177 6.2.1.3 Construction d'un maillage ,
, , p.181
,
, , p.185
, Restauration d'images de profondeur par des splines d'interpolation 195
,
204 7.3.1 Résultats préliminaires pour les familles de changements d'échelle, Construction des changements ,
,
,
, Rappels et notations
,
opérateur ?, au vecteur ? et au paramètre ? les solutions (si elles existent) ? ? (f ) du problème de minimisation suivant Trouver ? ? (f ) ? H m (?) tel que ?v ? H m (?) ,
La spline d'ajustement ? ? (f ) appartient à l'espace des fonctions splines S introduit en (6.3.11) et est également solution du problème variationnel suivant Trouver ? ? (f ) ? H m (?) tel que ?v ? H m (?) , ? ? ? (f ), ? v R N + ? ? ? (f ), v m,? = ? ,
, L'espace des fonctions splines S contient donc à la fois les fonctions splines d'interpolation et d'ajustement. Il ne dépend que de la semi-norme | · | m
On s'intéresse dans ce paragraphe à la convergence de l'approximation ,
, Le premier résultat montre que lorsque ? ? 0, la spline d'ajustement sur ?, ? ? (f ), relative à ?, ? et ? est une approximation de la spline d'interpolation sur ?, ?(f )
on peut montrer que comme dans le cas de l'interpolation, la spline d'ajustement ? ? (f ) converge ,
196 7.2 Principe général de la méthode d'approximation ,
204 7.3.1 Résultats préliminaires pour les familles de changements d'échelle, Construction des changements ,
,
,
, On peut montrer que le problème (7.4.4) admet une solution unique appelée spline d'interpolation sur
, Étape 2. La deuxième étape de la démonstration consiste à montrer que toute sous-suite extraite de la famille ? d ? ? d (? d ? f ) d?D , convergeant fortement dans H r?? (?), converge vers la fonction ? ? (? ? f )
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