The Scalar Auxiliary Variable(SAV) Approach For Gradient Flows In Digital Image Processing

In recent years,lots of researchers have spent time in a very hot field,digital image processing.Digital image processing uses good algorithms to process image on com-puter.Compared to traditional analog image processing,digital image processing have a lot of advantages.For example,digital image processing allows a much wider range of algorithms to be used and avoid problems such as distortion during processing.The fast development of digital image processing based on three main factors.Firstly,the development of computer has brought a huge increase in amount of data that can be processed.In the last century,many algorithms and theories can not be achieved,although they are correct and important.But now this problem is solved.Secondly,the mathemat-ical theories about digital image processing have also been sufficiently developed.Espe-cially the improvement of discrete mathematics theory.It's the most important theory for this paper.Computational mathematics,as a young field of mathematics,because of its strong applicability,has developed extremely fast in the past 100 years.At last,there is a large demand for digital image processing.Digital image processing is widely used in many fields.Like environment,industry,medicine and so on.Digital image processing includes many branches,including denoising,enhance-ment,restoration and segmentation.This paper will focus on image denoising,and appro-priately discuss others.As the most classic and common image processing problem,image denoising has lots of algorithms.When we use or design an algorithm,we have to consider the following two factors.Firstly,whether the algorithm can be implemented.The calculations of some com-plex algorithms are not be allowed.We don't consider an algorithm which takes too much time,even if it works well.Secondly,whether removing some detail is acceptable if it allows more noise to be eliminated.In other words,in our algorithm,we can ignore some unimportant features of the origin image to achieve better results.Making a better choice between the detail in the image and the characteristics of the noise is necessary.In this paper,digital image processing is based on partial differential equation theory.Benefit from traditional partial differential theory as an important basic field of mathemat-ics,there are a large number of mature and good theories and systems.By using experi-ence gained from traditional image processing,many people like to use partial differential equation models to process digital image.The main idea and basis of digital image processing by PDE is,on a continuous mathematical model of the image,changing image following specific PDE models.And the solution of the PDE is our result.There are two common modeling methods.Establishing energy functional and use total variation theory to get Euler-Lagrange equation.Another is establishing equation by comparing the image smoothing with diffusion of impurities.This paper mainly focus on the first way.After getting a model,we should solve our target partial differential equation by using a good enough algorithm.We have to face many difficulties just like discontinuity of image function,nonlinearity of partial differential equations and large amounts of data.Therefore,numerical implementation is the most important step.Our numerical methods need to consider stability,accuracy and efficiency.PDE models have two main advantages compared with others.This model has strong local adaptability.Because of PDE models coming from continuous image models,the change of a pixel over time depends only on an infinitely small area around this pixel.In addition.PDE models have high flexibility.If we build a good PDE model,we can easily get better or wider approach to solve related problem.For example,generalize from one dimensional problem to two dimensional problem and generalize from bitmap to vector graphics.In digital image processing,total variation denoising,also known as total variation regularization,is based on one important principle.That is,the image with high noise has high total variation.In other words,the integral of the absolute gradient of the image is high.According to this reason,reducing total variation can get us closer to the original image with lower noise.It still preserves some important information like the edge.This theory was pioneered by Rudin,Osher and Fatemi in 1992.So we call it ROF model.Through the ROF model,we need to solve a minimization problem.The Euler-Lagrange equation corresponding to this minimization problem is our target equation.Then we can get our results by using gradient descent.When we use gradient descent,we need to solve a nonlinear PDE.A scheme to solve gradient flows is evaluated from some aspects.For example,efficiency and whether thescheme is easy to implement.The most important aspect is whether the scheme keeps the energy dissipation.Some common energy-decaying nonlinearly implicit schemes include convex splitting methods,discrete-gradient methods and Runge-Kutta methods.We can use these methods to get high-order schemes.But they require solving a nonlinear system.It's very complex and time consuming.As contrast to above full implicit methods,we can consider linearly implicit methods.It only needs to solve a linear system at each time step.There are also some common energy-decaying linearly implicit methods.For example,Lagrange multiplier approaches,IEQ and SAV approaches.In this paper,SAV approaches are our main method.IEQ and SAV approaches transform general equations into an equivalent system of PDE.In the reconstruction system,we can express the energy in terms of Hilbert space norms of the new variables.It helps us construct the energy-decaying schemes by an easier way.In particular,we can easily get a second-order energy-decaying scheme from traditional second-order BDF method.But the high-order schemes based on SAV formulation has remained open.Due to the limited time and knowledge of research,the object of this article is not the most common color image.This article will start from a more basic point of view and explain the classification of images in detail.According to different imaging principles,images are divided into bitmap and vector graphics.Our object is bitmap.Each point of a bitmap is called a pixel,and the color information stored in a pixel can be represented by a number.For color bitmap,we use the RGB color model to store color information,display the color through the brightness of three color channels.RGB means red,green and blue.The example used in this article is to convert a color bitmap into a grayscale bitmap artificially for numerical experiments.The grayscale bitmap is actually a simple case with only one color channel.The conclusion can be generalized to the case of color bitmap.The degree of one color channel has 256 levels,that is,from 0 to 255.This number is due to the storage of the computer,that is 28.So the color information of each pixel of a grayscale bitmap is completely represented by a number between 0 and 255.Therefore,we can use a matrix of the same size in matlab to store all the information of a grayscale bitmap.For example,a 50 × 50 pixels bitmap can be store in a 50 × 50 matrix.The operations we do afterwards are all operations on the elements of the matrix.In this paper,I will introduce and state related theoretical foundations and build mod-els.After getting our target partial differential equation,design schemes by using SAV approach.Then perform some numerical experiments.I will compare results with oth-ers obtained by traditional methods.I will analyze the advantages and disadvantages of our schemes.There is also a brief discussion about SAV approach in other digital image processing fields.The examples presented in this article are the most classic and well-known examples in the field of the image denoising.There are many interesting stories behind this example.I will add several different types of noise to the original image.In the theoretical analysis,I discussed two more complex schemes based on the SAV approach.These two schemes have more advantages than the scheme I use in this article,but they are also complicated.Compared with the traditional explicit scheme,the scheme used in this article has nearly the same denoising effect.But the biggest advantage of the scheme of this article is to use the stability of the SAV approach.The traditional explicit scheme is easy to implement,but has strict requirements on the time step.We need a very small time step to find a stable solution.However,the scheme of the SAV approach is stable,which makes the selection of time steps much easier.We can choose a larger time step.The most direct benefit brought by big time steps is that the number of iterations we finally a steady state is greatly reduced.And in this article we have proved that the scheme constructed by SAV approach is easy to implement like the traditional explicit scheme.So the most important factor in determining the amount of calculation is the number of iterations.The biggest advantage of the scheme is that is greatly saves the calculation cost.The example we use is a small picture.If the denoising object is a high resolution image,the advantages of this scheme will be more obvious.