Per Wavelength Calculation

I give the volume rendering integral (Equation 1.2 derived in Section 1.1 on page ) as a scalar function. That is, I define the input luminance and attenuation variables as single scalar values, as is the output light intensity. Although such a calculation is correct for a monochromatic image, most images will benefit from full color.

Visible light actually comprises a continuous spectrum of light waves with varying wavelengths [1]. Each wavelength stimulates the receptors in the human eye differently. A cloud attains its color by having different responses to each wavelength of light. It is therefore often practical to define the luminance and attenuation properties of a volume on a per wavelength basis.

Given volume properties defined on a per wavelength basis, how does this affect the volume rendering integral? As light travels through or bounces off a material, it is generally true that the wavelength of a given ray of light does not change and that one ray of light has no effect on other rays of light. Thus, the produced light intensity of a particular wavelength, $I_\lambda$, depends only on volumetric properties for that particular wavelength, $L_\lambda$ and $\tau _\lambda$. Therefore, technically we should express the volume rendering integral as

$${ I_\lambda(D) = {I_\lambda}_0 e^{-\int_0^D \tau _\lambda(t) dt} ... ...\int_0^D L_\lambda(s) \tau _\lambda(s) e^{-\int_s^D \tau _\lambda(t) dt} ds } }$$ (3)

However, because we have defined everything on a per wavelength basis, and because all calculations are wavelength independent, the $\lambda$ subscript is redundant, and I drop it from further equations.

Given this ``new'' per wavelength form of the volume rendering integral, a question arises: For how many wavelengths should we compute the volume rendering integral? It is impractical and unnecessary to perform the calculation for the continuous spectra of visible light. Furthermore, because speed is paramount, the fewer wavelengths we have to compute the better.

The answer is three. Any color space needs only three parameters to represent any color perceptible by humans. The color photoreceptors of the human eye come in only three flavors, and we perceive color by the amount each receptor type is stimulated [26]. Therefore, we can combine light comprised of only three different wavelengths with different intensities to reproduce any color discernible with the human eye. The wavelengths for red, green, and blue light are a good choice [27]. This is the RGB color space.

In addition to representing colors in RGB color space, I shall perform calculations only on red, green, and blue wavelengths. Although we can represent every color with only red, green, and blue wavelengths, a volume may have material properties that we do not capture on only these wavelengths. This can introduce inaccuracies. For example, although equal parts of red and green light are perceived as yellow, a ray of yellow light passing through a volume may behave differently than equal parts of red and green. Although there has been a bit of exploration in capturing these effects [3,35], the effects are of limited value and most research (including this dissertation) ignores them.