The next part of the process is reflectance. That is where the incoming illuminance reflects off an object. We are very familiar with the ability of shiny surfaces or mirrors to reflect light.
In high school physics we learn about the rules of reflection from such a surface, where the angle of incidence equals the angle of reflection and we refer to this as specular reflection. Now this is not saying anything about the color change that occurs during reflection, it is simply talking about the geometry of the reflection process.
We refer to this as specular reflection, which is related to the Latin word for mirror. We met William Herschel before, who is the guy who discovered infra-red radiation and he made the big mirror in his telescope from a type of metal that was called speculum metal — it was a reflective alloy of copper and tin and it was used in the old days for making telescope mirrors.
A more realistic model for the way light is reflected off objects is what is called Lambertian reflection, and Lambertian reflection differs from specular reflection in that the incoming light is reflected off in all manner of directions. This is the sort of reflection that occurs from a dull or matt surface. Sometimes called diffuse reflection and it is named after Johann Lambert, who was a physicist in the 1700s. The intensity of the light that is reflected from the surface is related to the angle of reflection, and what is interesting with Lambertian reflection is that it leads to the brightness being invariant to the observer’s angle of view.
Now the moon is a really good example of Lambertian reflection. We know that the moon is a sphere, but when we look at it, it appears to be a fairly uniformly bright disk, apart from the craters and the seas. So the reason that it appears is to be uniformly bright even though the light has been reflected, is proportioned to the cos sign of the angle, is that the area of the region that we are looking at appears to change in size. Here is an exaggerated example: I take this particular patch of the moon and most of the light from that patch is being reflected directly up, it corresponds to an area something like this. If I take a patch that is down near the pole it is clearly going to be inclined with respect to my field of view. So it is going to have one of these rays down here associated, so it is going to be reflecting less light, but the area—the apparent area of this patch—is smaller because I am looking at it obliquely. So if I consider the brightness to be the ratio of the light being emitted to its surface area, this is invariant to the view angle.
So we have just looked at the geometry of the reflection, the direction of the rays being reflected form the surface as a function of the incoming ray. Let’s talk now about how reflection changes the color of incoming light. Here we see a red house brick, the plot here shows what we call the reflectance of the house brick; that is, the fraction of light that is reflected form the brick as a function of wavelength. And it is quite a complex curve, we see that there are some parts of the spectrum where it reflects a large amount of light and there are other parts of the spectrum where it reflects very little light.
The right hand graph shows a zoom into the visible part of the spectrum.
Now we can see that the red brick reflects not very much blue light in the region, say 400 to 500 nanometers, but it does reflect a fair amount of the red light that is falling down, say the light between 600 and 700 nanometers, and that is why it looks red to us, it reflects more red light then it reflects blue light.
So this is the spectrum of a red brick, different minerals have different reflectance spectrums and here we see the reflectance spectrum for a number of minerals. If we have an instrument that can measure the reflectance at a number of different wavelengths, an instrument perhaps like a spectrometer or a hyper spectral camera, then we can measure one of these particular curves and we can match it against a library of known minerals and use it to identify what minerals are within the scene.
So to recap, incoming light, we refer to as the illuminance and it has got a spectrum E, which has got a function of the wavelength lambda, and we have the reflectance of the object itself which we refer to as R and it is also a function of lambda.
So the light that leaves the apple, which we refer to as the illuminance of the apple, I am going to use the symbol L and it is also a function of the wavelength, now L at any particular lambda is equal to E at that same lambda multiplied by R at that particular lambda. That is the spectrum of the light that is leaving the apple, the function of the spectrum of the incoming light, and the function of the reflectance of the apple.
Most objects reflect the light that falls on them and there are two aspects to this reflection. The first is geometric and concerned with the directions of the light rays: it can be specular reflection from a mirror like surface, or scattered Lambertian reflection from a matte surface. The second is the reflectance function which describes how the amount of light reflected depends on the wavelength, a red apple reflects red light but not blue.