3.1. The Proposed Model
We start from the following assumed additive noise degradation model:
where denotes the observed raw image data, is the original good image, and is additive measurement noise. The goal of image denoising is to recover from the observed image data . The shrinkage algorithm on some multi-scale frame can be written as follows:
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].
Although traditional wavelets perform well only for representing point singularities, they become computationally inefficient for geometric features with line and surface singularities. To overcome this problem, we choose the curvelet as the tool of shrinkage algorithm. In general, the shrinkage operators are considered to be in the form of a symmetric function ; thus the coefficients are estimated by
Let denote the dual frame, and then a denoised image is generated by the reconstruction algorithm:
Following the wavelet shrinkage idea which was proposed by Donoho and Johnstone , the curvelet shrinkage operators can be taken as a soft thresholding function defined by a fixed threshold , that is,
or a hard shrinkage function
The major problem with wavelet shrinkage methods, as discussed, is that shrinking large coefficients entails an erosion of the spiky image features, while shrinking small coefficients towards zero yields Gibbs-like oscillation in the vicinity of edges and loss of texture. As a new MGA tool, Curvelet shrinkage can suppress this pseudo-Gibbs and preserve the image edge; however, some shapes of curve-like artifacts are generated (see Figure 4).
In order to suppress the staircasing effect and curvelet-like artifacts, we propose a new objective functional:
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
Let us denote
Then, the cost function is a new hybrid data fidelity term, and its corresponding Euler equation is
The Euler equation (20) equals to produce a new image whose Fourier transform is described as follows: if , then
Apply Fourier transform to the Euler equation (20) and we will get
According to the differential properties of the Fourier transform
If , then we get
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.
The energy functional is convex.
For all for all , on one hand we have the following conclusion:
On the other hand, we have the following conclusion:
Then, we have
According to (26) and (28), we get the
From Proposition 2, the convexity of the energy functional can guarantee the global optimizing and the existence of the unique solution, while Proposition 1 shows us that the solution has some special form in Fourier domain. Combining Propositions 1 and 2 together, we can remark that the unique solution of (19) is
Then, we can prove the following existence and uniqueness theorem.
Let be a positive, bounded function with ; then the minimizing problem of energy functional in (18) admits a unique solution satisfying
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 ).
3.3. Adaptive Parameters Estimation
For solving the minimizing energy functional , it often transforms the optimizing problem into the Euler-Lagrange equation. Using the standard computation of Calculus of Variation of with respect to , we can get its Euler-Lagrange equation:
where is the outward unit normal vector on the boundary , and . For a convenient numerical simulation of (31), we apply the gradient descent flow and get the evolution equation:
There are three parameters involved in the iterative procedure. For the threshold parameter in the curvelet coefficients shrinkage, the common one is to choose , where denotes the standard deviation of Gaussian white noise. The Monte-Carlo simulations can calculate an approximation value of the individual variance. In our experiments, we use the following hard-thresholding rule for estimating the unknown curvelet coefficients:
Here, we actually chose a scale-dependent value for : we have for the first scale (the finest scale) and for the others.
For the parameters and , they are very important to balance trade-off between the image fidelity term and the regularization term. An important prior fact is that the Gaussian distributed noise has the following restriction condition:
Therefore, we merely multiply the first equation of (32) by and integrate by parts over ; if the steady state has been reached, the left side of the first equation of (32) vanishes; thus we have
Obviously, the above equation is not sufficient to estimate the values of and simultaneously. This implies that we should introduce another prior knowledge. Borrowing the idea of spatial varying data fidelity from Gilboa , we compute the parameter by the formula:
where , and is the local power of the residue . The local power of the residue is given by
where is a normalized and radially symmetric Gaussian Function window, and is the expected value. After we compute the value of , we can estimate the value of using (38), that is,
3.4. Description of Proposed Algorithm
To discretize equation (32), the finite difference scheme in  is used. Denote the space step by and the time step by . Thus we have
where and is the regularized parameter chosen near 0.
The numerical algorithm for (32) is given in the following (the subscripts are omitted):
with boundary conditions
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).
Apply curvelet transform (the FDCT ) to noisy image and obtain the discrete coefficients .
Use robust method to estimate an approximation value , and then use the shrinkage operator in (33) to obtain the estimated coefficients .
Apply the inverse curvelet transform and obtain the initial restored image .
Iteration. WhileIterative-Steps Do
Compute according to (40).
Update the parameter according to (36).
Update the parameter according to (38).
End Do Output: .
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  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 .
Secondly, from the regularization theory, the gradient fidelity term works as Tikhonov regularization in Sobolev functional space The problem of admits a unique solution characterized by the Euler-Lagrange equation Moreover, the function is called harmonic (subharmonic, superharmonic) if it satisfies . Using the mean value theorems , for any ball , we have
However, in , the gradient fidelity term is chosen as
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  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.