Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Équipe A DAGIO LORIA Campus Scientifique - BP 239 54506 Vandoeuvre-lès-Nancy Cedex, France L AMA Bâtiment Chablais, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France 1 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Outline Introduction 1 Arc segmentation 2 Unsupervised Noise Detection 3 A framework for arc recognition along noisy curves 4 Experimentations 5 Conclusions and futur work 6 2 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Outline Introduction 1 Arc segmentation 2 Unsupervised Noise Detection 3 A framework for arc recognition along noisy curves 4 Experimentations 5 Conclusions and futur work 6 3 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Introduction Motivation Reel image Arc and circle are basic objects in discrete geometry. ⇒ The study of thes primitives are important. Arc and circle appear often also in images. Due to the effect of aquisition phase, there is often noise in images ⇒ The detection, recognition of these primitives in noisy condition are interesting topic in pattern recognition. 4 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Introduction Document graphic Motivation Arc and circle are basic objects in discrete geometry. ⇒ The study of thes primitives are important. Arc and circle appear often also in images. Due to the effect of aquisition phase, there is often noise in images ⇒ The detection, recognition of these primitives in noisy condition are interesting topic in pattern recognition. 4 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Discrete circle Discrete circle Définition Basic object in discrete geometry. A discrete circle ([Kim84]) is constructed from digital points that are are the most Based on the discretization of a reel nearest and interior in a discrete circle. circle. Existing definitions Kim’s definition Nakamura’s definition Andres’ definition C. E. Kim. Digital disks. Pattern Analysis and Machine Intelligence, IEEE Transactions on , PAMI-6(3) :372–374, May 1984. 5 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Discrete circle Discrete circle Définition Basic object in discrete geometry. A discrete circle ([Nakamura84]) is a sequenque of digital points that are Based on the discretization of a reel nearest a discrete circle. circle. Existing definitions Kim’s definition Nakamura’s definition Andres’ definition A. Nakamura and K. Aizawa. Digital circles. Computer Vision, Graphics, and Image Processing , 26(2) :242–255, 1984. 5 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Discrete circle Discrete circle Définition Basic object in discrete geometry. A digital circle ([Andres95]) is a sequence of digital points that verifies : Based on the discretization of a reel ( R − w 2 ) 2 ≤ ( x − x 0 ) 2 +( y − y 0 ) 2 < ( R + w 2 ) 2 } circle. Existing definitions Kim’s definition Nakamura’s definition Andres’ definition E. Andres. Discrete circles, rings and spheres. Computers & Graphics , 18(5) :695–706, 1994. 5 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Outline Introduction 1 Arc segmentation 2 Unsupervised Noise Detection 3 A framework for arc recognition along noisy curves 4 Experimentations 5 Conclusions and futur work 6 6 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Tangent space representation [Arkin91], [Latecki00] Input Output C = { C i } n i = 0 is a polygonal curve We consider the transform that associates polygon C of Z 2 to a polygon of R 2 constituted with α i = ∠ ( − C i − 1 C i , − − − − → − − − → by the segments T i 2 T ( i + 1 ) 1 , T ( i + 1 ) 1 T ( i + 1 ) 2 for i C i C i + 1 ) from 0 to n − 1 width l i is the length of segment T 02 = ( 0 , 0 ) C i C i + 1 . T i 1 = ( T ( i − 1 ) 2 . x + l i − 1 , T ( i − 1 ) 2 . y ) , pour i α i > 0 if C i + 1 is at the right side of − − − − → de 1 Ã n , C i − 1 C i , α i < 0 T i 2 = ( T i 1 . x , T i 1 . y + α i ) , pour i de 1 otherwise. Ã n − 1. C 1 α 1 C 3 y l 0 l 1 C 2 α 3 α 2 T 32 T 41 C 0 C 4 T 12 T 21 l 1 α 3 α 2 α 1 T 02 l 0 x 0 T 11 T 22 T 31 L. Latecki and R. Lakamper. 7 / 27 Shape similarity measure based on correspondence of visual parts. T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images Pattern Analysis and Machine Intelligence, IEEE Transactions on , 22(10) :1185–1190, Oct 2000.
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Property of an arc in tangent space Principal result Suppose that i = 0 is a polygon with α i = ∠ ( − C i − 1 C i , − − − − → − − − → C = { C i } n C i C i + 1 ) such that sin α i ≃ α i for i ∈ { 1 , . . . , n − 1 } T ( C ) its representation in the tangent space, constituted by segments T i 2 T ( i + 1 ) 1 , T ( i + 1 ) 1 T ( i + 1 ) 2 for i from 0 to n − 1 { M i } n − 1 i = 0 is a set of central point of { T i 2 T ( i + 1 ) 1 } n − 1 i = 1 . Therefore, C is a polygon that approximates an arc of circle if and only if { M i } n − 1 i = 0 is a set of collinear points. T ( i + 1 ) 2 Mi + 1 T ( i + 2 ) 1 y Ti 2 Mi Ii + 1 T ( i + 1 ) 1 Ii T ( i − 1 ) 2 Mi − 1 Ti 1 0 x 8 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
Introduction Arc segmentation Unsupervised Noise Detection A framework for arc recognition along noisy curves Experimentations Conclusions and futur work Consequence Interest of this result Reconnaissance Reconnaissance de cercle (arc) de droite Example (a) Entry curve (b) Approximated polygon 0 0 Tangent space representation Curve of midpoints in tangent space -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 -7 -7 0 100 200 300 400 500 600 0 100 200 300 400 500 600 (c) Tangent space represen- (d) Central curve in the tan- tation gent space 9 / 27 T. P . NGUYEN, B. KERAUTRET, I. DEBLED-RENNESSON, J. O. LACHAUD Unsupervised, Fast and Precise Recognition of Digital Arcs in Noisy Images
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