[PPT] - The Shannon Total Variation Rmy Abergel, joint work with Lionel PowerPoint Presentation

SLIDE 1

The Shannon Total Variation

Rémy Abergel, joint work with Lionel Moisan. CNRS, MAP5 Laboratory, Paris Descartes University. Conference on variational methods and optimization in imaging, The Mathematics of Imaging, IHP , February 6, 2019.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 1 / 34

SLIDE 2

The total variation

Given an image U : Ωc ⊂ R2 → R that belongs to the Sobolev space W1,1(Ωc), we note TV(U) =

Ωc

|DU(x, y)| dx dy . This definition can be extended to the space BV(Ωc) of functions with bounded variation that are non-necessarily differentiable.

Definition (discrete total variation)

The total variation of a discrete image u : Ω ⊂ Z2 → R is generally defined by TVd(u) =

(x,y)∈Ω

|∇u(x, y)| , where ∇ denotes a finite-differences scheme.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 2 / 34

SLIDE 3

The total variation

Given an image U : Ωc ⊂ R2 → R that belongs to the Sobolev space W1,1(Ωc), we note TV(U) =

Ωc

|DU(x, y)| dx dy . This definition can be extended to the space BV(Ωc) of functions with bounded variation that are non-necessarily differentiable.

Definition (discrete total variation)

The total variation of a discrete image u : Ω ⊂ Z2 → R is generally defined by TVd(u) =

(x,y)∈Ω

|∇u(x, y)| , where ∇ denotes a finite-differences scheme.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 2 / 34

SLIDE 4

The total variation

The use of the total variation in image processing has become popular with the work of Rudin, Osher et Fatemi1 (ROF), who proposed an image denoising model based on the minimization of the energy ∀u ∈ RΩ, EROF(u) = u − u02

2

data fidelity

+ λ TVd(u)

regularity (promoting sparsity)

, where u0 represents the noisy image and λ ∈ R+ a regularity parameter that must be set by the user.

1L. I. Rudin, S. Osher, and E. Fatemi. “Nonlinear total variation based noise removal

algorithms”. Physica D: Nonlinear Phenomena, 1992.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 3 / 34

SLIDE 5

Interpolation of an image denoised using TVd

We denoise an image u0 by computing the minimizer of the ROF energy:

(a) input image u0 Shannon zooming of (a) spectrum of (a) (b) denoised image Shannon zooming of (b) spectrum of (b)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 4 / 34

SLIDE 6

Interpolation of an image denoised using TVd

We denoise an image u0 by computing the minimizer of the ROF energy:

(a) input image u0 Bicubic zooming of (a) spectrum of (a) (b) denoised image Bicubic zooming of (b) spectrum of (b)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 5 / 34

SLIDE 7

Shannon Sampling Theorem

This Theorem states that a band-limited function can be recovered exactly from an infinite set of samples.

Theorem (Shannon Sampling Theorem)

Let U : Rd → R be an absolutely integrable function whose Fourier transform ∀ξ ∈ Rd,

U(ξ) =
Rd U(x)e−iξ,x dx ,

satisfies U(ξ) = 0 si ξ ∈ [−π, π]d. Then, U is uniquely determined by its values on Zd since we have ∀x ∈ Rd, U(x) =

k∈Zd

U(k) sinc(x − k) , where we have set sinc(x1, . . . , xd) =

d

j=1

sin (πxj ) πxj

and sin(0) = 1.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 6 / 34

SLIDE 8

Shannon interpolation of a discrete image

Definition (discrete Shannon interpolation (2D))

Given a discrete domain Ω = {0, . . . , M − 1} × {0, . . . , N − 1}, and an image u : Ω → R, we call discrete Shannon interpolation of u the (M, N)-periodical function U : R2 → R defined by ∀(x, y) ∈ R2, U(x, y) =

(k,ℓ)∈Ω

u(k, ℓ) sincdM(x − k) sincdN(y − ℓ) , where sincdM(x) =          sin (πx) M sin ( πx

M )

if M is odd, sin (πx) M tan ( πx

M )

if M is even.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 7 / 34

SLIDE 9

Shannon interpolation of a discrete image

We can show that the Shannon interpolate of a discrete image can be evaluated in the Fourier domain.

Proposition

The Shannon interpolate of a discrete image u : Ω → R satisfies U(x, y) = 1 MN

α,β∈Z

− M

2 ≤α≤ M 2

− N

2 ≤β≤ N 2

εM(α)εN(β) u(α, β) e

2iπ αx M + βy N

,

where εM et εN are given2 by εM(α) =

1

si |α| < M/2 1/2 si |α| = M/2 εN(β) =

1

si |β| < N/2 1/2 si |β| = N/2 . This interpolation formula is useful to apply precise subpixellic geometric transforms (rotations, translations, zoom) to discrete images.

2R. Abergel and L. Moisan: “The Shannon Total Variation”, Journal of Mathematical Imaging and Vision, 2017.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 8 / 34

SLIDE 10

Shannon interpolation of a discrete image

We can show that the Shannon interpolate of a discrete image can be evaluated in the Fourier domain.

Proposition

The Shannon interpolate of a discrete image u : Ω → R satisfies U(x, y) = 1 MN

α,β∈Z

− M

2 ≤α≤ M 2

− N

2 ≤β≤ N 2

εM(α)εN(β) u(α, β) e

2iπ αx M + βy N

,

where εM et εN are given2 by εM(α) =

1

si |α| < M/2 1/2 si |α| = M/2 εN(β) =

1

si |β| < N/2 1/2 si |β| = N/2 . This interpolation formula is useful to apply precise subpixellic geometric transforms (rotations, translations, zoom) to discrete images.

2R. Abergel and L. Moisan: “The Shannon Total Variation”, Journal of Mathematical Imaging and Vision, 2017.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 8 / 34

SLIDE 11

The Shannon total variation

We call Shannon total variation (STV) of the discrete image u the continuous totale variation of U.

Definition (STV∞)

STV∞(u) := TV(U) =

[0,M]×[0,N]

|DU(x, y)| dxdy . For practical implementations, we can estimage STV∞(u) using a Riemann sum (involving an oversampling factor n = 2 or 3 for the integration domain).

Definition (STVn)

For any integer n ≥ 1, set STVn(u) = 1 n2

(k,ℓ)∈Ωn
DU

k

n , ℓ n

= 1

n2

(k,ℓ)∈Ωn

|Dnu(k, ℓ)| , where Dnu(k, ℓ)=DU k

n , ℓ n

, and Ωn ={0, ... , nM−1}×{0, ... , nN−1}.

STVn and TVd share the same structure since TVd(u) =

(k,ℓ)∈Ω |∇u(k, ℓ)|

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 9 / 34

SLIDE 12

The Shannon total variation

We call Shannon total variation (STV) of the discrete image u the continuous totale variation of U.

Definition (STV∞)

STV∞(u) := TV(U) =

[0,M]×[0,N]

|DU(x, y)| dxdy . For practical implementations, we can estimage STV∞(u) using a Riemann sum (involving an oversampling factor n = 2 or 3 for the integration domain).

Definition (STVn)

For any integer n ≥ 1, set STVn(u) = 1 n2

(k,ℓ)∈Ωn
DU

k

n , ℓ n

= 1

n2

(k,ℓ)∈Ωn

|Dnu(k, ℓ)| , where Dnu(k, ℓ)=DU k

n , ℓ n

, and Ωn ={0, ... , nM−1}×{0, ... , nN−1}.

STVn and TVd share the same structure since TVd(u) =

(k,ℓ)∈Ω |∇u(k, ℓ)|

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 9 / 34

SLIDE 13

The Shannon total variation

We call Shannon total variation (STV) of the discrete image u the continuous totale variation of U.

Definition (STV∞)

STV∞(u) := TV(U) =

[0,M]×[0,N]

|DU(x, y)| dxdy . For practical implementations, we can estimage STV∞(u) using a Riemann sum (involving an oversampling factor n = 2 or 3 for the integration domain).

Definition (STVn)

For any integer n ≥ 1, set STVn(u) = 1 n2

(k,ℓ)∈Ωn
DU

k

n , ℓ n

= 1

n2

(k,ℓ)∈Ωn

|Dnu(k, ℓ)| , where Dnu(k, ℓ)=DU k

n , ℓ n

, and Ωn ={0, ... , nM−1}×{0, ... , nN−1}.

STVn and TVd share the same structure since TVd(u) =

(k,ℓ)∈Ω |∇u(k, ℓ)|

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 9 / 34

SLIDE 14

The Shannon total variation

We call Shannon total variation (STV) of the discrete image u the continuous totale variation of U.

Definition (STV∞)

STV∞(u) := TV(U) =

[0,M]×[0,N]

|DU(x, y)| dxdy . For practical implementations, we can estimage STV∞(u) using a Riemann sum (involving an oversampling factor n = 2 or 3 for the integration domain).

Definition (STVn)

For any integer n ≥ 1, set STVn(u) = 1 n2

(k,ℓ)∈Ωn
DU

k

n , ℓ n

= 1

n2

(k,ℓ)∈Ωn

|Dnu(k, ℓ)| , where Dnu(k, ℓ)=DU k

n , ℓ n

, and Ωn ={0, ... , nM−1}×{0, ... , nN−1}.

STVn and TVd share the same structure since TVd(u) =

(k,ℓ)∈Ω |∇u(k, ℓ)|

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 9 / 34

SLIDE 15

Related works

F. Malgouyres and F. Guichard. “Edge direction preserving image zooming: a

mathematical and numerical analysis.” SIAM Journal (2001).

L. Moisan. “How to discretize the total variation of an image?” PAMM

Proceedings (2007).

J. Preciozzi, P. Musé, A. Almansa, S. Durand, F. Cabot, Y. Kerr and B. Rougé.

“Sparsity-based restoration of SMOS images in the presence of outliers.” IGARSS (2012).

J. Preciozzi, P. Musé, A. Almansa, S. Durand, A. Khazaal and B. Rougé.

“SMOS images restoration from L1A data: A sparsity-based variational approach.” IGARSS Proceedings (2014). D.C. Soncco, C. Barbanson, M. Nikolova, A. Almansa and Y. Ferrec. “Fast and accurate multiplicative decomposition for fringe removal in interferometric images”. IEEE Transactions on Computational Imaging, (2017).

T. Briand and J. Vacher. “How to apply a filter defined in the frequency domain

by a continuous function”. Image Processing On Line, (2016).

R. Abergel and L. Moisan. “The Shannon Total Variation”, Journal of

Mathematical Imaging and Vision (2017).

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 10 / 34

SLIDE 16

Dual formulation

As in the discrete setting, a dual formulation of STVn can be easily derived.

Proposition (dual formulation of STVn)

STVn(u) = max

p:Ωn→R2 1 n2 Dnu, p − δB∗(p)

where δB∗(p) =    if max

(k,ℓ)∈Ωn |p(k, ℓ)| ≤ 1 ,

+∞

therwise .

Sketch of proof.

1. The Legendre-Fenchel transform of · 1,2 is · ⋆

1,2 = δB∗ ,

2. thus, STVn(u) = 1

n2 Dnu1,2 = 1 n2 Dnu⋆⋆ 1,2 = δ⋆ B∗( 1 n2 Dnu) ,

3. besides, the supremum involved in δ⋆

B∗ is a maximum.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 11 / 34

SLIDE 17

Numerical evaluation of STVn(u)

In order to evaluate STVn(u) = 1 n2

(k,ℓ)∈Ωn

|Dnu(k, ℓ)| , we need to compute Dnu over Ωn. The following proposition explain how Dnu can be computed efficiently in the Fourier domain.

Proposition (Fast evaluation of Dnu)

Let n > 1 and Ωn :=

− nM

2 , nM 2

×
− nN

2 , nN 2

∩ Z2 the cannonical frequency

domain associated to Ωn. For all (α, β) ∈ Ωn, we have

Dnu(α, β) = n2 εM(α)εN(β) Zn

u(α, β) 2iπ α/M β/N

,

where Zn u(α, β) = u(α, β) si |α| ≤ M

2 et |β| ≤ N 2

sinon. Besides, we have the upper-bound |||Dn||| ≤ πn √ 2 .

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 12 / 34

SLIDE 18

Numerical evaluation of STVn(u)

In order to evaluate STVn(u) = 1 n2

(k,ℓ)∈Ωn

|Dnu(k, ℓ)| , we need to compute Dnu over Ωn. The following proposition explain how Dnu can be computed efficiently in the Fourier domain.

Proposition (Fast evaluation of Dnu)

Let n > 1 and Ωn :=

− nM

2 , nM 2

×
− nN

2 , nN 2

∩ Z2 the cannonical frequency

domain associated to Ωn. For all (α, β) ∈ Ωn, we have

Dnu(α, β) = n2 εM(α)εN(β) Zn

u(α, β) 2iπ α/M β/N

,

where Zn u(α, β) = u(α, β) si |α| ≤ M

2 et |β| ≤ N 2

sinon. Besides, we have the upper-bound |||Dn||| ≤ πn √ 2 .

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 12 / 34

SLIDE 19

Numerical evaluation of STVn(u)

In order to evaluate STVn(u) = 1 n2

(k,ℓ)∈Ωn

|Dnu(k, ℓ)| , we need to compute Dnu over Ωn. The following proposition explain how Dnu can be computed efficiently in the Fourier domain.

Proposition (Fast evaluation of Dnu)

Let n > 1 and Ωn :=

− nM

2 , nM 2

×
− nN

2 , nN 2

∩ Z2 the cannonical frequency

domain associated to Ωn. For all (α, β) ∈ Ωn, we have

Dnu(α, β) = n2 εM(α)εN(β) Zn

u(α, β) 2iπ α/M β/N

,

where Zn u(α, β) = u(α, β) si |α| ≤ M

2 et |β| ≤ N 2

sinon. Besides, we have the upper-bound |||Dn||| ≤ πn √ 2 .

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 12 / 34

SLIDE 20

Example of Matlab implementation

Practical computations of the gradient field Dn and its adjoint D∗

n = −divn do

not raise particular difficulties. For instance, in the case of odd dimensions, More general implementations of Dn and its adjoint D∗

n = −divn are available

nline in Matlab or C language.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 13 / 34

SLIDE 21

Example of Matlab implementation

Practical computations of the gradient field Dn and its adjoint D∗

n = −divn do

not raise particular difficulties. For instance, in the case of odd dimensions, More general implementations of Dn and its adjoint D∗

n = −divn are available

nline in Matlab or C language.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 13 / 34

SLIDE 22

Image denoising

Given a noisy image u0, we consider the STVn variant of the ROF model argmin

u:Ω→R

u − u02

2 + λ STVn(u) .

We have the primal-dual reformulation argmin

u:Ω→R

max

p:Ωn→R2 u − u02 2 + λ n2 Dnu, p − δB∗(p) ,

which can be numerically computed using, for instance, the Chambolle-Pock algorithm3.

3A. Chambolle, T. Pock: “A first-order primal-dual algorithm for convex problems with

applications to imaging”, Journal of Mathematical Imaging and Vision, 2011.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 14 / 34

SLIDE 23

Image denoising

Given a noisy image u0, we consider the STVn variant of the ROF model argmin

u:Ω→R

u − u02

2 + λ STVn(u) .

We have the primal-dual reformulation argmin

u:Ω→R

max

p:Ωn→R2 u − u02 2 + λ n2 Dnu, p − δB∗(p) ,

which can be numerically computed using, for instance, the Chambolle-Pock algorithm3.

3A. Chambolle, T. Pock: “A first-order primal-dual algorithm for convex problems with

applications to imaging”, Journal of Mathematical Imaging and Vision, 2011.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 14 / 34

SLIDE 24

Image denoising

(a) noisy image u0 (σ = 20) (b) TVd (c) STV2 details of (b) details of (c)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 15 / 34

SLIDE 25

Image denoising

(a) noisy image u0 (σ = 20) (b) TVd (c) STV2 Bicubic zooming of (b) Bicubic zooming of (c)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 16 / 34

SLIDE 26

Image denoising

(a) noisy image u0 (σ = 20) (b) TVd (c) STV2 Shannon zooming of (b) Shannon zooming of (c)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 17 / 34

SLIDE 27

Inverse problems

We can also use STVn as a regularizer for inverse problems. Given a linear

perator A : RΩ → Rω, and u0 : ω → R, consider

argmin

u:Ω→R

Au − u02

2

f(Au)

+ λSTVn(u) , with primal-dual reformulation (use f(Au) = f ⋆⋆(Au)) argmin

u:Ω→R

max

p:Ωn→R2 q:ω→R

λ

n2 Dnu, Au

, (p, q)
−
δB∗(p) + q

2 + u02 2

.

and the Chambolle-Pock Algorithm can be used again.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 18 / 34

SLIDE 28

Motion deblurring

Consider that Au = k ∗ u is the convolution between u and a given motion blur kernel k.

(a) blurry & noisy image u0 (σ = 2) (b) discrete TV (TVd) (c) Shannon TV (STV2) details of (a) Shannon zooming of (b) Shannon zooming of (c)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 19 / 34

SLIDE 29

Spectrum extrapolation4

Now, A is a frequency masking operator of the type du type

Au(α, β) =

u(α, β) if (α, β) ∈ ω0 ,

therwise .

image u0 spectrum of u0

4F. Guichard and F. Malgouyres: “Total variation based interpolation”. In proceedings of the 9th

European Signal Processing Conference (EUSIPCO), 1998

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 20 / 34

SLIDE 30

Spectrum extrapolation

(a) image u0 (b) TVd (c) STV2 spectrum of (a) spectrum of (b) spectrum of (c)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 21 / 34

SLIDE 31

Spectrum extrapolation

(a) image u0 (b) TVd (Shannon zooming) (c) STV2 (Shannon zooming) spectrum of (a) spectrum of (b) spectrum of (c)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 22 / 34

SLIDE 32

A new model: the image “Shannonizer”

Given an input image u0 : Ω → R and a weight mapping γ : Ω → R+, we consider argmin

u:Ω→R

u −

u02

γ + λ STVn(u) ,

where

u −

u02

γ = 1

| Ω|

(α,β)∈

Ω

γ(α, β) · | u(α, β) − u0(α, β)|2 , is a weighted ℓ2 square distance between u et u0, which makes the regularization adaptative with respect to the frequency. A simple and interesting example of weighting: ∀(α, β) ∈ Ω, γ(α, β) = e

−2π2σ2

α2

M2 + β2 N2

low frequencies: γ(α, β) is high, we enforce

u(α, β) ≈ u0(α, β); high frequencies: γ(α, β) is low, the computed value u(α, β) is mostly driven by the regularity term STVn.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 23 / 34

SLIDE 33

A new model: the image “Shannonizer”

Given an input image u0 : Ω → R and a weight mapping γ : Ω → R+, we consider argmin

u:Ω→R

u −

u02

γ + λ STVn(u) ,

where

u −

u02

γ = 1

| Ω|

(α,β)∈

Ω

γ(α, β) · | u(α, β) − u0(α, β)|2 , is a weighted ℓ2 square distance between u et u0, which makes the regularization adaptative with respect to the frequency. A simple and interesting example of weighting: ∀(α, β) ∈ Ω, γ(α, β) = e

−2π2σ2

α2

M2 + β2 N2

low frequencies: γ(α, β) is high, we enforce

u(α, β) ≈ u0(α, β); high frequencies: γ(α, β) is low, the computed value u(α, β) is mostly driven by the regularity term STVn.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 23 / 34

SLIDE 34

A new model: the image “Shannonizer”

Given an input image u0 : Ω → R and a weight mapping γ : Ω → R+, we consider argmin

u:Ω→R

u −

u02

γ + λ STVn(u) ,

where

u −

u02

γ = 1

| Ω|

(α,β)∈

Ω

γ(α, β) · | u(α, β) − u0(α, β)|2 , is a weighted ℓ2 square distance between u et u0, which makes the regularization adaptative with respect to the frequency. A simple and interesting example of weighting: ∀(α, β) ∈ Ω, γ(α, β) = e

−2π2σ2

α2

M2 + β2 N2

low frequencies: γ(α, β) is high, we enforce

u(α, β) ≈ u0(α, β); high frequencies: γ(α, β) is low, the computed value u(α, β) is mostly driven by the regularity term STVn.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 23 / 34

SLIDE 35

A new model: the image “Shannonizer”

Given an input image u0 : Ω → R and a weight mapping γ : Ω → R+, we consider argmin

u:Ω→R

u −

u02

γ + λ STVn(u) ,

where

u −

u02

γ = 1

| Ω|

(α,β)∈

Ω

γ(α, β) · | u(α, β) − u0(α, β)|2 , is a weighted ℓ2 square distance between u et u0, which makes the regularization adaptative with respect to the frequency. A simple and interesting example of weighting: ∀(α, β) ∈ Ω, γ(α, β) = e

−2π2σ2

α2

M2 + β2 N2

low frequencies: γ(α, β) is high, we enforce

u(α, β) ≈ u0(α, β); high frequencies: γ(α, β) is low, the computed value u(α, β) is mostly driven by the regularity term STVn.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 23 / 34

SLIDE 36

A new model: the image “Shannonizer”

image credit: CNES

(a) initial image u0 details of (a) spectrum of (a)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 24 / 34

SLIDE 37

A new model: the image “Shannonizer”

image credit: CNES

(a) initial image u0 details of (a) spectrum of (a) (b) frequency attenuation details of (b) spectum of (b)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 24 / 34

SLIDE 38

A new model: the image “Shannonizer”

image credit: CNES

(a) initial image u0 details of (a) spectrum of (a) (c) Shannonized image (STV2) details of (c) spectrum of (c)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 25 / 34

SLIDE 39

A new model: the image “Shannonizer”

Shannon zoomings of the initial image Shannon zoomings of the Shannonized image

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 26 / 34

SLIDE 40

A new model: the image “Shannonizer”

(a) initial image spectrum of (a)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 27 / 34

SLIDE 41

A new model: the image “Shannonizer”

(b) Shannonized image spectrum of (b)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 28 / 34

SLIDE 42

A new model: the image “Shannonizer”

(a) initial image details of (a) Shannon zooming of (a) (b) Shannonized image details of (b) Shannon zooming of (b)

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 29 / 34

SLIDE 43

Isotropy of the STV scheme

Let us denoise a rotationally invariant image using the ROF model and its STV variant.

smooth sharp sharp smooth (a) reference (b) discrete TV (c) Shannon TV

We can compare the denoising models in terms of isotropy by plotting the gray levels of the denoised image as a function of their distance to the disk center.

0.2 0.4 0.6 0.8 1 28 30 32 34 36 38 40 intensity reference discrete TV distance from center 0.2 0.4 0.6 0.8 1 28 30 32 34 36 38 40 intensity distance from center reference

3

STV Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 30 / 34

SLIDE 44

Isotropy of the STV scheme

We can force the isotropy of the frequency support by adding the constraint Supp( u) ⊂ D

Ω, where

D

Ω =

(α, β) ∈

Ω, α M/2 2 + β N/2 2 ≤ 1

.

We consider the constrained problem argmin

u:Ω→R

u − u02

2 + λSTVn(u)

subject to Supp( u) ⊂ D

Ω .

0.2 0.4 0.6 0.8 1 28 30 32 34 36 38 40

reference

3

STV

(with constraints) 3

STV

distance from center intensity 0.2 0.4 0.6 0.8 1 28 30 32 34 36 38 40 distance from center intensity

reference

(with constraints) 3

STV Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 31 / 34

SLIDE 45

Isotropy of the STV scheme

We can compare this result to that obtain using the isotropic variant of TVd proposed by Condat5.

0.2 0.4 0.6 0.8 1 28 30 32 34 36 38 40

reference

3

STV

(with constraints) 3

STV TV by Condat

distance from center intensity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 30 31 32 33 34 35 36 37

3

STV

(with constraints) 3

STV TV by Condat

distance from center intensity

5L. Condat: “Discrete total variation: New definition and minimization,” SIAM Journal on Imaging

Sciences, 2017.

Rémy Abergel, conference on variational methods and optimization in imaging, IHP , February 6, 2019. 32 / 34

SLIDE 46