Comparison of orientated and spatially variant morphological filters - PowerPoint PPT Presentation

Comparison of orientated and spatially variant morphological filters vs mean/median filters for adaptive image denoising Rafael Verdú-Monedero 1 , Jesús Angulo 2 , Jorge Larrey-Ruiz 1 , Juan Morales-Sánchez 1 1 Dpto. de Tecnologías de la Información y Comunicaciones, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain. {rafael.verdu, jorge.larrey, juan.morales}@upct.es 2 Centre de Morphologie Mathématique (CMM), Ecole des Mines de Paris, Fontainebleau Cedex, France jesus.angulo@ensmp.fr ICIP 2010, International Conference on Image Processing

Outline Introduction 1 Orientation estimation 2 Spatially variant filters 3 Results 4 Conclusions 5 Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 2 / 22

Introduction Outline Introduction 1 Orientation estimation 2 Spatially variant filters 3 Results 4 Conclusions 5 Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 3 / 22

Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the absolute value and angle of the vector field is used to adapt the shape of the filter at each pixel of the image. Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 4 / 22

Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the absolute value and angle of the vector field is used to adapt the shape of the filter at each pixel of the image. (a) Original image (b) Orientation estimation Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 4 / 22

Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the absolute value and angle of the vector field is used to adapt the shape of the filter at each pixel of the image. (a) Spatially-invariant (b) Spatially-variant Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 4 / 22

Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the absolute value and angle of the vector field is used to adapt the shape of the filter at each pixel of the image. (a) Spatially-invariant (b) Spatially-variant Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 4 / 22

Introduction Objectives Obtain a vector field with orientation information in all pixels. 1 Perform a spatially-variant filtering: 2 the absolute value and angle of the vector field is used to adapt the shape of the filter at each pixel of the image. (a) Spatially-invariant (b) Spatially-variant Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 4 / 22

Orientation estimation Outline Introduction 1 Orientation estimation 2 Spatially variant filters 3 Results 4 Conclusions 5 Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 5 / 22

Orientation estimation Average Squared Gradient Vector Flow, ASGVF It consists of two steps: Obtain an initial estimation of the orientation 1 ( Average Squared Gradient , ASG) Regularize this vector field 2 ( Average Squared Gradient Vector Flow , ASGVF) (a) Step 1 - ASG (b) Step 2 - ASGVF Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 6 / 22

Orientation estimation Average squared gradient (ASG) ASG uses the following definition of the gradient � � ∂ X ( x , y ) � g 1 ( x , y ) � � � ∂ X ( x , y ) ∂ x g = = sign . ∂ X ( x , y ) g 2 ( x , y ) ∂ x ∂ y Then the gradient is squared and averaged in neighborhood W � g s , 1 ( x , y ) g 2 1 ( x , y ) − g 2 � � � � � � 2 ( x , y ) W g s = = . g s , 2 ( x , y ) � W ( 2 g 1 ( x , y ) g 2 ( x , y )) The directional field ASG is d = [ d 1 ( x , y ) , d 2 ( x , y )] ⊤ , where its angle is obtained as ∠ d = Φ 2 − sign (Φ) π 2 , being Φ = ∠ g s ; and the magnitude of d can be left as the magnitude of g s , or the squared root of g s or it can be set to unity. Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 7 / 22

Orientation estimation Average squared gradient (ASG) ASG uses the following definition of the gradient � � ∂ X ( x , y ) � g 1 ( x , y ) � � � ∂ X ( x , y ) ∂ x g = = sign . ∂ X ( x , y ) g 2 ( x , y ) ∂ x ∂ y Then the gradient is squared and averaged in neighborhood W � g s , 1 ( x , y ) g 2 1 ( x , y ) − g 2 � � � � � � 2 ( x , y ) W g s = = . g s , 2 ( x , y ) � W ( 2 g 1 ( x , y ) g 2 ( x , y )) The directional field ASG is d = [ d 1 ( x , y ) , d 2 ( x , y )] ⊤ , where its angle is obtained as ∠ d = Φ 2 − sign (Φ) π 2 , being Φ = ∠ g s ; and the magnitude of d can be left as the magnitude of g s , or the squared root of g s or it can be set to unity. Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 7 / 22

Orientation estimation Average squared gradient (ASG) ASG uses the following definition of the gradient � � ∂ X ( x , y ) � g 1 ( x , y ) � � � ∂ X ( x , y ) ∂ x g = = sign . ∂ X ( x , y ) g 2 ( x , y ) ∂ x ∂ y Then the gradient is squared and averaged in neighborhood W � g s , 1 ( x , y ) g 2 1 ( x , y ) − g 2 � � � � � � 2 ( x , y ) W g s = = . g s , 2 ( x , y ) � W ( 2 g 1 ( x , y ) g 2 ( x , y )) The directional field ASG is d = [ d 1 ( x , y ) , d 2 ( x , y )] ⊤ , where its angle is obtained as ∠ d = Φ 2 − sign (Φ) π 2 , being Φ = ∠ g s ; and the magnitude of d can be left as the magnitude of g s , or the squared root of g s or it can be set to unity. Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 7 / 22

Orientation estimation Regularization of the ASG: ASGVF The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [ v 1 ( x , y ) , v 2 ( x , y )] ⊤ that minimizes the energy functional: E ( v ) = D ( v ) + α S ( v ) . D represents a distance measure D ( v ) = 1 � || d || 2 || v − d || 2 dx dy . 2 E S determines the smoothness of the directional field �� ∂ v 1 � 2 � � 2 � 2 � 2 � ∂ v 1 � ∂ v 2 � ∂ v 2 S ( v ) = 1 � + + + dx dy . 2 ∂ x ∂ y ∂ x ∂ y E can be found by solving the following Euler equations ( v − d ) | d | 2 − α ∇ 2 v = 0 . Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 8 / 22

Orientation estimation Regularization of the ASG: ASGVF The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [ v 1 ( x , y ) , v 2 ( x , y )] ⊤ that minimizes the energy functional: E ( v ) = D ( v ) + α S ( v ) . D represents a distance measure D ( v ) = 1 � || d || 2 || v − d || 2 dx dy . 2 E S determines the smoothness of the directional field �� ∂ v 1 � 2 � � 2 � 2 � 2 � ∂ v 1 � ∂ v 2 � ∂ v 2 S ( v ) = 1 � + + + dx dy . 2 ∂ x ∂ y ∂ x ∂ y E can be found by solving the following Euler equations ( v − d ) | d | 2 − α ∇ 2 v = 0 . Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 8 / 22

Orientation estimation Regularization of the ASG: ASGVF The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [ v 1 ( x , y ) , v 2 ( x , y )] ⊤ that minimizes the energy functional: E ( v ) = D ( v ) + α S ( v ) . D represents a distance measure D ( v ) = 1 � || d || 2 || v − d || 2 dx dy . 2 E S determines the smoothness of the directional field �� ∂ v 1 � 2 � � 2 � 2 � 2 � ∂ v 1 � ∂ v 2 � ∂ v 2 S ( v ) = 1 � + + + dx dy . 2 ∂ x ∂ y ∂ x ∂ y E can be found by solving the following Euler equations ( v − d ) | d | 2 − α ∇ 2 v = 0 . Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 8 / 22

Orientation estimation Regularization of the ASG: ASGVF The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [ v 1 ( x , y ) , v 2 ( x , y )] ⊤ that minimizes the energy functional: E ( v ) = D ( v ) + α S ( v ) . D represents a distance measure D ( v ) = 1 � || d || 2 || v − d || 2 dx dy . 2 E S determines the smoothness of the directional field �� ∂ v 1 � 2 � � 2 � 2 � 2 � ∂ v 1 � ∂ v 2 � ∂ v 2 S ( v ) = 1 � + + + dx dy . 2 ∂ x ∂ y ∂ x ∂ y E can be found by solving the following Euler equations ( v − d ) | d | 2 − α ∇ 2 v = 0 . Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 8 / 22

Spatially variant filters Outline Introduction 1 Orientation estimation 2 Spatially variant filters 3 Results 4 Conclusions 5 Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 9 / 22

Spatially variant filters Mean and median filters Standard formulation: spatially-invariant The mean and median filter consists in computing for each pixel the mean and median, respectively, of the image values belonging to a fixed neighborhood (or window) W centered at the pixel. Proposed filters: spatially-variant At each pixel x a window W ( x ) is defined, whose shape depends on the orientation and homogeneity of the data around x , x → W ( x ) Spatially-variant mean filter � 1 � � µ W ( x ) ( X )( x ) = X ( y ) , y ∈ W ( x ) , ♯ W ( x ) ♯ W ( x ) is the number of pixels of the window centered at x Spatially-variant median filter m W ( x ) ( X )( x ) = { Median [ X ( z )] , z ∈ W ( x ) } . Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 10 / 22