Directional Total Generalized Variation - … · Directional Total Generalized Variation Imaging,...

Directional Total Generalized Variation

Imaging, Vision and Learning based on Optimization and PDEs, Bergen, Norway.

Rasmus Dalgas Kongskov (PhD student).

Joint work with: Yiqiu Dong and Kim Knudsen

HD-Tomo Group, Technical University of Denmark (DTU)

Total Generalized Variation• Regularization is used for wide range of optimization problems.• Choice of regularizer based on prior information of your object/solution.

• Within Imaging research: Total Variation (TV) [Rudin,Osher and Fatemi, ’92],• prior: piecewise constant function,• drawbacks: loss of contrast and stair-casing effects.

• Overcome stair-casing: use a higher order method - Total Generalized Variation(TGV) [Bredies, Kunisch and Pock, ’10],

• TGV2λ prior: piecewise affine function,

• drawbacks: higher computational cost and introduction of more parameters.Org CT Rec, TV Reg CT Rec, TGV Reg

1 of 3 DTU Compute DTGV Regularization 29.8.2016

Total Generalized Variation• Regularization is used for wide range of optimization problems.• Choice of regularizer based on prior information of your object/solution.• Within Imaging research: Total Variation (TV) [Rudin,Osher and Fatemi, ’92],

• prior: piecewise constant function,• drawbacks: loss of contrast and stair-casing effects.









• drawbacks: higher computational cost and introduction of more parameters.

Org CT Rec, TV Reg CT Rec, TGV Reg








DTV → DTGV• In Program: Abstract and title related to Directional Total Variation (DTV).• On Poster: Extended to TGV: Directional TGV (DTGV).• Directional regularization:

• Prior: object varies less along one direction than the orthogonal directions.• Here: an additional prior to e.g. piecewise constant.

• TV vs DTV:

= sup∫

Ωu div(v) dx

∣∣∣∣v ∈ C1c (Ω,R2),v(x) ∈ ∀x ∈ Ω

,

• TGV vs DTGV:

sup∫

Ωu divh(w) dx

∣∣∣∣w ∈ Chc (Ω,Symh(RN )), , l < k

,




• TV vs DTV:

TV(u) = sup∫

Ωu div(v) dx

∣∣∣∣v ∈ C1c (Ω,R2),v(x) ∈ B(0, 1) ∀x ∈ Ω

,

• TGV vs DTGV:

sup∫

Ωu divh(w) dx

∣∣∣∣w ∈ Chc (Ω,Symh(RN )), , l < k

,




• TV vs DTV:

DTV(u) = sup∫

Ωu div(v) dx

∣∣∣∣v ∈ C1c (Ω,R2),v(x) ∈ Eθ,a ∀x ∈ Ω

,

• TGV vs DTGV:

sup∫

Ωu divh(w) dx

∣∣∣∣w ∈ Chc (Ω,Symh(RN )), , l < k

,




• TV vs DTV:

DTV(u) = sup∫

Ωu div(v) dx

∣∣∣∣v ∈ C1c (Ω,R2),v(x) ∈ Eθ,a ∀x ∈ Ω

,

• TGV vs DTGV: TGVhλ(u) =

sup∫

Ωu divh(w) dx

∣∣∣∣w ∈ Chc (Ω,Symh(RN )), ‖divl(w)‖∞ ≤ λl, l < k

,




• TV vs DTV:

DTV(u) = sup∫

Ωu div(v) dx

∣∣∣∣v ∈ C1c (Ω,R2),v(x) ∈ Eθ,a ∀x ∈ Ω

,

• TGV vs DTGV: DTGVkλ(u) =

sup

∫

Ωu divh(w) dx

∣∣∣∣w ∈ Chc (Ω,Symh(RN )), divl(w) ∈ Eλla,θ × . . .× Eλla,θ︸︷︷︸

h−l times

, l < k

,


Properties and Numerical Experiments

• For DTGVhλ:(a

maxl∈(0,h−1)

λ2l

)h

TGVhλ(u) ≤ DTGVhλ(u) ≤ TGVhλ(u).

• 1-homogeneous,• convex,• lower semi-continuous.

• L2 − DTGV2λ :

argminu∈BGV 2

λ(Ω)

12‖G(u)− f‖2

L2(Ω) + DTGV2λ(u).

• existence of minimizer.

Directional Total Generalized VariationRasmus Dalgas Kongskov ([email protected]), Yiqiu Dong, Kim Knudsen

Department of Applied Mathematics and Computer Science, Technical University of Denmark

Total Generalized Variation Regularization

• Overcome ill-posedness of inverse problems: Regularization.

• Choice of RegularizerR according to Prior information about u.

• Focus here on linear inverse problems, e.g.:

• image processing problems,• tomography problems.

• Variational formulation

minu

12‖Au− b‖2

2 +R(u).• Classical regularization for Imaging problems: Tikhonov and Total Variation (TV).

• Total Generalized Variation (TGV), TGVhλ(u) =

sup∫

Ωu divh(w) dx

∣∣∣∣w ∈ Chc (Ω, Symh(RN)), ‖divl(w)‖∞ ≤ λl, l = 0, ..., k− 1

.

• Prior for second order TGV2λ: piecewise linear functions.

• TGV2λ is advantageous to TV, most notably because it overcomes stair-casing.

• Rule-of-thumb for TGV2λ: Chose regularization parameters as λ1

λ0= 2.


• Prior for DTGV2λ: Piecewise linear and

the object varies less along one direction opposed to orthogonal direction(s).

• Modify constraints of TGV2λ using ellipse Ea,θ.

• For v ∈ B(0, 1) we restrict v ∈ Ea,θ via

v = RθΛav, Rθ =

(cos θ − sin θsin θ cos θ

), Λa =

(1 00 a

).

• We restrict w in the tensor ellipse:

DTGV2λ(u) = sup

∫

Ωu div2(w) dx

∣∣∣∣ w ∈W

,

W =

C2c (Ω, Sym2(R2)), w ∈ Eλ0

a,θ × Eλ0a,θ, div(w) ∈ Eλ1

a,θ

.

• DTGV2λ(u) is 1-homogeneous, convex and lower semi-continuous.

• We can proof existence of a minimizer for

minu

12‖Au− b‖2

2 + DTGV2λ(u).

• DTGV2λ depends on two extra parameters a and θ, but θ can be estimated.

Regularizing Image Denoising Problems

• Remove additive Gaussian noise from images, A = I.• Estimate direction directly from noisy images:

smooth image→ gradient→ pixel-wise angles→mean direction.

• Primal-dual method (Chambolle-Pock) for minimization.• Peak Signal-to-noise (PSNR) ratio for a range of (a, θ):

Best PSNR (λ) for (a, θ)

-62 -52 -42 -32 -22

θ

0.01

0.05

0.15

0.25

0.35

0.45

a

30.5

31

31.5

32

32.5

33

33.5

Overall best PSNR = 33.6

• Lines: tested angles, estimated angle and best angle

Org

Noisy, psnr = 20.0

TV, psnr = 22.5 TGV, psnr = 22.7

DTV, psnr = 25.5 DTGV, psnr = 25.6

Org

Noisy, psnr = 14.0



Regularizing Computed Tomography Reconstructions

• Reconstruct object u from attenuated x-ray measurements b.• Filtered Back-projection (FBP) is a classical 2D-reconstruction method, but it requires

densely sampled measurements.• Estimation of main direction directly from measured noisy data:

Phantom with direction

var(

∇tb)

Sinogram / b

t

θ

30

60

90

120

150

• Examples: Underdetermined system with simulated Gaussian noise.• a = 0.1 and θ estimated from noisy measurements.

Org FBP, psnr = 12.9

TGV, psnr = 22.9 DTGV, psnr = 27.6



References

[1] I. Bayram and M. E. Kamasak.

A directional total variation.

Eur. Signal Process. Conf., 19(12):265–269, 2012.

[2] K. Bredies, K. Kunisch, and T. Pock.

Total Generalized Variation.

SIAM J. Imaging Sci., 3(3):492–526, 2010.

[3] A. Chambolle and T. Pock.A first-order primal-dual algorithm for convex problemswith applications to imaging.J. Math. Imaging Vis., 40(1):120–145, dec 2011.

[4] L. I. Rudin, S. Osher, and E. Fatemi.Nonlinear total variation based noise removal algorithms.Phys. D Nonlinear Phenom., 60(1-4):259–268, 1992.

Acknowledgement

This work is part of the project HD-Tomo funded by Ad-vanced Grant No. 291405, European Research Council.

Conclusions

• DTGV has same functional properties as TGV.• Directional information in images and CT data is noticeable and can be

estimated directly.• When the prior for DTGV is met, it is highly advantageous to TV and TGV.

Future Research

• Combined multiple main directions.• Directional information combined with segmentation.


Properties and Numerical Experiments

• For DTGVhλ:(a

maxl∈(0,h−1)

λ2l

)h

TGVhλ(u) ≤ DTGVhλ(u) ≤ TGVhλ(u).

• 1-homogeneous,• convex,• lower semi-continuous.

• L2 − DTGV2λ :

argminu∈BGV 2

λ(Ω)

12‖G(u)− f‖2

L2(Ω) + DTGV2λ(u).

• existence of minimizer.

Directional Total Generalized VariationRasmus Dalgas Kongskov ([email protected]), Yiqiu Dong, Kim Knudsen

Department of Applied Mathematics and Computer Science, Technical University of Denmark

Total Generalized Variation Regularization

• Overcome ill-posedness of inverse problems: Regularization.

• Choice of RegularizerR according to Prior information about u.

• Focus here on linear inverse problems, e.g.:

• image processing problems,• tomography problems.

• Variational formulation

minu

12‖Au− b‖2

2 +R(u).• Classical regularization for Imaging problems: Tikhonov and Total Variation (TV).

• Total Generalized Variation (TGV), TGVhλ(u) =

sup∫

Ωu divh(w) dx

∣∣∣∣w ∈ Chc (Ω, Symh(RN)), ‖divl(w)‖∞ ≤ λl, l = 0, ..., k− 1

.

• Prior for second order TGV2λ: piecewise linear functions.

• TGV2λ is advantageous to TV, most notably because it overcomes stair-casing.

• Rule-of-thumb for TGV2λ: Chose regularization parameters as λ1

λ0= 2.


• Prior for DTGV2λ: Piecewise linear and

the object varies less along one direction opposed to orthogonal direction(s).

• Modify constraints of TGV2λ using ellipse Ea,θ.

• For v ∈ B(0, 1) we restrict v ∈ Ea,θ via

v = RθΛav, Rθ =

(cos θ − sin θsin θ cos θ

), Λa =

(1 00 a

).

• We restrict w in the tensor ellipse:

DTGV2λ(u) = sup

∫

Ωu div2(w) dx

∣∣∣∣ w ∈W

,

W =

C2c (Ω, Sym2(R2)), w ∈ Eλ0

a,θ × Eλ0a,θ, div(w) ∈ Eλ1

a,θ

.

• DTGV2λ(u) is 1-homogeneous, convex and lower semi-continuous.

• We can proof existence of a minimizer for

minu

12‖Au− b‖2

2 + DTGV2λ(u).

• DTGV2λ depends on two extra parameters a and θ, but θ can be estimated.

Regularizing Image Denoising Problems

• Remove additive Gaussian noise from images, A = I.• Estimate direction directly from noisy images:

smooth image→ gradient→ pixel-wise angles→mean direction.

• Primal-dual method (Chambolle-Pock) for minimization.• Peak Signal-to-noise (PSNR) ratio for a range of (a, θ):

Best PSNR (λ) for (a, θ)

-62 -52 -42 -32 -22

θ

0.01

0.05

0.15

0.25

0.35

0.45

a

30.5

31

31.5

32

32.5

33

33.5

Overall best PSNR = 33.6

• Lines: tested angles, estimated angle and best angle

Org

Noisy, psnr = 20.0



Org

Noisy, psnr = 14.0



Regularizing Computed Tomography Reconstructions

• Reconstruct object u from attenuated x-ray measurements b.• Filtered Back-projection (FBP) is a classical 2D-reconstruction method, but it requires

densely sampled measurements.• Estimation of main direction directly from measured noisy data:

Phantom with direction

var(

∇tb)

Sinogram / b

t

θ

30

60

90

120

150

• Examples: Underdetermined system with simulated Gaussian noise.• a = 0.1 and θ estimated from noisy measurements.





References

[1] I. Bayram and M. E. Kamasak.

A directional total variation.

Eur. Signal Process. Conf., 19(12):265–269, 2012.

[2] K. Bredies, K. Kunisch, and T. Pock.

Total Generalized Variation.

SIAM J. Imaging Sci., 3(3):492–526, 2010.

[3] A. Chambolle and T. Pock.A first-order primal-dual algorithm for convex problemswith applications to imaging.J. Math. Imaging Vis., 40(1):120–145, dec 2011.

[4] L. I. Rudin, S. Osher, and E. Fatemi.Nonlinear total variation based noise removal algorithms.Phys. D Nonlinear Phenom., 60(1-4):259–268, 1992.

Acknowledgement

This work is part of the project HD-Tomo funded by Ad-vanced Grant No. 291405, European Research Council.

Conclusions

• DTGV has same functional properties as TGV.• Directional information in images and CT data is noticeable and can be

estimated directly.• When the prior for DTGV is met, it is highly advantageous to TV and TGV.

Future Research

• Combined multiple main directions.• Directional information combined with segmentation.


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