Monotonic sampling of a continuous closed curve from its Gauss digitization. Application to length estimation

Étienne Le Quentrec (1) , Loïc Mazo (1) , Étienne Baudrier (1) , Mohamed Tajine (1)

1 ICube - Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie

Our study is about perimeter estimation of continuous shapes based on their digitization (i.e. their pixelated representation). To study the estimation error, we are led to propose the notion of monotonically sampling back-digitization, a mapping between the boundary of a digitized Jordan curve and the Jordan curve itself. This mapping ensures that the digital curve and the continuous curve are traveled together in a cyclic order manner. Then, we use back-digitization to prove the multigrid convergence of perimeter estimators that are based on polygons inscribed in the digitization. Furthermore, convergence speed is given for this class of estimators. The convergence is guaranteed provided the polygon edge lengths tend toward 0 and their edge sizes relative to the grid step tends towards infinity (thus the sampling of the digital boundary has to be both tight and sparse). All the proofs are established in the framework of locally turn bounded curves (LTB) presented in the previous article Le Quentrec et al., Local turn-boundedness: A curvature control for a good digitization, DGCI 2019. If, moreover, the continuous curves also have a Lipschitz turn, an explicit error bound is calculated. Besides, an equivalence between LTB-curves with a Lipschitz turn and curves of class C 1,1 is subsequently established.

Mots clés: digitization; integral curvature; turn; Jordan curve; perimeter discrete estimation

Domaines: Traitement des images [eess.IV]; Mathématiques générales [math.GM]

Étienne Le Quentrec, Loïc Mazo, Étienne Baudrier, Mohamed Tajine. Monotonic sampling of a continuous closed curve from its Gauss digitization. Application to length estimation. 2021. ⟨hal-02987858v2⟩