Cell Boundary Smoothness

In this section, we analyze how the various methods for computing the volume rendering integral outlined in this dissertation behave across cell boundaries. As in the previous section, I solve the volume rendering integral for a set of parameters offline using the numerical solving capabilities of Mathematica [108] and compare those to computations performed on the actual graphics card. I used a Quadro FX 3000 graphics card for the GPU calculations.

The difference between the measurements in this section and those in the previous section is that those in this section take into account spatial effects. Viewers are unlikely to notice differences in color if they are all uniform. After all, there are uniform differences in color with different display media or with the parameters of the medium (for example the paper type in a printer or adjustment controls on a monitor). Furthermore, the response of a human's visual system is constantly changing with its environment.

Although the human visual system readily adjusts to uniform changes in light intensity, it is sensitive to changes in light intensity across its field of vision. This sensitivity is critical for segmenting a visual scene and helps us identify the size, shape, and orientation of objects. If the errors introduced by our approximations are not uniform, they may become noticeable. Therefore, in this section we analyze how the error may change spatially within the image.

The light receptors in the human eye are clustered together into ganglion cells [26]. The receptors in the center of each cell have a positive response to incoming light whereas those toward the edge have a negative response. When aimed at a constant field of light, the positive and negative receptors cancel each other out. When aimed at a varying field of light, the positive and negative receptors may contribute in the same way, which enhances the effect of the change. These enhancements generated by our visual system are Mach bands.

Figure 1.12: Model used to study Mach bands caused by approximation errors.

Figure 1.12 shows the model I used to study how approximation errors may cause fluctuations in colors across the viewing plane, which could induce the human visual system to create mach bands. The model is such that from the viewpoint the volume has a uniform length. The front and back faces of the model each have constant volume parameters. The color of the volume should be constant from the viewpoint, but approximations may cause the color to vary. For each approximation, I plot the output intensity across the face.

Figure 1.13: Ganglion receptor response.

I also plot the convolution of the light intensity with an example ganglion receptor response function shown in Figure 1.13.

Figure 1.14: Color output from model shown in Figure 1.12. The attenuation is constant. In the red plot, the luminance is constant. The luminance is full in the front and zero in the back for the green plot and vice versa for the blue plot. The dashed plots show the convolution with the ganglion response of Figure 1.13.

(a) Average Luminance and Attenuation.	(b) Partial Pre-Integration.	(c) Linear Luminance and Attenuation.

Figure 1.15: Cell boundaries with large attenuation in the back. In the red plot, the luminance is constant. The luminance is full in the front and zero in the back for the green plot and vice versa for the blue plot. The dashed plots show the convolution with the ganglion response of Figure 1.13.

(a) Average Luminance and Attenuation.	(b) Partial Pre-Integration.	(c) Linear Luminance and Attenuation.

Figure 1.16: Cell boundaries with large attenuation in the front. In the red plot, the luminance is constant. The luminance is full in the front and zero in the back for the green plot and vice versa for the blue plot. The dashed plots show the convolution with the ganglion response of Figure 1.13.

(a) Average Luminance and Attenuation.	(b) Partial Pre-Integration.	(c) Linear Luminance and Attenuation.

Figures 1.14, 1.15, and 1.16 show the output of our model with various ray integration methods that linearly interpolate attenuation and with various volume parameter combinations. The Average Luminance and Attenuation model has noticeable spikes when convolved with the ganglion response. The other two models have no noticeable fluctuations across the visual field.

Figure 1.17: Color output from model shown in Figure 1.12. The opacity is constant. In the red plot, the luminance is constant. The luminance is full in the front and zero in the back for the green plot and vice versa for the blue plot. The dashed plots show the convolution with the ganglion response of Figure 1.13.

(a) Average Luminance and Opacity.	(b) Linear Luminance, Average Opacity.	(c) Linear Luminance and Opacity Approx.

Figure 1.18: Cell boundaries with large opacity in the back. In the red plot, the luminance is constant. The luminance is full in the front and zero in the back for the green plot and vice versa for the blue plot. The dashed plots show the convolution with the ganglion response of Figure 1.13.

(a) Average Luminance and Opacity.	(b) Linear Luminance, Average Opacity.	(c) Linear Luminance and Opacity Approx.

Figure 1.19: Cell boundaries with large opacity in the front. In the red plot, the luminance is constant. The luminance is full in the front and zero in the back for the green plot and vice versa for the blue plot. The dashed plots show the convolution with the ganglion response of Figure 1.13.

(a) Average Luminance and Opacity.	(b) Linear Luminance, Average Opacity.	(c) Linear Luminance and Opacity Approx.

Figures 1.17, 1.18, and 1.19 show the output of our model with various ray integration methods that linearly interpolate opacity and with various volume parameter combinations. The Average Luminance and Opacity and Linear Luminance, Average Opacity Approx models both have significant spikes when convolved with the ganglion response. In contrast, the Linear Luminance and Opacity model has little fluctuation, even when convolved with the ganglion response.

Although I have addressed spatial effects of image error in this chapter, the errors I quantify may still not be indicative of the error perceived. Ganglion cells and Mach bands are but a small part of the human visual system. In fact, there is still much we do not understand about the human visual system. To get a true measure of how well humans are able to detect the errors discussed in this chapter, one must generate a set of images from different ray integration methods and compare them either by human experiment or with image quality metrics based off human experiment [23,65,96].