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Groupwise MDL Algorithm

A groupwise method which uses a Minimum Description Length (MDL) formulation [] was described in Chapter 4. The main idea is that the complete set of images can be encrypted as a coded message, and the description length [] in bits used as an objective function. Rather than encoding the raw images, the encoding uses an appearance model, built using the estimated correspondences, to approximate the data; the encoding needs also to include details of the model itself and of the discrepancy between each image and its model approximation. As the registration proceeds, the correspondences, and hence the appearance model, are continually updated so as to minimise the description length.

It is important to emphasise that MDL in this case is not the same as in Chapter 5 where only a determinant was used to approximate MDL. Here, as shown in [], groupwise MDL-based registration is established by summing up the description length of the reference image, the description length of the parameters, the description length of image-model mappings, and the description length of the residuals. This is a sensible extension of the method described in Section 4.3 but it is applied to images and it produces a total description length comprising intensity information too.

The description length, $L_{total}$, can be formulated as follows:

$$L_{total}=L_{reference}+L_{parameters}+L_{data}+L_{residuals}$$ (8.1)

The full derivation of $L_{parameters}$ is specified in Chapter 4 and $L_{data}$ remains as before.