Multiscale local multiple orientation estimation using Mathematical Morphology and B-spline interpolation Jesús Angulo 1 , Rafael Verdú 2 , Juan Morales 2 1 Centre de Morphologie Mathématique (CMM), Ecole des Mines de Paris, Fontainebleau Cedex, France jesus.angulo@ensmp.fr 2 Dpto. de Tecnologías de la Información y Comunicaciones, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain. {rafael.verdu, juan.morales}@upct.es Int. Symp. on Image and Signal Processing and Analysis
Outline Introduction 1 Modelling orientation using mathematical morphology 2 Proposed method 3 Results 4 Conclusions 5 Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 2 / 30
Introduction Outline Introduction 1 Modelling orientation using mathematical morphology 2 Proposed method 3 Results 4 Conclusions 5 Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 3 / 30
Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the shape of the filter at each pixel of the image depends on the absolute value and angle of the vector field. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 4 / 30
Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the shape of the filter at each pixel of the image depends on the absolute value and angle of the vector field. (a) Original image (b) Orientation estimation Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 4 / 30
Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the shape of the filter at each pixel of the image depends on the absolute value and angle of the vector field. (a) Spatially-invariant (b) Spatially-variant Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 4 / 30
Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the shape of the filter at each pixel of the image depends on the absolute value and angle of the vector field. (a) Spatially-invariant (b) Spatially-variant Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 4 / 30
Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the shape of the filter at each pixel of the image depends on the absolute value and angle of the vector field. (a) Spatially-invariant (b) Spatially-variant Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 4 / 30
Introduction Preliminaries Previous work of the authors is based on Average Squared Gradient , ASG, and its regularization Average Squared Gradient Vector Flow , ASGVF. It does not deal with the multiple orientation case. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 5 / 30
Introduction Preliminaries Previous work of the authors is based on Average Squared Gradient , ASG, and its regularization Average Squared Gradient Vector Flow , ASGVF. It does not deal with the multiple orientation case. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 5 / 30
Introduction Preliminaries Previous work of the authors is based on Average Squared Gradient , ASG, and its regularization Average Squared Gradient Vector Flow , ASGVF. It does not deal with the multiple orientation case. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 5 / 30
Introduction Preliminaries Previous work of the authors is based on Average Squared Gradient , ASG, and its regularization Average Squared Gradient Vector Flow , ASGVF. It does not deal with the multiple orientation case. (a) Gradient (b) ASG (c) ASGVF Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 5 / 30
Introduction Estimation of local orientation Mathematical morphology has shown an excellent performance for global and local orientation estimation: directional signature. Orientation of a pixel is defined as the angle associated to the directional opening which produces the maximal value of signature at this pixel. Proposed method: Determine all the significant orientations: multiple peak detection of the directional signature interpolated by b-splines. Multiscale approach using various lengths of structuring elements in directional openings. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 6 / 30
Introduction Estimation of local orientation Mathematical morphology has shown an excellent performance for global and local orientation estimation: directional signature. Orientation of a pixel is defined as the angle associated to the directional opening which produces the maximal value of signature at this pixel. Proposed method: Determine all the significant orientations: multiple peak detection of the directional signature interpolated by b-splines. Multiscale approach using various lengths of structuring elements in directional openings. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 6 / 30
Introduction Estimation of local orientation Mathematical morphology has shown an excellent performance for global and local orientation estimation: directional signature. Orientation of a pixel is defined as the angle associated to the directional opening which produces the maximal value of signature at this pixel. Proposed method: Determine all the significant orientations: multiple peak detection of the directional signature interpolated by b-splines. Multiscale approach using various lengths of structuring elements in directional openings. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 6 / 30
Introduction Estimation of local orientation Mathematical morphology has shown an excellent performance for global and local orientation estimation: directional signature. Orientation of a pixel is defined as the angle associated to the directional opening which produces the maximal value of signature at this pixel. Proposed method: Determine all the significant orientations: multiple peak detection of the directional signature interpolated by b-splines. Multiscale approach using various lengths of structuring elements in directional openings. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 6 / 30
Introduction Estimation of local orientation Mathematical morphology has shown an excellent performance for global and local orientation estimation: directional signature. Orientation of a pixel is defined as the angle associated to the directional opening which produces the maximal value of signature at this pixel. Proposed method: Determine all the significant orientations: multiple peak detection of the directional signature interpolated by b-splines. Multiscale approach using various lengths of structuring elements in directional openings. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 6 / 30
Introduction Estimation of local orientation Mathematical morphology has shown an excellent performance for global and local orientation estimation: directional signature. Orientation of a pixel is defined as the angle associated to the directional opening which produces the maximal value of signature at this pixel. Proposed method: Determine all the significant orientations: multiple peak detection of the directional signature interpolated by b-splines. Multiscale approach using various lengths of structuring elements in directional openings. Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 6 / 30
Modelling orientation using mathematical morphology Outline Introduction 1 Modelling orientation using mathematical morphology 2 Proposed method 3 Results 4 Conclusions 5 Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 7 / 30
Modelling orientation using mathematical morphology Directional opening The directional opening of an image f ( x ) by a linear (symmetric) structuring element (SE) of length l and direction θ is defined as γ L θ, l ( f )( x ) = δ L θ, l [ ε L θ, l ( f )] ( x ) where the directional erosion and dilation are, respectively, � ε L θ, l ( f )( x ) = { f ( x + h ) } , h ∈ L θ, l ( x ) � δ L θ, l ( f )( x ) = { f ( x − h ) } . h ∈ L θ, l ( x ) Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 8 / 30
Modelling orientation using mathematical morphology Directional opening The directional opening of an image f ( x ) by a linear (symmetric) structuring element (SE) of length l and direction θ is defined as γ L θ, l ( f )( x ) = δ L θ, l [ ε L θ, l ( f )] ( x ) where the directional erosion and dilation are, respectively, � ε L θ, l ( f )( x ) = { f ( x + h ) } , h ∈ L θ, l ( x ) � δ L θ, l ( f )( x ) = { f ( x − h ) } . h ∈ L θ, l ( x ) Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 8 / 30
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