Synthetic Test Datasets

To test the quality of the new interpolation method, the test function introduced by Marschner and Lobb [22] is used. This function, $\rho _{ml}$ in Equation 6.1, is defined in three-dimensional space and sampled at 20 samples per unit distance in each dimension. The range for $x, y, z$ is $-1 < x,y,z < 1$, with the constant values $f_M = 6$ and $\alpha = 0.25$. This continuous volume is discretized with $41$ by $41$ by $41$ samples, which captures $99.8\%$ of the signals energy.

$$\rho_{ml}(x,y,z) = \frac{1- \sin(\frac{1}{2}\pi z) + \alpha(1+ \rho_r(\sqrt{x^2 + y^2}))} {2(1+\alpha)}$$ (6.1)

with

$$\rho_r(r) = \cos(2\pi f_M\cos( \frac{1}{2}\pi r))$$ (6.2)

Figure 6.1 displays an image of the $\rho _{ml}$ function, rendered with an iso-value of $0.5$, a $10.0$ times zoom and a sampling distance along the viewing rays of $0.05$. The function $\rho _{ml}$ was evaluated for every sample point during the rendering of this image. A standard test to judge the quality of a reconstruction filter is to render the sampled $\rho _{ml}$ function and compare it to this reference image.

Figure 6.1: Characteristics of the synthetic datasets.

Description:

        Synthetic test function introduced by

        Marschner and Lobb [22].

        With a sampling of $41 \times 41 \times 41$

        $99.8\%$ of the signals energy is captured.

Dimension Original:

         $41 \times 41 \times 41$ double precision

Dimension Mirrored Version:

         $41 \times 41 \times 80$ double precision

The $\rho _{ml}$ dataset is sampled almost with Nyquist frequency. The conduct of the function in Z direction, with $x$ and $y$ set to the constant values $c_1$ and $c_2$, is a sine wave with low frequency (see Equation 6.3), only shifted by a constant value $c_3$ (depending on the values of $x$ and $y$).

$$\rho(c_1,c_2,z) = \frac{1- \sin(\frac{1}{2}\pi z) + \alpha(1+ \cos(2 \pi f_M cos(\frac{1}{2}\pi \sqrt{c_1^2 + c_2^2}))} {2(1+\alpha)}$$ (6.3)

$$= - \frac{\sin (\frac{1}{2}\pi z)}{2(1+\alpha)} + c_3$$ (6.4)

The assumption when using the discrete Fourier transform (DFT) (Section 3.1) is that the signal is periodic and discrete in spatial and frequency domain. Through the sampling of $\rho _{ml}$ we get one period of this signal, which is half the period of a $sine$ wave. In the zone between two of these periods (see Figure 6.2(a)), a jump in the signal is present. This discontinuity leads to high frequency components and a not band-limited behavior. In order to avoid this problem, and to maintain compatibility with the original signal $\rho _{ml}$, the sample points are mirrored along the $z = -1$ plane (see Figure 6.2(b)). The samples at $z = -1$ are not copied, because they are positioned at the plane of reflection. Further the samples at $z = +1$ are not copied as well to create smooth transitions in the periodic sequence of sine waves. This creates a extended version of the test dataset with the dimensions of $41 \times 41 \times 80$ samples. This extended test dataset is used for the render benchmarks and is referred to as $\rho _{mlext}$.

Figure: Density characteristic of the test function in Z direction, (a) in the original version $\rho _{ml}$ and (b) after mirroring the signal $\rho _{mlext}$.

(a)	(b)