[PPT] - Variational denoising for manifold-valued data Andreas Weinmann PowerPoint Presentation

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Variational denoising for manifold-valued data

Andreas Weinmann Helmholtz Center Munich & TU M¨ unchen Paris, le 21 novembre 2014

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Overview

An algorithm for TV minimization for manifold-valued data (joint work with

L. Demaret and M.Storath)

Second order TV type functionals for S1-valued data (joint work with R.

Bergmann, F. Laus, G. Steidl)

Potts and Blake-Zisserman functionals for manifold-valued signals with a few jumps (joint work with L. Demaret and M.Storath)

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Manifold-valued data in DTI

In diffusion tensor imaging (DTI)

(Basser et al. ’94) the data are

positive(-definite) matrices.

It is reasonable (cf. Pennec et al. ’2004) to

equip Posn with the Riemannian metric gP(A, B) = trace(P− 1

2 AP−1BP− 1 2 ),

P positive and A, B symmetric.

Positive Matrices visualized as ellipsoids.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Manifold-valued data in DTI

In diffusion tensor imaging (DTI)

(Basser et al. ’94) the data are

positive(-definite) matrices.

It is reasonable (cf. Pennec et al. ’2004) to

equip Posn with the Riemannian metric gP(A, B) = trace(P− 1

2 AP−1BP− 1 2 ),

P positive and A, B symmetric.

Posn with the metric gP is a Cartan

Hadamard manifold (complete, nonpositive sectional curvature, simply connected).

Positive Matrices visualized as ellipsoids.

log and exp can be computed explicitly by

logP Q = P

1 2 log(P− 1 2 QP− 1 2 )P 1 2 ,

expP A = P

1 2 exp(P− 1 2 AP− 1 2 )P 1 2 .

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

TV functionals for manifold-valued data

We consider the variational denoising problem given by the (discrete, anisotropic, bivariate) functionals Fα(u) =

i,j

dist(ui, fi)p + α

i,j

dist(uij, ui−1,j)q + α

i,j

dist(uij, ui,j−1)q, with data f and p, q ≥ 1.

Choosing q=1 corresponds to (anisotropic) TV minimization/ROF

model in Lagrange form (Rudin, Osher, Fatemi ’90).

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

TV functionals for manifold-valued data

We consider the variational denoising problem given by the (discrete, anisotropic, bivariate) functionals Fα(u) =

i,j

dist(ui, fi)p + α

i,j

dist(uij, ui−1,j)q + α

i,j

dist(uij, ui,j−1)q, with data f and p, q ≥ 1.

Choosing q=1 corresponds to (anisotropic) TV minimization/ROF

model in Lagrange form (Rudin, Osher, Fatemi ’90).

Choose the Riemannian distance dist to obtain the corresponding

functionals for manifold-valued data.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

TV functionals for manifold-valued data

We consider the variational denoising problem given by the (discrete, anisotropic, bivariate) functionals Fα(u) =

i,j

dist(ui, fi)p + α

i,j

dist(uij, ui−1,j)q + α

i,j

dist(uij, ui,j−1)q, with data f and p, q ≥ 1.

Choosing q=1 corresponds to (anisotropic) TV minimization/ROF

model in Lagrange form (Rudin, Osher, Fatemi ’90).

Choose the Riemannian distance dist to obtain the corresponding

functionals for manifold-valued data.

Increase anisotropy by additionally considering diagonals, knight

moves, . . .

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

TV denoising on real DTI data (Camino project, Cook et. al. ’06)

Real data Our method for ℓ2 − TV α = 0.11.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithms - TV problem

Idea: Write (for simplicity univariate, multivarite analogous):

F(u) = γ

i dist(ui, ui−1)q +
j dist(uj, fj)p =
i Fi(u) + G(u),

where Fi(u) = γ dist(ui, ui−1)q, G =

j dist(uj, fj)p.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithms - TV problem

Idea: Write (for simplicity univariate, multivarite analogous):

F(u) = γ

i dist(ui, ui−1)q +
j dist(uj, fj)p =
i Fi(u) + G(u),

where Fi(u) = γ dist(ui, ui−1)q, G =

j dist(uj, fj)p.
Apply the cyclic proximal point algorithm (Bacak, Bertsekas) : Iterate the

proximal mappings (Moreau) of G and Fi, i = l, . . . , r, proxλFi(u) = arg min

v 1 2dist(u, v)2 + λFi(v).

Central Point: The proximal mappings of Fi, G can be computed

explicitely (next slide).

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithms - TV problem

Minimize F(u) =

i Fi(u) + G(u),

Fi(u) = γ dist(ui, ui−1)q, G =

j dist(uj, fj)p.
The proximal mapping of G is explicitly given by

proxλG(u)i = [ui, fi]t, t =       

2λ (1+2λ)dist(ui, fi)

for p= 2, min(λ, dist(ui, fi)) for p= 1. (“Soft thresholding” for p = 1.)

The proximal mapping of Fi is explicitly given by (Demaret, Storath, W.)

proxλFi(u)j =              uj if j i, i − 1, [ui, ui−1]t if j = i, [ui−1, ui]t if j = i − 1, t =

γλ (2+2γλ)dist(ui, ui−1) for q=2, t = min(λγ, 1 2dist(ui, ui−1)) for q=1.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithms - TV problem

Minimize F(u) =

i Fi(u) + i Gi(u),

Fi(u) = γ dist(ui, ui−1)q, Gi = dist(ui, fi)p.

A parallel proximal point algorithm: Calculate the proximal

mappings of Fi, Gi at u(k) u(k+1)

i

= proxλFi(u(k)), u(k+1)

n+i

= proxλGi(u(k)), and then average them using intrinsic means (Cartan, Frechet, Karcher, . . .) u(k+1) = arg min

u

i dist(u, u(k+1)

i

)2.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithms - TV problem

Minimize F(u) =

i Fi(u) + i Gi(u),

Fi(u) = γ dist(ui, ui−1)q, Gi = dist(ui, fi)p.

A parallel proximal point algorithm: Calculate the proximal

mappings of Fi, Gi at u(k) u(k+1)

i

= proxλFi(u(k)), u(k+1)

n+i

= proxλGi(u(k)), and then average them using intrinsic means (Cartan, Frechet, Karcher, . . .) v = u(k+1) = arg min

u

i dist(u, u(k+1)

i

)2.

To compute the minimizer, we use the gradient descent (Karcher)

vnew = expvold( 1

N

i=1 logvold u(k+1) i

).

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithms - TV problem

Minimize F(u) =

i Fi(u) + i Gi(u),

Fi(u) = γ dist(ui, ui−1)q, Gi = dist(ui, fi)p.

A parallel proximal point algorithm: Calculate the proximal

mappings of Fi, Gi at u(k) u(k+1)

i

= proxλFi(u(k)), u(k+1)

n+i

= proxλGi(u(k)), and then average them using intrinsic means (Cartan, Frechet, Karcher, . . .) v = u(k+1) = arg min

u

i dist(u, u(k+1)

i

)2.

To compute the minimizer, we use the gradient descent (Karcher)

vnew = expvold( 1

N

i=1 logvold u(k+1) i

).

Fast Variant: Approximate the mean by iterated geodesic averages.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Synthetic DTI example

Synthetic DT image Rician noise, σ = 90. ℓ2-TV (our cyclic PPA) ℓ2-TV (our parallel PPA)

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Analytic Results

Theorem (Demaret, Storath, W.)

In a Cartan-Hadamard manifold (complete, simply connected, nonpositive sectional curvature) the proposed algorithms (cyclic, parallel and the parallel variant with approximative mean computation) for Lp-TV minimization converge towards a global minimizer.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Analytic Results

Theorem (Demaret, Storath, W.)

In a Cartan-Hadamard manifold (complete, simply connected, nonpositive sectional curvature) the proposed algorithms (cyclic, parallel and the parallel variant with approximative mean computation) for Lp-TV minimization converge towards a global minimizer. Sceleton of proof:

Proof that in a connected, complete Riemannian manifold, the

proximal mappings of the first differences and the distances are given by the formulas derived above.

For the cyclic PPA apply the convergence result of Bacak (Bacak ’14).
For the parallel PPAs base on techniques used in (Bacak ’14) and find

suitable modifications.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Denoising on the LCh color model (S1 × R2).

Synthetic image Gaussian noise (PSNR: 15.64). ℓ2-TV on RGB (PSNR:23.92) ℓ2-TV on LCh (PSNR:32.19)

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Denoising S2 data.

Original Von Mises-Fisher noise (κ = 12.7) ℓ1-TV regularization

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Denoising SO3 data.

10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100

Synthetic signal Fisher noise (κ = 75.)

10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100

ℓ2-TV ℓ2-Huber

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Denoising inSAR data

Real data L 2-TV denoising L 1-TV denoising TV with Huber data term

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Part II:

Second order TV type functionals for S1-valued data

(joint work with R. Bergmann, F. Laus, G. Steidl)

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Second order TV for S1 valued data

Second order TV type functional for real-valued data:

F(u) = u − f2

2 + α∇1u1 + β∇2u1.

Here, ∇2u(i) = u(i − 1) − 2u(i) + u(i + 1).

Question: What are second differences for S1 valued data?

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Second order TV for S1 valued data

Second order TV type functional for real-valued data:

F(u) = u − f2

2 + α∇1u1 + β∇2u1.

Here, ∇2u(i) = u(i − 1) − 2u(i) + u(i + 1).

Question: What are second differences for S1 valued data?
Idea: Translate

∇2u(i) = (u(i − 1) − u(i)) + (u(i − 1) − u(i)) to the manifold setting: ∇2u(i) = exp−1

u(i) u(i − 1) + exp−1 u(i) u(i + 1).

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Second order TV for S1 valued data

Second order TV type functional for real-valued data:

F(u) = u − f2

2 + α∇1u1 + β∇2u1.

Here, ∇2u(i) = u(i − 1) − 2u(i) + u(i + 1).

Question: What are second differences for S1 valued data?
Idea: Translate

∇2u(i) = (u(i − 1) − u(i)) + (u(i − 1) − u(i)) to the manifold setting: ∇2u(i) = exp−1

u(i) u(i − 1) + exp−1 u(i) u(i + 1).

Problem: These second differences are not continuous in

ui−1, ui, ui+1.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Second order TV for S1 valued data

Alternative: View ui ∈] − π, π] as real-valued data and define the

absolute cyclic difference d2(fi−1, fi, fi+1) = min

k,l,m=−1,0,1 |∇2(fi−1 + k2π, fi + l2π, fi+1 + m2π)|

These differences are continuous in fi−1, fi, fi+1.

Equivalent: Consider all liftings and take the minimal difference on

the lifted R-valued data.

For nearby fi−1, fi, fi+1 the manifold and the lifting definition agree.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Second order TV for S1 valued data

Alternative: View ui ∈] − π, π] as real-valued data and define the

absolute cyclic difference d2(fi−1, fi, fi+1) = min

k,l,m=−1,0,1 |∇2(fi−1 + k2π, fi + l2π, fi+1 + m2π)|

These differences are continuous in fi−1, fi, fi+1.

Equivalent: Consider all liftings and take the minimal difference on

the lifted R-valued data.

For nearby fi−1, fi, fi+1 the manifold and the lifting definition agree.
The proximal mappings for d2 can be computed explicitely (Bergmann,

Laus, Steidl, W. ’14): for w = (1, −2, 1)T, and |f, w| < π,

proxλd2(f) = (f − swm)2π, m = min

λ, f,w

w2

2

,

s = signf, w.

All ingredients for the cyclic proximal point algorithm are available.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Convergence of the cyclic proximal point algorithms

Theorem (Bergmann, Laus, Steidl, W., 2014)

For data f with nearby data items and small enough parameters α, β, the cyclic proximal point algorithm for second order TV type minimization converges to a minimizer.

What nearby means and α, β can be quantified.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Convergence of the cyclic proximal point algorithms

Theorem (Bergmann, Laus, Steidl, W., 2014)

For data f with nearby data items and small enough parameters α, β, the cyclic proximal point algorithm for second order TV type minimization converges to a minimizer.

What nearby means and α, β can be quantified.

Idea of proof:

Lift the setting to the covering space R.
For R-valued data we have convergence and the distance of the

iterates can be estimated basing on (Bacak, Bertsekas).

Lifting commutes with the proximal mappings and all other relevant
perations for the considered data.
Conclude nearness for S1 data and derive convergence.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Second order TV minimization - synthetic example.

π

π 2

− π

2

−π

Original

π

π 2

− π

2

−π π

π 2

− π

2

−π

Noisy Second order TV.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Denoising the H channel in HSV space.

Image Noisy hue hue denoising on R hue denoising on S1

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Second order TV for real SAR data of Mt. Vesuvius.

π

π 2

− π

2

−π π

π 2

− π

2

−π

Original Second order TV denoising.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Part III:

Potts and Blake-Zisserman functionals for manifold-valued signals with a few jumps

(joint work with L. Demaret and M.Storath)

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Potts and Blake-Zisserman functionals for manifold-valued data

Define (univariate) Potts functionals Pγ for manifold-valued data, Pγ(u) = γ #{i : ui ui−1} +

i dist(ui, fi)p,

and Blake-Zisserman functionals Bγ for manifold-valued data, Bγ(u) = γ

i min(sq, dist(ui, ui−1)q) +
i dist(ui, fi)p.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Example: L2-Potts minimization for DTI data.

Original (synthetic) signal: Noisy data (Rician noise with σ = 60): L2-Potts reconstruction:

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Example: Blake-Zisserman vs. Potts.

Original (synthetic) signal: Rician noise with σ = 50 : L2-Potts reconstruction: L2-Blake-Zisserman reconstruction:

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithm - Potts problem

Pγ(u) = γ #{i : ui ui−1} + n

i=1 dist(ui, fi)p → min,

Algorithm based on dynamic programming.
Most time consuming: For each subinterval [l, r] calculate

u = arg min r

i=l dist(u, fi)p,

(p=2: Riemannian center of mass; p=1: Riemannian median.) We use (sub-)gradient descent; e.g., for p = 1, uk+1 = expuk       τk

r

i=l

loguk fi loguk fi        . Converges for p = 1 when τ ∈ ℓ2 \ ℓ1 (Arnaudon et al. ’11).

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithm - Blake-Zisserman problem

Bγ(u) = γ

i min(sq, dist(ui, ui−1)q) +
j dist(uj, fj)p → min .
Algorithm based on dynamic programming.
For each subinterval [l, r] calculate the minimizer of

F(u) = γ

i dist(ui, ui−1)q +
j dist(uj, fj)p
This is a TV minimization problem (or, more general, ℓq variation

minimization) for manifold data which can be solved by the developed methods.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithms

Theorem (Demaret, Storath, W.)

Let p, q ≥ 1. In a Cartan-Hadamard manifold, our algorithm for the minimization of the (univariate) Potts functionals Pγ produces a minimizer.

Theorem (Demaret, Storath, W.)

Let p, q ≥ 1. In a Hadamard space, our algorithm for the minimization of the (inivariate) Blake-Zisserman functionals Bγ produces a minimizer.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Minimization algorithms

Theorem (Demaret, Storath, W.)

Let p, q ≥ 1. In a Cartan-Hadamard manifold, our algorithm for the minimization of the (univariate) Potts functionals Pγ produces a minimizer.

Theorem (Demaret, Storath, W.)

Let p, q ≥ 1. In a Hadamard space, our algorithm for the minimization of the (inivariate) Blake-Zisserman functionals Bγ produces a minimizer.

For multivariate data, the Potts and the Blake-Zisserman problem

are NP hard.

In this case, we use a splitting approach (cf. W., Demaret, Storath ’14).

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Segmentation: real data from the Camino project (Cook et al. ’06)

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Segmentation: real data from the Camino project (Cook et al. ’06)

Edge between neighbouring P, Q ⇐⇒ dist(P, Q) ≥ s (s B.-Z. parameter). Blake-Zisserman regularization (p, q = 1) with s = 0.67, γ = 4.3.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Summary

We have derived algorithms for TV minimization for manifolds.
We have shown convergence to a minimizer for Hadamard

manifolds.

We have seen the potential in various applications.
We have derived an algorithm for second order TV type functionals

for S1 data.

We have obtained convergence for nearby neighboring data and

shown applications.

We have obtained algorithms for Potts and Blake-Zisserman

problems for manifold valued data.

We have seen a segmentation of a real corpus callosum.

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TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals

Some References

M. Baˇ

c´ ak. Computing medians and means in Hadamard spaces. SIAM J. Optim. (to appear), 2014.

J. Lellmann, E. Strekalovskiy, S. Koetter, and D. Cremers.

Total variation regularization for functions with values in a manifold. In IEEE ICCV 2013, pages 2944–2951, 2013.

A. Weinmann, L. Demaret, and M. Storath.

Total variation regularization for manifold-valued data. SIAM J. Imaging Sci., 7(4):2226–2257, 2014.

R. Bergmann, F. Laus, G. Steidl, and A. Weinmann.

Second order differences of cyclic data and applications in variational denoising. SIAM J. Imaging Sci. (to appear), 2014.

A. Weinmann, L. Demaret, and M. Storath.

Mumford-Shah and Potts regularization for manifold-valued data with applications to DTI and Q-ball imaging. Preprint, 2014.

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