# An ill-posed problem of partial differential equation of turn image

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As a preprocessing method, image processing is almost a prelude to all image processing methods. In many cases, image filtering as a preprocessing method of image recognition, it needs to satisfy two restrictive conditions: the contrast is invariable and the affine inconvenience. Affine invariance can be decomposed into translation invariant, rotation invariant, European invariant, and telescopic invariant. There is only one partial differential equation satisfying the invariant of contrast and affine invariant, i.e. the AMSS (affine morphological scale Space) equation. L.alvarez,f.guichard,p.l.lions and J.m.morel, etc. in the literature: axioms and fundamental equations of image processing the entire derivation process is cleverly organized, Formed a system of kilometers. The publication of the article is considered to be a symbol of the formation of this discipline based on the partial differential equation of image processing. Mathematical morphology operators are also incorporated into the entire derivation system, so the classical filter is given a new meaning.

Image processing based on partial differential equation belongs to the category of low-level image processing, and its processing results are usually used as intermediate results for other image processing methods.

At present, many branches are derived from image processing based on partial differential equation, such as dynamic boundary, image processing based on level set (line), Image Distortion (deform), image model research, etc. The development of this field has been expanding in the field of application, for example, the CNES has adopted the AMSS operator as the standard method for image enhancement of aerial image, and secondly, with the development of this subject, people are digging more and more deeply the essence of image and image processing, and attempts to transform the existing image processing methods with strict mathematical theory, which is a challenge to the traditional image processing method based on using.

The nature of image processing is non-stationary

Lu Ying Professor (Jilin University) simple and comprehensive introduction of the basic knowledge of image processing, The main content and all levels, but also put forward a lot of problems to be solved. Professor Jiang Ming (Peking University) said two questions: first, the scale space theory, from the multi-scale representation of the image and basic invariance (causality, transformation invariance and morphological invariance) of these axioms to obtain the partial differential equation, so that the image processing problems into the problem of partial differential equations; and the statistical image processing, Based on Bayes inference, stochastic process, Markov random field theory and so on, the image processing Mumford and Shah's Model are obtained, which is a variational problem. So it seems that many of the problems in image processing seem to have a profound mathematical nature, so that mathematics can work and do many things in this field. Professor Zhang (xi ' an Jiaotong University) the partial differential equation is also obtained from the scale space and the retina model, and it is noteworthy that he can solve the clustering problem by using this model, that is to say, the application of partial differential equation in image processing has a deep biological background. The equations above are mainly diffusion equations (isotropic diffusion equations and anisotropic diffusion equations), and Professor Yun Jingxue and his PhD Wang Chunbong (Jilin University) have given a definite answer (in some sense) to the question of the existence of solutions to certain specific diffusion equations. Professor Zhou Shulin (Peking University) spoke of the existence and uniqueness of the solution to the variational problem and related theories. The problem of image processing has a great demand for the speed of computation, so the problem of solving the problem of these problems lies in front of us. Professor Sun Weiwei (City University of Hong Kong) The fast algorithm in partial differential equations is introduced, because many of the computations in partial differential equations are eventually transformed into matrix operations, so the main content is the calculation of special matrices (for example, cyclic matrices). Image can be regarded as a sampling of a continuous surface, so it can also be studied from the angle of geometry, Kuchangzheng (Northwestern University) and so on the current international research on the relatively many invariant geometry and curvature flow. Above are from the general mathematical point of view, in order to have a more in-depth understanding of image processing, but also some experts in some areas of expertise have experienced a number of specific issues. Professor Lu Ying (Jilin University) made a summary of fingerprint identification technology. Professor Peng Lizhong (Peking University) introduced the new progress of wavelet, especially the application of frame wavelet in digital watermarking and face recognition. Professor Wang Lisheng (Tsinghua University) made a summary of medical image processing. Professor Chen (Shanghai Jiaotong University) has talked about the issues of information security and image processing.

The research of ill-posed problem in image processing (ill posed problem) or inverse problem (inverse problem) has become a hot topic in the world since the end of 20th century, and has become a research field widely concerned by modern mathematicians, computer vision and image processing scholars. The study of the inverse Problem of mathematics and physics has long been, and the French mathematician Adamaz house the concept of ill-posed problems as early as 19th century: the existence, uniqueness and stability of the solution to the problem of a mathematical physical solution is said to be appropriate (well Posed). If one or more of the above criteria is not satisfied, the problem is said to be ill-defined. Typical image processing ill-posed problems include: Image de-noising (Images de-nosing), image restorsion, image zooming, image patching (images inpainting), Image de-Mosaic (image demosaicing), picture super-resolution, etc.
So far, many methods have been proposed to address the ill-defined nature of image processing. However, how to further describe the important visual geometry in the image such as edge, texture and angle, and to improve the ability of the method to maintain the structure and texture effectively on the basis of noise suppression, is a problem to be researched deeply.

Review on the status quo of 1 ill-posed image processing problems at home and abroad

Because the inverse problem in image processing is often ill-posed. The effective way to solve the ill-posed problem is to introduce the prior information about the image in the image processing. So the prior model of image is very important for image inverse problem and other computer vision or image processing problem. For the study of the prior model of image, the researchers studied from several angles, the representative mainly has "statistical method" and "regularization geometric Modeling method", "sparse representation method" three kinds of mainstream methods, and the recent rise of image morphological component analysis (MCA) method attracts a large number of researchers at home and abroad attention.
1.1 Regularization geometric models change rapidly
The "regularization geometry method" for natural image modeling is the subject of hot discussion in recent years. One of the methods is to use partial differential equation theory to establish image processing model, the current development trend is to design various low-order, high-order or low-order and high-order integrated partial differential equations from the angle of selective nonlinear diffusion, or the diffusion to the complex diffusion from the airspace to the space frequency domain and the synthesis of different singular structures [1 ]。
Another kind of method is based on the energy functional optimal variational method. In 1992, Rudin-osher-fatemi proposed that the image could be decomposed into a full variation model of the component and a component of the bounded variation space [2]. According to the international and my research, it is shown that the ROF model model can depict the important visual edge structure in image, but it can't describe the texture information. 2001 Meyer proposed the theory of oscillatory mode decomposition [2]: he thinks that the oscillation component can be expressed as the divergence form of a vector function, and the oscillation component can belong to 3 possible function spaces. Firstly, an approximate dual space of bounded variation (boundedvariational, BV) space is introduced to characterize the oscillation component of the image. Meyer further points out that the bounded mean oscillation space and the Besov space of John-nirenberg are the appropriate function spaces of oscillatory components, and derive the (BV,G) model, (BV,F) model and (BV,E) model of image decomposition. Meyer theoretically solves the theoretical frame of oscillatory components and becomes the cornerstone of the decomposition of the oscillation modes, but the original model is difficult to calculate. Later, most scholars work on the basis of Meyer work. Vese-osher proposes to approximate (bv,g) The divergence of the vector field modeled by the oscillating component [3], which essentially approximates the G space as a negative Soblev space [4]. L.lieu and L.vese further extended to fractional-order negative Soblev space [5]. Aujol,chamboll and others define a subspace in G-space, and based on the projection algorithm of ROF model proposed by Chamboll earlier, this paper proposes that the oscillation component of the image is the projection component on the subspace, and thus presents the famous BV space semi-norm + g space Norm + L2 norm constrained optimization A2BC model and subspace projection algorithm [6-7]. J.b.garnet,t.m.le,y.meyer,l.a.vese presents a more general homogeneous Besov space to characterize the oscillation component [8]. Recently, the j.aujol,a.chamboll of the TV norm, G-norm, F-norm, e-norm, L 2 norm, respectively, the mathematical Statistics and correlation analysis of the image cartoon image, texture component, Gaussian noise, and put forward the use of TV norm, G-norm and e-norm respectively to constrain the image's cartoon component, Three-component image decomposition model of texture component and noise component [9]. In 2007, G.gilboa and S.osher were presented with the concept of non-localized g-space, and the non-local ROF model, nonlocal Meyer model and nonlocal ROF+L1 model were proposed in general, and a new idea of image priori model was provided theoretically. But in the present study, the main deficiency of variational method is the characterization of texture and noise.is not fine enough.

1.2 Sparse indicates ascendant

The problem of sparse representation of images originated from the "effective coding hypothesis". Attneave first suggested that the goal of visual perception is to produce an effective representation of an external input signal. In the field of neurobiology, "effective coding hypothesis" is Barlow based on information theory, and the main function of primary visual cortex neurons is to remove the statistical correlation of input stimulation [11]. After the "Effective coding hypothesis" was put forward, many researchers put forward different theories according to its thought. The main ideas are divided into two major categories. The direct method is the mechanism test, that is, from the biological mechanism, the response characteristics of nerve cells are detected under the condition of natural image stimulation. Well-known work such as: 2001 in the "Nature" published in the study results show that in the redundancy measure and the natural stimulation of a group of retinal ganglion to external stimulation of the independent code [12];2000 published similar results in "science" [13] : By recording nerve cell responses in open natural scenes and simulating natural scenes in the short-tailed ape V1 region, the neurons in the visual cortex (V1 region) are effectively represented by sparse coding for natural scenes, and sparse coding transmits information with minimal redundancy. Another alternative method is the model simulation method, which uses the statistical characteristics of natural images to establish the processing mechanism of the early-stage visual processing system. For example, Olshausen and field[14] proposed sparse coding model, sparse coding theory shows that by looking for natural images of sparse coding, the neural network can learn to be similar to simple cell perception of the structure of the wild. Bell proposes an unsupervised algorithm based on information maximization, which expands the independent component analysis (ICA) method by measuring the joint information entropy of "factor", and successfully constructs an effective coding model and obtains similar results as above [15]. Hyvarinen further, a two-layer sparse coding model is used to construct a basic function similar to the response characteristics of complex cells, and the set of base functions forms a regular topological structure [16]. This part shows that the effective coding hypothesis can also be applied to the processing of neural cells in advanced area of visual system.
At present, the research on image sparse representation system is mainly carried out along two main lines. One of these is the theory of geometric analysis along the multi-scale. The authors consider that the non-stationarity and non-Gaussian properties of the images are difficult to deal with by linear algorithms, and that an appropriate image model capable of processing the geometric structure of the edges to the textures should be established. The traits in two-dimensional images in singular edges and in-D images of filaments (filaments) and tubes (tubes) Geometric features cannot be represented by isotropic "block bases" (such as wavelet bases), and the optimal or "least sparse" function representations should be characterized by the anisotropic "wedge". Therefore, the multi-scale geometric analysis [16-22], represented by Ridgelet, Curvelet and Bandlet,contourlet transformations, is an effective approach to sparse representation of images. Figure 2.1.1 (a) the process of approximating a two-dimensional separable wavelet at different resolutions is given, with the resolution increasing, the scale becomes thinner, and the result is the use of many "points" to approximate the curve.

Compared with wavelet, Contourlet not only has the multi-resolution characteristic and time-frequency localization of wavelet, but also has good directivity and anisotropy, that is, when the scale J, the support domain edge length of wavelet base is approximate, and the aspect ratio of the support domain of Contourlet in this scale can be arbitrarily selected. Figure 2.1.1 (b) is a more efficient and sparse representation of the process of approximating a curve with the support domain of the Contourlet basis function, because its support domain of the base function is expressed as "rectangle". With the isotropic difference of the direction support domain of the two-dimensional separable wavelet base function, the "rectangular" support field of the Contourlet is characterized by anisotropy (anisotropy).
These sparse representations are all based on a "single base", and the other is a sparse representation of the image: The base function is replaced by an over-complete redundancy system called the Atomic Library. [23] Mallat and Zhang first presented the idea of decomposition of the signal over the complete library (over-completedictionary) in 1993. By decomposition of the signal over the complete library, The base used to indicate the signal can be adaptively selected according to the characteristics of the signal itself to obtain a very sparse representation. Later, such as the base tracking algorithm, the matching tracking algorithm (MP), the orthogonal matching tracking algorithm (OMP), the hybrid matching tracking algorithm (HMP) and many variants were proposed. The atoms involved include the multiscale Gabor function, the anisotropic fine atoms, the wavelet and the sine function cascade [24-15], and the training method to obtain the structure and texture component sparse representation dictionary [26-28].
At present, the research of sparse representation of images has aroused the concern of many researchers in China. CAs says Mr Yang, Wang Yunxiu and other people, the Chinese Academy of Sciences Zhongzhi Shi researcher, engineering professor of Xidian University, Cheshenly professor of South China Institute of Science and Technology, Yun Zhongko professor, Jiaotong University, Nanjing University Wezhihui professor, Shaoliang and Dr. At present, the research of sparse representation of images has become a hot issue for many researchers in China in recent 3 years, according to the << Chinese journal full-text database >> Search, there were few related reports before 2004, and from January 2004 to February 2008, There are 187 papers on sparse representation and multi-scale analysis and application in Chinese periodicals, including about Ridgelet56, about Contourlet 63, about Curvelet34, and about over-complete sparse representation of 34 articles. Professor Engineering of Xidian University, Professor Cheshenly of South China Institute of Technology, Professor of Yun Zhongko Jiaotong University, Jinguang, professor and member of the National Defense Science and Engineering Institute have launched a study on the application of sparse representation (29-33). In this paper, based on multi-scale geometric analysis of image enhancement, denoising, fusion, edge detection, perceptual compression and digital watermarking, the research results show that the morphological component decomposition theory based on sparse representation can capture the geometric features of images very well, and has inherent superiority in image modeling and processing. However, the survey of domestic research and foreign original results have a large gap. Especially in the sparse representation of the structure of the dictionary, efficient sparse decomposition algorithm, sparse reconstruction and other levels have a lot of work to do.

1.3 Morphological component analysis Temporary dew angle

The MCA method is an internationally renowned scholar, J.-l Starck, M. Elad, in the 2004, a method of decomposing images into "geometrical structure", "texture" and "noise" in the form of component decomposition [34]. The method is almost identical to the blind separation of aliasing signals, and is closely related to independent component Analysis (ICA). Before the MCA was presented, the study of image decomposition was in full swing. Mainly includes "image decomposition based on sparse representation" and "image decomposition based on variational method". The MCA method is a good combination of the variational method and the sparse representation of the two types of image decomposition advantages, for the ill-posed image processing problem provides a favorable processing mechanism.
First of all, from the variational method of image morphological component decomposition, the research in the world is moving toward the development of more detailed characterization of morphological components, such as image structure and texture, and more simple calculation. Image decomposition (BV,G) model, (BV,F) model and (BV,E) model are essentially a form component analysis method.
And the method of image decomposition based on variational methods is different, j.l.stack,m. An important hypothesis in the MCA framework of Elad and D.l.donoho is that the geometry and texture components of an image are sparse within a particular kikuyus or over-complete sub-dictionary, and that the Kikuyus or over-complete sub-dictionaries of the sparse representations of the morphological components are irrelevant. The effective separation of image morphological components is achieved by sparse representation of sparsity (l0-norm or L1-norm metric) by classifying sparse representations of structural components and texture components. There are three types of sparse representation systems involved: orthogonal systems (such as DCT,DWT), redundant systems (such as Curvelet,contoulet), and over-complete dictionaries (such as Ar-gauss mixed dictionaries). With the development of sparse representation theory, different algorithm of image morphological component analysis can be derived by different classification sparse representation dictionary, sparsity measure and regularization method [35]. After that, the researchers studied the sparsity and diversity of morphological components [36], Adaptive projection threshold selection [37] in MCA, and extended it to multi-channel cases [38].
1.4 Statistical Models Enduring
The research on "statistical modeling method" of natural image has been long-standing. The early work was largely driven by an important discovery by Davidfield in the the mid 1980s: the filter response of natural images always showed a statistical property of Greater Kurtosis [39]. The most important factor of the classical wavelet analysis in signal and image processing is that the research on the statistical model of wavelet transform domain has made great progress, including the MRF model in wavelet domain, the hidden Markov model in wavelet domain and the layered hidden Markov model. With the rise of multi-scale geometric analysis, people are concerned about the statistical models of Ridgelet, Curvelet and Bandlet,contourlet transform domains. In fact, the histogram of sparse representation coefficients is much more than 3, showing obvious non-Gaussian, which indicates that non-Gaussian contains the intrinsic characteristics of the image.
For example, the contourlet coefficients of cameraman images are analyzed. Observing the above distribution, we can find that the Contourlet coefficient shows a significant heavy-tailed distribution. Investigate the histogram kurtosis (Kurtosis)

By calculation, the kurtosis value is much larger than the typical Gaussian distribution Kurtosis value (approximately 3).
Many univariate transcendental models, such as generalized Gaussian models and student t-distribution models, have been successfully applied to the non-Gaussian statistical properties of the wavelet coefficients of natural images. Recently, scholars have begun to pay attention to the joint statistical behavior of natural image filter responses. ZHUS.C has discussed four kinds of mainstream statistical research methods of natural image visual mode, and discussed including several Markov random models and their variants from the sparse representation of signals [40]. In this paper, 10 statistical models, including hidden Malkov (HMT) and background hidden Markov model (CHMM), are compared and analyzed by engineering, [41].

An ill-posed problem of partial differential equation of turn image

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