Windowing

If a signal is sampled below the Nyquist frequency the replicas of the fundamental period in the frequency domain overlap. Thus the perfect reconstruction of the original function is not possible. For more details see Section 2.1.2. The errors introduced when reconstructing a signal sampled below the Nyquist frequency are called aliasing. Unfortunately in practice it is often the case that signals are not band limited and therefore the Nyquist frequency is infinite. Every discrete sampling of such a not band limited signal introduces aliasing artifacts by definition. One possible appearance of these aliasing artifacts is the Gibbs phenomenon [27]. The Gibbs phenomenon is an overshooting of the reconstructed function which appears around discontinuities of the sampled function. It is also referred to as ringing. The appearance of the Gibbs phenomenon can be decreased through a multiplication of the Fourier series representation of the signal with a weighting window function.

Some commonly used windows are:

Rectangular

$$\hat{W}_N[\nu] = \left\{ \begin{array}{ll} 1 & 0 \leq \nu \leq M \\ 0 & otherwise \end{array} \right.$$ (3.19)

Bartlett(triangular)

$$\hat{W}_N[\nu] = \left\{ \begin{array}{ll} \frac{2\nu}... ...rac{M}{2} < \nu \leq M \\ 0 & otherwise \end{array} \right.$$ (3.20)

Hanning

$$\hat{W}_N[\nu] = \left\{ \begin{array}{ll} 0.5-0.5 \co... ...) & 0 \leq \nu \leq M \\ 0 & otherwise \end{array} \right.$$ (3.21)

Hamming

$$\hat{W}_N[\nu] = \left\{ \begin{array}{ll} 0.54 - 0.46... ...) & 0 \leq \nu \leq M \\ 0 & otherwise \end{array} \right.$$ (3.22)

Figure 3.1: (a)The Stanford Bunny rendered with the new Fourier based rendering algorithm. Ringing artifacts are especially visible parallel to the hind leg. An application of the Hamming window in all three dimensions in the frequency domain representation of the dataset before starting the rendering process smooths these effects. (b)demonstrates the obtained rendering result after windowing. The images of the first row were rendered with a zoom factor of 4 and the enlarged sections with a zoom factor of 20.

(a)	(b)

Figure 3.1 displays two images which where rendered with the rendering algorithm introduced in this work. Figure 3.1(a) shows the rendering result obtained by direct application of the proposed algorithm. Strong ringing artifacts are visible parallel to the hind leg. The image in Figure 3.1(b) was rendered after an application of the Hamming window in all three space directions in the frequency domain representation of the dataset. The ringing artifacts are very much reduced, but also some detail in the image is smoothed out. This new volume rendering algorithm is presented in detail in the following section. The Fourier transforms, theories and methods introduced in this section are put together to perform important parts of this new method.