Although most of our notation is compliant with the international standard ISO 31-11 (183), we feel that it is appropriate to give the details of the notation most commonly used in this manuscript. We also give some basic definitions concerning, for instance, some standard operators or matrices.

(2.1) |

(2.2) |

In practice, the dependency on is omitted when it is obvious from the context. Consequently, is often shortened to or even .

For a vector-valued function , the counterpart of the gradient is the Jacobian matrix, denoted . If the components of are denoted then the Jacobian matrix is defined as:

(2.3) |

As for the gradient, the point where the Jacobian matrix is evaluated is omitted when it is clear from the context.

The Hessian matrix of a scalar-valued function is the matrix of the partial derivatives of second order. This matrix is denoted and it is defined by:

(2.4) |

As for the gradient and the Jacobian matrix, we consider that the notation is equivalent to the notation when the vector is obvious from the context.

*Real intervals* are denoted using brackets: is the set of all the real numbers such that
.
The scalars and are called the endpoints of the interval.
We use outwards-pointing brackets to indicate the exclusion of an endpoint: for instance,
.

*Integer intervals* (also known as *discrete intervals*) are denoted using either the double bracket notation or the `three dots' notation.
For instance, the integer interval
may be written
or
.

*Collections*, *i.e.* grouping of heterogeneous or `complicated' elements such as point correspondences are denoted using fraktur fonts (*e.g.*
).

Matrices and Vectors

(2.5) |

The bracket notation is used when the matrix is defined with `blocks',

(2.6) |

(2.7) |

The matrix of size filled with zeros is denoted . The subscripts in the notation and are often omitted when the size can be easily deduced from the context.

(2.8) |

The operator deals with diagonal matrices. The effect of this operator is similar to the one of the diag function in Matlab. Applied to a vector , it builds a matrix such that:

(2.9) |

Conversely, when applied to a square matrix , the operator builds a vector that contains the diagonal coefficients of :

(2.10) |

(2.11) |

(2.12) |

Note that the 1-norm is also known as the

(2.13) |

The

(2.14) |

Note that the maximum norm corresponds to the -norm when .

(2.15) |

The Frobenius norm of a matrix is related to the Euclidean norm of a vector in the sense that they are both defined as the square root of the sum of the squared coefficients. In fact, we have the following equality:

(2.16) |

Contributions to Parametric Image Registration and 3D Surface Reconstruction (Ph.D. dissertation, November 2010) - Florent Brunet

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