Research On Reverse Solving Of Image Sampling And Its Application In 3D Representation

Image-based 3D representation is to establish mathematical model for image data which is suitable for computer representation and processing. The procedure is to retrieve the original scene from the image data to generate 3D model suitable for various processing. Therefore, image-based 3D representation is not only the basis for image processing, image operation and image property analysis in computer-based environment, but also the key technique to create the virtual reality for expressing objective world. Image-based 3D representation is the common scientific issue and core technology of CAGD, CG, computer animation, computer vision, medical image processing, scientific computing and virtual reality, digital content creation, etc. With the rapid development of information technology, it has become a key problem of many important applications in our nation, society and technological development to analyze, mine, retrieve, and use of 3D shape information with the help of high capabilities of computers in data storage, compression, computation and transmission, while 3D shape information of significant values is generated by constructing the virtual, massive information space on computer in real-time instantaneously. Also, image-based 3D representation and processing is critical and has been recognized as one of the common basic problems in many fields that are related with significant national requirements, such as Digital City, product advertising design, DCC and medical image processing.The research of this thesis is centered around the NSF project "Research on the key issues in mass data points based 3D representation" and 973 sub-project "Visual media theory and method of calculation", and is aimed at the reverse solution of image sampling model and its application in 3D representation. Four issues are discussed in this paper:(1) image resizing and corresponding 3D representation; (2) reverse solution of image sampling model, together with its application in MC; (3) reverse solution of image sampling model and quadratic spline fitting; (4) 3D representation of imaged-based shape-preserving fitting. Aimed at the solution of the four problems, this thesis presents novel theories and methods, of which the achievements and innovations are as follows.1. For the problem of image resizing and corresponding 3D representation, we propose one novel method to construct the fitting surface based on image data, in which the given image data can be seen as sampling from one original scene. Scene is supposed to be represented by one surface defined by piecewise quadratic polynomials, and the surface representing the original scene is called original surface. Based on this supposition, the main idea of the novel method is to reproduce the original surface by retrieving the inverse procedure of image sampling, and image resizing is implemented by re-sampling of the reproduced surface. Due to the limited image data, it is impossible to reproduce the original surface based on the given data, and approximate surface is retrieved in the novel method.The main innovation of the method is that the idea of reproducing the original surface by retrieving the inverse procedure of image sampling is proposed, and the original surface is appropriated by piecewise polynomials. Based by this idea, the inverse solution of image sampling can be transformed from one complex non-linear problem to simple linear one, which not only improves the computation efficiency, but presents new method for constructing the fitting surface of the image. Based on this idea, the procedure of constructing fitting surface becomes the inversing one of image sampling. The procedure to produce the surface is as follows. First, for the neighbor region of each pixel, a quadratic polynomial patch to approximate the original surface in adjacent areas is constructed. On each quadrilateral corresponding four adjacent pixels, a bi-cubic patch is constructed as the approximate surface, which is the weighted combination of the quadratic polynomial patches associated with the four pixels. The whole approximate surface of the original surface is composed by all bi-cubic patches. The experiment results showed that compared with current image resizing methods, the images based on the novel method are of high precision and good quality.2. For the application of reverse solution of image sampling model in MC, this thesis improves the interpolation precision of MC algorithm by regenerating the interpolation points used in MC. As tri-linear interpolation being used in MC, the main factor to raise MC precision is to improve the accuracy of interpolation points. The main idea of which is as follows. The new method is based on a supposition that the given image data are sampled from an original 3D scene that can be approximated by piecewise tri-bi-quadratic polynomials, the original scene can be approximated by piecewise tri-bi-quadratic polynomials, and hence, we can reconstruct the volume function of the original 3D scene through the inverse procedure of image sampling. Since the original volume function is approximated by tri-bi-quadratic polynomials, the complex, non-linear problem of the inverse procedure of 3D image sampling is transformed to a linear problem. In the inverse procedure, one needs to solve a equation set with 27 unknowns, to reduce the computation cost and complex, a new method is proposed to subdivide the equation set with 27 unknowns into a sets of the equation sets, each of which has 3 unknowns. Theoretically speaking, when MC algorithm is used, the precision of tri-interpolation by using the new method for computing interpolation points is higher than that of directly using the image data as interpolation points. The experiments also showed that when MC algorithm is used to construct 3D model using the interpolation points produced by the new method, the model constructed has the higher precision and offers more detailed information.3. For the problem of reverse solution of image sampling model and quadratic spline fitting, this thesis proposes a new method to construct 3D model of higher precision by the quadratic spline. The main innovation is that original scene surface can be approximated by a surface defined by quadratic spline, and hence, the problem of reverse solving original scene changes into a problem of solving a set of linear equations with n×n unknowns. To simplify the complexity of solving the set of equations, the new method proposes a simple way to solve equations with n×n unknowns. The equations with n×n unknowns is reduced to a sets of equations with n unknown, and hence, the stability and efficiency of the process of solving equations are improved. Experiment results indicate that compared with the piecewise polynomial surface constructed by local way, the quadratic spline surface has higher precision and better visual effect.4. Due to the human visual characteristics, the quality of image edges plays an important role to the visual quality of the image. Dealing with image using continuous polynomial function, one could not get the satisfactory result at and near the edge of image, the edges of the resulting image is of the blurred and serrated shape often. In order to improve the quality of the interpolation image at the edges, the idea of shape-preserving fitting is applied into image resizing, and one method to enlarge the image twice is proposed. The new method supposes that the original surface can be approximately piecewise by bilinear polynomial functions, then, the certain correlation between adjacent pixels can be obtained. The correlation can be used as the constraints of adjacent pixels in the enlarged image. As it can be seen, the number of constraints is more than that of pixels. Hence, the solution of constraints can be determined by weighted least square method. The weight functions in the least square method could make the edge pixels have a greater impact on the generation of the bilinear polynomial surface. In addition, if the image edges change linearly, the generated surface will also change linearly along the edge, which will make the enlarged image not only have bi-linear polynomial precision, but also a good visual result. The new method will enlarge the image of multiples. Experiment results show that the resized image by the novel method has high precision and good quality.