PPCA and BPCA, see [35], [37]. For simplicity, here we use PPCA to update the model parameters for the fixed latentspace dimension, i.e., rk = r, ∀k ∈ [1, . . . , K ]. C. Computational Complexity In the clustering step of PMDE and PMDSE, the computational complexity is dominated by the matrix inversions of Ck j in (5). Since Ck j is a Q × Q positive definite, the matrix inversion can be implemented with O(Q 3 /3 + Q 2 ) using Cholesky factorization [38]. In the step of MAP estimates based on morphological diversity, the estimates of morphological components are basically calculated with a sequence of linear filters, a major part of computational cost lies in the calculation of C−1 k j and the filters k j T C−1 k j , as shown in (15), in PMDE , wherein

C−1 k j has been given in first step. In fact, the filters can be pre-computed for the K Gaussian distributions due to the nonoverlapped sampling for the different patches with the same sensing matrix . Calculating (15) with the precomputed filter requires only O(D Q). Similar calculations are performed by PMDSE in (34), in which the main cost is the matrix inversion of Dk j , whose computational compleity is O(r 3 /3 + r 2 ), with r D. These filters also can be pre-computed before performing MAPMD algorithm. Thus calculating (34) with the pre-computed filter requires only O(Dr ). During the process of performing MAPMD algorithm, with the J selected morphological components,

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Fig. 1. Natural images of six groups: (a) People, (b) Animal, (c) Building, (d) Landscape, (e) Truck/Plane, and (f) Texture.

the complexities per iteration are therefore dominated by O(J D Q) and O(J Dr ) for PMDE and PMDSE, respectively. If the number of iterations is I when MAPMD algorithm stops, then the complexities are O(I J D Q) or O(I J Dr ) for both methods with I, J D, Q, r . In the last steps of PMDE and PMDSE, the calculation of K covariance matrices (20) and (38) requires O(K D 2 ). For PMDSE, the complexity per iteration of PPCA is the matrix inversion Mk−1 in (39) and (σk2 Ir + Mk−1 WkT Sk Wk )−1 in (36), whose computational complexities are both O(r 3 /3 + r 2 ). In the following experiments, the iteration number of PPCA is set to 100. IV. E XPERIMENTS AND D ISCUSSION The performances of the proposed PMDE and PMDSE are evaluated in this section. The experiments are performed on an image database, which includes 60 gray-level natural images (available at http://decsai.ugr. es/cvg/dbimagenes/index.php) with size 512 × 512. These test images are divided into six groups: people, animal, building, landscape, plane/truck, and texture. Each group contains ten images, as √ in Fig. 1. The test images √ illustrated are partitioned into D × D = 8 × 8 nonoverlapped sub-patches, and each image patch is vectorized to form a set of sub-patches {xn }1≤n≤N . The corresponding CS measurements are produced as {yn = xn }1≤n≤N . The same sensing matrix , that is used to sample the different image patches, is generated by randomly selecting Q rows from an D × D orthogonal matrix with i.i.d. draws of a Gaussian distribution N (0, 1). The reconstructed performance is evaluated in terms of peak signal-to-noise ratio (PSNR), structural similarity (SSIM) [39] and visually. The following experiments contain two parts: the effects of the number of morphological models and the latent-space dimension on the reconstructed performances of PMDE and PMDSE are discussed in the first part of experiments. Then we give the reconstructed performance comparisons on compressed sensing of natural images with other state-of-theart methods, including KSVD-OMP [17], [18],2 EPLL [26],3 PLE-SCS [24], [25], MMLE-GMM and MMLE-MFA [33].4 The dictionaries used for KSVD-OMP is learned by the KSVD algorithm from 50,000 sample image patches, extracted 2 The source code of KSVD-OMP is available at: http://www.cs.technion. ac.il/~ronrubin/software.html. 3 The source code of EPLL is available at: https://people. csail.mit.edu/danielzoran/. 4 The source code of MMLE-GMM and MMLE-MFA are available at: https://sites.google.com/site/jbysite/.

from the entire standard Berkeley segmentation database containing 300 natural images [40],5 and the size of KSVD dictionary is set to that can provide the best result within {128, 256, 512}. The OMP algorithm is used to solving the sparse recovery problem of image patches. MMLE-GMM and MMLE-MFA are two algorithms related with PLE-SCS. Given the K Gaussian models, these two algorithms recover the image patches from compressed measurements by learning a Gaussian mixture model and a mixture of factor analyzer to depict the patch at global level, respectively, however PLE-SCS represents the patch with a single Gaussian at local level (i.e., within a cluster). Our algorithms, PMDE and PMDSE, are also based on a collection of Gaussian models. All of these algorithms starting with a collection of initial Gaussian models, then alternate between estimating the image patches and updating the model parameters. Thus the same initialization method, the geometry-motivated Gaussian models [24], is used for them. For PLE-SCS, MMLE-GMM, and PMDE, the K − 1 = 18 unconstrained Gaussian models are learned from the synthetic directional image patches, which correspond to 18 angles that uniformly sampled from 0◦ to 180◦ , and a DCT basis is employed as the K th Gaussian model to capture the isotropic image patterns. The mean vectors of K Gaussians {μk }1≤k≤K are set to zeros. We make the eign-decomposition for each covariance matrix k , and take r eigenvectors corresponding to the first r largest eigenvalues as the columns of the factor loading matrix of MMLE-MFA. For PMDSE, the parameters {Wk , σk2 }1≤k≤K are initialized as follows D 1/2 1 , σk2 = λk,i (40) Wk = Uk k − σk2 Ir D −r i=r+1

where Uk is composed of the first rk eigenvectors of k , which correspond to the first r eigenvalues arranged in the diagonal D are the last D − r eigenvalues matrix k , and {λk,i }i=r+1 of k . In fact, (40) are the ML estimates of parameters Wk and σk2 . See [35] for a more detailed discussion. In addition, the variance for measurement noise, i.e., σ02 is set to 5e − 4 for PMDE , and 1e − 4 for PLE-SCS, MMLE-GMM, MLE-MFA, and PMDSE. The fixed variance parameter γ = 1e − 3 is used in MMLE-MFA to indicate that each Gaussian is well approximated by a low-rank factor model, however in our algorithm PMDSE, the variance parameter σk2 for each low-rank Gaussian is updated in each iteration by learning parameters from the recovered image patches. PLE-SCS, MMLE-GMM, MMLE-MFA, PMDE, and PMDSE all perform five outer iterations, the reconstructed result is the best one. Although a collection of Gaussian models is also used in EPLL, this GMM is learned from a set of natural image patches, and the model parameters are not be updated when performing EPLL algorithm. Indeed, our algorithms also work well with the trained GMM without needing to re-estimate model parameters. Therefore, in the following experiments, we compare PMDE with EPLL based on the GMM learned from a train set of 100,000 natural image patches, sampled 5 The database is available at: http://www.eecs.berkeley.edu/Research/ Projects/CS/vision/grouping/resources.html


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from the entire standard Berkeley segmentation database. The models used for comparisons are with learned 20, 40, and 100 Gaussian mixture components, respectively. The parameters of EPLL are set to the default values used in the noise free case. A. Parameter Setting of PMDE and PMDSE The number of morphological models is an important parameter for PMDE and PMDSE. In addition, the performance of PMDSE is affected by the latent-space dimension (i.e., the rank r of the constrained Gaussian). Thus we discuss their impact on the algorithm performances by taking Barbara image as an example. We divide Barbara image into four sub-images with the different texture features, shown in Fig. 2 (a)-(d), respectively. We can see that, from the subimage Barbara (a) to Barbara (d), the contained textures from weak become strong. The left column of Fig. 2(a)-(d) show the reconstructed PSNRs of one realization as the function of the number of morphological models J (the same value of J is used for all image patches for simplicity) with six sampling rates Q/D = 12.5%, 25%, 37.5%, 50%, 62.5%, 75%. The resulted PNSRs are generated on about 1,024 nonoverlapping patches extracted from four sub-images, respectively. Here PMDSE uses the same latent-space dimension for each lowrank Gaussians, and the performances are tested with four latent-space dimensions, r = 18, 24, 36, 54. In each test, PMDE and PMDSE using the fixed number of morphological models are denoted as PMDE-f and PMDSE-f, respectively. For the first two sub-images that contain few texture patches, the reconstructed performances of PMDE-f and PMDSE-f improve with the increasing number of morphological models specially when the sampling rate Q/D ≥ 25%. For the third sub-image that contains almost the same proportion of smooth and texture patches, the best performances PDME-f and PMDSE-f are obtained by using 3-4 morphological models at the sampling rate Q/D ≥ 37.5%. Only two morphological models are used by PDME-f and PMDSE-f to obtain the better performances when the sampling rate Q/D is less than 37.5%. The textures are dominant in the fourth sub-image, we can see that PDME-f and PMDSE-f with two morphological models produce the best reconstructed results, however the increasing of the number of morphological components can result in the decreased performances. The above experimental results demonstrate that the image patch that has the smooth feature or homogeneous textures can be represented by using multiple Gaussian morphological models, while only the few morphological models (e.g., only two Gaussians) are needed to depict those patches with the complex and strong textures. A natural image usually contains many patches with the different texture features, using the same number of morphological models for different patches cannot provide the best results on the whole. Can we select the different number of morphological models for different patches during the implementation of PMDE and PMDSE? According to the analysis about the above experimental results, we propose to determine the number of the morphological models by roughly classifying the image patches according to the statistical measures extracted from the gray level co-occurrence

Fig. 2. Reconstructed PSNRs (dB) versus the number of morphological components for PMDE and PMDSE (r = 18, 24, 36, 54) with six sampling rates Q/D = 12.5%, 25%, 37.5%, 50%, 62.5%, 75%. The reconstructed results for four sub-images of Barbara shown in the left column of (a)-(d) are given by PMDE-f and PMDSE-f with the fixed number of morphological models. The results shown in the right column are given by PMDE-a and PMDSE-a, which adaptively determine the number of morphological models.

matrix (GLCM). Assume that the patch x has been assigned to the kth Gaussian cluster in the clustering step, we calculate the GLCM of its MAP estimate under the kth Gaussian, i.e., (˜x)k = k T C−1 k (y − μk ) + μk , and then extract two statistical measures: Entropy and Homogeneity, denoted by E and H , respectively. The criterion for determining the number of morphological models is given by J1 , if E ≥ e and H < h; J= (41) J2 , if (E < e) or (E ≥ e and H ≥ h). where e and h are the threshold parameters, and set to 0.2 and 0.6, respectively. J1 and J2 are the numbers of morphological models corresponding the different variation ranges of threshold parameters. The Entropy indicator is used

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to measure the disorder or complexity of image patch. The value of entropy is higher, the stronger non-uniformity and complexity the image patch has. The Homogeneity indicator measures the local homogeneity of image patch. The high value of it demonstrates that the image patch has the uniform textures. Generally, the smooth patch has a very low entropy value (E < 0.2), and the patch with the high entropy value (E > 1) contains the strong textures. We roughly distinguish the uniform and non-uniform textures by setting h = 0.6. J1 = 2 morphological models are selected for the non-smooth image patch with the non-uniform textures (i.e., E ≥ 0.2 and H < 0.6). For the smooth patch (i.e., E < 0.2) and the patch with the uniform textures (i.e., E ≥ 0.2 and H ≥ 0.6), we further test the algorithm performances with different values of J2 . Here the algorithms, which adaptively determine the number of morphological models with (41), are denoted as PMDE-a and PMDSE-a, respectively. The reconstructed results of PMDE-a and PMDSE-a are demonstrated in the right column of Fig. 2(a)-(d), respectively. For the first sub-image, we can see that most of patches present the smooth and homogeneous features, and there are no significant changes in the reconstructed PSNRs of PMDE-a and PMDSE-a compared with the results of PMDE-f and PMDSE-f, i.e., the reconstructed qualities improve with the increasing number of morphological models. There are only a small number of texture patches in the second sub-image, the reconstructed results are slightly inferior than those given by PMDE-f and PMDSE-f, when restricting to use only two morphological models to depict these few texture patches. The number of texture patches increases obviously in the third sub-image, in contrast with the results shown in the left column, the reconstructed PSNRs of PMDE-a and PMDSE-a are improved continuously by adaptively restricting the number of morphological models for the texture patches. This implies that using fewer morphological models for the non-uniform texture patches and more morphological models for the smooth and homogeneous patches does improve the reconstructed performance of PMDE and PMDSE. As mentioned earlier, too many morphological models can lead to a rapid decrease of the reconstructed quality of the non-uniform texture patches. However, from the results of the forth sub-image, we can see that the reconstructed qualities are no longer getting worse, when the number of morphological models increases. This further suggests that when we use multiple morphological models to obtain a good estimate for the smooth image, we can still maintain a better estimate for the texture patch. Taking into account the trade-off between the reconstructed qualities of the different patches and the complexity of algorithms, we set J2 = 5 in the following experiments. In order to illustrate the universality of the proposed method, we compare the performances of PMDE and PMDSE with the fixed and adaptive number of morphological models on the image database shown in Fig. 1. The average reconstructed PNSRs of six groups as the function of the sampling rates, shown in Fig. 3(a)-(f), are given by PMDE-f and PMDSEf(r = 24, 36) with the fixed number of morphological models J = 5, PMDE-a and PMDSE-a(r = 24, 36), respectively. We can see that, in the average sense, the reconstructed results

Fig. 3. Comparison average reconstructed PSNRs (dB) for six image groups using PMDE-f and PMDSE-f (r = 24, 36) with 5 morphological models, and PMDE-a and PMDSE-a (r = 24, 36) under the varying sampling rates. (a) People, (b) Animal, (c) Building, (d) Landscape, (e) Truck/Plane, and (f) Texture.

have no significant differences for the algorithms using the different methods to determine the number of morphological models, except for group People. Fig. 3(a) show that PMDE-a and PMDSE-a(r = 24, 36) produce the superior performances than the corresponding PMDE-f and PMDSE-f(r = 24, 36). The reason may be that there are many non-smooth and nonuniform texture patches contained in group People. As mentioned above, PMDE-a and PMDSE-a can effectively improve the reconstructed qualities of these patches by restricting the number of morphological models, while ensuring the qualities of the patches contained the smooth and uniform textures. B. Image Compressed Sensing In this section, we give the reconstruction comparisons of EPLL, KSVD-OMP, PLE-SCS, MMLE-GMM, MMLE-MFA and our algorithms for the test images. The reconstructions are performed on 4,096 nonoverlapping patches extracted from the image with the size of 512 × 512, and 1,024 nonoverlapping patches extracted from the image with the size of 256 × 256. As mentioned earlier, EPLL works based on a GMM model, which is learned from a set of natural image patches in advance, and does not need to update the model parameters during the execution of algorithm. The proposed PMDE algorithm can also be performed well in this framework. Taking Lena and Boat for examples, Fig. 4 (a)-(b) demonstrate the reconstructed PSNRs (shown in the left column) and SSIMs (shown in the right column) of EPLL and PMDE using the GMM with 20, 40, and 100 Gaussian mixture components, respectively. We can see that the performances of PMDE are significantly better than those of EPLL at the sampling rate


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TABLE I R ECONSTRUCTED PSNR AND SSIM C OMPARISONS ON S IX I MAGES G ROUPS W ITH T HREE AVAILABLE S AMPLING R ATES

Fig. 4. Performance comparisons of EPLL and PMDE based on the GMM model with the sampling rates Q/D varying from 12.5% to 75.0%. The GMMs with 20, 40, and 100 mixture components are learned from a set of 100,000 natural image patches. The reconstructed PSNRs of (a) Lena and (b) Boat are shown in the left column, and the SSIMs of reconstructed images are shown in the right column.

Q/D ≤ 37.5%. PMDE is slightly better than EPLL at the sampling rate Q/D > 50%. EPLL exceeds PMDE only at the higher sampling rate Q/D = 75%. It is worth noting that we can obtain better reconstructed results by using the pre-training GMM model, however, training the GMM with the above training set (100,000 patches) is a time-consuming process, e.g., it takes around 2-8h under MATLAB R2016b on PCs with Xeon E3-1505M processor.

Given the GMM model, running times of EPLL20 (with 20 Gaussian components) and PMDE(20,40,100) are around 4s per image (with the size of 512 × 512), and EPLL40 and EPLL100 run around 7s and 20s, respectively. The iteratively updating Gaussian models and estimating image patches of PMDE and PMDSE based on the self-trained model may reduce the computational efficiency of the reconstruction process, however, the algorithms converge fast for image recover problem, typically in 3 to 5 iterations, as PLE-SCS. More importantly, we don’t need to spend a large amount of time training a GMM from a large image database. Next, we give the performance comparisons of PMDE and PMDSE with KSVD-OMP, PLE-SCS, MMLE-GMM, and MMLE-MFA. Table I presents the reconstructed performance comparisons on six image groups of all algorithms with three sampling rates. We can see that PMDE and PMDSE provide the superior reconstructed PSNRs and SSIMs than other algorithms. For all image groups, the significant performance gains are provided by our algorithms with the increase of sampling rates, the average PSNRs are improved from 0.7dB at low sampling rates to 3 dB at high sampling rates. Table II shows the reconstructed PSNRs and SSIMs of six test images at three sampling rates. Except for Cameraman at the sampling rate Q/D > 25% and for House (with the size 256×256) at the sampling rate Q/D = 75%, the performances of KSVD-OMP are inferior to those of the other algorithms. MMLE-GMM and MMLE-MFA are slightly superior than PLE-SCS. For Barbara image, except at the sampling rate 25% where MMLE-GMM gives the best reconstructed result, our algorithms considerably outperform the other methods in all

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TABLE II R ECONSTRUCTED PSNR AND SSIM C OMPARISONS ON S IX T EST I MAGES W ITH T HREE AVAILABLE S AMPLING R ATES

the cases. In addition, PMDSE based on the low rank (r = 18) model provides the higher PSNR and SSIM than other algorithm for Cameraman image, e.g., it improves the PSNRs about 0.8-2.8dB than PMDE. This implies that the morphological representation with the low rank Gaussian models is effective for the sparse image. PMDSE with the moderate size rank (r = 24, 36) can provide the higher PSNRs for some compressible images. However, from the reconstructed results shown in Table I, it seems that PMDE is more suitable for the images contained more complex scenes (e.g., image groups Animal, Building, Landscape), because it produces the higher reconstructed PSNRs and SSIMs than other algorithms. Comparing the running times, PLE-SCS is fastest which only needs around 1-2s for an image with the size 512 × 512. KSVD-OMP runs about 1-5s, but it needs to a pre-trained dictionary, which is learned from a large image database. MMLE-GMM runs about 5-20s, and MMLE-MFA needs 20-70s. Running times of PMDE and PMDSE are around 5-20s and 5-60s, respectively. The visual comparisons of the reconstructed images of Lena, Boat, and Barbara images at the sampling rate Q/D = 37.5% are shown in Fig. 5, 6 and 7 respectively. The first six reconstructed images are given by KSVD-OMP, PLE-SCS, MMLE-GMM, MMLE-MFA, PMDE and PMDSE, respectively. The last results are given by EPLL-20 and PMDE-20, these two algorithms are based on a pretrained GMM model contained 20 mixture components. It can be observed that all of the reconstructed images by KSVD-OMP look like to be corrupted by the noticeable noises. PMDE and PMDSE seem to deliver the best visual qualities for each image, by contrast, the reconstructed images by PLE-SCS, MMLE-GMM, and MMLE-MFA all suffer from the blocking artifacts. For example, in some smooth regions, such

Fig. 5. Reconstruction performance comparison on the Lena image at the sampling rates Q/D = 37.5%. (a) Original image; The reconstructed images by (b) KSVD-OMP(PSNR = 30.09dB, SSIM = 0.9295); (c) PLE-SCS(PSNR = 30.72dB, SSIM = 0.9501); (d) MMLE-GMM (PSNR = 31.7261dB, SSIM = 0.9621); (e) MMLE-MFA(r = 36)(PSNR = 31.7282dB, SSIM = 0.9559); (f) PMDE(PSNR = 33.3485dB, SSIM = 0.9694); (g) PMDSE(r = 24)(PSNR = 33.5938dB, SSIM = 0.9683); (h) EPLL20(PSNR = 32.9242dB; SSIM = 0.9666); (i) PMDE20(PSNR = 33.7043dB, SSIM = 0.9701).

as Lena and Barbaras face, and Barbaras arm, the blocking artifacts of PLE-SCS, MMLE-GMM and MMLE-MFA are more obvious. Comparing the recovered edge and contour regions, PMDE and PMDSE are better than the other three algorithms significantly, e.g. the edges exist in Lena’s face and hat, Barbara’s face and arm, Boat’s masts, and the background of Lena and Barbara. For the coarse texture region, e.g. the feathers on Lena’s hat and the ground in Boat image, PMDE and PMDSE also do well than PLE-SCS, MMLEGMM and MMLE-MFA, the blocking artifacts in recovered images are weakened by our algorithms. For the texture region


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V. C ONCLUSION

Fig. 6. Reconstruction performance comparison on the Boat image at the sampling rates Q/D = 37.5%. (a) Original image; The reconstructed images by (b) KSVD-OMP(PSNR = 26.6325dB, SSIM = 0.8746); (c) PLE-SCS (PSNR = 26.9613dB, SSIM = 0.9049); (d) MMLE-GMM(PSNR = 27.9029dB, SSIM = 0.9303); (e) MMLE-MFA(r = 18)(PSNR = 28.1166dB, SSIM = 0.9186); (f) PMDE(PSNR = 30.2492dB, SSIM = 0.9533); (g) PMDSE(r = 36)(PSNR = 30.3579dB, SSIM = 0.9516); (h) EPLL20(PSNR = 30.5025dB, SSIM = 0.9554); (i) PMDE20(PSNR = 30.9290dB, SSIM = 0.9567).

Fig. 7. Reconstruction performance comparison on the Barbara image at the sampling rates Q/D = 37.5%. (a) Original image; The reconstructed images by (b) KSVD-OMP(PSNR = 21.8578dB, SSIM = 0.8054); (c) PLE-SCS (PSNR = 21.8578dB, SSIM = 0.8054); (d) MMLE-GMM(PSNR = 26.9861dB, SSIM = 0.9346); (e) MMLE-MFA(r = 18)(PSNR = 27.0019dB, SSIM = 0.9270); (f) PMDE(PSNR = 27.6137dB, SSIM = 0.9368); (g) PMDSE(r = 18)(PSNR = 28.0285dB, SSIM = 0.9399); (h) EPLL20(PSNR = 26.1996dB, SSIM = 0.9141); (i) PMDE20(PSNR = 27.6137dB, SSIM = 0.9368).

on Barbaras trousers, it seems that the recovered textures by PLE-SCS and MMLE-GMM are clear than our algorithms. This confirms the observed results obtained in Section IVA, i.e. only a few morphological models are needed to represent these strong textures which have the regular shape. However, we can see that our algorithms produce the better reconstructed qualities than PLE-SCS and MMLE-GMM for the fold and edge regions on Barbara’s trousers and scarf. From the last two reconstructed images, we can see that EPLL20 and PMDE20 provide the good visual qualities for Lena and Boat, however, the recovered textures on Barbaras trousers and scarf are interior to those of PMDE and PMDSE. For the smooth regions, EPLL20 and PMDE20 produce the better visual effects than others.

In this paper, we proposed a nonlocally multi-morphological representation model for image, the developed piecewise morphological diversity (sparse) estimation algorithms are applied to the image reconstruction based on the nonoverlapping patches from compressed measurements. The proposed nonlocally multi-morphological representation model attempts to incorporate the multi-morphological representation of image patch into the nonlocal image recovery method. Under the probabilistic framework, the different morphological features of image patch are represented by the unconstrained and constrained Gaussian models respectively, and the MAP and sparse estimates for Gaussian morphological components are effectively calculated by using the alternating iterations, thus leading to the nonlinear estimation for each patch. The Gaussian model parameters can be effectively estimated by using the ML and PPCA estimation from the patches with nonlocal similarities. The experimental results show that the developed PMDE and PMDSE algorithms can provide the superior reconstructions than other competing approaches in the case of sensing nonoverlapping patches with Gaussian random matrix. This implies that a single Gaussian model is not enough to well represent a collection of image patches with nonlocal similarities, and the multi-morphological representation for patches can further improve the reconstructed performance. As mentioned above, the different number of morphological models are used to represent the patches with different feature. In this work, we only develop a simple method to automatically determine the number of morphological models for different patches, and this approach does help for improving the reconstructed quality. Therefore, how to classify the image patches with different features from the compressed measurements and determine the appropriate number of morphological models for them deserve more study. In addition, the image patches are treated as an independent individual by the above algorithms, the correlations among the image patches have not been considered. This is worth studying and exploring in future work. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable suggestions, which have helped to greatly improve this paper. R EFERENCES [1] D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, Apr. 2006. [2] E. J. Candès 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. [3] E. J. Candès, 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. [4] E. J. Candès and T. Tao, “Ecoding by linear programming,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203–4215, Dec. 2005. [5] G. G. Peyré, “Best basis compressed sensing,” IEEE Trans. Signal Process., vol. 58, no. 5, pp. 2613–2622, 2010. [6] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33–61, 1999. [7] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani, “Least angle regression,” Ann. Statist., vol. 32, no. 2, pp. 407–499, 2004.

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Feilong Cao received the Ph.D. degree in applied mathematics from Xi’an Jiaotong University, China, in 2003. He was a Research Fellow with the Center of Basic Sciences, Xi’an Jiaotong University, from 2003 to 2004. From 2004 to 2006, he was a Post-Doctoral Research Fellow with the School of Aerospace, Xi’an Jiaotong University. In 2011 and from 2013 to 2014, he was a Visiting Professor with the Department of Computer Science, Chonbuk National University, South Korea, and the Department of Computer Sciences and Computer Engineering, La Trobe University, Melbourne, VIC, Australia, respectively. He is currently a Professor with China Jiliang University, Hangzhou, China. He has authored or co-authored over 180 scientific papers in refereed journals. His current research interests include pattern recognition, neural networks, and approximation theory. Juncheng Yin received the B.S. degree and the M.S. degree in applied mathematics from Shanxi Normal University, Linfen, China, in 1998 and 2002, respectively, and the Ph.D. degree in basic mathematics from Shaanxi Normal University, Xi’an, in 2013. He is currently a Lecturer with the College of Sciences, China Jiliang University, Hangzhou, China. His research interests include image processing, machine learning, and artificial intelligence computing.