Reconstruction of Connected Digital Lines Based on Constrained Regularization

This paper presents a new approach for reconstruction of disconnected digital lines (DDLs) based on a constrained regularization model which ensures connectivity of the digital lines (DLs) in the discrete image plane. The first step in this approach is to determine the order of given pixels of the DDL. To determine connectivity of pixels, we use the usual 8-neighbor connectivity in discrete images. For any neighboring pixels of the DDL that are not connected, we determine a number of new pixel values that need to be reconstructed between these pixels. Next, the integer-valued <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>- and <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula>-coordinates of the location of the pixels of the DDLs are segregated into two 1D signal vectors. Then the <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>- and <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula>-coordinates of the missing pixels of the DDLs are estimated using a new constrained regularization. While the solution of this constrained minimization problem provides real values for the <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>- and <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula>-coordinates of pixels positions, the imposed constraint ensures connectivity of the resulting DLs in the image plane after transforming the computed values from <inline-formula> <tex-math notation="LaTeX">$\mathbb {R}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$\mathbb {N}$ </tex-math></inline-formula>. The proposed regularization approach forces connected lines with small curvature. The experimental results demonstrate that the proposed technique improves DL intersection detection, as well. Moreover, this technique has a high potential to be used as a fast approach in binary image inpainting particularly overcoming the shortcomings of conventional methods which cause destruction of thin objects and blurring in the recovered regions.

[1]  Fang Liu,et al.  Global Low-Rank Image Restoration With Gaussian Mixture Model , 2018, IEEE Transactions on Cybernetics.

[2]  Hossein Rabbani,et al.  Missing Surface Estimation Based on Modified Tikhonov Regularization: Application for Destructed Dental Tissue , 2018, IEEE Transactions on Image Processing.

[3]  Jie Qin,et al.  A comparative study of new and current methods for dental micro-CT image denoising. , 2016, Dento maxillo facial radiology.

[4]  John C. Russ,et al.  The Image Processing Handbook , 2015 .

[5]  Martin Stoll,et al.  A Fractional Inpainting Model Based on the Vector-Valued Cahn-Hilliard Equation , 2015, SIAM J. Imaging Sci..

[6]  Hossein Rabbani,et al.  A fast and accurate dental micro-CT image denoising based on total variation modeling , 2015, 2015 IEEE Workshop on Signal Processing Systems (SiPS).

[7]  Junyan Yi,et al.  An Adaptive Elastic Net Method for Edge Linking of Images , 2015, Int. J. Interdiscip. Telecommun. Netw..

[8]  Jung-Il Choi,et al.  Fast local image inpainting based on the Allen-Cahn model , 2015, Digit. Signal Process..

[9]  Michael Elad,et al.  Sparsity Based Methods for Overparametrized Variational Problems , 2014, SIAM J. Imaging Sci..

[10]  Martin Stoll,et al.  Fast Solvers for Cahn-Hilliard Inpainting , 2014, SIAM J. Imaging Sci..

[11]  Haklin Kimm,et al.  Edge Detection and Linking Pattern Analysis Using Markov Chains , 2013, 2013 IEEE 16th International Conference on Computational Science and Engineering.

[12]  Ivan W. Selesnick,et al.  Least Squares with Examples in Signal Processing 1 , 2013 .

[13]  Allen R. Tannenbaum,et al.  Self-Crossing Detection and Location for Parametric Active Contours , 2012, IEEE Transactions on Image Processing.

[14]  Long Xiang,et al.  Remaining edges linking method of motion segmentation based on edge detection , 2012, 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery.

[15]  Yi Liu,et al.  A three-dimensional gap filling method for large geophysical datasets: Application to global satellite soil moisture observations , 2012, Environ. Model. Softw..

[16]  Damien Garcia,et al.  Robust smoothing of gridded data in one and higher dimensions with missing values , 2010, Comput. Stat. Data Anal..

[17]  Lin He,et al.  Cahn--Hilliard Inpainting and a Generalization for Grayvalue Images , 2009, SIAM J. Imaging Sci..

[18]  Liu Wei,et al.  An improved sequential edge linking model for contour detection in medical images , 2009, 2009 4th IEEE Conference on Industrial Electronics and Applications.

[19]  Ricardo H. Nochetto,et al.  A Variational Shape Optimization Approach for Image Segmentation with a Mumford--Shah Functional , 2008, SIAM J. Sci. Comput..

[20]  Hong Zhang,et al.  Edge linking using geodesic distance and neighborhood information , 2008, 2008 IEEE/ASME International Conference on Advanced Intelligent Mechatronics.

[21]  Andrea L. Bertozzi,et al.  Analysis of a Two-Scale Cahn-Hilliard Model for Binary Image Inpainting , 2007, Multiscale Model. Simul..

[22]  Yan Xu,et al.  Local discontinuous Galerkin methods for the Cahn-Hilliard type equations , 2007, J. Comput. Phys..

[23]  Folkmar Bornemann,et al.  Fast Image Inpainting Based on Coherence Transport , 2007, Journal of Mathematical Imaging and Vision.

[24]  Hanspeter Schaub,et al.  Efficient polygonal intersection determination with applications to robotics and vision , 2005, 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[25]  Andrea Bonci,et al.  A Bayesian approach to the Hough transform for line detection , 2005, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[26]  T. Chan,et al.  Variational image inpainting , 2005 .

[27]  Lydia E. Kavraki,et al.  Fast intersection checking for parametric deformable models , 2005, SPIE Medical Imaging.

[28]  Frank Y. Shih,et al.  Adaptive mathematical morphology for edge linking , 2004, Inf. Sci..

[29]  Patrick Pérez,et al.  Region filling and object removal by exemplar-based image inpainting , 2004, IEEE Transactions on Image Processing.

[30]  Nikolaos Papanikolopoulos,et al.  Closed dynamic contour models that split and merge , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[31]  Eero P. Simoncelli,et al.  Image quality assessment: from error visibility to structural similarity , 2004, IEEE Transactions on Image Processing.

[32]  Guillermo Sapiro,et al.  Simultaneous structure and texture image inpainting , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[33]  Ling Guan,et al.  Perceptual image processing for digital edge linking , 2003, CCECE 2003 - Canadian Conference on Electrical and Computer Engineering. Toward a Caring and Humane Technology (Cat. No.03CH37436).

[34]  Jianhong Shen,et al.  Digital inpainting based on the Mumford–Shah–Euler image model , 2002, European Journal of Applied Mathematics.

[35]  O. Ghita,et al.  A computationally effcient method for edge thinning and linking using endpoints , 2002 .

[36]  Tony F. Chan,et al.  Nontexture Inpainting by Curvature-Driven Diffusions , 2001, J. Vis. Commun. Image Represent..

[37]  Ulrich Eckhardt,et al.  Shape descriptors for non-rigid shapes with a single closed contour , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[38]  Gregory W. Cook,et al.  A Gaussian mixture model for edge-enhanced images with application to sequential edge detection and linking , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[39]  Robert I. Damper,et al.  An empirical comparison of neural techniques for edge linking of images , 1997, Neural Computing & Applications.

[40]  Sanjoy K. Mitter,et al.  A hierarchical approach to high resolution edge contour reconstruction , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[41]  Qiuming Zhu,et al.  Edge linking by a directional potential function (DPF) , 1996, Image Vis. Comput..

[42]  Gregory W. Cook,et al.  Multiresolution sequential edge linking , 1995, Proceedings., International Conference on Image Processing.

[43]  Dinesh Manocha,et al.  I-COLLIDE: an interactive and exact collision detection system for large-scale environments , 1995, I3D '95.

[44]  Junji Maeda,et al.  Template based method of edge linking using a weighted decision , 1993, Proceedings of 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS '93).

[45]  Luc Vincent,et al.  Morphological grayscale reconstruction in image analysis: applications and efficient algorithms , 1993, IEEE Trans. Image Process..

[46]  Wesley E. Snyder,et al.  Closing gaps in edges and surfaces , 1992, Image Vis. Comput..

[47]  Ming Xie,et al.  Edge linking by using causal neighborhood window , 1992, Pattern Recognit. Lett..

[48]  Aly A. Farag,et al.  Edge linking by sequential search , 1992, Other Conferences.

[49]  Aly A. Farag,et al.  A path metric for sequential search and its application in edge linking , 1991, Conference Proceedings 1991 IEEE International Conference on Systems, Man, and Cybernetics.

[50]  Robert M. Haralick,et al.  Edge Linking by Ellipsoidal Clustering , 1990, Other Conferences.

[51]  Wei-Chung Lin,et al.  An iterative edge linking algorithm with noise removal capability , 1988, [1988 Proceedings] 9th International Conference on Pattern Recognition.

[52]  Vishvjit S. Nalwa,et al.  Edgel aggregation and edge description , 1987, Comput. Vis. Graph. Image Process..

[53]  D. Bertsekas.  Enlarging the region of convergence of Newton's method for constrained optimization , 1982 .

[54]  K. Ramesh Babu,et al.  Linear Feature Extraction and Description , 1979, IJCAI.

[55]  Ramakant Nevatia,et al.  Locating Object Boundaries in Textured Environments , 1976, IEEE Transactions on Computers.

[56]  A. Bertozzi,et al.  Unconditionally stable schemes for higher order inpainting , 2011 .

[57]  Wei Wang,et al.  An Adaptive Clonal Selection Algorithm for Edge Linking Problem , 2009 .

[58]  Andrea L. Bertozzi,et al.  Inpainting of Binary Images Using the Cahn–Hilliard Equation , 2007, IEEE Transactions on Image Processing.

[59]  Zoltan Tomori,et al.  Border detection of the object segmented by the pyramid linking method , 1995, IEEE Trans. Syst. Man Cybern..

[60]  Azriel Rosenfeld,et al.  Edge segment linking based on gray level and geometrical compatibilities , 1982, Pattern Recognit..