Introduction to image denoising Algorithms

This is an assignment of my introduction to intelligent science. It cannot be said that my homework has copied my blog !!!

 

I. background

 

With the popularization of various digital instruments and digital products, images and videos have become the most common information carriers in human activities. They contain a large amount of information about objects, it has become the main way for people to obtain original external information. However, in the process of image acquisition, transmission and storage, the image is often affected by various kinds of noise.AlgorithmThe quality of the image is directly related to the effect of subsequent image processing, such as segmentation, target recognition, and edge extraction. Therefore, it is necessary to reduce noise in order to obtain high-quality digital images, the original information integrity (the main feature) can be maintained as much as possible, while useless information in the signal can be removed. Therefore, noise reduction has always been a hot topic in image processing and computer vision research.

The ultimate goal of image and video denoising is to improve the given image and solve the problem of image quality reduction caused by noise interference of the actual image. The de-noise technique can effectively improve the image quality, increase the signal-to-noise ratio, and better reflect the information carried by the original image. As an important pre-processing method, image Denoising algorithms have been extensively studied. In the existing de-noise algorithms, some de-noise algorithms have good effects in low-dimensional signal image processing, but they are not suitable for high-dimensional signal image processing; or the de-noise effect is better, however, some of the image edge information is lost, or we are committed to researching and detecting the image edge information and retaining the image details. How to find a better balance between noise resistance and detail preservation has become the focus of research in recent years.

 

Ii. Theoretical Basis of Image Denoising

 

2.1Concept of Image Noise

Noise can be understood as a factor that hinders people's sensory organs from understanding the received source information ". For example, if the plane brightness distribution of a black-and-white image is assumed to be f (x, y), then the brightness distribution R (x, y) that interferes with the image reception ), it is called image noise. However, in theory, noise can be defined as "unpredictable random errors that can only be recognized by Probability Statistics ". Therefore, it is appropriate to regard image noise as a multi-dimensional random process. Therefore, the method to describe noise can be completely described by the random process, that is, its probability distribution function and probability density distribution function. However, in many cases, such a description method is complex or even impossible. The actual application is often unnecessary. It usually uses its Digital features, such as mean variance and correlation functions. Because these digital features can reflect noise characteristics in some aspects.

 

2.2Common Image Noise

Common noises in our images are as follows:

(1), Adding noise

The addition of noise and image signal strength are irrelevant, such as the channel noise introduced during transmission. The TV camera scans the image noise. This type of Noisy Image g can be seen as the sum of F and noisy N, that is:

(2), Multiplication Noise

Multiplication and image signals are related and often change with the changes of image signals, such as noise in the Flying Point scan image, television scan grating, and film particles, the relationship between such noises and images is:

(3), Quantization noise

The quantize voice is the main noise source of digital images. Its size shows the differences between digital images and original images, the best way to reduce such audible alarms is to select the optimal level based on the gray-level probability density function.

(4),"Salt and pepper"Noise

The white points on the black image, the Black Point noise on the white image, the error introduced in the conversion field, and the transformed noise caused by the reversed image.

 

2.3Image Noise Model

The noise contained in the obtained image can be classified according to different categories. Based on the probability score of noise, it can be divided into Gaussian noise, Riley noise, gamma noise, exponential noise and even noise. Their probability density functions (PDF) are as follows:

(1), Gaussian Noise

In spatial and frequency domains, Gaussian noise (also known as normal noise) models are often used in practice due to their mathematical ease of processing. The following PDF of Gaussian random variable Z is provided:

Where, Z indicates the gray value, μ indicates the average value or expected value of Z, and α indicates the standard deviation of Z. When Z is subject to the above distribution, its value falls within the range of [(μ-2 σ), (μ + 2 σ.

(2) Impulsive noise (salt and pepper noise)

A pdf of (bipolar) pulse noise can be provided below:

If B> A, the grayscale value B is displayed as a bright spot in the image, and vice versa. If Pa or Pb is zero, the pulse is called a single pole pulse. If both PA and Pb cannot be zero, especially when they are approximately the same, the pulse noise value will be similar to the pepper and salt particles randomly distributed on the image. For this reason, bipolar pulse noise is also called salt and pepper noise.

(3) Riley Noise

The mean and variance are:

(4) Gamma noise

The mean and variance of the density are:

(5) Exponential distribution Noise

Where a> 0, the expected values and variance of the probability density function are:

(6) Uniform noise

The mean and variance are:

 

2.4Image denoising algorithm Classification

(1), Spatial domain filtering

Airspace filtering directly performs data operations on the original image to process the gray-scale values of pixels. Common spatial domain image denoising algorithms include the neighborhood, median filter, and low-pass filter.

(2), Transform domain filtering

The de-noise method of the image transform domain is to transform the image from the spatial domain to the transform domain, and then process the transform coefficient in the transform domain, then, the image is transformed from the transform domain to the spatial domain to achieve the goal of removing the image noise. There are many ways to transform an image from a spatial domain to a transform domain, such as Fourier transform, Fourier transform-hadama transform, cosine transform, K-L transform and wavelet transform. Fourier transform and wavelet transform are common methods for image denoising.

(3), Partial differential equation

Partial Differential Equation is an image processing method that has emerged in recent years. It mainly targets low-level image processing and has achieved good results. The partial differential equation has the characteristics of the opposite sex. It can be used in image denoising to ensure edge well while removing noise. The Application of partial differential equations can be divided into two types: one is the basic iteration format, and the image gradually approaches the desired effect through time-varying updates, this algorithm represents the equations of perona and Malik [27], as well as the subsequent work after its improvement. This method has a great choice for determining the diffusion coefficient. It has the backward diffusion function while the forward diffusion function. Therefore, it has the ability to smooth images and visualize edges. Partial differential equations have achieved good results in low-noise density image processing, but the de-noise effect is not good when processing high-noise density images, and the processing time is significantly higher.

(4), Variational method

Another method of image de-noise using mathematics is to determine the energy function of the Image Based on the variational method. By minimizing the energy function, the image is smooth, the fully-variational TV model, which has been widely used, is such a type. The key to this method is to find an appropriate energy equation, ensure the stability of evolution, and obtain the desired results.

(5), Morphological Noise Filter

Combining open and closed features can be used to filter out noise. First, an open operation is performed on a noisy image. You can select a structure element matrix that is larger than the noise size. Therefore, the background noise is removed; close the image obtained in the previous step to remove the noise from the image. Based on this, we can see that the image type used in this method is that the object size in the image is relatively large, and there are no small details, the noise removal effect for such images will be better.

 

Iii. Introduction to several image denoising Algorithms

 

3.1Spatial Domain-based Median Filtering

A median filter is a commonly used non-linear smoothing filter. The basic principle is to replace the values of a point in a digital image or a digital sequence with the values of each point in the neighborhood of the point. F (x, y) indicates the gray value of the pixel (x, y) of the digital image. The median filter with the filter window of a can be defined:

When N is an odd number, N numbers are x1, x2 ,... The mean value of XN is the number in the middle of the order of values. When n is an even number, we define the mean value of the two intermediate numbers as the mean value.

 

3.2Wavelet-based threshold de-noise

The wavelet shrinking method is currently the most widely studied method. The wavelet shrinking method is divided into the following two categories: the 1st category is the threshold shrinking, because the threshold shrinking is mainly based on the following facts, that is, the larger wavelet coefficients are generally dominated by actual signals, while the smaller coefficients are largely noise. Therefore, you can set an appropriate threshold to first set the coefficients smaller than the threshold value to zero, while retaining the wavelet coefficients greater than the closed value. Then, the estimated coefficients are obtained through threshold function ing; at last, we can achieve de-noise and reconstruction by performing inverse transformation on the estimation coefficient. Another method of shrinking is different. It judges the degree of noise pollution of the coefficient, various measurement methods (such as probability and membership) are introduced to this degree to determine the proportion of shrinkage. Therefore, this method is also called proportional shrinkage. The two basic elements of the threshold shrinking method are the threshold and threshold functions.

Threshold Value Selection:

The determination of the threshold is the most critical in the shrinking threshold. Currently, thresholds can be divided into global and local adaptive thresholds. The global threshold is consistent for all the wavelet coefficients in each layer or the wavelet coefficients in the same layer. The local adaptive threshold is determined based on the local conditions around the current coefficient. Currently, the following global thresholds are proposed:

(1), donoho and johastone unified threshold (DJ threshold ):

Where σ is the standard noise variance, and N is the signal size or length.

(2) Confidence Interval threshold based on zero-mean normal distribution:

(3) Bayes shrink threshold and map shrink threshold. Under the assumption that the wavelet coefficients are subject to the Generalized Gaussian distribution, Chang et al. obtain the threshold value:

Where, (R is the standard variance of noise, and Rb is the standard variance of Generalized Gaussian distribution ).

(4) Minimum maximization threshold: this is the minimum maximization threshold that donoho and John Stone get in the sense of minimum maximization. It is dependent on signals and does not have an explicit expression, you need to know the original signal before obtaining it.

(5) Ideal threshold: the ideal threshold is the optimal threshold value under the mean variance criterion. Like the maximum minimum threshold, there is no explicit expression, and the calculation of this threshold usually also requires the prophet signal itself.

Threshold function:

Bruce and Gao. A semi-soft threshold function is proposed:

By selecting the appropriate thresholds T1 and 12, the soft threshold method and the hard threshold method can reach a good compromise. In addition, Zhang et al. proposed another threshold function to perform gradient-based optimization search for sijre error criterion functions, the difference between this threshold function and the edge closed value function is that it has a higher derivative order, so the reconstructed image is smoother. However, the author attributed the improvement of the de-noise effect to the search method, actually, donoh. And Johnstone proposed in the current wavelet coefficient set, the optimal threshold search method, for the current is already excellent, it can be seen that the improvement of the noise reduction effect is due to the selection of threshold function.

 

3.3Based onPDDEImage de-noise

At present, the study of Image Processing Methods Based on pdpa is also a hot topic in image denoising, in addition, some theory and practical applications have been obtained. The process of de-noise is to establish the noise image as the initial condition of a non-linear partial de-code, and then solve the problem, the result of the filter is obtained at different times. Perona and Malik proposed a non-linear diffusion filtering method (P-M) based on PVDF. The de-noise model of the opposite sex determines the Diffusion Speed Based on the gradient value of the image, it can meet both the noise elimination and edge persistence requirements.

This method, represented by the P-M model, has been widely used in image enhancement, image segmentation and edge detection, and has achieved good results.

P-M is a kind of non-linear method of the opposite sex, the purpose is to overcome the shortcomings of the linear filtering method in fuzzy edge and edge position movement. The basic idea is to reduce the diffusion coefficient when the image features are strong, and enhance the diffusion coefficient when the image features are weak. The equation is as follows:

Here, u (x, y, T) is a time-varying image, a gradient model, and the diffusion coefficient function is used to control the diffusion speed. The ideal diffusion coefficient should make the diffusion of the opposite sex quickly in areas with gentle gray-scale changes, and at a low-speed diffusion or even non-diffusion function at a location with sharp gray-scale changes (that is, image features, therefore, it should have the following properties:

Based on the above two properties, the P-M proposed the following diffusion coefficient function:

K is the edge threshold, which is used to determine the edge area and flat area. The introduced flux function is mainly used to clarify the role of Threshold K in the diffusion operation. Its function definition is as follows:

Although the P-M equation has achieved a certain effect in suppressing noise and retaining the important features of the image, it is abnormal and unstable. Catt and others have improved the equation. They first perform convolution with the image using the Gaussian Kernel, and then use its gradient model to estimate the edge information of the image. This paper proposes to use an optimized symmetric index filter to smooth the image, and then use its gradient model to estimate the edge information of the image. The basic idea of these two estimation methods is to reduce noise interference and extract the edge feature information more realistically, so that the edge information can better control the diffusion behavior of P-M equations.

 

3.4Total Variation (TV) Image Denoising

The TV method is proposed by Rudin Osher and Fatemi. Based on the variational method, it determines the image energy function and minimizes the image energy function to achieve smooth noise removal. It is a popular image restoration method. The energy function equation of the image is:

In [2] of the literature, the Energy Functional of the total variational denoising is:

To minimize the number of energy functions, the Euler's-Laplace equation is:

Where, the gradient operator:

Regularizedtype:

Used to reduce degradation of flat areas. After converting the entire left to the local coordinate system of any pixel in the image, the equation can be divided into two directions: edge direction and edge orthogonal direction, the coefficient in the next direction controls the diffusion intensity in this direction. The diffusion direction is actually a linear diffusion equation of the opposite sex. The diffusion operator only spreads along the orthogonal direction of the image gradient, and the diffusion coefficient is 1/| ▽ μ |, but it does not spread toward a gradient. In this way, the edge position can be determined through the image gradient to minimize the edge diffusion coefficient, thus reducing the degree of blur to the edge. However, because the edge diffusion coefficient is small, the noise is not well restrained, in addition, when | μ |> λ, the potential energy function is non-convex, making the processing at the edge unstable. Therefore, how to determine the value of the diffusion parameter is a problem.

 

Iv. Summary

 

With the development of science and technology and the needs of work and life, digital image filtering will become more and more widely used and require more and more applications. So far, there are still a lot of new ideas and methods in terms of noise reduction, constantly enriching the image denoising methods. Furthermore, the research scope of noise is constantly expanding, from gaussian noise to non-Gaussian noise. Noise reduction technology has a wide range of applications and research prospects, and the research fields are constantly expanding.

The main content of this article is a brief introduction to the image denoising technology. The full text outlines the image denoising technology, including the concept and principle of noise, and introduces some basic image denoising methods. Due to the time relationship, and this is the essay assignment of the introductory course, there is no in-depth and meticulous research.

 

5. References

 

[1] gonzaresrc, woodsre. digitalimageproeessing, seeondedition Beijing eleetronieand industrial Press, 2002

[2] Leonid I. Rudin 1, Stanley Osher and Emad Fatemi nonlinear total variation based Noise Removal algorithms 1992

[3] W. Luo, an efficient detail preserving approach for removing impulse noise in images, IEEE signal proeess. Lett., 413 (7): 416.