Volume-matching methodologies

Methodologies for matching volume data can be loosely divided into two categories:

Direct comparison

Volumes are represented as a grid of density values. One volume is rotated and translated with respect to the other, and the overlapping density values are compared, typically via a correlation coefficient. As the grids do not, in general, lie on top of each other, some interpolation of density values is required. The comparison may be done within a smaller masked region representing the overlap between the volumes (e.g. local correlation coefficient).

Pros:

• Works with original volume data as derived from experiment
• Full resolution of volume data is retained

Cons:

• Computational intensive, as map values at a large number of grid points (potentially a few million) need to be compared, for each trial rotation/translation.

Reduced representation

Alternatively, volume data can be coarse-grained in some way to make calculations faster. One method represents the volume as a set of anisotropic overlapping Gaussian functions. Another expands the volume as a series of orthogonal functions, for example spherical harmonics, and compares the coefficients of the expansion. The first step of volume matching involves finding the optimum representation in the reduced form, while the second step involves fast matching of the individual terms.

Pros:

• Matching the reduced representations of two volumes is considerably faster than matching the original maps

Cons:

• Need to decide the level of detail required for each volume
• Generating the optimum reduced representation may become the rate limiting step