Paint, moondust, and electrons

Why do objects look the way they do? Why does a basketball look round, while the moon looks like a flat disk? If white objects supposedly reflect all light, why don't white-painted walls look like mirrors?

And when we see pictures taken with scanning electron beams, why is it so easy to interpret them as landscapes and artifacts? Why is it so much harder to see what's going on under a standard light microscope?

The answer is that different materials reflect radiation in different ways. In our daily life, we come across several different familiar types of materials and lighting conditions. For example:

And of course there are less familiar materials and lighting conditions:

Minnaert's reflectance family

The astronomer Marcel Minnaert invented a family of functions that captures the reflectance behaviors of materials as diverse as paint, moondust, and objects viewed under an electron microscope. The function models how radiation (e.g. visible light or electrons) impinging on a surface from one direction gets transmitted in another direction, based on the material the surface is made of. It's a function relating four variables, giving the radiance (officially, the energy flux per foreshortened area per solid angle) emitted in a particular emission angle based on the energy coming in from the incident angle, as a result of the surface material. The function is: $$L = \frac{k+1}{2\pi}[\cos{\theta_i}]^{k}\cdot [\cos{\theta_e}]^{k-1}$$ where \(L\) is the radiance, \(\theta_i\) and \(\theta_e\) are the angles at which the light is incoming and emitted relative to the surface normal direction, and \(k\) is a parameter that encodes something about the material and lighting conditions. The important benchmarks for the material constant \(k\) are:

A four-variable nomogram for Minnaert's reflectance family

Here, I've plotted Minnaert's reflectance family as an oriented nomogram with transparency. (For details, see some of the other nomograms of this type which I've made.) This nomogram relates radiance, incident and emitted angles, and the surface parameter. You can use this nomogram to solve for any one parameter given the other three, and also to see qualitatively how the parameters affect each other. For example, to solve for radiance given the other three parameters:
  1. On the vertical axis, locate the material parameter \(k\). Click and drag the teal (radiance) axis to that height.
  2. At the top of the nomogram, locate the dark curve corresponding to the cosine of the incident angle.
  3. At the bottom of the nomogram, locate the red curve corresponding to the cosine of the emission angle.
  4. Follow the two curves until they reach the height of the teal (radiance) axis.
  5. The horizontal separation between the two curves corresponds to the radiance. We can use the transparent movable overlay to determine the radiance quantitatively.
  6. Slide the teal (radiance) axis left or right until the red calibration dot is on the grey incident angle curve.
  7. Find where the teal scale intersects the red emitted angle curve. Read off the teal scale's value at that point.
For example, the following picture shows how to find the radiance given \(k=0.5\) and the incident and emitted angles both have a cosine of 0.3. First, you find the material parameter on the vertical axis (0.5). Next, you drag the teal (radiance) axis to that height. Third, you find the grey curve corresponding to the incident angle (0.3), and the red curve corresponding to the emitted angle (0.3). Follow those curves to where they meet the teal (radiance) axis. Fourth, you slide the teal scale until the red calibration dot is over the grey incident angle curve, as shown. Fifth, you find where the teal axis meets the red curve. Reading off the teal scale at that point, you solve for the luminance: it is 3/(4π), or around 0.23. As for qualitative trends, you can explore the following: For more information, see Prof. Berthold Horn's detail-rich textbook Robot Vision, which uses physical models to find closed-form analytic and/or computationally inexpensive solutions to problems in vision.