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In this paper, we propose a new time dependent model for solving total variation (TV) minimization problem in image denoising. The main idea is to apply a priori smoothness on the solution image. This is a constrained optimization type of numerical algorithm for removing noise from images. The constraints are imposed using Lagrange’s multipliers and the solution is obtained using the gradient projection method. 1D and 2D numerical experimental results by explicit numerical schemes are discussed.

In many image processing problems, a denoising step is required to remove noise or spurious details from corrupted images. The presence of noise in images is unavoidable. It may be introduced at the stage of image formation like image recording, image transmission, etc. These random distortions make it difficult to perform any required image analysis. For example, the feature oriented enhancement introduced in [

In practice, to estimate a true signal in noise, the most frequently used methods are based on the least squares criteria. This procedure is L^{2}-norm dependent. L^{2}-norm based regularization is known to remove high frequency components in denoised images and make them appear smooth.

Most of the classical image deblurring or denoising techniques, due to linear and global approach, are contaminated by Gibb’s phenomenon resulting into smearing near edges. In order to preserve edges Rudin et al. [^{1} norms derivatives, hence L^{1} estimation procedures are more appropriate for the subject of image restoration. For more details we refer to [

In this paper we present a new time dependent model constructed by evolving the Euler-Lagrange equations of the optimization problem. We propose to apply priori smoothness on the solution image and then denoise it by minimizing the total variation norm of the estimated solution. We have tested our algorithm on various types of signals and images and found our model (11) better than previously known model (10). To quantify results, the experimental values in terms of PSNR are given in Tables 1-3.

Formation of a noisy image is typically modeled as

where

We wish to reconstruct u from

subject to constraints involving the mean

and standard deviation

The resulting linear system is now easy to solve using modern numerical techniques.

The total variation based image denoising model, which is based on the constrained minimization problem appeared in [

subject to constraints

and

The first constraint corresponds to the assumption that the noise has zero mean, and the second constraint uses a priori information that the standard deviation of the noise

The Euler-Lagrange equation is given by,

in

Since (8) is not well defined at points where

where

The solution procedure uses a parabolic equation with time as an evolution parameter, or equivalently, the gradient descent method. This means that we solve

for

Applying a priori smoothness on the solution image, our new time dependent model becomes,

for

be noticed that (11) only replaces u in (10) by its estimate

Witkin [

The first constraint (8) is dropped because it is automatically enforced by the evolution procedure, i.e., the mean of

To compute

This gives us a dynamic value

We still write

The modified initial data are chosen so that the constraints are satisfied initially, i.e.,

The explicit partial derivatives of model (10) and model (11) can be expressed as:

We define the derivative terms as,

We let,

and

Then (14) reads as follows:

with boundary conditions

The explicit method is stable and convergent for

The 2D model described before is more regular than the corresponding 1D model because the 1D original optimization problem is barely convex. For the sake of understanding the numerical behavior of our schemes, we also discuss the 1D model. The Euler-Lagrange equation in the 1D case reads as follows:

This equation can be written either as

using the small regularizing parameter

using the

Our model in 1D will be

where

We can also state our model in terms of the

In this paper, we approximate

These evolution models are initialized with the noisy signal

We have estimated

CFL

The following is the explicit numerical scheme of model (22).

Let

We let,

Then (22) reads as follows:

We, as an example, have taken 1D signals

noise is added to them, we get noisy signals.

In our test, we will use the signal to noise ratio (SNR) of the signal u to measure the level of noise, defined as

where

The standard deviation of noisy signals (given in

We use

We have performed many other experiments on 1D signals obtaining similar results.

In our tests, we use peak signal to noise ratio (PSNR) as a criteria for the quality of restoration. This quality is usually expressed in terms of the logarithmic decibel scale:

where

and R is the maximum fluctuation in the input image data.

When Gaussian white noise with mean zero and variance

We have used three gray scale images, Goldhill

The values of PSNR obtained using model (11) given in Tables 1-3 are larger than that of using model (10) at the same iteration number. Thus based on PSNR values and also on human perception, we conclude that the model (11) gives better denoised images than that of model (10).

We have presented a new time dependent model (11) to solve the nonlinear total variation problem for image denoising. The main idea is to apply a priori smoothness on the solution image. Nonlinear explicit schemes are

Images | PSNR | Images | PSNR | Images | PSNR |
---|---|---|---|---|---|

(Noisy images) | (Model-10) | (Model-11) | |||

13.18 | 18.79 | 19.30 | |||

12.23 | 17.43 | 18.06 | |||

11.52 | 16.35 | 17.10 | |||

- | - | No. of iterations | 5 | No. of iterations | 5 |

Images | PSNR | Images | PSNR | Images | PSNR |
---|---|---|---|---|---|

(Noisy Images) | (Model-10) | (Model-11) | |||

13.36 | 19.10 | 19.39 | |||

12.38 | 17.85 | 18.26 | |||

11.64 | 16.86 | 17.38 | |||

- | - | No. of iterations | 5 | No. of iterations | 5 |

Images | PSNR | Images | PSNR | Images | PSNR |
---|---|---|---|---|---|

(Noisy images) | (Model-10) | (Model-11) | |||

12.96 | 17.07 | 17.50 | |||

11.97 | 15.75 | 16.28 | |||

11.27 | 14.73 | 15.33 | |||

- | - | No. of iterations | 5 | No. of iterations | 5 |

used to discretize models (10) and (11). The model (11) gives larger PSNR values than that of model (10), at the same iteration numbers. Besides, a new time dependent model (22) to solve the signal denoising in 1D has also been given.