Theory: Error

What is Error and Where does It Come From?

Read CW[10] pp 152-166.

  

Figure 2: The various discontinuities that are important to handle correctly.

General features of radiance functions include large areas that are smoothly changing, and the occasional discontinuity formed either by objects in contact with the receiver (D⁰), or occluders between the light and the receiver (D¹ and D²) (see Figure  3.1.1. D¹ discontinuities are formed by an edge of the light lining up exactly with the edge of an occluder. D² discontinuities are formed by a single vertex of the light or occluder lining up with an edge of the occluder or light. See Lischinski etal.[23] for more details.

The approximation doesn't (and usually can't) match the real function. In order to determine how far off from the real function we are, we need to estimate the norm of the function representing the difference between the real function and the approximation $\left\vert \left\vert L - \mbox{$\widetilde{L}$ } \right\vert \right\vert$ which tells us the magnitude of the difference.

Note that different norms give different magnitudes to the same difference, such as total variation in energy (L₁ norm) or maximum variation in radiance ($L_{\infty}$ norm).

The value returned by the norm will be called the error.

How do We Estimate It?

Bounding Norms for constant basis functions

Norms(metrics) for Linear basis functions. Practice and theory.

Global error versus local error.

Perceptual error metrics such as contrast and more sophisticated [37].

Problems with using metrics instead of norms. Norms relate to actual error. Metrics may only be able to give an ordering. This is due to norms being linear with respect to scalar multiplication, and metrics not requiring this.

More Reading

The paper by Walters[37] deals with creating a good mesh from a set of densities. It's not the same as assignment 1, but it does have some of the same issues. It can be found online here.