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

▶
comparison of orientated and spatially variant
SMART_READER_LITE
LIVE PREVIEW

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

Sep 26, 2023 252 likes •780 views

Comparison of orientated and spatially variant morphological filters vs mean/median filters for adaptive image denoising Rafael Verd-Monedero 1 , Jess Angulo 2 , Jorge Larrey-Ruiz 1 , Juan Morales-Snchez 1 1 Dpto. de Tecnologas de la

slide-1
SLIDE 1

Comparison of orientated and spatially variant morphological filters vs mean/median filters for adaptive image denoising

Rafael Verdú-Monedero1, Jesús Angulo2, Jorge Larrey-Ruiz1, Juan Morales-Sánchez1

  • 1Dpto. de Tecnologías de la Información y Comunicaciones,

Universidad Politécnica de Cartagena, 30202 Cartagena, Spain. {rafael.verdu, jorge.larrey, juan.morales}@upct.es

2Centre de Morphologie Mathématique (CMM),

Ecole des Mines de Paris, Fontainebleau Cedex, France jesus.angulo@ensmp.fr

ICIP 2010, International Conference on Image Processing

slide-2
SLIDE 2

Outline

1

Introduction

2

Orientation estimation

3

Spatially variant filters

4

Results

5

Conclusions

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 2 / 22

slide-3
SLIDE 3

Introduction

Outline

1

Introduction

2

Orientation estimation

3

Spatially variant filters

4

Results

5

Conclusions

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 3 / 22

slide-4
SLIDE 4

Introduction

Objectives

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

slide-5
SLIDE 5

Introduction

Objectives

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

slide-6
SLIDE 6

Introduction

Objectives

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

slide-7
SLIDE 7

Introduction

Objectives

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

slide-8
SLIDE 8

Introduction

Objectives

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

slide-9
SLIDE 9

Orientation estimation

Outline

1

Introduction

2

Orientation estimation

3

Spatially variant filters

4

Results

5

Conclusions

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 5 / 22

slide-10
SLIDE 10

Orientation estimation

Average Squared Gradient Vector Flow, ASGVF It consists of two steps:

1

Obtain an initial estimation of the orientation (Average Squared Gradient, ASG)

2

Regularize this vector field (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

slide-11
SLIDE 11

Orientation estimation Average squared gradient (ASG)

ASG uses the following definition of the gradient g = g1(x, y) g2(x, y)

  • = sign
  • ∂X(x,y)

∂x

∂X(x,y)

∂x ∂X(x,y) ∂y

  • .

Then the gradient is squared and averaged in neighborhood W gs = gs,1(x, y) gs,2(x, y)

  • =

W

  • g2

1(x, y) − g2 2(x, y)

  • W (2 g1(x, y) g2(x, y))
  • .

The directional field ASG is d = [d1(x, y), d2(x, y)]⊤, where its angle is obtained as ∠d = Φ 2 − sign(Φ)π 2, being Φ = ∠gs; and the magnitude of d can be left as the magnitude of gs, or the squared root of gs or it can be set to unity.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 7 / 22

slide-12
SLIDE 12

Orientation estimation Average squared gradient (ASG)

ASG uses the following definition of the gradient g = g1(x, y) g2(x, y)

  • = sign
  • ∂X(x,y)

∂x

∂X(x,y)

∂x ∂X(x,y) ∂y

  • .

Then the gradient is squared and averaged in neighborhood W gs = gs,1(x, y) gs,2(x, y)

  • =

W

  • g2

1(x, y) − g2 2(x, y)

  • W (2 g1(x, y) g2(x, y))
  • .

The directional field ASG is d = [d1(x, y), d2(x, y)]⊤, where its angle is obtained as ∠d = Φ 2 − sign(Φ)π 2, being Φ = ∠gs; and the magnitude of d can be left as the magnitude of gs, or the squared root of gs or it can be set to unity.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 7 / 22

slide-13
SLIDE 13

Orientation estimation Average squared gradient (ASG)

ASG uses the following definition of the gradient g = g1(x, y) g2(x, y)

  • = sign
  • ∂X(x,y)

∂x

∂X(x,y)

∂x ∂X(x,y) ∂y

  • .

Then the gradient is squared and averaged in neighborhood W gs = gs,1(x, y) gs,2(x, y)

  • =

W

  • g2

1(x, y) − g2 2(x, y)

  • W (2 g1(x, y) g2(x, y))
  • .

The directional field ASG is d = [d1(x, y), d2(x, y)]⊤, where its angle is obtained as ∠d = Φ 2 − sign(Φ)π 2, being Φ = ∠gs; and the magnitude of d can be left as the magnitude of gs, or the squared root of gs or it can be set to unity.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 7 / 22

slide-14
SLIDE 14

Orientation estimation Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 8 / 22

slide-15
SLIDE 15

Orientation estimation Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 8 / 22

slide-16
SLIDE 16

Orientation estimation Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 8 / 22

slide-17
SLIDE 17

Orientation estimation Regularization of the ASG: ASGVF

The Average Squared Gradient Vector Flow (ASGVF) is the vector field v = [v1(x, y), v2(x, y)]⊤ that minimizes the energy functional: E(v) = D(v) + αS(v). D represents a distance measure D(v) = 1 2

  • E

||d||2||v − d||2 dx dy. S determines the smoothness of the directional field S(v) = 1 2

  • E

∂v1 ∂x 2 + ∂v1 ∂y 2 + ∂v2 ∂x 2 + ∂v2 ∂y 2 dx dy. can be found by solving the following Euler equations (v − d)|d|2 − α∇2v = 0.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 8 / 22

slide-18
SLIDE 18

Spatially variant filters

Outline

1

Introduction

2

Orientation estimation

3

Spatially variant filters

4

Results

5

Conclusions

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 9 / 22

slide-19
SLIDE 19

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 µW(x)(X)(x) =

  • 1

♯W(x)

  • X(y), y ∈ W(x)
  • ,

♯W(x) is the number of pixels of the window centered at x Spatially-variant median filter mW(x)(X)(x) = {Median[X(z)], z ∈ W(x)} .

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 10 / 22

slide-20
SLIDE 20

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 µW(x)(X)(x) =

  • 1

♯W(x)

  • X(y), y ∈ W(x)
  • ,

♯W(x) is the number of pixels of the window centered at x Spatially-variant median filter mW(x)(X)(x) = {Median[X(z)], z ∈ W(x)} .

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 10 / 22

slide-21
SLIDE 21

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 µW(x)(X)(x) =

  • 1

♯W(x)

  • X(y), y ∈ W(x)
  • ,

♯W(x) is the number of pixels of the window centered at x Spatially-variant median filter mW(x)(X)(x) = {Median[X(z)], z ∈ W(x)} .

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 10 / 22

slide-22
SLIDE 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 µW(x)(X)(x) =

  • 1

♯W(x)

  • X(y), y ∈ W(x)
  • ,

♯W(x) is the number of pixels of the window centered at x Spatially-variant median filter mW(x)(X)(x) = {Median[X(z)], z ∈ W(x)} .

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 10 / 22

slide-23
SLIDE 23

Spatially variant filters Morphological filters

Premises When the variation of a structuring function (which plays an equivalent role to the window in the mean/medial filter) follows a law based on the data, i.e., x → B(x), the complement of the dilation and of the adjoint opening can not be theoretically calculated Then the four basic operations (dilation, erosion, opening and closing) must be expressed exclusively by using the datum of the structuring function B(x), without resorting to complement, or equivalently, to reciprocal dilation.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 11 / 22

slide-24
SLIDE 24

Spatially variant filters Morphological filters

Premises When the variation of a structuring function (which plays an equivalent role to the window in the mean/medial filter) follows a law based on the data, i.e., x → B(x), the complement of the dilation and of the adjoint opening can not be theoretically calculated Then the four basic operations (dilation, erosion, opening and closing) must be expressed exclusively by using the datum of the structuring function B(x), without resorting to complement, or equivalently, to reciprocal dilation.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 11 / 22

slide-25
SLIDE 25

Spatially variant filters Morphological filters

Premises When the variation of a structuring function (which plays an equivalent role to the window in the mean/medial filter) follows a law based on the data, i.e., x → B(x), the complement of the dilation and of the adjoint opening can not be theoretically calculated Then the four basic operations (dilation, erosion, opening and closing) must be expressed exclusively by using the datum of the structuring function B(x), without resorting to complement, or equivalently, to reciprocal dilation.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 11 / 22

slide-26
SLIDE 26

Spatially variant filters Morphological filters

Formulation for gray-level images based on pulse functions The pulse function ix,t of level t at point x is introduced as ix,t(z) = t z = x z = x which is associated with the numerical function X(x) under study and the set structuring function x → B(x). Dilating it,x by the structuring function B(x) results in the cylinder CB(x),t of base B(x) and height t. Image X can be decomposed into the supremum of its pulses, i.e., X = ∨{ix,X(x), x ∈ E},

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 12 / 22

slide-27
SLIDE 27

Spatially variant filters Morphological filters

Formulation for gray-level images based on pulse functions The pulse function ix,t of level t at point x is introduced as ix,t(z) = t z = x z = x which is associated with the numerical function X(x) under study and the set structuring function x → B(x). Dilating it,x by the structuring function B(x) results in the cylinder CB(x),t of base B(x) and height t. Image X can be decomposed into the supremum of its pulses, i.e., X = ∨{ix,X(x), x ∈ E},

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 12 / 22

slide-28
SLIDE 28

Spatially variant filters Morphological filters

Formulation for gray-level images based on pulse functions The pulse function ix,t of level t at point x is introduced as ix,t(z) = t z = x z = x which is associated with the numerical function X(x) under study and the set structuring function x → B(x). Dilating it,x by the structuring function B(x) results in the cylinder CB(x),t of base B(x) and height t. Image X can be decomposed into the supremum of its pulses, i.e., X = ∨{ix,X(x), x ∈ E},

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 12 / 22

slide-29
SLIDE 29

Spatially variant filters Morphological filters

Morphological operators using exclusively the structuring function B Dilation δB(x)(X)(x) = ∨{CB(x),X(x), x ∈ E}. Erosion εB(x)(X)(x) = ∨{ix,t | CB(x),t ≤ X, x ∈ E}. Opening γB(x)(X)(x) = ∨{CB(x),t ≤ X, x ∈ E} Closing ϕB(x)(X) = M − γB(x)(M − X)(x)

[1] Verdú-Monedero, R.; Angulo, J.; Serra, J.; “Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields”, IEEE Trans Image Processing, DOI: 10.1109/TIP .2010.2056377

Averaged Alternated Filter (AAF) ξB(x)(X) = ϕB(x)γB(x)(X) + γB(x)ϕB(x)(X) 2

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 13 / 22

slide-30
SLIDE 30

Spatially variant filters Morphological filters

Morphological operators using exclusively the structuring function B Dilation δB(x)(X)(x) = ∨{CB(x),X(x), x ∈ E}. Erosion εB(x)(X)(x) = ∨{ix,t | CB(x),t ≤ X, x ∈ E}. Opening γB(x)(X)(x) = ∨{CB(x),t ≤ X, x ∈ E} Closing ϕB(x)(X) = M − γB(x)(M − X)(x)

[1] Verdú-Monedero, R.; Angulo, J.; Serra, J.; “Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields”, IEEE Trans Image Processing, DOI: 10.1109/TIP .2010.2056377

Averaged Alternated Filter (AAF) ξB(x)(X) = ϕB(x)γB(x)(X) + γB(x)ϕB(x)(X) 2

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 13 / 22

slide-31
SLIDE 31

Spatially variant filters Morphological filters

Morphological operators using exclusively the structuring function B Dilation δB(x)(X)(x) = ∨{CB(x),X(x), x ∈ E}. Erosion εB(x)(X)(x) = ∨{ix,t | CB(x),t ≤ X, x ∈ E}. Opening γB(x)(X)(x) = ∨{CB(x),t ≤ X, x ∈ E} Closing ϕB(x)(X) = M − γB(x)(M − X)(x)

[1] Verdú-Monedero, R.; Angulo, J.; Serra, J.; “Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields”, IEEE Trans Image Processing, DOI: 10.1109/TIP .2010.2056377

Averaged Alternated Filter (AAF) ξB(x)(X) = ϕB(x)γB(x)(X) + γB(x)ϕB(x)(X) 2

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 13 / 22

slide-32
SLIDE 32

Spatially variant filters Morphological filters

Morphological operators using exclusively the structuring function B Dilation δB(x)(X)(x) = ∨{CB(x),X(x), x ∈ E}. Erosion εB(x)(X)(x) = ∨{ix,t | CB(x),t ≤ X, x ∈ E}. Opening γB(x)(X)(x) = ∨{CB(x),t ≤ X, x ∈ E} Closing ϕB(x)(X) = M − γB(x)(M − X)(x)

[1] Verdú-Monedero, R.; Angulo, J.; Serra, J.; “Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields”, IEEE Trans Image Processing, DOI: 10.1109/TIP .2010.2056377

Averaged Alternated Filter (AAF) ξB(x)(X) = ϕB(x)γB(x)(X) + γB(x)ϕB(x)(X) 2

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 13 / 22

slide-33
SLIDE 33

Spatially variant filters Morphological filters

Morphological operators using exclusively the structuring function B Dilation δB(x)(X)(x) = ∨{CB(x),X(x), x ∈ E}. Erosion εB(x)(X)(x) = ∨{ix,t | CB(x),t ≤ X, x ∈ E}. Opening γB(x)(X)(x) = ∨{CB(x),t ≤ X, x ∈ E} Closing ϕB(x)(X) = M − γB(x)(M − X)(x)

[1] Verdú-Monedero, R.; Angulo, J.; Serra, J.; “Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields”, IEEE Trans Image Processing, DOI: 10.1109/TIP .2010.2056377

Averaged Alternated Filter (AAF) ξB(x)(X) = ϕB(x)γB(x)(X) + γB(x)ϕB(x)(X) 2

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 13 / 22

slide-34
SLIDE 34

Spatially variant filters Morphological filters

Morphological operators using exclusively the structuring function B Dilation δB(x)(X)(x) = ∨{CB(x),X(x), x ∈ E}. Erosion εB(x)(X)(x) = ∨{ix,t | CB(x),t ≤ X, x ∈ E}. Opening γB(x)(X)(x) = ∨{CB(x),t ≤ X, x ∈ E} Closing ϕB(x)(X) = M − γB(x)(M − X)(x)

[1] Verdú-Monedero, R.; Angulo, J.; Serra, J.; “Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields”, IEEE Trans Image Processing, DOI: 10.1109/TIP .2010.2056377

Averaged Alternated Filter (AAF) ξB(x)(X) = ϕB(x)γB(x)(X) + γB(x)ϕB(x)(X) 2

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 13 / 22

slide-35
SLIDE 35

Results

Outline

1

Introduction

2

Orientation estimation

3

Spatially variant filters

4

Results

5

Conclusions

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 14 / 22

slide-36
SLIDE 36

Results

(a) Image “Lenna” (b) Image “circles”

Spatially-variant (SV) operators vs spatially-invariant (SI) The proposed operators are µW(x)(X), mW(x)(X) y ξB(x)(X). Results are measured in terms of peak signal-to-noise ratio (PSNR), correlation ratio (CR) and mutual information (MI). Original 256×256 images have been corrupted with additive gaussian white noise of mean 0 and variance 0.01

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 15 / 22

slide-37
SLIDE 37

Results

(a) Image “Lenna” (b) Image “circles”

Spatially-variant (SV) operators vs spatially-invariant (SI) The proposed operators are µW(x)(X), mW(x)(X) y ξB(x)(X). Results are measured in terms of peak signal-to-noise ratio (PSNR), correlation ratio (CR) and mutual information (MI). Original 256×256 images have been corrupted with additive gaussian white noise of mean 0 and variance 0.01

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 15 / 22

slide-38
SLIDE 38

Results

(a) Image “Lenna” (b) Image “circles”

Spatially-variant (SV) operators vs spatially-invariant (SI) The proposed operators are µW(x)(X), mW(x)(X) y ξB(x)(X). Results are measured in terms of peak signal-to-noise ratio (PSNR), correlation ratio (CR) and mutual information (MI). Original 256×256 images have been corrupted with additive gaussian white noise of mean 0 and variance 0.01

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 15 / 22

slide-39
SLIDE 39

Results

(a) Image “Lenna” (b) Image “circles”

Spatially-variant (SV) operators vs spatially-invariant (SI) The proposed operators are µW(x)(X), mW(x)(X) y ξB(x)(X). Results are measured in terms of peak signal-to-noise ratio (PSNR), correlation ratio (CR) and mutual information (MI). Original 256×256 images have been corrupted with additive gaussian white noise of mean 0 and variance 0.01

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 15 / 22

slide-40
SLIDE 40

Results

(a) Image “Lenna” (b) Image “circles”

Mask/Structuring Element used with spatially-variant operators Image “Lenna”: orientated rectangle

9×1 in the edges and a neighborhood of two pixels, 7×3 near the edges (between three and four pixel far), 5×5 in homogeneous areas.

Image “circles”: 9×1 orientated segment.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 15 / 22

slide-41
SLIDE 41

Results Image “Lenna”

(a) Original image X(x) (b) SI median, W 5×5 (c) SI AAF , se 5×5 (d) SV median (e) SV AAF

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 16 / 22

slide-42
SLIDE 42

Results Image “Lenna”

PSNR (dB) CR (%) MI SI SV SI SV SI SV Image “Lenna” 28.7 79.4 1.43 Mean 32.5 32.3 93.4 94.7 2.31 2.31 Median 31.7 31.5 93.3 93.7 2.25 2.19 AAF 32.2 32.3 91.2 94.2 2.22 2.29

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 17 / 22

slide-43
SLIDE 43

Results Image “circles”

(a) Original image X(x) (b) ASGVF , 45×45 window, α=100

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 18 / 22

slide-44
SLIDE 44

Results Image “circles”

(a) Original image (b) SV mean (c) SV median (d) SV AAF

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 19 / 22

slide-45
SLIDE 45

Conclusions

Outline

1

Introduction

2

Orientation estimation

3

Spatially variant filters

4

Results

5

Conclusions

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 20 / 22

slide-46
SLIDE 46

Conclusions

This papers shows a method to estimate the orientation of the

  • bjects contained in an image.

The orientation information is diffused from the edges of the

  • bjects (where the gradient is high) to homogeneous areas

(where the gradient is near zero) with a regularization process. After the diffusion, the orientation estimation is provided at all

  • pixels. This is necessary in order to apply spatially-variant

filters. Results show the ability of spatially variant filters to reduce the noise in images, adapting shape and orientation according to the data contained in the image. Morphological operators can also preserve main structures as well as link elongated structures that due to noise could be disconnected or broken.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 21 / 22

slide-47
SLIDE 47

Conclusions

This papers shows a method to estimate the orientation of the

  • bjects contained in an image.

The orientation information is diffused from the edges of the

  • bjects (where the gradient is high) to homogeneous areas

(where the gradient is near zero) with a regularization process. After the diffusion, the orientation estimation is provided at all

  • pixels. This is necessary in order to apply spatially-variant

filters. Results show the ability of spatially variant filters to reduce the noise in images, adapting shape and orientation according to the data contained in the image. Morphological operators can also preserve main structures as well as link elongated structures that due to noise could be disconnected or broken.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 21 / 22

slide-48
SLIDE 48

Conclusions

This papers shows a method to estimate the orientation of the

  • bjects contained in an image.

The orientation information is diffused from the edges of the

  • bjects (where the gradient is high) to homogeneous areas

(where the gradient is near zero) with a regularization process. After the diffusion, the orientation estimation is provided at all

  • pixels. This is necessary in order to apply spatially-variant

filters. Results show the ability of spatially variant filters to reduce the noise in images, adapting shape and orientation according to the data contained in the image. Morphological operators can also preserve main structures as well as link elongated structures that due to noise could be disconnected or broken.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 21 / 22

slide-49
SLIDE 49

Conclusions

This papers shows a method to estimate the orientation of the

  • bjects contained in an image.

The orientation information is diffused from the edges of the

  • bjects (where the gradient is high) to homogeneous areas

(where the gradient is near zero) with a regularization process. After the diffusion, the orientation estimation is provided at all

  • pixels. This is necessary in order to apply spatially-variant

filters. Results show the ability of spatially variant filters to reduce the noise in images, adapting shape and orientation according to the data contained in the image. Morphological operators can also preserve main structures as well as link elongated structures that due to noise could be disconnected or broken.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 21 / 22

slide-50
SLIDE 50

Conclusions

This papers shows a method to estimate the orientation of the

  • bjects contained in an image.

The orientation information is diffused from the edges of the

  • bjects (where the gradient is high) to homogeneous areas

(where the gradient is near zero) with a regularization process. After the diffusion, the orientation estimation is provided at all

  • pixels. This is necessary in order to apply spatially-variant

filters. Results show the ability of spatially variant filters to reduce the noise in images, adapting shape and orientation according to the data contained in the image. Morphological operators can also preserve main structures as well as link elongated structures that due to noise could be disconnected or broken.

Verdú,Angulo,Larrey,Morales (UPCT-CMM) ICIP 2010 Monday, 27.09.2010 21 / 22

slide-51
SLIDE 51

Comparison of orientated and spatially variant morphological filters vs mean/median filters for adaptive image denoising

Rafael Verdú-Monedero1, Jesús Angulo2, Jorge Larrey-Ruiz1, Juan Morales-Sánchez1

  • 1Dpto. de Tecnologías de la Información y Comunicaciones,

Universidad Politécnica de Cartagena, 30202 Cartagena, Spain. {rafael.verdu, jorge.larrey, juan.morales}@upct.es

2Centre de Morphologie Mathématique (CMM),

Ecole des Mines de Paris, Fontainebleau Cedex, France jesus.angulo@ensmp.fr

ICIP 2010, International Conference on Image Processing