Multiscale local multiple orientation estimation using Mathematical - - PowerPoint PPT Presentation

Multiscale local multiple orientation estimation using Mathematical Morphology and B-spline interpolation

Jesús Angulo1, Rafael Verdú2, Juan Morales2

1Centre de Morphologie Mathématique (CMM),

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

• 2Dpto. 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

1

Introduction

2

Modelling orientation using mathematical morphology

3

Proposed method

4

Results

5

Conclusions

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 2 / 30

Introduction

Outline

1

Introduction

2

Modelling orientation using mathematical morphology

3

Proposed method

4

Results

5

Conclusions

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 3 / 30

Introduction

Objectives

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

1

Obtain a vector field with orientation information in all pixels.

2

Perform a spatially-variant filtering: 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

• f 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

• f 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

• f 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

• f 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

• f 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

• f 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

1

Introduction

2

Modelling orientation using mathematical morphology

3

Proposed method

4

Results

5

Conclusions

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 γLθ,l(f)(x) = δLθ,l [εLθ,l(f)] (x) where the directional erosion and dilation are, respectively, εLθ,l(f)(x) =

• h∈Lθ,l(x)

{f(x + h)} , δLθ,l(f)(x) =

• h∈Lθ,l(x)

{f(x − h)} .

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 γLθ,l(f)(x) = δLθ,l [εLθ,l(f)] (x) where the directional erosion and dilation are, respectively, εLθ,l(f)(x) =

• h∈Lθ,l(x)

{f(x + h)} , δLθ,l(f)(x) =

• h∈Lθ,l(x)

{f(x − h)} .

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 8 / 30

Modelling orientation using mathematical morphology Orientation field

Orientation information Let us consider the orientation of image f from its gradient: g(f)(x) = ∇f(x) =

• ∂f(x,y)

∂x

2 +

• ∂f(x,y)

∂y

2 . Image orientation model Based on a multiscale decomposition of the gradient information g(f)(x) ≈ aΘ

l1 (g)(x) + aΘ l2 (g)(x) + aΘ l3 (g)(x),

where aΘ

lj (g)(x) =

• i∈Ij

γLθi ,lj (r Θ

lj−1)(g)(x).

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 9 / 30

Modelling orientation using mathematical morphology Orientation field

Orientation information Let us consider the orientation of image f from its gradient: g(f)(x) = ∇f(x) =

• ∂f(x,y)

∂x

2 +

• ∂f(x,y)

∂y

2 . Image orientation model Based on a multiscale decomposition of the gradient information g(f)(x) ≈ aΘ

l1 (g)(x) + aΘ l2 (g)(x) + aΘ l3 (g)(x),

where aΘ

lj (g)(x) =

• i∈Ij

γLθi ,lj (r Θ

lj−1)(g)(x).

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 9 / 30

Modelling orientation using mathematical morphology Orientation decomposition

The error of the approximation of the orientation information r Θ

lj (g)(x)

can be obtained as r Θ

l1 (g)(x) = r Θ l0 (g)(x) − aΘ l1 (g)(x),

r Θ

l2 (g)(x) = r Θ l1 (g)(x) − aΘ l2 (g)(x),

r Θ

l3 (g)(x) = r Θ l2 (g)(x) − aΘ l3 (g)(x),

with r Θ

l0 (g)(x) being the original gradient intensity g(x).

It can be easily shown that g(x) =

3

• j=1

aΘ

lj (g)(x) + r Θ l3 (g)(x).

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 10 / 30

Modelling orientation using mathematical morphology Orientation decomposition

The error of the approximation of the orientation information r Θ

lj (g)(x)

can be obtained as r Θ

l1 (g)(x) = r Θ l0 (g)(x) − aΘ l1 (g)(x),

r Θ

l2 (g)(x) = r Θ l1 (g)(x) − aΘ l2 (g)(x),

r Θ

l3 (g)(x) = r Θ l2 (g)(x) − aΘ l3 (g)(x),

with r Θ

l0 (g)(x) being the original gradient intensity g(x).

It can be easily shown that g(x) =

3

• j=1

aΘ

lj (g)(x) + r Θ l3 (g)(x).

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 10 / 30

Proposed method

Outline

1

Introduction

2

Modelling orientation using mathematical morphology

3

Proposed method

4

Results

5

Conclusions

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 11 / 30

Proposed method Orientation signature and multiple peaks detection

Orientation signature For the block centred on pixel xp, at each stage j sxp;lj(i) =

• y∈WN(xp) γLθi ,lj (r Θ

lj−1)(g)(y)

• y∈WN(xp) r Θ

lj−1(g)(y)

. which contains, for each discrete angle θi, the sum of the pixels in window WN(xp) of the image opened by Lθi,lj.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 12 / 30

Proposed method Orientation signature and multiple peaks detection

Multiple peak detection The orientation signature is interpolated using cubic b-splines: ˆ Sxp;lj(α) =

2(lj−1)−1

• i=0

sxp;lj(i) b3(α −

90 lj−1i),

The peaks are found by searching the angles αp where d ˆ Sxp;lj(α) dα

• α=αp

= 0, and d2 ˆ Sxp;lj(α) d2α

• α=αp

< 0.

• nly those angles αp whose peak value is greater than a given

threshold will be considered.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 13 / 30

Proposed method Orientation signature and multiple peaks detection

Multiple peak detection The orientation signature is interpolated using cubic b-splines: ˆ Sxp;lj(α) =

2(lj−1)−1

• i=0

sxp;lj(i) b3(α −

90 lj−1i),

The peaks are found by searching the angles αp where d ˆ Sxp;lj(α) dα

• α=αp

= 0, and d2 ˆ Sxp;lj(α) d2α

• α=αp

< 0.

• nly those angles αp whose peak value is greater than a given

threshold will be considered.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 13 / 30

Results

Outline

1

Introduction

2

Modelling orientation using mathematical morphology

3

Proposed method

4

Results

5

Conclusions

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 14 / 30

Results Image #1

Figure: Image #1 used in the study: 194×152 pixels. The square depicts the block of 32×32 pixels which has been analyzed.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 15 / 30

Results Image #1

rl

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sx

p;l 1

, length of SE: 31

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 16 / 30

Results Image #1

rl

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sx

p;l 1

, length of SE: 31 20 40 60 80 100 120 140 160 180 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 16 / 30

Results Image #1

rl

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sx

p;l 1

, length of SE: 31 20 40 60 80 100 120 140 160 180 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 16 / 30

Results Image #1

rl

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sx

p;l 1

, length of SE: 31 20 40 60 80 100 120 140 160 180 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 16 / 30

Results Image #1

rl

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Sx

p;l 1

, length of SE: 31 20 40 60 80 100 120 140 160 180 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 16 / 30

Results Image #1

rl

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 2

, length of SE: 15

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 17 / 30

Results Image #1

rl

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 2

, length of SE: 15 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 17 / 30

Results Image #1

rl

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 2

, length of SE: 15 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 17 / 30

Results Image #1

rl

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 2

, length of SE: 15 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 17 / 30

Results Image #1

rl

1

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 2

, length of SE: 15 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 17 / 30

Results Image #1

rl

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

3

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 3

, length of SE: 7

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 18 / 30

Results Image #1

rl

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

3

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 3

, length of SE: 7 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 18 / 30

Results Image #1

rl

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

3

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 3

, length of SE: 7 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 18 / 30

Results Image #1

rl

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

3

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 3

, length of SE: 7 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 18 / 30

Results Image #1

rl

2

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 al

3

Θ(x)

5 10 15 20 25 30 5 10 15 20 25 30 50 100 150 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sx

p;l 3

, length of SE: 7 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 18 / 30

Results Image #1

Figure: Image #1 used in the study: 194×152 pixels. The square depicts the blocks of 32×32 pixels which has been analyzed.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 19 / 30

Results Image #1

(a) r Θ

l0 (g)(x)

(b) r Θ

l1 (g)(x)

(c) rΘ

l2 (g)(x)

(d) aΘ

l1 (g)(x)

(e) aΘ

l2 (g)(x)

(f) aΘ

l3 (g)(x)

Figure: Block of Image #1.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 20 / 30

Results Image #1 20 40 60 80 100 120 140 160 180 0.05 0.1 0.15 0.2 0.25 0.3 0.35 SE length: 31 SE length: 15 SE length: 7

(a) Block of Image #1.

Figure: (left) Orientation signature sxp;lj(i) and (right) multiscale multiple

• rientations estimated.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 21 / 30

Results Image #2

Figure: Image #2 used in the study: 640×480 pixels. The squares depicts the block of 32×32 pixels which has been analyzed.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 22 / 30

Results Image #2

(a) r Θ

l0 (g)(x)

(b) r Θ

l1 (g)(x)

(c) rΘ

l2 (g)(x)

(d) aΘ

l1 (g)(x)

(e) aΘ

l2 (g)(x)

(f) aΘ

l3 (g)(x)

Figure: Block of Image #2.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 23 / 30

Results Image #2 20 40 60 80 100 120 140 160 180 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 SE length: 31 SE length: 15 SE length: 7

(a) Block of Image #1.

Figure: (left) Orientation signature sxp;lj(i) and (right) multiscale multiple

• rientations estimated.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 24 / 30

Results Image #3

Figure: Image #3 used in the study: 128×128 pixels. The squares depicts the block of 32×32 pixels which has been analyzed.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 25 / 30

Results Image #3

(a) r Θ

l0 (g)(x)

(b) r Θ

l1 (g)(x)

(c) rΘ

l2 (g)(x)

(d) aΘ

l1 (g)(x)

(e) aΘ

l2 (g)(x)

(f) aΘ

l3 (g)(x)

Figure: Block of Image #3.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 26 / 30

Results Image #3 20 40 60 80 100 120 140 160 180 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 SE length: 31 SE length: 15 SE length: 7

(a) Block of Image #1.

Figure: (left) Orientation signature sxp;lj(i) and (right) multiscale multiple

• rientations estimated.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 27 / 30

Conclusions

Outline

1

Introduction

2

Modelling orientation using mathematical morphology

3

Proposed method

4

Results

5

Conclusions

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 28 / 30

Conclusions

This paper proposes a multiscale approach to estimate local multiple orientations. The method relies on two keypoints:

1

Multiscale directional signature, obtained by means of morphological openings.

2

Multiple peak detection, using cubic b-spline interpolation.

Results show the accuracy of the proposed approach to estimate the multiple orientations in textured images. The authors are working on the use of multiple orientation information to implement spatially-variant filters. In this line of research, the kernels of the filter (linear and/or nonlinear) vary over space, adapting their shape and orientation according to the data contained in the image under study.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 29 / 30

Conclusions

This paper proposes a multiscale approach to estimate local multiple orientations. The method relies on two keypoints:

1

Multiscale directional signature, obtained by means of morphological openings.

2

Multiple peak detection, using cubic b-spline interpolation.

Results show the accuracy of the proposed approach to estimate the multiple orientations in textured images. The authors are working on the use of multiple orientation information to implement spatially-variant filters. In this line of research, the kernels of the filter (linear and/or nonlinear) vary over space, adapting their shape and orientation according to the data contained in the image under study.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 29 / 30

Conclusions

This paper proposes a multiscale approach to estimate local multiple orientations. The method relies on two keypoints:

1

Multiscale directional signature, obtained by means of morphological openings.

2

Multiple peak detection, using cubic b-spline interpolation.

Results show the accuracy of the proposed approach to estimate the multiple orientations in textured images. The authors are working on the use of multiple orientation information to implement spatially-variant filters. In this line of research, the kernels of the filter (linear and/or nonlinear) vary over space, adapting their shape and orientation according to the data contained in the image under study.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 29 / 30

Conclusions

This paper proposes a multiscale approach to estimate local multiple orientations. The method relies on two keypoints:

1

Multiscale directional signature, obtained by means of morphological openings.

2

Multiple peak detection, using cubic b-spline interpolation.

Results show the accuracy of the proposed approach to estimate the multiple orientations in textured images. The authors are working on the use of multiple orientation information to implement spatially-variant filters. In this line of research, the kernels of the filter (linear and/or nonlinear) vary over space, adapting their shape and orientation according to the data contained in the image under study.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 29 / 30

Conclusions

This paper proposes a multiscale approach to estimate local multiple orientations. The method relies on two keypoints:

1

Multiscale directional signature, obtained by means of morphological openings.

2

Multiple peak detection, using cubic b-spline interpolation.

Results show the accuracy of the proposed approach to estimate the multiple orientations in textured images. The authors are working on the use of multiple orientation information to implement spatially-variant filters. In this line of research, the kernels of the filter (linear and/or nonlinear) vary over space, adapting their shape and orientation according to the data contained in the image under study.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 29 / 30

Conclusions

This paper proposes a multiscale approach to estimate local multiple orientations. The method relies on two keypoints:

1

Multiscale directional signature, obtained by means of morphological openings.

2

Multiple peak detection, using cubic b-spline interpolation.

Results show the accuracy of the proposed approach to estimate the multiple orientations in textured images. The authors are working on the use of multiple orientation information to implement spatially-variant filters. In this line of research, the kernels of the filter (linear and/or nonlinear) vary over space, adapting their shape and orientation according to the data contained in the image under study.

Angulo,Verdú,Morales (CMM-UPCT) ISPA 2011 Friday, 02.09.2011 29 / 30

Multiscale local multiple orientation estimation using Mathematical Morphology and B-spline interpolation

Jesús Angulo1, Rafael Verdú2, Juan Morales2

1Centre de Morphologie Mathématique (CMM),

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

• 2Dpto. 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