Spatially-variant anisotropic morphological filters driven by gradient fields Rafael Verdú-Monedero 1 Jesús Angulo 2 Jean Serra 3 1 Department of Information Technologies and Communications, Technical University of Cartagena, 30202, Cartagena, Spain, rafael.verdu@upct.es 2 Centre de Morphologie Mathématique (CMM), Ecole des Mines de Paris, Fontainebleau Cedex, France jesus.angulo@ensmp.fr 3 Laboratoire A2SI - ESIEE, B.P . 99, 93162 Noisy-le-Grand, France serraj@esiee.fr ISMM 2009, 9 th International Symposium on Mathematical Morphology
Outline Introduction 1 Spatially-variant morphology 2 Dilation/erosion and opening/closing Dilation for numerical functions Directional field modelling 3 Average squared gradient (ASG) Regularization of the ASG: ASGVF Applications 4 Conclusions and perspectives 5 Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 2 / 27
Introduction Outline Introduction 1 Spatially-variant morphology 2 Dilation/erosion and opening/closing Dilation for numerical functions Directional field modelling 3 Average squared gradient (ASG) Regularization of the ASG: ASGVF Applications 4 Conclusions and perspectives 5 Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 3 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Introduction “Spatially variant” encompasses: Two level structure of a space E , and of all subsets P ( E ) , and functions on E . Some variable processing over space E e.g. geometrical deformation of the Euclidean space by perspective (e.g. actual application of traffic monitoring) by rotation invariance Proposed approach Spatially variant morphological operators are computed from their direct geometric definitions. Structuring elements are linear orientated segments. Orientation information is extracted from the image under study itself. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 4 / 27
Spatially-variant morphology Outline Introduction 1 Spatially-variant morphology 2 Dilation/erosion and opening/closing Dilation for numerical functions Directional field modelling 3 Average squared gradient (ASG) Regularization of the ASG: ASGVF Applications 4 Conclusions and perspectives 5 Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 5 / 27
Spatially-variant morphology Operations with data-based variation The four basic operations must be expressed exclusively by means of the structuring function. It is not possible to resort to complement, or equivalently, to reciprocal dilation. Notation E → arbitrary set (discrete or continuous space) or any graph x = ( x , y ) ∈ E → points of E X ⊆ E → subsets of E . P ( E ) is the set of all these subsets. A structuring function δ : E → P ( E ) is an arbitrary family { δ ( x ) } of sets indexed by the points of E . The transform of a point is a set. T → numerical axis [ 0 , M ] (e.g. [ 0 , + ∞ ] , [ 0 , 255 ] , etc.) The family of all numerical functions f : E → T is denoted by F ( E , T ) . Both sets P ( E ) and F ( E , T ) are complete lattices. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 6 / 27
Spatially-variant morphology Operations with data-based variation The four basic operations must be expressed exclusively by means of the structuring function. It is not possible to resort to complement, or equivalently, to reciprocal dilation. Notation E → arbitrary set (discrete or continuous space) or any graph x = ( x , y ) ∈ E → points of E X ⊆ E → subsets of E . P ( E ) is the set of all these subsets. A structuring function δ : E → P ( E ) is an arbitrary family { δ ( x ) } of sets indexed by the points of E . The transform of a point is a set. T → numerical axis [ 0 , M ] (e.g. [ 0 , + ∞ ] , [ 0 , 255 ] , etc.) The family of all numerical functions f : E → T is denoted by F ( E , T ) . Both sets P ( E ) and F ( E , T ) are complete lattices. Verdú, Angulo & Serra (UPCT, CMM & A2SI) ISMM 2009 August 24-27 6 / 27
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