Denoising Tensors via Lie Group Flows

Abstract

The need to regularize tensor fields arise recently in various applications. We treat in this paper tensors that belong to
matrix Lie groups. We formulate the problem of these SO(N) flows in terms of the principal chiral model (PCM) action. This action is defined over a Lie group manifold. By minimizing
the PCM action with respect to the group element, we obtain the equations of motion for the group element (or the corresponding
connection). Then, by writing the gradient descent equations we obtain the PDE for the Lie group flows. We use these flows
to regularize in particular the group of N-dimensional orthogonal matrices with determinant one i.e. SO(N). This type of regularization
preserves their properties (i.e., the orthogonality and the determinant). A special numerical scheme that preserves the Lie
group structure is used. However, these flows regularize the tensor field isotropically and therefore discontinuities are
not preserved. We modify the functional and thereby the gradient descent PDEs in order to obtain an anisotropic tensor field
regularization. We demonstrate our formalism with various examples.

Full text

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Denoising Tensors via Lie Group Flows
Y. Gur and N. Sochen
Department of Applied Mathematics, Tel-Aviv university, Ramat-Aviv, Tel-Aviv
69978, Israel
yanivg@post.tau.ac.il, sochen@post.tau.ac.il
Abstract. The need to regularize tensor fields arise recently in various
applications. We treat in this paper tensors that belong to matrix Lie
groups. We formulate the problem of these SO(N) flows in terms of
the principal chiral model (PCM) action. This action is defined over a
Lie group manifold. By minimizing the PCM action with respect to the
group element, we obtain the equations of motion for the group element
(or the corresponding connection). Then, by writing the gradient descent
equations we obtain the PDE for the Lie group flows. We use these
flows to regularize in particular the group of N-dimensional orthogonal
matrices with determinant one i.e. SO(N). This type of regularization
preserves their properties (i.e., the orthogonality and the determinant).
A special numerical scheme that preserves the Lie group structure is
used. However, these flows regularize the tensor field isotropically and
therefore discontinuities are not preserved. We modify the functional
and thereby the gradient descent PDEs in order to obtain an anisotropic
tensor field regularization. We demonstrate our formalism with various
examples.
1Introduction
For more than a decade PDE’s are widely used to tackle many image processing
problems such as image restoration, segmentation, image enhancing and much
more. Especially interesting are the nonlinear PDE’s which in the context of
image restoration has been proved to have remarkable denoising, deblurring as
well as edges preserving properties. We will mention some of these works such
as the pioneering work by Perona and Malik [19] on image denoising, the work
by Osher and Rudin [16] on image enhancement and many others which are
discussed extensively in [1,31,11,23]. Some of the image processing problems
may be formulated in terms of Lagrangian actions where the variation of the
Lagrangian leads to the equations of motion. The gradient descent equations
then defines the PDE’s that we wish to apply to images in order to obtain the
desired result [15,21] (i.e, segmentation, denoising, etc).
Earlier studies dealt with scalar valued images. It was later generalized to
vector-valued images (see for example [35,2,24,22,29] and references therein).
Works on constrained regularization of vector-valued image were treated in the
literature as well in [20,28,3,26,12].

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2Y. Gur, N. Sochen
In the last years new methods which consider tensor-valued images have
emerged. In these new methods at each point of the two (or three) dimensional
image space a tensor is attached rather than a scalar or a vector. This tensor
field might be noisy and therefore one has to regularize it in order to extract its
original texture. Moreover, the tensor field has certain properties that we wish
to preserve along the flow (e.g., orthogonality, unit norm, etc) and it might lie on
a non-flat manifold. In order to regularize these fields and preserve their original
properties one has to adopt new methods, both analytical and numerical.
A solution to the problem of orthogonal tensor field regularization was pro-
posed by Deriche et al. [30,5]. In their formalism, the orthogonality of the tensor
field is preserved by adding a constraint term to the unconstraint gradient de-
scent equation using Lagrange multipliers. The constraint term preserve the
orthonormality of the vector basis along the flow. The unconstraint gradient
descent equation was obtained by minimizing the unconstrained φ functional.
Different methods for regularization of tensor fields were proposed recently by
[32,4,18,17,13]. Their work is mainly in relation with the DT-MRI application.
In this work we suggest a novel and natural framework to the problem of
tensor field regularization. We assume here that the tensor at each point is a
Lie group element and construct a regularization flow that respects the group’s
structure. The constrained gradient descent equation will be derived directly
from a Lagrangian without any additional constraint e.g. without a lagrange
multiplier. The functional is defined directly on the Lie group manifold. In order
to solve the PDE numerically such that the Lie group field evolves on the group
manifold we use the Lie group integrating methods introduced in [8,10]. Our
main example is the SO(N) group which is of relevant in DT-MRI yet the
formalism is of general applicability.
The plan of the paper is as follows: In section 2 we will give some math-
ematical preliminaries that will be used in this work. In section 3 we present
the generalized Principal Chiral Model (PCM) action and derive the gradient
descent equations. The gradient descent equations for this action will define the
PDE flow on the group manifold. In section 4 we describe how to implement
the flow to evolve on group manifold in general and on SO(N) in particular.
We will present and use modern Lie-group numerical integration methods. Re-
sults are given in section 5 where we demonstrate regularization of noisy three-
dimensional orthogonal tensor field. Finally, concluding remarks are given in
section 6.
2A bit about Lie groups
For our discussion it is essential to introduce some of the basic definitions con-
sidering Lie groups and Lie algebra.
Definition 1 A Lie group is a group G which is a differentiable manifold equipped
with smooth product G × G ?→ G.

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Denoising Tensors via Lie Group Flows3
Definition 2 The Lie algebra g of Lie group G is defined as the linear vector
space of all tangent vectors to G at the identity. This tangent space is denoted
TIG.
Definition 3 A real matrix Lie group is a smooth subset G ⊆ IRN×Nclosed
under matrix product and matrix inversion. The identity matrix is denoted I ∈ G.
Definition 4 A Lie algebra of a matrix Lie group is a linear subspace g ⊆
IRN×Nequipped with the operation g × g ?→ g which is the Lie bracket (the
commutator) [A,B] = AB − BA. This operation is bilinear, skew-symmetric
([A,B] = −[B,A]), and satisfies the Jacobi identity
[A,[B,C]] + [C,[A,B]] + [B,[C,A]] = 0.
(1)
Definition 5 The elements which span the Lie algebra space are called the gen-
erators of the Lie group or the infinitesimal operators of the group. Let ta, tb
and tcbe the generators of the Lie group, then their algebra is close under the
commutator operation
[ta,tb] = fc
where fc
lower indices fc
abtc,
(2)
abare the structure constants of the group and are antisymmetric in their
ab= −fc
We will demonstrate our study on the special orthogonal matrix Lie group,
SO(N). Its elements are N ×N orthogonal matrices with determinant one. This
group is a subgroup of O(N) which is the orthogonal group and its elements are
N × N orthogonal matrices. The Lie algebra of SO(N) and O(N) is denoted
so(n) and consists of N × N skew-symmetric matrices. O(N) and SO(N) are
special cases of quadratic Lie group which takes the form
ba.
G = {X|XTPX = P},
(3)
where P is a constant matrix (for O(N) and SO(N), P is identity matrix). The
corresponding Lie algebra is given by g = {A|PA + ATP = 0}.
In order to map elements of the Lie algebra into the Lie group one may use
the following maps
Definition 6 The exponential mapping expm : g ?→ G is defined as
expm(A) =
∞
?
k=0
Ak
k!,
(4)
where expm(0) = I. Note that for A which is sufficiently near o ∈ g the exponen-
tial mapping has a smooth inverse given by the matrix logarithm logm : G ?→ g.
For quadratic groups one may also use the Cayley mapping
Definition 7 The Cayley mapping Cay : g ?→ G is defined as
Cayρ(x) = (I − ρx)−1(I + ρx),
(5)

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4Y. Gur, N. Sochen
where ρ is a non-negative constant. When ρ = 1/2 the Cayley map is a special
case of the Pad´ e approximant to the exponential, Cay1/2(x) = expm(x)+O(x3).
Definition 8 The inverse of the Cayley mapping invcay : G ?→ g is defined as
invcayρ(X) =1
ρ(I + X)−1(X − I).
(6)
Note that if X has an eigenvalue -1, this transform is undefined.
Definition 9 The adjoint representation, Ad, and its derivative, ad, are given
by the formulae
AdA(B) = BAB−1,
adA(B) = [A,B] = AB − BA.
(7)
(8)
3The Generalized Principal Chiral Model
The principal chiral models (PCMs) which are known also as the sigma models
arise in many branches of physics (e.g., classical and quantum physics, condensed
matter, high-energy physics, etc...). These models are known to be integrable
[33,34,7]. We consider a variation of the sigma models which is the generalized
principal chiral model (GPCM) and is given by the action [26,9]
?
where g takes values in the Lie group G, η is the spatial metric and Hab(g) is
invertible symmetric dimG × dimG matrix such that
Hab(g) = H(g)Kab,
L =
d2xηµνHab(g)(g−1∂µg)a(g−1∂νg)b,
(9)
(10)
where Kabis the bilinear Killing form
Kab= Tr(tatb), ta,tb∈ g.
(11)
The bilinear form is considered as the metric over the Lie group manifold.
Since we are interested in tensor fields which are attached to two-dimensional
flat image space, we will take the metric ηµνto be the Euclidean metric ηµν=
δµν. The integration is taken over the two-dimensional image space. The term
Aµ= g−1∂µg is known as the flat-connection and also as the Yang-Mills gauge
field. The flat-connection is an element of the Lie algebra and therefore it may
be represented in terms of the generators of the Lie algebra such that
Aµ= g−1∂µg = Aa
µta.
(12)
Also, it obeys the Bianchi identity
∂µAν− ∂νAµ+ [Aµ,Aν] = 0.
(13)

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Denoising Tensors via Lie Group Flows5
In order to obtain the equations of motion we vary the GPCM action with
respect to g−1δg to obtain
−HadδL
δρd= ∂µAµa+ Γa
bcAb
µAµc= 0,
(14)
where we have used the relation
δAa
µ= ∂νδρa− fa
bcAb
νδρc,
(15)
and where δρ = g−1δg. The connection Γa
bcis a sum of two parts
Γa
bc= Sa
bc+ γa
bc,
(16)
where Sa
bcis defined as
Sa
bc=1
2(Fa
bc+ Fa
cb),
(17)
Fa
bc= (H−1)apfq
pbHqc.
The second part are the Christoffel symbols for the metric Hab
γa
bc=1
2(H−1)ad(∂bHcd+ ∂cHbd− ∂dHbc).
(18)
Taking Hab to be constant on the group manifold (i.e., Hab = Kab) we have
γa
negative definite and is given by Kab= Tr(tatb) = −2δab. Plugging Habinto Eq.
(17) we have,
Fa
However, since the structure constants are antisymmetric in their indices (i.e.,
fa
bc= 0. The bilinear form over the SO(N) group manifold, for example, is
bc= 2(fa
bc+ fa
cb).
(19)
bc= −fa
cb), Fa
bc= 0 and we are left with the equation of motion
∂µAµa= 0.
(20)
Contracting this equation with the group generators tafrom the right we have
∂µAµata= ∂µAµ= 0.
(21)
Since Aµ= g−1∂µg we may write the equation of motion in the following form
∂µ(g−1∂µg) = 0.
(22)
In order to write the gradient descent equations we have to remember that the
term ∂µ(g−1∂µg) is in the Lie algebra and therefore the left hand side (LHS)
of the gradient descent equation should contain a term which is also in the Lie
algebra. Therefore, we suggest the following expression
g−1∂g
∂t= ∂µ(g−1∂µg)
= ∂x(g−1∂xg) + ∂y(g−1∂yg).
(23)