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Numerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and Statistics Mississippi State University December 13, 2006 Hyeona Lim Numerical Methods on the Image Processing Problems
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Page 1: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Numerical Methods on the Image ProcessingProblems

Hyeona Lim

Department of Mathematics and StatisticsMississippi State University

December 13, 2006

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 2: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Objective

I Develop efficient PDE (partial differential equations) basedmathematical models and their numerical algorithms for

1 Noise removal

Enhance the quality of images

2 Image segmentation

Edge (2D) or surface (3D) detection

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 3: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Objective

I Develop efficient PDE (partial differential equations) basedmathematical models and their numerical algorithms for

1 Noise removal

Enhance the quality of images

2 Image segmentation

Edge (2D) or surface (3D) detection

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 4: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Objective

I Develop efficient PDE (partial differential equations) basedmathematical models and their numerical algorithms for

1 Noise removal

Enhance the quality of images

2 Image segmentation

Edge (2D) or surface (3D) detection

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 5: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Applications - Noise Removal

Image with 20% impulse noise (left) and denoised image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 6: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Applications - Noise Removal

Image with 10% impulse noise (left) and denoised image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 7: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Applications - Noise Removal

Color image with impulse noise (left) and denoised image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 8: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Applications - Edge detection

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 9: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Applications - Edge detection

Medical Imaging

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 10: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Applications - Edge detection

Image Analysis in Materials Science

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 11: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Outline

1 History

- PDE based Mathematical Image Processing2 Image Denoising

I Conventional approach

- Total variation (TV) minimization

I New models and their numerical procedure

- Non-convex diffusion model - Kim and Lim (’05)- Anisotropic diffusion model - Lim and Williams (’06)

3 Image Segmentation

I Conventional ApproachI Method of Background Subtraction (MBS) for Medical Image

Segmentation - Kim and Lim (’05)

4 Conclusions

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 12: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Outline

1 History

- PDE based Mathematical Image Processing2 Image Denoising

I Conventional approach

- Total variation (TV) minimization

I New models and their numerical procedure

- Non-convex diffusion model - Kim and Lim (’05)- Anisotropic diffusion model - Lim and Williams (’06)

3 Image Segmentation

I Conventional ApproachI Method of Background Subtraction (MBS) for Medical Image

Segmentation - Kim and Lim (’05)

4 Conclusions

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 13: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Outline

1 History

- PDE based Mathematical Image Processing2 Image Denoising

I Conventional approach

- Total variation (TV) minimization

I New models and their numerical procedure

- Non-convex diffusion model - Kim and Lim (’05)- Anisotropic diffusion model - Lim and Williams (’06)

3 Image Segmentation

I Conventional ApproachI Method of Background Subtraction (MBS) for Medical Image

Segmentation - Kim and Lim (’05)

4 Conclusions

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 14: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Outline

1 History

- PDE based Mathematical Image Processing2 Image Denoising

I Conventional approach

- Total variation (TV) minimization

I New models and their numerical procedure

- Non-convex diffusion model - Kim and Lim (’05)- Anisotropic diffusion model - Lim and Williams (’06)

3 Image Segmentation

I Conventional ApproachI Method of Background Subtraction (MBS) for Medical Image

Segmentation - Kim and Lim (’05)

4 Conclusions

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 15: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Outline

1 History

- PDE based Mathematical Image Processing2 Image Denoising

I Conventional approach

- Total variation (TV) minimization

I New models and their numerical procedure

- Non-convex diffusion model - Kim and Lim (’05)- Anisotropic diffusion model - Lim and Williams (’06)

3 Image Segmentation

I Conventional ApproachI Method of Background Subtraction (MBS) for Medical Image

Segmentation - Kim and Lim (’05)

4 Conclusions

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 16: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

History of PDE based Mathematical Image Processing

I Short history, but has strong impact

I Image denoising, deconvolution (deblurring), segmentation(edge/surface detection)

I Image denoising

- Total variation (TV) minimization (Osher (’92), Lions (’97),Chan (’98), Kim (’01))

- Weakness: Edges of images can be easily smeared out due todiffusion property of TV minimization

original image (left) and smeared image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 17: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

History of PDE based Mathematical Image Processing

I Short history, but has strong impact

I Image denoising, deconvolution (deblurring), segmentation(edge/surface detection)

I Image denoising

- Total variation (TV) minimization (Osher (’92), Lions (’97),Chan (’98), Kim (’01))

- Weakness: Edges of images can be easily smeared out due todiffusion property of TV minimization

original image (left) and smeared image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 18: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

History of PDE based Mathematical Image Processing

I Short history, but has strong impact

I Image denoising, deconvolution (deblurring), segmentation(edge/surface detection)

I Image denoising

- Total variation (TV) minimization (Osher (’92), Lions (’97),Chan (’98), Kim (’01))

- Weakness: Edges of images can be easily smeared out due todiffusion property of TV minimization

original image (left) and smeared image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 19: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

History of PDE based Mathematical Image Processing

I Short history, but has strong impact

I Image denoising, deconvolution (deblurring), segmentation(edge/surface detection)

I Image denoising

- Total variation (TV) minimization (Osher (’92), Lions (’97),Chan (’98), Kim (’01))

- Weakness: Edges of images can be easily smeared out due todiffusion property of TV minimization

original image (left) and smeared image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 20: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

History of PDE based Mathematical Image Processing

I Color image denoising (Kim (’02), Osher (’03))- Use RGB color component

Color image with 15% impulse noise (left) and denoised image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 21: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising

Conventional approach

TV minimization model

ut − σ|∇u|γ+1∇ ·(

∇u

‖∇u‖

)= β (uo − u)

I Efficiently removes noiseI Image loses sharpness since the frequency of noise and edges

are similarI Produces a staircasing (locally constant) effect and

nonphysical dissipation

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 22: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising

Conventional approach

TV minimization model

ut − σ|∇u|γ+1∇ ·(

∇u

‖∇u‖

)= β (uo − u)

I Efficiently removes noiseI Image loses sharpness since the frequency of noise and edges

are similarI Produces a staircasing (locally constant) effect and

nonphysical dissipation

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 23: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising

Conventional approach

TV minimization model

ut − σ|∇u|γ+1∇ ·(

∇u

‖∇u‖

)= β (uo − u)

I Efficiently removes noiseI Image loses sharpness since the frequency of noise and edges

are similarI Produces a staircasing (locally constant) effect and

nonphysical dissipation

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 24: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising

Conventional approach

TV minimization model

ut − σ|∇u|γ+1∇ ·(

∇u

‖∇u‖

)= β (uo − u)

I Efficiently removes noiseI Image loses sharpness since the frequency of noise and edges

are similarI Produces a staircasing (locally constant) effect and

nonphysical dissipation

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 25: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - NC Model

Non-convex diffusion model

Control of nonphysical dissipationI Consider min

uFε,p(u), where

Fε,p(u) =∫Ω|∇εu|pdx + λ

2 ‖f − u‖2. Then

−p∇ ·(

∇u

|∇εu|2−p

)− λ(f − u) = 0,

where |∇εu| = (u2x + u2

y + ε2)1/2.

Sharp (left) and blurry image (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 26: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - NC Model

Non-convex diffusion model

Fε,p Sharp image Blurry image

F0,2 0 + 0 + 12 + 0 = 1 0 + 0.52 + 0.52 + 0 = 0.5F0,1 0 + 0 + 1 + 0 = 1 0 + 0.5 + 0.5 + 0 = 1

F0.1,1 0.1 + 0.1 +√

1.01 + 0.1 ≈ 1.31 0.1 +√

0.26 +√

0.26 + 0.1 ≈ 1.22F0,0.9 0 + 0 + 10.9 + 0 = 1 0 + 0.50.9 + 0.50.9 + 0 ≈ 1.07

I The strictly convex minimization (p > 1) makes imagesblurrier.

I The TV model itself (p = 1 and ε = 0) may not introduce“blur” but its regularization (p = 1 and ε > 0) does.

I When p < 1(non-convex), the model can make the imagesharper.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 27: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - NC Model

Non-convex diffusion model

Fε,p Sharp image Blurry image

F0,2 0 + 0 + 12 + 0 = 1 0 + 0.52 + 0.52 + 0 = 0.5F0,1 0 + 0 + 1 + 0 = 1 0 + 0.5 + 0.5 + 0 = 1

F0.1,1 0.1 + 0.1 +√

1.01 + 0.1 ≈ 1.31 0.1 +√

0.26 +√

0.26 + 0.1 ≈ 1.22F0,0.9 0 + 0 + 10.9 + 0 = 1 0 + 0.50.9 + 0.50.9 + 0 ≈ 1.07

I The strictly convex minimization (p > 1) makes imagesblurrier.

I The TV model itself (p = 1 and ε = 0) may not introduce“blur” but its regularization (p = 1 and ε > 0) does.

I When p < 1(non-convex), the model can make the imagesharper.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 28: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - NC Model

Non-convex diffusion model

Fε,p Sharp image Blurry image

F0,2 0 + 0 + 12 + 0 = 1 0 + 0.52 + 0.52 + 0 = 0.5F0,1 0 + 0 + 1 + 0 = 1 0 + 0.5 + 0.5 + 0 = 1

F0.1,1 0.1 + 0.1 +√

1.01 + 0.1 ≈ 1.31 0.1 +√

0.26 +√

0.26 + 0.1 ≈ 1.22F0,0.9 0 + 0 + 10.9 + 0 = 1 0 + 0.50.9 + 0.50.9 + 0 ≈ 1.07

I The strictly convex minimization (p > 1) makes imagesblurrier.

I The TV model itself (p = 1 and ε = 0) may not introduce“blur” but its regularization (p = 1 and ε > 0) does.

I When p < 1(non-convex), the model can make the imagesharper.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 29: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - NC Model

Non-convex diffusion model

Fε,p Sharp image Blurry image

F0,2 0 + 0 + 12 + 0 = 1 0 + 0.52 + 0.52 + 0 = 0.5F0,1 0 + 0 + 1 + 0 = 1 0 + 0.5 + 0.5 + 0 = 1

F0.1,1 0.1 + 0.1 +√

1.01 + 0.1 ≈ 1.31 0.1 +√

0.26 +√

0.26 + 0.1 ≈ 1.22F0,0.9 0 + 0 + 10.9 + 0 = 1 0 + 0.50.9 + 0.50.9 + 0 ≈ 1.07

I The strictly convex minimization (p > 1) makes imagesblurrier.

I The TV model itself (p = 1 and ε = 0) may not introduce“blur” but its regularization (p = 1 and ε > 0) does.

I When p < 1(non-convex), the model can make the imagesharper.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 30: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - NC Model

Non-convex diffusion model

New non-convex (NC) model

ut−|∇εu|1+ω∇·(

∇u

|∇εu|1+ω

)= β (f − u) , ω ∈ (−1, 1), β ≥ 0

Numerical proceduresI Linearized θ- method.I Alternating Direction Implicit (ADI) method

Theorem (Stability)

The θ- method for the new NC model is stable and holds themaximum principle.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 31: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - NC Model

Non-convex diffusion model

New non-convex (NC) model

ut−|∇εu|1+ω∇·(

∇u

|∇εu|1+ω

)= β (f − u) , ω ∈ (−1, 1), β ≥ 0

Numerical proceduresI Linearized θ- method.I Alternating Direction Implicit (ADI) method

Theorem (Stability)

The θ- method for the new NC model is stable and holds themaximum principle.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 32: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - NC Model

Non-convex diffusion model

New non-convex (NC) model

ut−|∇εu|1+ω∇·(

∇u

|∇εu|1+ω

)= β (f − u) , ω ∈ (−1, 1), β ≥ 0

Numerical proceduresI Linearized θ- method.I Alternating Direction Implicit (ADI) method

Theorem (Stability)

The θ- method for the new NC model is stable and holds themaximum principle.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 33: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Numerical Experiments - NC Model

Image with 20% mean zero noise (left), conventional method (middle), new NC method (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 34: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Numerical Experiments - NC Model

Image with 20% mean zero noise (left), conventional method (middle), new NC method (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 35: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Numerical Experiments - NC Model

Original brain image (left) and enhanced image by new NC method (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 36: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Numerical Experiments - NC Model

Horizontal line cuts of brain image and its restored images

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 37: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - AD Model

Anisotropic diffusion model

I Speckle noise is multiplicative and it can be modeled as(Krissian et. al. ’04, ’05):

f = u +√

un,

I Corresponding time marching equation:

∂u

∂t− u2

f + u|∇u|∇·

( ∇u

|∇u|

)= λ |∇u| (f − u).

• u2

f +u ≈ u/2 makes the diffusion faster in the lighter region(where the image values are high) and slower in the darkerregion (where the image values are low).⇒ unrealistic and ineffective in practice!

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 38: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - AD Model

Anisotropic diffusion model

I Speckle noise is multiplicative and it can be modeled as(Krissian et. al. ’04, ’05):

f = u +√

un,

I Corresponding time marching equation:

∂u

∂t− u2

f + u|∇u|∇·

( ∇u

|∇u|

)= λ |∇u| (f − u).

• u2

f +u ≈ u/2 makes the diffusion faster in the lighter region(where the image values are high) and slower in the darkerregion (where the image values are low).⇒ unrealistic and ineffective in practice!

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 39: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - AD Model

Anisotropic diffusion model

I Speckle noise is multiplicative and it can be modeled as(Krissian et. al. ’04, ’05):

f = u +√

un,

I Corresponding time marching equation:

∂u

∂t− u2

f + u|∇u|∇·

( ∇u

|∇u|

)= λ |∇u| (f − u).

• u2

f +u ≈ u/2 makes the diffusion faster in the lighter region(where the image values are high) and slower in the darkerregion (where the image values are low).⇒ unrealistic and ineffective in practice!

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 40: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - AD Model

Anisotropic diffusion model

I Speckle noise is multiplicative and it can be modeled as(Krissian et. al. ’04, ’05):

f = u +√

un,

I Corresponding time marching equation:

∂u

∂t− u2

f + u|∇u|∇·

( ∇u

|∇u|

)= λ |∇u| (f − u).

• u2

f +u ≈ u/2 makes the diffusion faster in the lighter region(where the image values are high) and slower in the darkerregion (where the image values are low).⇒ unrealistic and ineffective in practice!

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 41: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - AD Model

I Considerf = u +

(√u − us

)n,

where us : smoothed version of the noised image f .

New Anisotropic Diffusion (AD) Model

∂u

∂t− C |u − us |α|∇u|∇·

( ∇u

|∇u|

)= β (f − u), C > 0, 1/2 < α < 2

• On the regions where noise is present, |u − us | is relatively big.⇒ Diffusion is big enough to reduce the noise efficiently.

• On the regions where noise is not present, |u − us | is small.⇒ Diffusion is relatively slower.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 42: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - AD Model

I Considerf = u +

(√u − us

)n,

where us : smoothed version of the noised image f .

New Anisotropic Diffusion (AD) Model

∂u

∂t− C |u − us |α|∇u|∇·

( ∇u

|∇u|

)= β (f − u), C > 0, 1/2 < α < 2

• On the regions where noise is present, |u − us | is relatively big.⇒ Diffusion is big enough to reduce the noise efficiently.

• On the regions where noise is not present, |u − us | is small.⇒ Diffusion is relatively slower.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 43: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - AD Model

I Considerf = u +

(√u − us

)n,

where us : smoothed version of the noised image f .

New Anisotropic Diffusion (AD) Model

∂u

∂t− C |u − us |α|∇u|∇·

( ∇u

|∇u|

)= β (f − u), C > 0, 1/2 < α < 2

• On the regions where noise is present, |u − us | is relatively big.⇒ Diffusion is big enough to reduce the noise efficiently.

• On the regions where noise is not present, |u − us | is small.⇒ Diffusion is relatively slower.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 44: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Image Denoising - AD Model

I Considerf = u +

(√u − us

)n,

where us : smoothed version of the noised image f .

New Anisotropic Diffusion (AD) Model

∂u

∂t− C |u − us |α|∇u|∇·

( ∇u

|∇u|

)= β (f − u), C > 0, 1/2 < α < 2

• On the regions where noise is present, |u − us | is relatively big.⇒ Diffusion is big enough to reduce the noise efficiently.

• On the regions where noise is not present, |u − us | is small.⇒ Diffusion is relatively slower.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 45: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Numerical Procedure - AD Model

1 TFR (texture-free residual) parametrization

1 Set β as a constant: β(x, 0) = β0.2 For n = 2, 3, · · ·

I Compute the absolute residual and a quantity G n−1Res :

Rn−1 = |f − un−1|,

G n−1Res = max

0, Sm(Rn−1)− Rn−1

,

where Sm is a smoothing operator and Rn−1 denotes theL2-average of Rn−1.

I Update:βn = βn−1 + γn G n−1

Res ,

where γn is a scaling factor having the property: γn → 0 asn →∞.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 46: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Numerical Experiments - AD Model

Cuba Missile Crisis: The original (left above) and restored images by using ITV (right

above), ITV-TFR (left below), and AD-TFR (right below) model

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 47: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Medical Image Segmentation - MBS

Motivation

I Medical images can involve noise, diverse artifacts, andunclear edges.

I Conventional segmentation methods show difficulties whenapplied to medical imagery.

I When an appropriate background is subtracted from the givenimage, the residue can be considered as an essentially binaryimage.

Hyeona Lim Numerical Methods on the Image Processing Problems

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Medical Image Segmentation - MBS

Procedure of method of background subtraction

Hyeona Lim Numerical Methods on the Image Processing Problems

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Medical Image Segmentation - MBS

Procedure of the construction of background

1 Select a coarse mesh Ωij for the image domain Ω andchoose a coarse image Uc on Ωij. Each element Ωij in thecoarse mesh corresponds to mx ×my pixels.

2 Smooth Uc .

3 Prolongate Uc to the original mesh Ω, for Uf .

4 Smooth Uf . Assign the result for the background U.

Strategies for background construction

I In step I, choose Uc on Ωij as raij + (1− r)mij , 0 ≤ r ≤ 1,aij : arithmetic average, mij : minimum of U0 on Ωij .

I U must contain only background Information, not objectsinformation. Thus select m = mx = my such that number ofblocks in Uc corresponding to objects are smaller than thenumber of smoothing iterations in step II.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 50: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Medical Image Segmentation - MBS

Procedure of the construction of background

1 Select a coarse mesh Ωij for the image domain Ω andchoose a coarse image Uc on Ωij. Each element Ωij in thecoarse mesh corresponds to mx ×my pixels.

2 Smooth Uc .

3 Prolongate Uc to the original mesh Ω, for Uf .

4 Smooth Uf . Assign the result for the background U.

Strategies for background construction

I In step I, choose Uc on Ωij as raij + (1− r)mij , 0 ≤ r ≤ 1,aij : arithmetic average, mij : minimum of U0 on Ωij .

I U must contain only background Information, not objectsinformation. Thus select m = mx = my such that number ofblocks in Uc corresponding to objects are smaller than thenumber of smoothing iterations in step II.

Hyeona Lim Numerical Methods on the Image Processing Problems

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Numerical Experiments - MBS

Heart: Original (left), Conventional Method (middle), Conventional approach with MBS (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

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Numerical Experiments - MBS

Hand: Original (left), Conventional Method (middle), Conventional approach with MBS (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

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Numerical Experiments - MBS

Leukemia: Original (left), Conventional Method (middle), Conventional approach with MBS (right)

Hyeona Lim Numerical Methods on the Image Processing Problems

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Conclusions

1 Regularization of |∇u| can be a significant source ofnonphysical dissipation in TV-based models.

2 Non-convex (NC) diffusion model has been introduced inorder to simultaneously suppress the noise and enhance edges.

3 Anisotropic diffusion (AD) model is more efficient on SARimage denoising than conventional models.

4 Numerical procedures of NC and AD models are stable.5 The method of background subtraction (MBS) can be

efficiently used as a pre-process of various segmentationmethods for medical image segmentation.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 55: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Conclusions

1 Regularization of |∇u| can be a significant source ofnonphysical dissipation in TV-based models.

2 Non-convex (NC) diffusion model has been introduced inorder to simultaneously suppress the noise and enhance edges.

3 Anisotropic diffusion (AD) model is more efficient on SARimage denoising than conventional models.

4 Numerical procedures of NC and AD models are stable.5 The method of background subtraction (MBS) can be

efficiently used as a pre-process of various segmentationmethods for medical image segmentation.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 56: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Conclusions

1 Regularization of |∇u| can be a significant source ofnonphysical dissipation in TV-based models.

2 Non-convex (NC) diffusion model has been introduced inorder to simultaneously suppress the noise and enhance edges.

3 Anisotropic diffusion (AD) model is more efficient on SARimage denoising than conventional models.

4 Numerical procedures of NC and AD models are stable.5 The method of background subtraction (MBS) can be

efficiently used as a pre-process of various segmentationmethods for medical image segmentation.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 57: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Conclusions

1 Regularization of |∇u| can be a significant source ofnonphysical dissipation in TV-based models.

2 Non-convex (NC) diffusion model has been introduced inorder to simultaneously suppress the noise and enhance edges.

3 Anisotropic diffusion (AD) model is more efficient on SARimage denoising than conventional models.

4 Numerical procedures of NC and AD models are stable.5 The method of background subtraction (MBS) can be

efficiently used as a pre-process of various segmentationmethods for medical image segmentation.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 58: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Conclusions

1 Regularization of |∇u| can be a significant source ofnonphysical dissipation in TV-based models.

2 Non-convex (NC) diffusion model has been introduced inorder to simultaneously suppress the noise and enhance edges.

3 Anisotropic diffusion (AD) model is more efficient on SARimage denoising than conventional models.

4 Numerical procedures of NC and AD models are stable.5 The method of background subtraction (MBS) can be

efficiently used as a pre-process of various segmentationmethods for medical image segmentation.

Hyeona Lim Numerical Methods on the Image Processing Problems

Page 59: Numerical Methods on the Image Processing Problemshl107.math.msstate.edu/pdfs/khu_06.pdfNumerical Methods on the Image Processing Problems Hyeona Lim Department of Mathematics and

Conclusions

1 Regularization of |∇u| can be a significant source ofnonphysical dissipation in TV-based models.

2 Non-convex (NC) diffusion model has been introduced inorder to simultaneously suppress the noise and enhance edges.

3 Anisotropic diffusion (AD) model is more efficient on SARimage denoising than conventional models.

4 Numerical procedures of NC and AD models are stable.5 The method of background subtraction (MBS) can be

efficiently used as a pre-process of various segmentationmethods for medical image segmentation.

Hyeona Lim Numerical Methods on the Image Processing Problems