# Image Variational Denoising Using Gradient Fidelity on Curvelet Shrinkage

- Liang Xiao
^{1, 2}Email author, - Li-Li Huang
^{1, 3}and - Badrinath Roysam
^{2}

**2010**:398410

https://doi.org/10.1155/2010/398410

© Liang Xiao et al. 2010

**Received: **27 December 2009

**Accepted: **7 June 2010

**Published: **30 June 2010

## Abstract

A new variational image model is presented for image restoration using a combination of the curvelet shrinkage method and the total variation (TV) functional. In order to suppress the staircasing effect and curvelet-like artifacts, we use the multiscale curvelet shrinkage to compute an initial estimated image, and then we propose a new gradient fidelity term, which is designed to force the gradients of desired image to be close to the curvelet approximation gradients. Then, we introduce the Euler-Lagrange equation and make an investigation on the mathematical properties. To improve the ability of preserving the details of edges and texture, the spatial-varying parameters are adaptively estimated in the iterative process of the gradient descent flow algorithm. Numerical experiments demonstrate that our proposed method has good performance in alleviating both the staircasing effect and curvelet-like artifacts, while preserving fine details.

## Keywords

## 1. Introduction

Image denoising is a very important preprocessing in many computer vision tasks. The tools for attracting this problem come from computational harmonic analysis (CHA), variational approaches, and partial differential equations (PDEs) [1]. The major concern in these image denoising models is to preserve important image features, such as edges and texture, while removing noise.

In the direction of multi-scale geometrical analysis (MGA), the shrinkage algorithms based on the CHA tools, such as contourlets [2] and curvelets [3–5], are very important in image denoising because they are simple and have efficient computational complexity and promising properties for singularity analysis. Therefore, the pseudo-Gibbs artifacts caused by the shrinkage methods based on Fourier transform and wavelets attempt to be overcame by the methods based on MGA at least partially. However, there are still some curve-like artifacts in MGA-based shrinkage methods [6].

Algorithms designed by variatinal and PDE models are free from the above lacks of MGA but cost heavy computational burden that is not suitable for time critical application. In addition, the PDE-based algorithms tend to produce a staircasing effect [7], although they can achieve a good trade-off between noise removal and edge preservation. For instance, the total variation (TV) minimizing [8] method has some undesirable drawbacks such as the staircasing effect and loss of texture, although it can reduce pseudo-Gibbs oscillations effectively. Similar problem can be found in many other nonlinear diffusion models such as the Perona-Malik model [9] and the mean curvature motion model [10]. In this paper, we will focus on the hybrid variational denoising method. Specificity, we will emphasis on the improvement on the TV model, and we will propose a novel gradient fidelity term based on the curvelet shrinkage algorithm.

### 1.1. Related Works and Analysis

where the first term is the image fidelity term, which penalizes the inconsistency between the under-estimated recovery image and the acquired noisy image, while the second term is the regularization term which imposes some priori constraints on the original image and to a great degree determines the quality of the recovery image, and is the regularization parameter which balances trade-off between the image fidelity term and the regularization term .

Here could be some higher-order norm, for example, . More complex higher-order norms were brought to variational method in order to alleviate the staircasing effect [13].

where is the Gaussian kernel with scale , and the symbol " " denotes the convolution operator. Their studies show that this gradient fidelity term can alleviate the staircasing effect. However, classical Gaussian filtering technique is the uniform smoothing in all direction of images and fine details are easily destroyed with these filters. Hence, the gradient of smoothed image is unreliable near the edges, and the gradient fidelity cannot preserve the gradient and thereby image edges.

The third approach is the combination of the variational model and the MGA tools. In [14], the TV model has been combined with wavelet to reduce the pseudo-Gibbs artifacts resulted from wavelet shrinkage [15]. Nonlinear diffusion has been combined with wavelet shrinkage to improve the rotation invariance [16]. The author in [17] presented a hybrid denoising methods in which the complex ridgelet shrinkage was combined with total variation minimization [6]. From their reports, the combination of MGA and PDE methods can improve the visual quality of the restored image and provides a good way to take full advantages of both methods.

### 1.2. Main Contribution and Paper's Organization

In this paper, we add a new gradient fidelity term to the TV model to some second-order nonlinear diffusion PDEs for avoiding the staircasing effect and curvelet-like artifacts. This new gradient fidelity term provides a good mechanism to combine curvelet shrinkage algorithm and the TV regularization.

This paper is organized as follows. In Section 2, we introduce the curvelet transform. In Section 3, we propose a new hybrid model for image smoothing. In this model, we have two main contributions. We propose a new hybrid fidelity term, in which the gradient of multi-scale curvelet shrinkage image is used as a feature fidelity term in order to suppress the staircasing effect and curvelet-like artifacts. Secondly, we propose an adaptive gradient descent flow algorithm, in which the spatial-varying parameters are adaptively estimated to improve the ability of preserving the details of edges and texture of the desired image. In Section 4, we give numerical experiments and analysis.

## 2. Curvelet Transform

## 3. Combination TV Minimization withGradient Fidelity on Curvelet Shrinkage

### 3.1. The Proposed Model

where is the corresponding MGA operator, that is, , for all , and " " is a set of indices. The rational is that the noise is nearly Gaussian. The principles of the shrinkage estimators which estimate the frame coefficients from the observed coefficients have been discussed in different frameworks such as Bayesian and variational regularization [20, 21].

In the cost functional (18), the term is called curvelet shrinkage-based gradient data fidelity term and is designed to force the gradient of to be close to the gradient estimation and to alleviate the staircase effect. And the parameters and control the weights of each term. For the sake of simplicity for description, we always let and in the following sections.

### 3.2. Basic Properties of Our Model

Proposition 1.

Proof.

where and are parameters in the frequency domain.

Proposition 1 tells us that the Euler equation (20) equals to compute a new image whose Fourier frequency spectrum is the interpolation of and . The weight coefficients of and are and , respectively.

Proposition 2.

The energy functional is convex.

Proof.

According to (26) and (28), we get the

Then, we can prove the following existence and uniqueness theorem.

Theorem 1.

Proof.

Using the lower semicontinuity and compactness of and the convexity of , the proof can be made following the same procedure of [23, 24] (for detail, see appendix in [24]).

### 3.3. Adaptive Parameters Estimation

Here, we actually chose a scale-dependent value for : we have for the first scale (the finest scale) and for the others.

### 3.4. Description of Proposed Algorithm

where and is the regularized parameter chosen near 0.

for . The parameters are chosen like this: , while the parameters are computed dynamically during the iterative process according to the formulae (33), (36), and (38).

In summary, according to the gradient descent flow and the discussion of parameters choice, we now present a sketch of the proposed algorithm (the pipeline is shown in Figure 2).

Initialization

- (1)
Apply curvelet transform (the FDCT [5]) to noisy image and obtain the discrete coefficients .

- (2)
Use robust method to estimate an approximation value , and then use the shrinkage operator in (33) to obtain the estimated coefficients .

- (3)

### 3.5. Analysis of Staircase and Curve-Like Effect Alleviation

The essential idea of denoising is to obtain the cartoon part of the image , preserve more details of the edge and texture parts , and filter out the noise part . In classical TV algorithm and Curvelet threshold algorithm, the staircase effects and curve-like artifacts are often generated in the restored cartoon part , respectively. Our model provides a similar ways to force gradient to be close to an approximation. However, our model provides a better mechanism to alleviate the staircase effects and curve-like artifacts.

Firstly, the "TVGF" model in [7] uses the Gaussian filtering to approximation. However, because Gaussian filter is uniform smoothing in all directions of an image, it will smooth the image too much to preserve edges. Consequently, their gradient fidelity term cannot maintain the variation of intensities well. Differing from the TVGF model, our model takes full advantage of curvelet transform. The curvelets allow an almost optimal sparse representation of object with -singularities. For a smooth object with discontinuities along -continuous curves, the best -term approximation by curvelet thresholding obeys , while for the wavelets the decay rate is only .

Comparing the above two results, we can understand the difference between two gradient fidelity term smoothing mechanism. We remark that the gradient fidelity term in [7] will tend to produce more edge blurring effect and remove more texture components with the increasing of the scale parameter of Gaussian kernel, although it helps to alleviate the staircase effect and produces some smoother results. However, our model tends to produce the curvelet shrinkage image and can remain the curve singularities in images; thus it will obtain good edge preserving performance.

In addition, another rationale behind our proposed model is that the spatially varying fidelity parameters and are incorporated into our model. In our proposed algorithm, as described in (36), we use the measure ; here is the local power of the residue . In the flat area where (basic cartoon model without textures or fine-scale details), the local power of the residue is almost constant and hence . We get a high-quality denoising process so that the noise, the staircase effect, and the curve-like artifacts are smoothed. In the texture area, since the noise is uncorrelated with the signal, thus the total power of the residue can be approximated as , the sum of local powers of the noncartoon part and the noise, respectively. Therefore, textured regions are characterized by high local power of the residue. Thus, our algorithm will reduce the level of filtering so that it will preserve the detailed structure of such regions.

## 4. Experimental Results and Analysis

In this section, experimental results are presented to demonstrate the capability of our proposed model. The results are compared with those obtained by using the curvelet shrinkage method [26], the "TV" model (2) proposed by Rudin et al. [8], and the "TVGF" model proposed by Zhu and Xia [7].

In the curvelet shrinkage method, denoising is achieved by hard-thresholding of the curvelet coefficients. We select the thresholding at for all but the finest scale where it is set at ; here is the noise level of a coefficient at scale and angle . In our experiments, we actually use a robust estimator to estimate noise level using the following formula: , here, represents the corresponding curvelet coefficients at scale and angle , and MED represents the medium operator to calculate the medium value for a sequence coefficients.

Here and are set to be 0.02 and 0.01, respectively. The regularization parameter is dynamically updated to satisfy the noise variance constrains according to (4).

where is set to be 0.01, and is the noise variance.

### 4.1. Image Quality Assessment

For the following experiments, we compute the quality of restored images by the signal-to-noise ratio (SNR) to compare the performance of different algorithms. Because of the limitation of SNR on capturing the subjective appearance of the results, the mean structure similarity (MSSIM) index as defined in [27] is used to measure the performance of the different methods. As shown by theoretical and experimental analysis [27], the MSSIM index intends to measure the perceptual quality of the images.

where is the mean of the image ; is the standard deviation of the image ; is the covariance of the image and image ; are the constants.

In order to evaluate the performance of alleviation of staircase effect and curve-like artifacts, the difference image between the restored image and the original clean image is used to visually judge image quality.

The stopping criterion of both the proposed method, "TV" method, and the "TVGF" method is that the MSSIM reached maximum or the total iteration number reached the maximal iteration number 3000.

#### 4.1.1. Qualitative and Quantitative Results

The SNR, MSSIM of the restored "Lena" images by using four models. The numbers in the bracket under the "MSSIM" column refer to the total iteration number of the algorithm.

Noise standard | Noisy image | Curvelet shrinkage method | "TV" Model | "TVGF" Model | Proposed model | |||||
---|---|---|---|---|---|---|---|---|---|---|

deviation | SNR (dB) | MSSIM | SNR (dB) | MSSIM | SNR (dB) | MSSIM | SNR (dB) | MSSIM | SNR (dB) | MSSIM |

20 | 7.59 | 0.34 | 16.62 | 0.86 | 16.76 | 0.85 (1923) | 15.68 | 0.82 (969) | 17.26 | 0.88 (486) |

25 | 5.63 | 0.27 | 15.70 | 0.84 | 15.68 | 0.82 (2290) | 14.75 | 0.77 (1115) | 16.36 | 0.86 (615) |

30 | 4.04 | 0.22 | 14.95 | 0.82 | 14.81 | 0.80 (2991) | 13.79 | 0.71 (1135) | 15.58 | 0.85 (763) |

35 | 2.70 | 0.18 | 14.29 | 0.81 | 13.92 | 0.76 (3000) | 12.88 | 0.67 (1142) | 14.92 | 0.84 (898) |

40 | 1.54 | 0.15 | 13.72 | 0.79 | 12.94 | 0.71 (3000) | 12.01 | 0.61 (1151) | 14.36 | 0.82 (1033) |

The SNR, MSSIM of the restored "Barbara" images by using four models. The numbers in the bracket under the "MSSIM" column refer to the total iteration number of the algorithm.

Noise standard | Noisy image | Curvelet shrinkage method | "TV" Model | "TVGF" Model | Proposed model | |||||
---|---|---|---|---|---|---|---|---|---|---|

deviation | SNR (dB) | MSSIM | SNR (dB) | MSSIM | SNR (dB) | MSSIM | SNR (dB) | MSSIM | SNR (dB) | MSSIM |

20 | 8.73 | 0.48 | 12.05 | 0.77 | 12.74 | 0.77 (2032) | 11.39 | 0.72 (724) | 13.15 | 0.81 (427) |

25 | 6.79 | 0.40 | 11.42 | 0.74 | 11.83 | 0.72 (2634) | 11.05 | 0.68 (836) | 12.26 | 0.78 (567) |

30 | 5.21 | 0.34 | 11.02 | 0.71 | 11.17 | 0.69 (3000) | 10.69 | 0.64 (894) | 11.64 | 0.75 (687) |

35 | 3.91 | 0.30 | 10.72 | 0.70 | 10.63 | 0.65 (3000) | 10.32 | 0.60 (936) | 11.19 | 0.73 (783) |

40 | 2.71 | 0.26 | 10.45 | 0.68 | 10.12 | 0.61 (3000) | 9.93 | 0.56 (953) | 10. 81 | 0.70 (928) |

## 5. Conclusion

In this paper, a curvelet shrinkage fidelity-based total variation regularization is proposed for discontinuity-preserving denoising. We propose a new gradient fidelity term, which is designed to force the gradients of desired image to be close to the curvelet approximation gradients. To improve the ability of preserving the details of edges and texture, the spatial-varying parameters are adaptively estimated in the iterative process of the gradient descent flow algorithm. We carry out many numerical experiments to compare the performance of various algorithms. The SNR and MSSIM improvements demonstrate that our proposed method has the best performance than the TV algorithm, the curvelet shrinkage, and the TVGF algorithm. Our future work will extend this new gradient fidelity term to other PDE-based methods.

## Declarations

### Acknowledgments

The authors would like to express their gratitude to the anonymous referees for making helpful and constructive suggestions. The authors also thank financial support from the National Natural Science Foundation of China (60802039) and the National 863 High Technology Development Project (2007AA12Z142), Specialized Research Fund for the Doctoral Program of Higher Education (20070288050), and NUST Research Funding under Grant no. 2010ZDJH07.

## Authors’ Affiliations

## References

- Buades A, Coll B, Morel JM: A review of image denoising algorithms, with a new one.
*Multiscale Modeling and Simulation*2005, 4(2):490-530. 10.1137/040616024MathSciNetView ArticleMATHGoogle Scholar - Do MN, Vetterli M: The contourlet transform: an efficient directional multiresolution image representation.
*IEEE Transactions on Image Processing*2005, 14(12):2091-2106.MathSciNetView ArticleGoogle Scholar - Candès E, Donoho D: Curvelets-a surprisingly effective nonadaptive representation for objects with edges. In
*Curves and Surface Fitting: Saint-Malo 1999*. Edited by: Cohen A, Rabut C, Schumaker L. Vanderbilt University Press, Nashville, Tenn, USA; 2000:105-120.Google Scholar - Candès EJ, Donoho DL:New tight frames of curvelets and optimal representations of objects with piecewise
singularities.
*Communications on Pure and Applied Mathematics*2004, 57(2):219-266. 10.1002/cpa.10116MathSciNetView ArticleMATHGoogle Scholar - Candès E, Demanet L, Donoho D, Ying L: Fast discrete curvelet transforms.
*Multiscale Modeling and Simulation*2006, 5(3):861-899. 10.1137/05064182XMathSciNetView ArticleMATHGoogle Scholar - Ma J, Plonka G: Combined curvelet shrinkage and nonlinear anisotropic diffusion.
*IEEE Transactions on Image Processing*2007, 16(9):2198-2206.MathSciNetView ArticleGoogle Scholar - Zhu L, Xia D: Staircase effect alleviation by coupling gradient fidelity term.
*Image and Vision Computing*2008, 26(8):1163-1170. 10.1016/j.imavis.2008.01.008MathSciNetView ArticleGoogle Scholar - Rudin LI, Osher S, Fatemi E: Nonlinear total variation based noise removal algorithms.
*Physica D*1992, 60(1–4):259-268.View ArticleMathSciNetMATHGoogle Scholar - Perona P, Malik J: Scale-space and edge detection using anisotropic diffusion.
*IEEE Transactions on Pattern Analysis and Machine Intelligence*1990, 12(7):629-639. 10.1109/34.56205View ArticleGoogle Scholar - Alvarez L, Lions P, Morel J: Image selective smoothing and edge detection by nonlinear diffusion. II.
*SIAM Journal on Numerical Analysis*1992, 29(3):845-866. 10.1137/0729052MathSciNetView ArticleMATHGoogle Scholar - Chan T, Esedoglu S, Park F, Yip A: Recent developments in total variation image restoration. In
*Handbook of Mathematical Models in Computer Vision*. Edited by: Paragios N, Chen Y, Faugeras O. Springer, Berlin, Germany; 2004.Google Scholar - Lysaker M, Tai X-C: Iterative image restoration combining total variation minimization and a second-order functional.
*International Journal of Computer Vision*2006, 66(1):5-18. 10.1007/s11263-005-3219-7View ArticleMATHGoogle Scholar - Li F, Shen C, Fan J, Shen C: Image restoration combining a total variational filter and a fourth-order filter.
*Journal of Visual Communication and Image Representation*2007, 18(4):322-330. 10.1016/j.jvcir.2007.04.005View ArticleGoogle Scholar - Gilboa G, Zeevi YY, Sochen N: Texture preserving variational denoising usingan adaptive fidelity term.
*Proceedings of the 2nd IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision (VLSM '03), 2003, Nice, France*137-144.Google Scholar - Ma J, Fenn M: Combined complex ridgelet shrinkage and total variation minimization.
*SIAM Journal of Scientific Computing*2006, 28(3):984-1000. 10.1137/05062737XMathSciNetView ArticleMATHGoogle Scholar - Plonka G, Steidl G: A multiscale wavelet-inspired scheme for nonlinear diffusion.
*International Journal of Wavelets, Multiresolution and Information Processing*2006, 4(1):1-21. 10.1142/S0219691306001063MathSciNetView ArticleMATHGoogle Scholar - Ma J, Fenn M: Combined complex ridgelet shrinkage and total variation minimization.
*SIAM Journal of Scientific Computing*2006, 28(3):984-1000. 10.1137/05062737XMathSciNetView ArticleMATHGoogle Scholar - Ying L, Demanet L, Candès E: 3D discrete curvelet transform.
*Wavelets XI, August 2005, San Diego, Calif, USA, Proceedings of SPIE*1-11.Google Scholar - Demanet L, Ying L: Curvelets and wave atoms for mirror-extended images.
*Wavelets XII, August 2007, San Diego, Calif, USA, Proceedings of SPIE*Google Scholar - Achim A, Tsakalides P, Bezerianos A: SAR image denoising via Bayesian wavelet shrinkage based on heavy-tailed modeling.
*IEEE Transactions on Geoscience and Remote Sensing*2003, 41(8):1773-1784. 10.1109/TGRS.2003.813488View ArticleGoogle Scholar - Durand S, Nikolova M:Denoising of frame coefficients using
data-fidelity term and edge-preserving regularization.
*Multiscale Modeling & Simulation*2007, 6(2):547-576. 10.1137/06065828XMathSciNetView ArticleMATHGoogle Scholar - Donoho DL, Johnstone IM: Ideal spatial adaptation by wavelet shrinkage.
*Biometrika*1994, 81(3):425-455. 10.1093/biomet/81.3.425MathSciNetView ArticleMATHGoogle Scholar - Kornprobst P, Deriche R, Aubert G: Image sequence analysis via partial differential equations.
*Journal of Mathematical Imaging and Vision*1999, 11(1):5-26. 10.1023/A:1008318126505MathSciNetView ArticleMATHGoogle Scholar - Xiao L, Huang L-L, Wei Z-H: A weberized total variation regularization-based image multiplicative noise removal algorithm.
*EURASIP Journal on Advances in Signal Processing*2010, 2010:-15.Google Scholar - Gilbarg D, Trudinger NS:
*Elliptic Partial Differential Equation of Second Order*. Springer, Berlin, Germany; 2003.MATHGoogle Scholar - Starck J-L, Candès EJ, Donoho DL: The curvelet transform for image denoising.
*IEEE Transactions on Image Processing*2002, 11(6):670-684. 10.1109/TIP.2002.1014998MathSciNetView ArticleMATHGoogle Scholar - Wang Z, Bovik AC, Sheikh HR, Simoncelli EP: Image quality assessment: from error visibility to structural similarity.
*IEEE Transactions on Image Processing*2004, 13(4):600-612. 10.1109/TIP.2003.819861View ArticleGoogle Scholar - Dong FF, Liu Z: A new gradient fidelity for avoiding staircasing effect.
*Journal of Computer Science and Technology*2009, 24(6):1162-1170. 10.1007/s11390-009-9289-1MathSciNetView ArticleGoogle Scholar

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