- ... triplet'
^{1.1} - This is a term that the reader will probably not found in the literature but that we nonetheless find quite appropriate.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... parameters
^{1.2} - An infinite number of parameters could even be considered but is not treated in this work.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... numbers
^{2.1} - Historically, these sets were typed using simple bold font but since it was difficult to make bold fonts on black boards, mathematicians created the `double-bar' letters which were later adopted by typographers.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... algorithm
^{2.2} - Note that in the recent literature, the Levenberg-Marquardt algorithm is often confused with the Levenberg algorithm.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...#tex2html_wrap_inline34867#
^{2.3} - From now on, multiple knots are allowed.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...#tex2html_wrap_inline35207#
^{2.4} - Or
if we consider the full definition domain instead of the natural definition domain.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... positive
^{2.5} - Although it is unusual, it could be negative. Either way, it must be different of zero.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... residual
^{3.1} - The residuals and the errors must not be confused: the
*residuals*are the discrepancies between the the measurements and the fitted model (*i.e.*the predictions) while the*errors*are the discrepancies between the measurements and the actual underlying model (which is generally unknown in practice).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... distribution
^{3.2} - For many M-estimators, equation (3.20) does not define an actual probability density function since it does not sum up to one. In this case, we may talk of
*pseudo-distribution*but, for the sake of simplicity, we still refer to it as a distribution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... point
^{3.3} - The breakdown point of an M-estimator is a value between 0 and 1 that indicates the proportion of outliers the dataset can contain without altering the accuracy of the M-estimator. Even though the breakdown point may lie in , it is not really possible to have a breakdown point greater than 0.5 for obvious reasons.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... less
^{3.4} - Some of these definitions are completely specific to the hyperparameters while some others are more generic but conveniently explained in this section.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... `number'
^{3.5} - `Number' is probably not the right term since it includes, for instance, the strength of a regularization term.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... hyperparameters
^{3.6} - Practical considerations such as non-convexity of the criterion make this problem even more difficult even though such properties are really desirable for automatic optimization.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Allen
^{3.7} - The PRESS statistic is similar to the LOOCV criterion but for a cost function with a data term only (3,12)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... `distance'
^{3.8} - The term `distance' is abusive in this case since the Kullback-Leibler divergence is not an actual distance in the mathematical definition of it.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... necessity''
^{3.9} *entia non sunt multiplicanda praeter necessitatem*(as it was said in a time where English had not overruled everything else). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... same
^{3.10} - Except for a multiplicative positive constant which plays no role since the goal is to minimize the criterion.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... information
^{3.11} - Note that here we do not bother with the precision of the representation of which may be infinite since it is a real number but which may be reasonably approximated.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... grid
^{4.1} - As for standard digital pictures, extensions to non-uniform grids could be envisaged but we take the side of not considering them in this manuscript.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... invasive
^{4.2} - Although there are now active sensors that illuminates the scene using infra-red (or `near infra-red') wavelengths which are not visible to the human eye.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... PMDTec
^{4.3} `http://www.pmdtec.com`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Imaging
^{4.4} `http://www.mesa-imaging.ch`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... scene
^{4.5} - Contrarily to other approaches such as shape-from-texture or shape-from-shading which work only under some quite strong assumptions on the scene.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... map
^{4.6} - The data were retrieved from
`www.geoportail.fr`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... norm
^{4.7} - The word
*norm*is a bit abusive here: is not an actual norm of since the matrix is rank deficient.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... image
^{5.1} - The fact that a warp is naturally a function from the source image to the target image is revealed when we use a visualisation grid to display the warp.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Cairo
^{5.2} `http://cairographics.org/`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... points
^{6.1} - This algebraic distance is valid only if the point coordinates are normalized according to (94).
We did not introduced it in our equations for the sake of clarity.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... images
^{6.2} - Besides, it seems natural that the NURBS-Warp estimate is better than the other approaches since its parameter estimation starts from an initial solution which corresponds to the parameters of the best estimate among the three other approaches.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Gay-Bellile
^{A.1} - CEA LIST, Embedded Vision Systems Laboratory, Point Courrier 94, Gif-sur-Yvette, F-91191 France
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... functions
^{A.2} - In practice, images are
functions where is the number of channels. For the sake of simplicity and without loss of generality, we consider that in all the derivations of this article.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... warp
^{A.3} - Actually, it can also be a very close approximation, depending on how the matrix
is defined.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... parameters
^{A.4} - The original ESM algorithm uses a compositional update. The Feature-Driven framework naturally extends it to non-rigid warps.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... videos
^{A.5} - These videos can be downloaded at
`http://www.florentbrunet.com/ijcv2010`.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... above
^{A.6} - Except LOOP which naturally includes this additional step.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... comparison
^{A.7} - The compositional update of the warp is not approximated since an homography belongs to a group.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .