The correlation coefficient, invariant under linear transformations of the voxel intensities, is widely used in mono-modal image registration. The inter-subject variability of diffusivity values motivates the introduction of a generalized correlation coefficient, invariant under these differences. In DTI, this variability has been reported and partially accounted for in some registration methods [

14]. The correlation coefficient between blocks

*F* and

*G* is defined as the scalar product of the normalized blocks:

where μ

_{F} is the mean of the image values in the block. It is invariant if

*F* (and/or

*G*) is replaced by

*aF* +

*b*. It has been generalized to vector images by redefining the means μ

_{F} and μ

_{G} as the projection of the block onto a constant block

*T* [

9]:

The corresponding generalized correlation coefficient is invariant if

*F* is replaced by

*aF* +

*bT* where

*T* is now any constant vector block. The definition of a scalar product between two blocks of mixture models seems impractical if not impossible. We therefore further generalize the correlation coefficient by substituting the inner product by a more general scalar mapping,

:

where

is a generalization of the norm. This definition does not guarantee the invariance property of the metric for any scalar mapping. One can show that the invariance is preserved as long as the scalar mapping is linear with respect to the constant block

:

To preserve the interpretability of ρ as a similarity metric, it needs to be symmetric, equal to one in case of perfect match and lower than one in any other case. These constraints on ρ translate into the following constraints on

*m*:

The latter is a generalized form of the Cauchy-Schwartz inequality for inner products. Conditions (

6–

9), the choice of

and the definition of the multiplication by a scalar and addition of the block

, stand together as a model to generate correlation coefficients in potentially any space. For DTI, if

is an isotropic tensor block

,

*m* is the log-Euclidean scalar product, and the log-Euclidean algebra is used, then ρ is invariant under linear transformations of the eigenvalues in the log-domain. For multi-tensor images, we fix

, and we define the addition,

, and the multiplication by a scalar,

, component-wise in the log-domain. The scalar mapping

is defined by pairing the tensors in each voxel to maximize the linear combination of pairwise scalar products. Let

and

defined on a domain Ω, we have:

where π is a pairing function associating one tensor of

to each tensor of

. This scalar mapping satisfies conditions (

6–

9). Interestingly, the resulting generalized correlation coefficient is invariant under any global (within the block) linear transformation of all eigenvalues in the log-domain. This similarity metric is therefore robust to the inter-subject variability of diffusivities.