MaxEnt 2014
An entropy model for diffusion MRI Pierre Mar´echal U NIVERSITY DE T OULOUSE
Septembre 24, 2014
– p. 1/30
Outline • • • •
Diffusion MRI Building a general entropy model Partially infinite convex programming Application to Diffusion MRI
– p. 2/30
Outline • • • •
Diffusion MRI Building a general entropy model Partially infinite convex programming Application to Diffusion MRI
– p. 2/30
What is dMRI ? A non-invasive imaging technique of the diffusion of water in biological tissues, for the study of the connectivity in the brain
– p. 3/30
What is dMRI ? A non-invasive imaging technique of the diffusion of water in biological tissues, for the study of the connectivity in the brain Probability P of particle displacements in each voxel of the volume to be imaged within a given time interval
– p. 3/30
What is dMRI ? A non-invasive imaging technique of the diffusion of water in biological tissues, for the study of the connectivity in the brain Probability P of particle displacements in each voxel of the volume to be imaged within a given time interval dP (x) = p(x) dx
– p. 3/30
Overview of the process Step 1 From MRI Fourier data Z zj = p(x)e−2iπhqj ,xi dx,
j = 1, . . . , m
R3
reconstruct • either diffusion tensors, under the assumption that P is gaussian [DTI] • or Orientation Diffusion Functions (ODF) via probability distributions Z ψ(s) := p(rs)r2 dr, s ∈ S 2 R
– p. 4/30
Overview of the process Step 1 From MRI Fourier data Z zj = p(x)e−2iπhqj ,xi dx,
j = 1, . . . , m
R3
reconstruct • either diffusion tensors, under the assumption that P is gaussian [DTI] • or Orientation Diffusion Functions (ODF) via probability distributions Z ψ(s) := p(rs)r2 dr, s ∈ S 2 R
Step 2 Image fibers [fiber tracking] – p. 4/30
Fiber tracking
Source: Human Connectome Project http://www.humanconnectomeproject.org/gallery/ – p. 5/30
Purpose of this work D.C. A LEXANDER, Maximum entropy spherical deconvolution for diffusion MRI, Information Processing in Medical Imaging, 19:76-87, 2005
– p. 6/30
Purpose of this work D.C. A LEXANDER, Maximum entropy spherical deconvolution for diffusion MRI, Information Processing in Medical Imaging, 19:76-87, 2005 P is assumed to be supported on a sphere
– p. 6/30
Purpose of this work D.C. A LEXANDER, Maximum entropy spherical deconvolution for diffusion MRI, Information Processing in Medical Imaging, 19:76-87, 2005 P is assumed to be supported on a sphere Our aim is to write a flexible model:
– p. 6/30
Purpose of this work D.C. A LEXANDER, Maximum entropy spherical deconvolution for diffusion MRI, Information Processing in Medical Imaging, 19:76-87, 2005 P is assumed to be supported on a sphere Our aim is to write a flexible model: ֒→ probability P supported in R3
– p. 6/30
Purpose of this work D.C. A LEXANDER, Maximum entropy spherical deconvolution for diffusion MRI, Information Processing in Medical Imaging, 19:76-87, 2005 P is assumed to be supported on a sphere Our aim is to write a flexible model: ֒→ probability P supported in R3 ֒→ possibility to account for moment constraints
– p. 6/30
Purpose of this work D.C. A LEXANDER, Maximum entropy spherical deconvolution for diffusion MRI, Information Processing in Medical Imaging, 19:76-87, 2005 P is assumed to be supported on a sphere Our aim is to write a flexible model: ֒→ probability P supported in R3 ֒→ possibility to account for moment constraints ֒→ use of more general entropies
– p. 6/30
Outline • • • •
Diffusion MRI Building a general entropy model Partially infinite convex programming Application to Diffusion MRI
– p. 7/30
Fourier data zj =
Z
e−2iπhqj ,xi dP (x),
j = 1, . . . , m
R3
– p. 8/30
Fourier data zj =
Z
e−2iπhqj ,xi dP (x),
j = 1, . . . , m
R3
yj =
Z
γj (x) dP (x),
j = 1, . . . , 2m
R3
with
cos 2πhq[(j+1)/2] , xi if j is even, γj (x) = sin 2πhq if j is odd [(j+1)/2] , xi
– p. 8/30
Fourier data zj =
Z
e−2iπhqj ,xi dP (x),
j = 1, . . . , m
R3
yj =
Z
γj (x) dP (x),
j = 1, . . . , 2m
R3
with
cos 2πhq[(j+1)/2] , xi if j is even, γj (x) = sin 2πhq if j is odd [(j+1)/2] , xi
⊤ y = EP [γ] with γ(x) := γ1 (x), . . . , γ2m (x)
– p. 8/30
Fourier data zj =
Z
e−2iπhqj ,xi dP (x),
j = 1, . . . , m
R3
yj =
Z
γj (x) dP (x),
j = 1, . . . , 2m
R3
with
cos 2πhq[(j+1)/2] , xi if j is even, γj (x) = sin 2πhq if j is odd [(j+1)/2] , xi
⊤ y = EP [γ] with γ(x) := γ1 (x), . . . , γ2m (x) Normalization: 1 = EP [1] =
Z
dP (x)
R3 – p. 8/30
Optional moment constraints From a physical viewpoint, it seems reasonable to assume in addition that the random variable x is centered or almost centered
– p. 9/30
Optional moment constraints From a physical viewpoint, it seems reasonable to assume in addition that the random variable x is centered or almost centered
EP [x] =
Z
x dP (x) = 0
R3
– p. 9/30
Entropy model Minimize K (P kν) s.t. (1, y) = EP [(1, γ)]
– p. 10/30
Entropy model Minimize K (P kν) s.t. (1, y) = EP [(1, γ)] K (P kν) :=
Z u(x) ln u(x) dν(x) if P ≺≺ ν
∞
otherwise
– p. 10/30
Entropy model Minimize K (P kν) s.t. (1, y) = EP [(1, γ)] K (P kν) :=
Z u(x) ln u(x) dν(x) if P ≺≺ ν
dP u := dν
∞
otherwise
(Radon-Nikodym derivative) – p. 10/30
Rewriting Fourier data (1, y) = EP [(1, γ)]
– p. 11/30
Rewriting Fourier data (1, y) = EP [(1, γ)] = A◦ u
– p. 11/30
Rewriting Fourier data (1, y) = EP [(1, γ)] = A◦ u Notation: Au = EP [γ] =
Z
γ(x) u(x) dν(x)
R3
Iu = EP [1] =
Z
u(x) dν(x)
R3
Mu = EP [x] =
Z
x u(x) dν(x)
R3
– p. 11/30
Rewriting Fourier data (1, y) = EP [(1, γ)] = A◦ u Notation: Au = EP [γ] =
Z
γ(x) u(x) dν(x)
R3
Iu = EP [1] =
Z
u(x) dν(x)
R3
Mu = EP [x] =
Z
x u(x) dν(x)
R3
A◦ u = (Iu, Au) – p. 11/30
An equivalent formulation (P)
Z Minimize Hν (u) := h u(x) dν(x) s.t. u ∈ L1ν (R3 ) 1 = Iu, y = Au
– p. 12/30
An equivalent formulation (P)
Z Minimize Hν (u) := h u(x) dν(x) s.t. u ∈ L1ν (R3 ) 1 = Iu, y = Au t ln t si t > 0 h(t) := 0 si t = 0 ∞ si t < 0
– p. 12/30
An equivalent formulation (P)
Z Minimize Hν (u) := h u(x) dν(x) s.t. u ∈ L1ν (R3 ) 1 = Iu, y = Au t ln t si t > 0 h(t) := 0 si t = 0 ∞ si t < 0
Minimizing Hν (u) corresponds to the desire to introduce as little prior information as possible. The reference measure may be chosen as an isotropic gaussian measure, the one we would have in an isotropic medium with no fiber – p. 12/30
Relaxation
(P )
Minimize Hν (u) 3 1 (R ) s.t. u ∈ L ν 1 = Iu, y = Au
– p. 13/30
Relaxation
(Pα )
1 Minimize Hν (u)+ ky − Auk2 2α 3 1 (R ) s.t. u ∈ L ν 1 = Iu
– p. 13/30
Relaxation
(Pα )
1 Minimize Hν (u)+ ky − Auk2 2α 3 1 (R ) s.t. u ∈ L ν 1 = Iu
(Pα )
Minimize H (u) − g(Au) ν 3 1 (R ) s.t. u ∈ L ν 1 = Iu
– p. 13/30
Relaxation
(Pα )
1 Minimize Hν (u)+ ky − Auk2 2α 3 1 (R ) s.t. u ∈ L ν 1 = Iu
(Pα )
Minimize H (u) − g(Au) ν 3 1 (R ) s.t. u ∈ L ν 1 = Iu
1 g(η) := − ky − ηk2 2α
– p. 13/30
Relaxation (Pα )
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (R3 ), 1 = Iu
– p. 14/30
Relaxation (Pα )
Minimize Hν (u) − g(Au)+δ(Iu|{1}) s.t. u ∈ L1ν (R3 )
– p. 14/30
Relaxation (Pα )
Minimize Hν (u) − g(Au)+δ(Iu|{1}) s.t. u ∈ L1ν (R3 ) δ(x|S) =
0 if x ∈ S ∞ otherwise
– p. 14/30
Relaxation (Pα )
Minimize Hν (u) − g(Au)+δ(Iu|{1}) s.t. u ∈ L1ν (R3 ) δ(x|S) =
(Pα )
0 if x ∈ S ∞ otherwise
Minimize Hν (u) − g◦ (A◦ u) s.t. u ∈ L1ν (R3 ) – p. 14/30
Relaxation (Pα )
Minimize Hν (u) − g(Au)+δ(Iu|{1}) s.t. u ∈ L1ν (R3 ) δ(x|S) =
(Pα )
0 if x ∈ S ∞ otherwise
Minimize Hν (u) − g◦ (A◦ u) s.t. u ∈ L1ν (R3 )
g◦ (η◦ , η) = g(η) − δ(η◦ |{1})
– p. 14/30
Outline • • • •
Diffusion MRI Building a general entropy model Partially infinite convex programming Application to Diffusion MRI
– p. 15/30
Duality (main issues)
(1) Write the dual problem of (Pα ) (2) Study the constraint qualification conditions (3) Establish the primal-dual relationship
– p. 16/30
Fenchel duality Theorem A
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i • A : L → Rd linear
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i • A : L → Rd linear • A⋆ : Rd → L⋆ its (formal) adjoint
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i • A : L → Rd linear • A⋆ : Rd → L⋆ its (formal) adjoint • H : L → (−∞, ∞] proper convex
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i • A : L → Rd linear • A⋆ : Rd → L⋆ its (formal) adjoint • H : L → (−∞, ∞] proper convex • H ⋆ : L⋆ → (−∞, ∞] its convex conjugate
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i • A : L → Rd linear • A⋆ : Rd → L⋆ its (formal) adjoint • H : L → (−∞, ∞] proper convex • H ⋆ : L⋆ → (−∞, ∞] its convex conjugate • g : Rd → [−∞, ∞) proper concave
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i • A : L → Rd linear • A⋆ : Rd → L⋆ its (formal) adjoint • H : L → (−∞, ∞] proper convex • H ⋆ : L⋆ → (−∞, ∞] its convex conjugate • g : Rd → [−∞, ∞) proper concave • g⋆ : Rd → [−∞, ∞) its concave conjugate
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i • A : L → Rd linear • A⋆ : Rd → L⋆ its (formal) adjoint • H : L → (−∞, ∞] proper convex • H ⋆ : L⋆ → (−∞, ∞] its convex conjugate • g : Rd → [−∞, ∞) proper concave • g⋆ : Rd → [−∞, ∞) its concave conjugate Assume ri(A dom H) ∩ ri(dom g) 6= ∅. Then ⋆ ⋆ η := inf H(p) − g(Ap) = max g⋆ (λ) − H (A λ) p∈L
λ∈Rd
– p. 17/30
Fenchel duality Theorem A • L, L⋆ vector spaces, paired by h·, ·i • A : L → Rd linear • A⋆ : Rd → L⋆ its (formal) adjoint • H : L → (−∞, ∞] proper convex • H ⋆ : L⋆ → (−∞, ∞] its convex conjugate • g : Rd → [−∞, ∞) proper concave • g⋆ : Rd → [−∞, ∞) its concave conjugate Assume ri(A dom H) ∩ ri(dom g) 6= ∅. Then ⋆ ⋆ η := inf H(p)−g(Ap) = max g⋆ (λ) − H (A λ) {z } p∈L λ∈Rd | D(λ)
– p. 17/30
Primal dual relationship Theorem B
– p. 18/30
Primal dual relationship Theorem B With the notation and assumptions of the previous theorem, assume that (⋆)
ri dom g⋆ ∩ ri dom(H ⋆ ◦ A⋆ ) 6= ∅
– p. 18/30
Primal dual relationship Theorem B With the notation and assumptions of the previous theorem, assume that (⋆)
ri dom g⋆ ∩ ri dom(H ⋆ ◦ A⋆ ) 6= ∅
and that (a) H ⋆⋆ = H and g⋆⋆ = g
– p. 18/30
Primal dual relationship Theorem B With the notation and assumptions of the previous theorem, assume that (⋆)
ri dom g⋆ ∩ ri dom(H ⋆ ◦ A⋆ ) 6= ∅
and that (a) H ⋆⋆ = H and g⋆⋆ = g ¯ a dual solution, and u¯ in ∂H ⋆ (A⋆ λ) ¯ (b) there exist λ, ¯ such that H ⋆ ◦ A⋆ has A¯ u as gradient at λ
– p. 18/30
Primal dual relationship Theorem B With the notation and assumptions of the previous theorem, assume that (⋆)
ri dom g⋆ ∩ ri dom(H ⋆ ◦ A⋆ ) 6= ∅
and that (a) H ⋆⋆ = H and g⋆⋆ = g ¯ a dual solution, and u¯ in ∂H ⋆ (A⋆ λ) ¯ (b) there exist λ, ¯ such that H ⋆ ◦ A⋆ has A¯ u as gradient at λ Then u¯ is a primal solution
– p. 18/30
Back to our entropy problem (Pα )
Minimize Hν (u) − g◦ (A◦ u) s.t. u ∈ L1ν (R3 )
g◦ (η◦ , η) = g(η) − δ(η◦ |{1})
– p. 19/30
Back to our entropy problem (Pα )
Minimize Hν (u) − g◦ (A◦ u) s.t. u ∈ L1ν (R3 )
g◦ (η◦ , η) = g(η) − δ(η◦ |{1})
The previous framework may be a powerful tool provided it is possible to compute the conjugate functions Hν⋆ and (g◦ )⋆
– p. 19/30
Back to our entropy problem (Pα )
Minimize Hν (u) − g◦ (A◦ u) s.t. u ∈ L1ν (R3 )
g◦ (η◦ , η) = g(η) − δ(η◦ |{1})
The previous framework may be a powerful tool provided it is possible to compute the conjugate functions Hν⋆ and (g◦ )⋆ α (g◦ )⋆ (λ◦ , λ) = λ◦ + g⋆ (λ) = λ◦ + hλ, yi − kλk2 2
– p. 19/30
Back to our entropy problem (Pα )
Minimize Hν (u) − g◦ (A◦ u) s.t. u ∈ L1ν (R3 )
g◦ (η◦ , η) = g(η) − δ(η◦ |{1})
The previous framework may be a powerful tool provided it is possible to compute the conjugate functions Hν⋆ and (g◦ )⋆ α (g◦ )⋆ (λ◦ , λ) = λ◦ + g⋆ (λ) = λ◦ + hλ, yi − kλk2 2 The computation of Hν⋆ is more tricky: it involves conjugacy through the integral sign – p. 19/30
Conjugacy through the integral Paired spaces
– p. 20/30
Conjugacy through the integral Paired spaces • (X, A , ν) complete measure space
– p. 20/30
Conjugacy through the integral Paired spaces • (X, A , ν) complete measure space • ν positive and σ-finite
– p. 20/30
Conjugacy through the integral Paired spaces • (X, A , ν) complete measure space • ν positive and σ-finite • L, Λ are 2 spaces of measurable functions
– p. 20/30
Conjugacy through the integral Paired spaces • • • •
(X, A , ν) complete measure space ν positive and σ-finite L, Λ are 2 spaces of measurable functions Assume: ∀f ∈ L, ∀ϕ ∈ Λ, f ϕ ∈ L1 (X)
– p. 20/30
Conjugacy through the integral Paired spaces • • • •
(X, A , ν) complete measure space ν positive and σ-finite L, Λ are 2 spaces of measurable functions Assume: ∀f ∈ L, ∀ϕ ∈ Λ, f ϕ ∈ L1 (X) (f, ϕ) 7→ hf, ϕi :=
Z
f (x)ϕ(x) dν(x)
X
– p. 20/30
Conjugacy through the integral Paired spaces • • • •
(X, A , ν) complete measure space ν positive and σ-finite L, Λ are 2 spaces of measurable functions Assume: ∀f ∈ L, ∀ϕ ∈ Λ, f ϕ ∈ L1 (X) (f, ϕ) 7→ hf, ϕi :=
Z
f (x)ϕ(x) dν(x)
X
Example The case where L = L1ν and Λ = L∞ ν is a classical example for which the above pairing is well-defined – p. 20/30
Conjugacy through the integral Definition A space L of A -measurable functions is said to be decomposable if it contains all functions of the form 1A f0 + 1AC f where A ∈ A is such that ν(A) < ∞, f0 is a measurable function such that f0 (A) is bounded and f is any function in L
– p. 21/30
Conjugacy through the integral Definition A space L of A -measurable functions is said to be decomposable if it contains all functions of the form 1A f0 + 1AC f where A ∈ A is such that ν(A) < ∞, f0 is a measurable function such that f0 (A) is bounded and f is any function in L Example The Lp -spaces are decomposable (for every p ∈ [1, ∞])
– p. 21/30
Conjugacy through the integral Theorem 1 [Rockafellar]
– p. 22/30
Conjugacy through the integral Theorem 1 [Rockafellar] • (X, A , ν) a complete measure space
– p. 22/30
Conjugacy through the integral Theorem 1 [Rockafellar] • (X, A , ν) a complete measure space • ν positive and σ-finite
– p. 22/30
Conjugacy through the integral Theorem 1 [Rockafellar] • (X, A , ν) a complete measure space • ν positive and σ-finite • h : R × X → (−∞, ∞] measurable, with h(·, x) l.s.c. for every x
– p. 22/30
Conjugacy through the integral Theorem 1 [Rockafellar] • (X, A , ν) a complete measure space • ν positive and σ-finite • h : R × X → (−∞, ∞] measurable, with h(·, x) l.s.c. for every x Then, the conjugate integrand h⋆ , defined by h⋆ (·, x) = [h(·, x)]⋆ , is a measurable integrand
– p. 22/30
Conjugacy through the integral Theorem 1 [Rockafellar] • (X, A , ν) a complete measure space • ν positive and σ-finite • h : R × X → (−∞, ∞] measurable, with h(·, x) l.s.c. for every x Then, the conjugate integrand h⋆ , defined by h⋆ (·, x) = [h(·, x)]⋆ , is a measurable integrand Corollary For every measurable function ϕ, the function x 7→ h⋆ (ϕ(x), x) is measurable and the integral Z h⋆ (ϕ(x), x) dν(x) is well defined with the convention ∞ − ∞ = ∞ – p. 22/30
Conjugacy through the integral Theorem 2 [Rockafellar]
– p. 23/30
Conjugacy through the integral Theorem 2 [Rockafellar] Let L, Λ be spaces of measurable functions, paired by Z (f, ϕ) 7→ hf, ϕi := f (x)ϕ(x) dν(x) X
– p. 23/30
Conjugacy through the integral Theorem 2 [Rockafellar] Let L, Λ be spaces of measurable functions, paired by Z (f, ϕ) 7→ hf, ϕi := f (x)ϕ(x) dν(x) X
With the notation and assumptions of the previous theorem, assume that L is decomposable, and that {f ∈ L|H(f ) ∈ R} 6= ∅. Then H ⋆ is given on Λ by Z ⋆ ⋆ H (ϕ) = h (ϕ(x), x) dν(x)
– p. 23/30
Primal solution
with
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (X) Hν (u) =
Z
h u(x), x dν(x)
– p. 24/30
Primal solution
with
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (X) Hν (u) =
Theorem
Z
h u(x), x dν(x)
– p. 24/30
Primal solution
with
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (X) Hν (u) =
Z
h u(x), x dν(x)
Theorem Assume: • ri A dom Hν ∩ ri dom g 6= ∅
– p. 24/30
Primal solution
with
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (X) Hν (u) =
Z
h u(x), x dν(x)
Theorem Assume: • ri A dom Hν ∩ ri dom g 6= ∅ • g⋆⋆ = g
– p. 24/30
Primal solution
with
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (X) Hν (u) =
Z
h u(x), x dν(x)
Theorem Assume: • ri A dom Hν ∩ ri dom g 6= ∅ • g⋆⋆ = g • h(·, x) convex l.s.c.
– p. 24/30
Primal solution
with
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (X) Hν (u) =
Z
h u(x), x dν(x)
Theorem Assume: • • • •
ri A dom Hν ∩ ri dom g 6= ∅ g⋆⋆ = g h(·, x) convex l.s.c. h⋆ (·, x) ∈ C 1 (R) for almost all x
– p. 24/30
Primal solution
with
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (X) Hν (u) =
Theorem Assume: • • • • •
Z
h u(x), x dν(x)
ri A dom Hν ∩ ri dom g 6= ∅ g⋆⋆ = g h(·, x) convex l.s.c. h⋆ (·, x) ∈ C 1 (R) for almost all x ¯ ∈ int dom D, dual optimal, such that ∃λ ⋆ ′ ⋆¯ u¯(x) := (h ) A λ(x), x ∈ L1 (X) ν
– p. 24/30
Primal solution
with
Minimize Hν (u) − g(Au) s.t. u ∈ L1ν (X) Hν (u) =
Theorem Assume: • • • • •
Z
h u(x), x dν(x)
ri A dom Hν ∩ ri dom g 6= ∅ g⋆⋆ = g h(·, x) convex l.s.c. h⋆ (·, x) ∈ C 1 (R) for almost all x ¯ ∈ int dom D, dual optimal, such that ∃λ ⋆ ′ ⋆¯ u¯(x) := (h ) A λ(x), x ∈ L1 (X) ν
Then u¯ is a primal solution
– p. 24/30
Back to our entropy problem Hν (u) = with
Z
h u(x) dν(x)
t ln t si t > 0 h(t) := 0 si t = 0 ∞ si t < 0
– p. 25/30
Back to our entropy problem Hν (u) = with
Z
h u(x) dν(x)
t ln t si t > 0 h(t) := 0 si t = 0 ∞ si t < 0 h⋆ (τ ) = exp(τ − 1)
– p. 25/30
Back to our entropy problem Hν (u) = with
Z
h u(x) dν(x)
t ln t si t > 0 h(t) := 0 si t = 0 ∞ si t < 0 h⋆ (τ ) = exp(τ − 1)
Since L1ν (R3 ) is decomposable, Z exp(ϕ(x) − 1) dν(x), Hν⋆ (ϕ) = R3
3 (R ) ϕ ∈ L∞ ν – p. 25/30
The dual problem (Dα )
Maximize D(λ , λ) ◦ s.t. (λ◦ , λ) ∈ R1+2m
– p. 26/30
The dual problem (Dα )
Maximize D(λ , λ) ◦ s.t. (λ◦ , λ) ∈ R1+2m
D(λ◦ , λ) = (g◦ )⋆ (λ◦ , λ) − (Hν⋆ ◦ A⋆◦ )(λ◦ , λ)
– p. 26/30
The dual problem (Dα )
Maximize D(λ , λ) ◦ s.t. (λ◦ , λ) ∈ R1+2m
D(λ◦ , λ) = (g◦ )⋆ (λ◦ , λ) − (Hν⋆ ◦ A⋆◦ )(λ◦ , λ) α = λ◦ + hλ, yi − kλk2 2 Z ⋆ − exp A◦ (λ◦ , λ)(x) − 1 dν(x) | {z } λ◦ +hλ,γ(x)i
– p. 26/30
The dual problem (Dα )
Maximize D(λ , λ) ◦ s.t. (λ◦ , λ) ∈ R1+2m
D(λ◦ , λ) = (g◦ )⋆ (λ◦ , λ) − (Hν⋆ ◦ A⋆◦ )(λ◦ , λ) α = λ◦ + hλ, yi − kλk2 2 Z ⋆ − exp A◦ (λ◦ , λ)(x) − 1 dν(x) | {z } λ◦ +hλ,γ(x)i
α = λ◦ + hλ, yi − kλk2 Z2 −exp(λ◦ − 1)
exphλ, γ(x)i dν(x) – p. 26/30
Optimality system • D is concave, with effective domain R1+2m
– p. 27/30
Optimality system • D is concave, with effective domain R1+2m • D ∈ C 1 R1+2m
– p. 27/30
Optimality system • D is concave, with effective domain R1+2m • D ∈ C 1 R1+2m Dual optimality reads Z ¯ 0 = 1 − exp(λ◦ − 1)
¯ γ(x)i dν(x) exphλ, R3 Z ¯ γ(x)i dν(x) ¯ − exp(λ ¯ ◦ − 1) 0 = y − αλ γ(x) exphλ, R3
– p. 27/30
Optimality system • D is concave, with effective domain R1+2m • D ∈ C 1 R1+2m Dual optimality reads Z ¯ 0 = 1 − exp(λ◦ − 1)
¯ γ(x)i dν(x) exphλ, R3 Z ¯ γ(x)i dν(x) ¯ ◦ − 1) ¯ − exp(λ 0 = y − αλ γ(x) exphλ, R3
– p. 27/30
Optimality system • D is concave, with effective domain R1+2m • D ∈ C 1 R1+2m Dual optimality reads Z ¯ 0 = 1 − exp(λ◦ − 1)
¯ γ(x)i dν(x) exphλ, R3 Z ¯ γ(x)i dν(x) ¯ ◦ − 1) ¯ − exp(λ 0 = y − αλ γ(x) exphλ, R3
which reduces to
¯− 0 = y − αλ
Z
¯ γ(x)i dν(x) γ(x) exphλ,
3 RZ
¯ γ(x)i dν(x) exphλ,
R3 – p. 27/30
Optimality system Observe that ¯− 0 = y − αλ
Z
¯ γ(x)i dν(x) γ(x) exphλ,
3 RZ
¯ γ(x)i dν(x) exphλ,
R3
– p. 28/30
Optimality system Observe that ¯− 0 = y − αλ
Z
¯ γ(x)i dν(x) γ(x) exphλ,
3 RZ
¯ γ(x)i dν(x) exphλ,
R3
is also the optimality system of Z α Maximize hλ, yi − kλk2 − ln exphλ, γ(x)i dν(x) 2 (D˜α ) s.t. λ ∈ R2m
– p. 28/30
Optimality system Observe that ¯− 0 = y − αλ
Z
¯ γ(x)i dν(x) γ(x) exphλ,
3 RZ
¯ γ(x)i dν(x) exphλ,
R3
is also the optimality system of Z α Maximize hλ, yi − kλk2 − ln exphλ, γ(x)i dν(x) 2 (D˜α ) s.t. λ ∈ R2m
Proposition The function R α 2 ˜ D(λ) := hλ, yi − 2 kλk − ln exphλ, γ(x)i dν(x) is concave and smooth (on R2m )
– p. 28/30
Algorithm ¯ ˜ • Maximze D(λ) ֒→ λ
– p. 29/30
Algorithm ¯ ˜ • Maximze D(λ) ֒→ λ • Compute ¯ ◦ − 1) = exp(λ
Z
−1 ¯ γ(x)i dν(x) exphλ,
– p. 29/30
Algorithm ¯ ˜ • Maximze D(λ) ֒→ λ • Compute ¯ ◦ − 1) = exp(λ
Z
−1 ¯ γ(x)i dν(x) exphλ,
• Compute ODF from ¯ ◦ − 1) exphλ, ¯ γ(x)i u¯(x) = exp(λ
– p. 29/30
Algorithm ¯ ˜ • Maximze D(λ) ֒→ λ • Compute ¯ ◦ − 1) = exp(λ
Z
−1 ¯ γ(x)i dν(x) exphλ,
• Compute ODF from ¯ ◦ − 1) exphλ, ¯ γ(x)i u¯(x) = exp(λ
The optimal u¯ is searched for in a smooth manifold of dimension 2m in L1ν (R3 ) – p. 29/30
Thank you for your attention !
– p. 30/30
