Mathematical method of image processing

Mathematical method of image processing

In the process of image processing, mathematics always plays an important role, and permeates all the branches of image processing.

By the 670 's, the linear processing method represented by Fourier analysis occupies almost the whole field of digital image processing. In the meantime, with the aid of the stochastic process theory, the image model is established through the probability theory and the information theory established on this basis, and the linear filter (Wiener filter, Kalman Filter) method provides a powerful theoretical support for low-level image processing. , and the FFT is widely used in almost all branches of image processing. These mathematical tools greatly promote the development and application of image processing.

Since the 80 's, non-linear science has gradually penetrated into the image processing method, and many new mathematical tools have been introduced into the field of image processing to make the related theories become more diversified. In particular, the information processing method, which is represented by wavelet and Multiscale analysis, inherits and develops the Fourier analysis, applies the newest results of function theory and approximation theory to engineering application, and establishes a complete system framework, which is used in image coding, image segmentation, texture recognition, image filtering, edge detection, In the application of feature extraction and analysis, remarkable achievements have been made. At present, wavelet analysis method has become one of the basic theory of signal processing.

At the same time, the application of other non-linear mathematical tools has achieved fruitful results: such as the application of fractal in image coding and texture recognition, the application of Lie groups in the recognition of dynamic image elastic deformation, the application of Multiscale analysis in image retrieval and recognition, the application of nonlinear programming in vector quantization and image coding, etc. In addition, the establishment of image deterministic model (BV model), the evaluation system of fuzzy mathematics to image quality, and the study of image distance by meaningful theory are further engraved on the essence of image, so that the computer can describe the human visual system more appropriately.

At the same time, the application of other non-linear mathematical tools has achieved fruitful results: such as the application of fractal in image coding and texture recognition, the application of Lie groups in the recognition of dynamic image elastic deformation, the application of Multiscale analysis in image retrieval and recognition, the application of nonlinear programming in vector quantization and image coding, etc. In addition, the establishment of image deterministic model (BV model), the evaluation system of fuzzy mathematics to image quality, and the study of image distance by meaningful theory are further engraved on the essence of image, so that the computer can describe the human visual system more appropriately.

In particular, the image processing method based on the nonlinear development (partial differential) equation has become a hotspot in image research in recent years. Based on the analysis of the mechanism of image denoising and the mathematical tools such as differential geometry and projective geometry of mathematical morphology, the axiom system of filtering and partial differential equations is established. In addition, it has been applied in image reconstruction, image segmentation, image recognition, remote sensing image processing, image analysis, edge detection, image interpolation, medical image processing, dynamic image patching, stereoscopic vision depth detection, motion analysis and so on. In the course of the study, people introduced some concepts, such as active coutour (snake), level set, and so on, to associate mathematics and images organically.

On the other hand, the actual demand of image processing and engineering background also stimulated the development of some branches of mathematics, such as the research power of wavelet theory is derived from the demand of time-frequency localization analysis in signal processing, and it has been widely used before the theory system is established. The concept of viscous solution of partial differential equation is also presented because of the image processing The application conditions in the application are not satisfied with the assumptions in the differential calculus, and the research on the projection geometry is also due to the need of the image moisaic.

In recent years, the mathematics department of our university has set up the major of information and scientific calculation, even such as the Department of Information Science of Peking University Mathematics Academy. As a new subject of rapid development in recent years, it uses modern mathematical methods and computer technology to solve the problems in the field of information science, which is widely used. Image processing is one of the most important directions, and many schools take image processing as a key development direction. However, there are still some problems: on the one hand, the researchers of the mathematics department have not enough understanding of the development of the image and the background, on the other hand, the communication of the new professional and image processing field in the mathematics department is not very smooth, so the understanding of some hot issues in the field of image processing is not enough timely and comprehensive.

After entering this century, with the rapid development of computer and Internet network technology, the development of image processing has entered a new leap stage. At the same time, there are still many problems to be solved in the engineering application of image processing and computer vision.

Mathematical method of image processing