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Bidimensional empirical mode decomposition (BEMD) algorithm, with high adaptive ability, provides a suitable tool for the noisy image processing, and, however, the edge effect involved in its operation gives rise to a problem—how to obtain reliable decomposition results to effectively remove noises from the image. Accordingly, we propose an approach to deal with the edge effect caused by BEMD in the decomposition of an image signal and then to enhance its denoising performance. This approach includes two steps, in which the first one is an extrapolation operation through the regression model constructed by the support vector machine (SVM) method with high generalization ability, based on the information of the original signal, and the second is an expansion by the closed-end mirror expansion technique with respect to the extrema nearest to and beyond the edge of the data resulting from the first operation. Applications to remove the Gaussian white noise, salt and pepper noise, and random noise from the noisy images show that the edge effect of the BEMD can be improved effectively by the proposed approach to meet requirement of the reliable decomposition results. They also illustrate a good denoising effect of the BEMD by improving the edge effect on the basis of the proposed approach. Additionally, the denoised image preserves information details sufficiently and also enlarges the peak signal-to-noise ratio.

In these years, digital image processing technology has been widely used in space exploration, remote sensing, biomedical and industrial inspection, and various other fields. Although it has made a great progress in many aspects, there are still some new challenges. One of them is closely related to image denoising, since noises can be introduced by almost all of the factors in the process of signal acquisition, conversion, and transfer, not to mention those involved in the original image from the instruments themselves, and environments. In order to obtain real information for a piece of image, denoising generally constitutes a necessary task before the implementation of the postanalysis (such as compression, split, and merge), aiming at improving the signal-to-noise ratio (SNR) and restoring true information of the image.

In the process of image denoising, it is necessary to eliminate the noise contaminating a given image as far as possible and also to preserve all important details of the image. To meet these requirements, a number of theoretical methods have been proposed and developed [

With the development of artificial intelligence algorithms, there have appeared a number of new attempts in image denoising. From a new view, the application of artificial neural network (ANN) fully shows advantages of parallel computing, nonlinear mapping, and adaptive capacity in image processing [

For the adaptive ability of image denoising, one of its most important senses addresses the idea that its denoised version can preserve the image features as far as possible, with little influence on the following analysis and processing of the image. However, such influence, even the image distortion, can be found after the actual denoising operation. It is just a view of the limitation of the techniques based on traditional Fourier transform, wavelet transform, and modern artificial intelligence algorithms. To be worthy of note, Huang et al. [

About ten years ago, Nunes et al. [

In fact, for the decomposition of a one-dimensional signal by the EMD, the edge effects or end effects may be deleted more or less by some strategies such as mirror expansion [

To this end, we propose an approach of improving the edge effect of the BEMD to enhance its image denoising performance. This approach applies the SVM method [

Similar to the EMD of the one-dimensional case, the BEMD uses the extrema that are found in the original image or obtained from the first derivative of the original or the higher-order derivative, to achieve the decomposition of the image signal. Distances between extrema may provide the information for characterizing the image on intrinsic length scales.

For a two-dimensional image denoted by

Initialize the image under consideration,

Initialize the parameters,

Extract extrema points of

By cubic spline, interpolate between local maxima and between minima, respectively, to get two envelope surfaces

Calculate the mean envelope surface in terms of these two envelope surfaces, given by

Update the original signal and designate a new one for iteration, given by

Calculate the standard deviation, SD, given by

Repeat steps

Update this signal and obtain the residual signal

Repeat steps

Once the decomposition process has finished, the original image can be expressed as the sum of all IMFs and the residual

Stopping criterion directly decides the end of the decomposition process and also determines the corresponding final residual term. Therefore, the choice of the value of SD has a close relation with the number of the IMFs obtained by the BEMD. It is necessary to properly choose the value of SD on the basis of the actual situation. In order to portray details of the image, it should be a best choice of a lower value of the SD, generally taken as 0.2–0.3.

In decomposing a one-dimensional signal through the EMD, the residual presents a trend varying with time, which can be ignored in the following analyses. However, the residual resulting from the decomposition of a two-dimensional signal through the BEMD often contains some information of the original image characteristics. It cannot be ignored in the following works, and its influence must be considered for some contributions to the original image.

It is an open problem that there is a certain edge effect in the process of the BEMD. Generally, an actual image signal is not so long or even very short, and serious edge effects can be seen sometimes for its decomposition. Such decomposition is a process screening one after another for many times. With the progression of this process, the edge effect will be more serious, leading to distortions of the IMFs of the image signal decomposed by the BEMD. To this end, some treatments should be taken to mitigate or eliminate edge effects occurring in the screening process of the BEMD operation. Combining advantages of the SVM method with those of the mirror expansion, a new approach to deal with the edge effect caused by BEMD in the decomposition of an image signal is proposed as follows.

Given training sampling data

The optimization problem presented by (

Since the regression model is driven by and based on characteristics of the training data, an obvious adaptive ability can be effectively guaranteed in the regression process if appropriate input nodes and related parameters are chosen. Therefore, the SVM regression modeling may provide a good foundation for the expansion of the image data in the following operation.

In order for support vector machine (SVM) algorithm to have a better effect, this paper uses the particle swarm algorithm to improve support vector machine (SVM) parameter, such as

The algorithm divides particles into two parts according to the fitness value; the fitness value is the standard of measuring the particle’s quality. It distinguishes the particle’s quality of the wheat from the chaff through comparing a single particle fitness value and the group average fitness value and then adjusts reasonably the particle’s search ability

In the above formula,

When

For a two-dimensional image

For all of the sampling data in the matrix, indexes of the column are regarded as discrete variable

Choosing these data

The regression model trained in step

Further, indexes of the row are regarded as the variable

The sampling data

The obtained regression model trained from step

All the forecasting values resulting from steps

After the extrapolation expansion of the SVM regression model is carried out, the closed-end mirror expansion method will be applied for the two-dimensional image. As a further expansion, it makes full use of the known information involved in the original data for the extrapolation has provided larger reliable expansion regions. So it may have a better performance for mitigating the edge effect caused by the decomposition process of BEMD.

In operating the closed-end mirror expansion of the extrapolation version resulting from the SVM regression model, the procedure can be briefly summarized as follows:

At first, identify if the forecasting values in the

Similarly, identify the forecasting values in the

Finally, put “mirrors” on those positions of local extrema close to the edges determined in steps

In the process of image acquisition, compression, conversion, and transmission, some noises may be mixed up with pure image information. Therefore, the real signal of the image may be generally regarded as a compound of the noise and pure image information, which can be expressed as

If the BEMD is operated to these two types of signals, assuming that

The noisy image can be decomposed into some IMFs by the BEMD. These IMFs contain high frequency components in space involved in the original pure image and also include contributions from the IMFs of the noise signal. Accordingly, it is a key problem for the image denoising to remove the latter from the decomposed results of the image. It is known that the BEMD algorithm can adaptively break the original image signal into finite components on different intrinsic scales. That is to say, it should be of benefit to the extraction of information details of the image, to lay a good foundation for the following denoising processing. In general, the noise has major contributions to some IMFs, but quite few to the others. Thus, only by removing those IMFs full of the noise contents can we reconstruct the real image from the original noisy one by synthesizing the remaining IMFs.

In order to obtain reliable decomposition results of the BEMD, the treatment of edge effects is introduced. To some extent, it provides a modified version of the BEMD with edge treatment for enhancing the image denoising performance. The flowchart of its image denoising may be schemed as shown in Figure

Firstly, input the image signal waiting for processing, and set

Expand each edge of the image by means of the proposed approach to get the expansion signal. After that, decompose this signal by the BEMD, which is referred to as the first decomposition. In this decomposition, the noisy image signal is just broken down into one component of IMF and a residual

Further decompose the obtained IMF by the modified version of the BEMD, which is referred to as the second decomposition. In this decomposition, one can extract the residual, designated as

Synthesize two residual components resulting from above two decomposition operations and obtain the reconstruction image

Flowchart of image denoising by the modified BEMD algorithm.

Consider a (256 × 256-pixel) gray Lean image given in Figure

Gray Lean image and its contaminated version by Gaussian white noise.

Original image

Noisy image

The image with Gaussian white noise, as shown in Figure

Comparison of denoising results for the image with Gaussian white noise.

The proposed approach

Wiener method

Free of edge treatment

Edge treatment by mirror expansion

Compared with the result denoised by the Wiener method, as shown in Figure

When the salt and pepper noise is added to the gray Lean image in Figure

Gray Lean image and its contaminated version by salt and pepper noise.

Original image

Noisy image

Similarly to the first example of the denoising of the image with Gaussian white noise, with regard to the noisy image in Figure

Comparison of denoising results for the image with salt and pepper noise.

The proposed approach

Wiener method

Free of edge treatment

Edge treatment by mirror expansion

The original gray Lean image is presented in Figure

Gray Lean image and its contaminated version by 20% random noise.

Original image

Noisy image

As expected, the proposed approach gives the best result of removing the random noise; see Figure

Comparison of denoising results for the image with 20% random noise.

The proposed approach

Wiener method

Free of edge treatment

Edge treatment by mirror expansion

To quantitatively interpret the effectiveness of the proposed denoising method, let us have a look at the peak signal-to-noise ratio (PSNR). The PSNR is an expression for the ratio between the maximum possible power of a signal and the power of corrupting noise that affects the fidelity of its representation, and it is most commonly used to measure the effect of image enhancement algorithm on image quality [

According to the definition of PSNR given by (

Peak signal to noise ratios (PSNRs) of the noisy and denoising images.

Noise type | Before denoising | Wiener method | The proposed approach | Free of edge treatment |
Edge treatment by | |||||
---|---|---|---|---|---|---|---|---|---|---|

MSE | PSNR | MSE | PSNR | MSE | PSNR | MSE | PSNR | MSE | PSNR | |

Gaussian white noise | 616.07 | 20.23 | 373.65 | 22.41 | 191.32 | 25.31 | 204.40 | 24.03 | 256.88 | 23.03 |

Salt and pepper noise | 975.80 | 18.24 | 398.16 | 22.13 | 62.49 | 30.17 | 99.96 | 28.13 | 137.25 | 26.76 |

20% random noise | 400.43 | 22.11 | 100.95 | 28.09 | 3.97 | 42.15 | 15.98 | 36.09 | 16.81 | 35.88 |

Image denoising by the BEMD is an interesting concept, introduced about ten years ago, where filtering is accomplished through the decomposed results on different intrinsic scales instead of the image itself. However, to enhance the BEMD performance for noise removal and other image processing applications, the edge effect is an essential problem worthy of notice, in addition to the development of fast and more effective methods. This fact motivated the formulation of the proposed approach to improving the edge effect to enhance the denoising performance of the BEMD. The edge treatment includes two steps, in which the first one is an extrapolation operation through the model constructed by the SVM based on the information of the original signal and the second is an expansion by the closed-end mirror expansion technique with respect to the extrema nearest to and beyond the edge of the data resulting from the first operation.

With consideration of the fact that the highest noise content of an image is extracted into the first IMF and the remaining fewer noise contents are extracted into subsequent IMFs, this work introduces a procedure of the image denoising through the modified version of the BEMD. The first decomposition just breaks down the noisy image into one component of IMF and a residual, and the second decomposition extracts a residual from the IMF of the first decomposition. Synthesizing these two residual components obtains the reconstruction image, which is the denoised result.

Applications to the gray Lean images contaminated by Gaussian white noise, salt and pepper noise, and random noise show that the proposed approach has an amazing ability for enhancing the image denoising performance of the BEMD. It not only overcomes some drawbacks in the BEMD free of edge treatment, but also improves the performance of the BEMD with edge treatment by mirror expansion. Compared with the Wiener method, the BEMD coupled with edge treatment of the proposed approach shows better results for the given three types of noises, in a significant reduction of noise without loss of desired image contents. Further, the PSNRs of the denoised images confirm the best performance of the BEMD by improving the edge effect.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Fundamental Research Funds for the Central Universities of China.