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chen l digital functions data reconstruction digital discrete methods 2013

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Textbook in PDF format Before Newton-Leibnitzs time, mathematics was basically discrete. Since then, continuous mathematics has dominated the literature. But discrete mathematics has found new life with the appearance and widespread use of the digital computer. However, we still prefer to use the thinking involved in continuous mathematics. For example, if we had discrete information on some samples, we would assume a continuous model to do the calculation. Sometimes, we need a discrete output from the continuous solution, and it is not hard to re-digitize the continuous results. For some problems, going from discrete to continuous back to discrete may not always be necessary. In such instances, we can directly employ a methodology to go from discrete input to discrete output. The tractability and practice of the methodology using such a philosophy is certainly valid. Let’s consider an example. In seismic data processing, the seismic data sets consist of synchronous records of reflected seismic signals registered by a large number of geophones (seismic sensors) placed along a straight line or in the nodes of a rectangular lattice on the earth’s surface. A series of explosions serve as the source of the initial seismic pulse, responses to which are averaged in a special manner. The vertical time axis forming the resulting two- or three-dimensional picture is identified with depth, so that the peculiarities of the reflected signal under the respective sensor carries information on the local properties of the rock mass at the respective point of the underground medium. In contrast to the above-lying sedimentary cover, the absence of pronounced reflecting surfaces in a crystalline body makes it difficult to infer the geological information from the basement interval of the seismic picture. We can see that layer description (or modeling) becomes a central problem. If we know a target layer in each horizontal or vertical (line) profile, we can get the entire layer in the 3D stratum. It can be transferred into a surface fitting problem where we can use a Coones surface, Bezier polynomial, or B-spline to fit the surfaces. Based on the boundary values to fill the interior, the most suitable technique is a Coones surface. However, for a layer, one must make two surfaces, one for the top of the layer and one for the bottom. The Coons surfaces have no property of preserving a fitted surface in the convex of a guiding point set. That is to say, the upper surface may intersect with the lower surface. That is not a desired solution. Since there are many sampled points on measured lines, the Bezier polynomial is also not a good choice. One cannot make the order of the polynomial very high. B-spline is a very good choice for the problem, but we need to do a pre-partition and coordinate transformations. In fact, for the problem, we have no special requirements for the smoothness, and we just need two reasonable surfaces to cover the layer. Another example is from computer vision. In observing an image, if you extract an object from the image, a representation of the object can sometimes be described by its boundary curve. If all values on the boundary are the same, then we can just fill the region. If the values on the boundary are not the same, and if we assume that the values are continuous on the boundary, then one needs a fitting algorithm to find a surface. How do we fill it? Its solution will directly relates to a famous mathematical problem called the Dirchet’s Problem and have direct application to in data compression. If the boundary is irregular, the 2D B-spline needs to partition the boundary into four segments to form a XY-plane vs. UV-plane translation. The different partitions may yield different results. Practically, the procedure of a computation is a set of discrete actions. The input of a curve is also discrete, and the output is discrete. We can, therefore, make the following arguments. Do we always need a continuous technique for surface fitting? Is it possible to have a discrete fitting algorithm to get a reasonable surface for the above problems? In 1989, L. Chen developed an algorithm to do such discrete surface fitting in 2D. The algorithm is called gradually varied fitting. Gradually varied fitting was based on so called gradually varied functions that is a type of digital functions in general sense. In 1986, A. Rosenfeld invented a basic type of the digital continuous function for the purpose of image segmentation where one is to find a continuous-looking part in a digital image, a digital space. This is book is written to different interest groups of readers. Chapters 1–3 are foundations for the entire book; Chapters 4 and 5 are for senior students, graduate students, or researchers who are interested in digital geometry and topology. Chapter 6 is a knowledge foundation for data reconstruction. Chapters 7 and 8 is for senior students in scientific computing. Chapter 9 will not be difficult for graduate students in computer science or senior students in mathematics with computer graphics background. Chapter 10 is for senior students in mathematics. Chapters 11 and 12 deal with future topics. For the Chapters marked with * may need some advanced knowledge. Digital Functions. Functions and Relations. Functions in Digital and Discrete Space. Gradually Varied Extensions. Digital and Discrete Deformation. Digital-Discrete Data Reconstruction. Basic Numerical and Computational Methods. Digital-Discrete Approaches for Smooth Functions. Digital-Discrete Methods for Data Reconstruction. Harmonic Functions for Data Reconstruction on 3D Manifolds. Advanced Topics. Gradual Variations and Partial Differential Equations. Gradually Varied Functions for Advanced Computational Methods. Digital-Discrete Method and Its Relations to Graphics and AI Methods

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chen l digital functions data reconstruction digital discrete methods 2013

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