A major problem in volume visualization is the 3D reconstruction of the sampled function. A lot of researchers from various fields like signal processing, image processing and recently also volume visualization are working on this problem. Some literature which is important for our purposes is reviewed in Chapter 2, to see what is the state of the art already. In this chapter we will see that interpolation (i.e., reconstruction) can be interpreted as linear filtering. So we can assess the reconstruction quality by investigating the corresponding filter, which will be the basis of our work in later chapters.
Although the process of reconstruction seems to be more important from a volume visualization point of view, the sampling process and its consequences are important for an overall understanding also. Chapter 3 therefore deals with sampling on its own. It will turn out that the interpretation of the sampling process in frequency domain is useful for understanding and analysis. So several functions which describe sampling mathematically and the Fourier transform to move between spatial and frequency domain will be subject of discussion there.
In Chapter 4 we will eventually take a closer look at the reconstruction process. We will see that (under certain conditions) the sampled function can be (theoretically) perfectly reconstructed by convolving it with the sinc function. However, in practice this is usually not possible so that we will take a closer look at practical reconstruction methods. First we will discuss simple but nevertheless (or probably just therefore) quite popular interpolation methods like nearest neighbor and linear interpolation as well as the also very popular and a little bit more sophisticated cubic splines. After that we will examine a method to use the sinc filter in practice, which is called windowing. These windowed sinc functions (or better its derivatives) will also be our primary concern in later chapters. Further, a method to interpolate in frequency domain (by zero insertion) and to asses reconstruction quality of various filters in spatial domain (by a Taylor series expansion of the convolution sum) will be explained in this chapter.
Derivative reconstruction will be discussed in Chapter 5. The derivative of a three-dimensional scalar function at a particular point, called the gradient, is a quite important concept because it can be interpreted as normal to an iso-surface passing through the point of interest. The normal is quite often used in volume visualization algorithms for shading and classification and it is shown by Möller et al. that it has a ``much higher impact on the rendered volume than the interpolation filter'' [40]. The derivative of the function could also (theoretically) be perfectly reconstructed by the cosc function, which is the derivative of the sinc function. However, in practice other gradient estimation schemes like central differences (with some kind of interpolation) or cubic spline derivatives are more often used. Nevertheless, it is also possible to window the cosc function which we will do and show that it often gives visually more appealing results than the standard techniques. These derivative reconstruction filters then are evaluated in three ways. First, by simply comparing the results visually. Second, by comparing their frequency responses with the ideal one, and third, by evaluating their numerical accuracy using a Taylor series expansion of the convolution sum. Further, we will derive the cosc function yielding the curc function which will, anyway, turn out to be not so suitable for reconstruction of curvature properties.
As some of the investigated filters we discuss are quite expensive to evaluate, we will take a closer look at the consequences of sampling the filter and using this sampled filter for reconstruction in Chapter 6. First we will give an object-oriented implementation scheme. Then we will evaluate these sampled filters again in three ways, by comparing results visually, by taking a look at the frequency response and by assessing the numerical error in spatial domain. As a result we will give two criteria how to properly sample a filter so that interpolation and (anti-)symmetry properties are retained.
The implementation will shortly be discussed in Chapter 7, a short summary of the whole thesis is given in Chapter 8 and, eventually, the conclusions drawn throughout this work will be summarized in Chapter 9. Further, some perspectives for future work will be suggested in Chapter 10. In Appendix A a glossary of important terms used throughout this thesis is given and additional results to Chapters 4 and 5 are shown in Appendix B and C, respectively.
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