A hybrid variational model combined first and secondorder total variation for image reconstruction from its finite number of noisy compressive samples is proposed in this paper. Inspired by majorizationminimization scheme, we develop an efficient algorithm to seek the optimal solution of the proposed model by successively minimizing a sequence of quadratic surrogate penalties. Both the nature and magnetic resonance (MR) images are used to compare its numerical performance with four stateoftheart algorithms. Experimental results demonstrate that the proposed algorithm obtained a significant improvement over related stateoftheart algorithms in terms of the reconstruction relative error (RE) and peak signal to noise ratio (PSNR).
Traditional approaches to sampling signals or images follow the basic principle of the NyquistShannon sampling theorem; the sampling rate must be at least twice of the frequency bandwidth. In many practical applications, including image and video cameras, MRI scanners, and radar receivers, requirements of high resolution imaging lead to very high Nyquist sampling rate which brings a series of realistic difficult problems in the field of data conversion (e.g., analogdigital conversion), storage, and transmission. The technique called compressive sensing (CS), which goes against conventional wisdoms in data acquisition, has recently been developed to overcome those problems.
CS theory designs nonadaptive sampling techniques that condense the information in sparse or compressible images into a small amount of data and yet reconstruct them accurately. The general framework of compressive sensing consists of two phases: encoding and decoding. In the encoding phase, a sparse or compressible image
Total variation regularization was firstly introduced by Rudin et al. [
In this paper, we propose a hybrid compressive sensing method using a new hybrid TV measure by a combined first and secondorder TV penalty for recovering a piecewise smooth image containing all possible sharper edges from limited compressive samples. To seek the optimal solution of the proposed model, we develop an efficient image reconstruction algorithm by using the majorizationminimization scheme. The novelty in this work is that our hybrid TV regularization method is able to utilize the best properties of first and secondorder TV regularization and manage to overcome the weaknesses of both.
The rest of the paper is organized as follows. Section
In this section, we reinterpret classic TV regularizer and extend it to secondorder TV regularizer. We denote firstorder directional derivative of
In order to extend the standard TV scheme to higherorder TV regularizer, we reinterpret the TV regularizer as a group separable
Hence, TV regularizer defined in (
Based on the above reinterpretation, we introduce secondorder TV regularizer by replacing firstorder directional derivative
Substituting (
Since the secondorder TV regularizer sums the square magnitude of the directional derivatives of the image along all directions and orientations, this penalty is invariant to rotations and translations and is also convex. The secondorder derivatives are steerable which enables us to obtain analytical expressions for the new regularizer as the standard TV regularizer. Furthermore, the minimization of secondorder TV regularizer will preserve the strong directional derivatives and attenuate the small ones at other directions; thus, it can provide better preservation along linelike features.
Combining firstorder and secondorder TV regularizers, we present a hybrid TV regularization model to reconstruct original image from its noisy compressive samples. Given noisy compressive samples
In the firstorder TV regularization method, images are often assumed to be piecewiseconstant, while the secondorder TV regularization method implicitly adopts a piecewiselinear assumption. The piecewiselinear approximation of images usually yields a lesser approximation error than piecewiseconstant representation. Therefore, assuming that the images under consideration can be better decomposed into piecewiseconstant and piecewiselinear functions, we expect the combination of first and secondorder regularization to be more accurate in terms of reconstruction quality.
Let us now consider the discrete formulation of two regularization terms in (
Using local firstorder differences to approximate the two orthogonal components of the gradient, we obtain a discrete version of firstorder TV regularizer. Consider
For the sake of simplicity, we denote
In this subsection, we derive an efficient optimization algorithm which belongs to the class of majorizationminimization (MM) approaches [
Let
To develop a MMbased algorithm, we derive a quadratic majorizer for the proposed hybrid TV regularizer. The choice of a quadratic majorizer is motivated by the fact that the minimization of a quadratic function amounts to solving a system of linear equations, a task for which there are excellent methods available in the literature. We define the following majorizer:
Using the inequality
Equation (
Notice that, for fixed
Let
Similarly, for the second term in (
Substituting (
Since the objective function in optimization (
In summary, the proposed algorithm for our hybrid TV model (
Input
Compute
Compute
Solve EulerLagrange equation (
Terminate if relative change
Here, we make one comment about implementation of this algorithm. In Step
In this section, we present some numerical results to illustrate the performance of our method. We compare our results with those obtained by four stateoftheart methods: (a) TVAL3 [
To compare the performance fairly, all parameters of those algorithms are set as the suggestion values by the authors in [
In our experiments, we consider a common task of reconstructing an image from their undersampled Fourier samples which is an important problem in magnetic resonance imaging (MRI). We generate our test sets using four images (see Figure
Test images.
The reconstructed images of “Lena” image with sampling ratio 25% and noise level 30 dB are shown in Figure
Reconstructed results of “Lena” image from noisy and undersampled Fourier data (noise level: 30 dB, sampling ratio: 25%). (a) Actual image; (b) TVAL3; (c) RecPF; (d) hybrid TVL1; (e) secondorder TV; (f) our method.
In order to highlight the differences, we zoom in the region (marked by the red box in Figure
Zoom into a region of images shown in Figure
Figure
Reconstructed results of “Brain” image from noisy and undersampled Fourier data (noise level: 40 dB, sampling ratio: 20%). (a) Actual image; (b) TVAL3; (c) RecPF; (d) hybrid TVL1; (e) secondorder TV; (f) our method.
Zoom into a region of images shown in Figure
To characterize the performance quantitatively, we show the PSNRs of the reconstructed images at various sampling ratios and SNR levels in Table
PSNRs of the reconstructed images.
Sampling ratio (%)  20  25  33  50  

Noise level (dB)  30  40  30  40  30  40  30  40 
Lena (256 × 256)  
TVAL3  26.99  27.14  28.12  27.65  29.03  29.19  31.51  32.34 
RecPF  29.93  30.92  30.76  32.06  32.01  34.06  34.03  37.94 
Hybrid TVL1  30.12  31.13  30.97  32.26  32.24  34.43  34.26  38.41 
Secondorder TV  30.57  31.52  31.89  33.87  32.83  36.09 

40.08 
Our method 






34.43 

 
Pepper (256 × 256)  
TVAL3  30.08  30.41  30.75  32.23  32.78  33.93 

38.08 
RecPF  30.92  31.95  31.72  33.79  33.01  35.68  34.91  39.25 
Hybrid TVL1  31.10  32.29  31.93  34.10  33.24  36.05  35.06  39.65 
Secondorder TV  31.72  33.76  31.72  35.79  33.67  37.67  35.07  41.07 
Our method 






34.82 

 
Barbara (256 × 256)  
TVAL3  27.25  27.57  27.78  28.27  29.14  29.45  31.42  32.45 
RecPF  28.70  29.29  29.45  30.28  30.67  31.80  33.07  35.62 
Hybrid TVL1  28.86  29.50  29.63  30.55  30.93  32.20  33.36  36.20 
Secondorder TV  29.25  30.21  30.02  31.54  31.41  33.44  33.86  37.98 
Our method 








 
Brain (256 × 256)  
TVAL3  32.63  33.22  33.76  34.49  35.23  36.32 

40.18 
RecPF  31.49  32.11  32.51  33.50  33.89  35.27  36.31  39.12 
Hybrid TVL1  31.69  32.38  32.75  33.83  34.23  35.75  36.63  39.71 
Secondorder TV  32.65  33.64  33.66  35.13  35.20  37.29 

41.11 
Our method 






37.35 

Relative errors of various methods versus sampling ratio.
Lena, Noise level: 40 dB
Pepper, Noise level: 40 dB
Barbara, Noise level: 40 dB
Brain, Noise level: 40 dB
We compare the average central processing unit (CPU) times of the different methods in Table
Average CPU times of different methods (s).
Sampling ratio (%)  20  25  0.33  0.5  

Noise level (dB)  30  40  30  40  30  40  30  40 
TVAL3  2.21  2.15  2.12  1.90  1.82  1.81  1.81  1.76 
RecPF 








Hybrid TVL1  1.76  1.72  1.68  1.67  1.56  1.56  1.51  1.50 
Secondorder TV  81.25  78.36  77.18  75.63  75.02  75.01  74.45  74.56 
Our method  29.50  26.79  26.37  24.13  24.01  23.76  23.24  23.12 
From Table
This paper presents a hybrid variational model for image compressive sensing. The model minimizes the sum of three terms corresponding to least squares data fitting, firstorder TV and secondorder TV. We propose an efficient majorizationminimization algorithm to determine the solution of our model. We test our method with nature images and MR image. Comparisons of the proposed regularization method with four stateoftheart algorithms demonstrate the significant improvement in the quality of the reconstructed images. Although achieving better performance in terms of RE and PSNR, our algorithm is slower than FFTbased method such as RecPF. How to accelerate the algorithm is an important problem which we will investigate in forthcoming research. We hope that our method is useful in relevant areas of image compressive sensing such as SAR/ISAR imaging and MR images reconstruction.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the anonymous reviewers for their careful reading and valuable comments on their paper, which helped them to improve that paper. This research is supported by the National Nature Science Foundation of China (61171165, 60802039), the Nature Science Foundation of Jiangsu Province (BK2010488), and the Qinglan Outstanding Scholar Project of Jiangsu Province.