Theory: Matrices

Read CW[10] chapter 5.

Full Matrix

We can invert the matrix explicitly or use iterative techniques to solve the system of equations without fully inverting.

Common iterative techniques are:

• Jacobi
  
  $$\widehat{L}^{(k + 1)} = \widehat{M}\widehat{L}^{(k)} + \widehat{E}$$
  
  

• Gauss-Seidel
  
  $$\widehat{L}^{(k + 1)}_i = \sum_{j, j \neq i} \widehat{M}_{ij} \widehat{L}^{{\rm newer}(k \vert k+1)}_j + \widehat{E}$$
  
  

• More Sophisticated (aren't needed)

As the matrices are very expensive to store (${\cal O}(n^2)$), the matrix elements were computed as they were needed. The matrix needed to be accessed row by row, and (not by coincidence) the hemicube method computes an entire row at once.

Progressive Radiosity

Notice that both Gauss-Seidel and Jacobi look at a row (i) of the matrix and compute a new entry for $\widehat{L}_i$. The physical explanation for this is that we are computing a new estimate for basis function (surface) i by gathering energy from the rest of the environment. Initially, most of the environment is dark, cause much of the work in the initial iteration(s) to be wasted. Computing a single new entry requires ${\cal O}(n\log n)$ work assuming efficient visibility checking. Therefore it takes a lot of work to get to a reasonably converged solution. There are too many insignificant transfers of energy when doing a gather at each element. We want to reorder the compuation so that significant transfers are accounted for first.

Rather than gathering energy at each reciever, shoot from each source. Order the ``shots'' so that the ones with the most energy occur first.

• Surfaces can gather energy after they have shot.
• You can never have shot all of the energy in the environment.

Each patch (or basis function) holds to values (colors), one stores the radiance for the surface, the other holds the radiance on the surface that has not yet been shot. When energy is shot to a patch, both the total radiance and the unshot radiance are incremented by the amount of radiance striking it. After a patch shoots it's energy, the value of the unshot gets set to zero.

Patches are sorted by energy, not radiance, because a very tiny, yet bright surface will have less effect on an environment than a very large yet less bright surface.

As the solution progresses, the overall intensity of the environment increases. Ambient Approximation

Structured and Implicit Matrices

A matrix as a set of links.

• Link as a transfer
• Link can hold state
• Links can be stored with recievers.

Links to different levels of a subdivision hierarchy.

More Reading

Cohen Progressive Radiosity [8]

Gortler Radiosity and Relaxation Methods: Progressive Radiosity is Southwell Iteration [14]