A Riemannian approach to anisotropic filtering of tensor fields

Abstract

Tensors are nowadays an increasing research domain in different areas, especially in image processing, motivated for example by diffusion tensor magnetic resonance imaging (DT-MRI). Up to now, algorithms and tools developed to deal with tensors were founded on the assumption of a matrix vector space with the constraint of remaining symmetric positive definite matrices. On the contrary, our approach is grounded on the theoretically well-founded differential geometrical properties of the space of multivariate normal distributions, where it is possible to define an affine-invariant Riemannian metric and express statistics on the manifold of symmetric positive definite matrices. In this paper, we focus on the contribution of these tools to the anisotropic filtering and regularization of tensor fields. To validate our approach we present promising results on both synthetic and real DT-MRI data.

Full text

Page 1

A Riemannian Approach to
Anisotropic Filtering of Tensor Fields
C.A. Casta˜ no-Moragaa,b, C. Lengleta, R. Dericheaand
J. Ruiz-Alzolab
aI.N.R.I.A., Projet Odyss´ ee, 2004 route des lucioles, 06902 Sophia-Antipolis, France.
bCenter for Technology in Medicine, Signals and Communications Department, Building
B, University of Las Palmas de Gran Canaria. Campus de Tafira s/n. 35017 Tafira, Spain.
Abstract
Tensors are nowadays an increasing research domain in different areas, especially in image
processing, motivated for example by DT-MRI (Diffusion Tensor Magnetic Resonance
Imaging). Up to now, algorithms and tools developed to deal with tensors were founded
on the assumption of a matrix vector space with the constraint of remaining symmetric
positive definite matrices. On the contrary, our approach is grounded on the theoretically
well-founded differential geometrical properties of the space of multivariate normal
distributions, where it is possible to define an affine-invariant Riemannian metric and
express statistics on the manifold of symmetric positive definite matrices. In this paper,
we focus on the contribution of these tools to the anisotropic filtering and regularization of
tensor fields. To validate our approach we present promising results on both, synthetic and
real DT-MRI data.
Key words: Differential geometry, Riemannian manifold, Geodesic distance, DT-MRI,
Tensor fields, Regularization, Anisotropic filtering, Nonlinear diffusion.
PreprintsubmittedtoElsevierScience2March2006

Page 2

1Introduction
DT-MRI (Diffusion Tensor Magnetic Resonance Imaging) is a relatively new
medical imaging modality [3,34] from which a great deal of research on tensors
has stemmed. But this is not the only domain where tensors arise. Image structure
analysis through the Local Structure Tensor [4], stress and strain analysis of
materials [5] or electromagnetism [35] are other examples. Whatever domain we
are referring to, working with these tensor fields may require some regularization to
reduce the noise arising, for instance, from the acquisition process. Regularization
and filtering schemes of tensor fields are widely studied in the literature, especially
in the domain of DT-MRI. As an example, [29] deals with the problem of
regularizing noisy fields of diffusion tensors, considered as symmetric and positive
definite n×n matrices, through a PDE-based scheme and a spectral decomposition.
A complementary work is that presented in [11] which provides a geometric
interpretationofconstrainedflowsformatrix-valuedfunctions.Thisyields,through
the use of exponential maps, suitable numerical schemes that are also constraints
preserving. Another approach presented in [33] provides a generalization of
anisotropic and nonlinear diffusion process to matrix-valued data. More recently,
[12] proposes a scheme, based on variational methods, restoring the main direction
of the tensors and using the resulting direction to regularize the eigenvalues by an
anisotropic diffusion process. However, tensor eigenvalues tend to regularize faster
than the associated eigenvectors. This phenomenon is known as the eigenvalue
swelling effect for long regularization time, as shown in [30], while noise removal
Email addresses: ccasmor@ctm.ulpgc.es (C.A. Casta˜ no-Moraga ),
Christophe.Lenglet@sohpia.inria.fr (C. Lenglet),
Rachid.Deriche@sophia.inria.fr (R. Deriche), jruiz@ctm.ulpgc.es (J.
Ruiz-Alzola).
2

Page 3

is not quite significant for short time regularization.
Other works try to couple the regularization with the tensors estimation process
from diffusion-weighted images. For example, [31] presents a constrained
variational principle which involves the minimization of a regularization term,
based on Lpnorms and subject to a nonlinear constraint on the data to obtain
positive definite tensors.
In this paper, we will rely on [17,18] where the authors proposed to use the
differential geometrical properties of the space of multivariate normal distributions
to define statistics on diffusion tensors, i.e. covariance matrices. This work is based
on the information geometry derived from the Fisher information, as proposed for
instance by Amari in [1].
Independently, other recent papers also addressed the issue of defining statistical
quantities and filtering tools for tensor fields. [14] first analyzes the space of
symmetric, positive definite matrices from a Lie groups perspective and shows
that this space does not form a vector space but can be regarded as a Riemannian
symmetric space. The authors used these ideas to develop efficient and elegant
methods for computing statistics and modes of variation of diffusion tensor data
in this non linear space.
More recently, and in parallel to our work initiated earlier in [17], [25] developed
also a nice and elegant computational framework for tensor processing with a
particular emphasize on interpolation, regularization and restoration of noisy tensor
fields.
Adding to these works where similar tools have been developed in order to study
thestatisticalvariabilityofdiffusiontensorimages,onecanalsorefertotheoriginal
point of view based on the information geometry recently developed in [18]. It is
thus very interesting tonote that comparable results were obtained throughdifferent
means while studying the statistical variability of diffusion tensor images. In our
3

Page 4

work, we use the mathematical framework, developed in [28] and used in [18] to
take into account the particular geometry of the set of symmetric, positive definite
matrices in the smoothing process. We introduce an anisotropic filtering algorithm
controlled by the magnitude of the spatial gradient of the tensor field. On the
contrary of [14], where a statistical analysis of diffusion tensors was proposed but
without addressing the problem of smoothing, or [25], where the regularization
task was addressed with a PDE point of view, our smoothing method, developed
in parallel to [25] and based on our previous work developed earlier in [17], only
relies on simple and local anisotropic averaging. Adding to that, our method is
favorably compared to a state-of-the-art approach [33]. A detailed analysis of the
performances of our approach is performed and qualitative and quantitative results
obtained on noisy and synthetic data show that our approach outperforms the one
proposed in [33]. The emphasize of this article is not on the derivation of the
Riemannian framework, which we developed earlier in [17,18], but rather on its
clear and efficient application to solve the important problem of anisotropically
smoothing a set of noisy tensor data using the right concepts and tools, extending
our previous PDE based approaches in [29].
Section 2 recalls basic facts on the differential geometry of the space of multivariate
normal distributions and section 3 shows how they can be used to compute local
averages and spatial gradient of diffusion tensor images. Section 4 introduces the
filtering process. Finally, section 5 presents and discusses numerical experiments.
We will show that our Riemannian anisotropic filtering method yields better results
on both synthetic and real DT-MRI datasets when compared to other approaches
such as the nonlinear diffusion proposed in [33].
4

Page 5

2Geometry of the Space of Multivariate Normal Distributions
We hereafter review the necessary material related to the Riemannian geometry
of the multivariate normal model. As in [18], we consider the family of three-
dimensional normal distributions with 0-mean as the 6-dimensional parameter
space of variances and covariances. We identify it with S+(3), the set of 3 × 3
real symmetric positive definite matrices, e.g. covariance matrices. Following the
work by Rao [27] and Burbea-Rao [7], where a Riemannian metric was introduced
for S+(3) in terms of the Fisher information matrix, it is possible to define notions
suchasthegeodesicdistance,thecurvature,themean,andthecovariancematrixfor
elements of S+(3). The basis of the tangent space TΣS+(3) = SΣ(3) at Σ ∈ S+(3) is
taken to be as in [28] and denoted by Ei, i = 1,...,6. The local coordinates are given
by θ1= Σ11,θ2= Σ12,...,θ6= Σ33. The fundamental mathematical tools needed to
derive our numerical schemes were detailed in [28], [8], [13], [9], [15] and [22].
Without employing the information geometry associated to the Fisher information
matrix but instead, identifying S+(3) with the quotient space GL+(3)/SO(3), (with
GL+(3) the general linear group of positive definite 3 × 3 matrices and SO(3) the
special orthogonal group of 3 × 3 matrices), other works such as [14] and [25]
recently used similar ideas to derive statistical or filtering tools on tensors fields.
By extending the symmetry group to SL(3), [20] studied a slightly different metric
which turns out to exhibit an additional term in one direction. This extension was
also recently formulated by [21] in terms of a weighted Fisher information matrix.
They also generalized this approach to a wider class of elliptical densities. The
metric tensor for S+(3), derived from the Fisher information matrix is given by the
following theorem, which is proved in [28]:
Theorem 1 The Riemannian metric for the space S+(3) of multivariate normal
5

Page 6

distributions with zero mean is given, ∀Σ ∈ S+(3) by:
gij= g(Ei,Ej) = ?Ei,Ej?Σ=1
2tr
?
Σ−1EiΣ−1Ej
?
i, j = 1,...,6(1)
In practice, this means that for any tangent vectors A, B, their inner product relative
?
[t1,t2] ⊂ R denotes a curve segment in S+(3) between two normal distributions
parameterized by the matrices Σ1and Σ2, its length is expressed as:
to Σ is ?A, B?Σ =
1
2tr
Σ−1AΣ−1B
?
. We recall that, if Σ : t ?→ Σ(t) ∈ S+(3), ∀t ∈
LΣ(Σ1,Σ2) =
?t2
t1

6 ?
i,j=1
gij(Σ(t))dθi(t)
dt
dθj(t)
dt

1/2
dt
(2)
As stated for example in [22], the geodesic starting from Σ(t1) ∈ S+(3) in the
direction V =˙Σ(t1) of the tangent space TS+(3) = S(3), is given by the exponential
map:
Σ(t) = Σ(t1)1/2exp
?
(t − t1)Σ(t1)−1/2VΣ(t1)−1/2?
Σ(t1)1/2∀t ∈ [t1,t2](3)
We recall that the geodesic distance D between any two elements Σ1and Σ2is the
length of the minimizing geodesic between Σ1and Σ2:
D(Σ1,Σ2) = inf
Σ{LΣ(Σ1,Σ2) : Σ1= Σ(t1),Σ2= Σ(t2)}
It is given by the following theorem, whose original proof is available in an
appendix of [2] but different versions can also be found in [28] and [15].
Theorem 2 (S.T. Jensen, 1976) Consider the family of multivariate normal
distributions with common mean vector but different covariance matrices. The
geodesic distance between two members of the family with covariance matrices
Σ1and Σ2is given by
D(Σ1,Σ2) =
?
1
2tr
?
log2?
Σ−1/2
1
Σ2Σ−1/2
1
??
=
?
?
1
2
3 ?
i=1
log2(ηi)(4)
6

Page 7

where ηidenote the 3 eigenvalues of the matrix Σ−1/2
1
Σ2Σ−1/2
1
.
3Local Average and Spatial Gradient of Diffusion Tensor Fields
Now, we can recall how the local average and spatial gradient of a diffusion tensor
image can be computed.
3.1Intrinsic mean
To estimate the intrinsic mean distribution of a set of zero-mean multivariate
normal distributions we resort to a gradient descent algorithm. As proposed in
[18], we use the classical definition of the Riemannian center of mass and the
geodesic equations to derive a manifold constrained numerical integrator and thus
ensure that each step forward of the gradient descent stays within the space S+(3).
The normal distribution parametrized byˆΣ ∈ S+(3) and defined as the empirical
mean of N distributions Σ1,Σ2,...,ΣN, achieves a local minimum of the function
µ : S+(3) → R+given as:
µ(ˆΣ,Σ1,Σ2,...,ΣN) =1
N
N ?
k=1
D2(ˆΣ,Σk)(5)
It has been proved in [16] that such a local minimum, known as the Riemannian
barycenter, exists and is unique for manifolds of non-positive sectional curvature,
which is the case for S+(3). A closed-form expression cannot be obtained, but
it is possible to derive a gradient descent algorithm for the computation of the
intrinsic mean as shown in [18]. To derive a flow evolving an initial guessˆΣ(0)
toward the Riemannian barycenter of the set Σ1,Σ2,...,ΣN∈ S+(3), we denote by
ˆΣ(s), s ∈ [0,∞) the family of solution of ∂sˆΣ(s) = V(ˆΣ(s)), where V is the direction
7

Page 8

of evolution. Then, we have:
ˆΣ(s) ∈ S+(3),∀s > 0 ⇔ {ˆΣ(0) ∈ S+(3) and V(ˆΣ(s)) ∈ TˆΣ(s)S+(3),∀s > 0}
(6)
We thus identify V with the opposite of the gradient of our objective function
µ(Σ1,Σ2,...,ΣN), which is [22]:
∇µ(ˆΣ,Σ1,Σ2,...,ΣN) =
ˆΣ(s)
N
N ?
k=1
log(Σ−1
kˆΣ(s))(7)
Hence, the evolution is:
∂sˆΣ(s) = −
ˆΣ(s)
N
N ?
k=1
log(Σ−1
kˆΣ(s)) (8)
Moreover, numerical implementation has to be dealt with carefully and we have
to build a step-forward operator such that the discrete flow provides an intrinsic
approximation of the evolution in Eq. 8. As proved in [18], this can be achieved for
any tangent vector V = −∇µ ∈ S(3), by resorting to the geodesic equation:
ˆΣl+1=ˆΣ1/2
l
exp(−dtˆΣ−1/2
l
∇µˆΣ−1/2
l
)ˆΣ1/2
l
(9)
Finally, using Eq. 7 in Eq. 9, we obtain the following evolution to calculate the
intrinsic mean:
ˆΣl+1=ˆΣ1/2
l
exp
−dt
ˆΣ1/2
l
N

N ?
k=1
log(Σ−1
kˆΣl)
ˆΣ−1/2
l
ˆΣ1/2
l, (l = 0,...,Niter)(10)
Numerical experimentation demonstrates that the evolution toward the desired
meanˆΣ converges in no more than Niter = 4 or 5 iterations for any initial guess
ˆΣ0.
8

Page 9

3.2Weighted intrinsic mean
In the same way the intrinsic mean is defined, we can also calculate a weighted
intrinsic mean which ponderates a set of N normal distributions Σ1,Σ2,...,ΣN ∈
S+(3) with the weights ω1,ω2,...,ωN,ωi∈ R+In this case, the normal distribution
parameterized byˆΣw∈ S+(3) and defined as the weighted empirical mean of N
distributions Σ1,Σ2,...,ΣN, achieves a minimum of the weighted sum of squared
distances defined by
µw(ˆΣw,Σ1,Σ2,...,ΣN) =
?N
k=1ωkD2(ˆΣw,Σk)
?N
k=1ωk
(11)
Following the same steps as in section 3.1, the following weighted step-forward
operator is readily obtained:
ˆΣw
l+1=ˆΣw 1/2
l
exp
−dt
ˆΣw 1/2
l
??N
k=1ωklog(Σ−1
?N
kˆΣw
l)
?ˆΣw −1/2
l
k=1ωk
ˆΣw 1/2
l
(12)
3.3Spatial gradient of diffusion tensor fields
It is possible to estimate the magnitude of the gradient of the tensor field Σ(x) :
Ω ⊂ R3?→ S+(3) at x ∈ Ω through the sum of squared geodesic distances between
tensors in orthogonal directions on a discrete grid, as indicated by the following
expression:
| ∇Σ(x) |2?
3 ?
k=1
D2(Σ(x),Σ(x ± ek))(13)
where ekdenotes the elements of the canonical basis in R3. To derive Eq. 13, we use
the explicit formulation of geodesic presented in Eq. 3, which allows us to calculate
the geodesic starting at Σ0= Σ(x) in the direction V =˙Σ(0) as:
Σ(t) = Σ1/2
0exp(tΣ−1/2
0
VΣ−1/2
0
)Σ1/2
0
(14)
9

Page 10

Hence, as we know that Σ(1) = Σ(x ± ek) = Σ1, we obtain that:
V = Σ1/2
0log(Σ−1/2
0
Σ1Σ−1/2
0
)Σ1/2
0
=˙Σ(0)(15)
It can be shown that this quantity is actually equivalent to the opposite of the
gradient of the squared geodesic distance ∇D2(Σ0,Σ1) whose expression was given
by Moakher [22]:
∇D2(Σ0,Σ1) = Σ0log
0log(Σ−1/2
?
Σ−1
1Σ0
?
)Σ1/2
0
Indeed, it is easy to see that Σ1/2
0
Σ1Σ−1/2
0
can be rewritten as
Σ0Σ−1/2
0
log(Σ−1/2
0
Σ1Σ−1/2
0
)Σ1/2
0. Using the following property:
M−1
1log(M2)M1= log(M−1
1M2M1), ∀M1, M2∈ GL(m)(16)
we get
V = Σ0log(Σ−1
0Σ1) = −Σ0log(Σ−1
1Σ0)
The symmetric matrix V can thus be used to approximate the spatial directional
derivative of a tensor field. To better understand why this is correct, we can consider
the Euclidean distance between tensor
D2
E(Σ0,Σ1) = |Σ0− Σ1|2
F= tr
?
(Σ0− Σ1)(Σ0− Σ1)T?
∀X,Y ∈ GL(m), we havefor which, using the fact that ∇tr(XY) = YT,
∇D2
vector, usually used in finite difference schemes to approximate a spatial gradient.
E(Σ0,Σ1) = Σ0− Σ1. In other words, this corresponds to the difference tangent
V generalizes this notion by taking into account the Riemannian structure of the
space S+(3).
All we need to prove now is that the magnitude of the tangent vector V, taking into
account the Riemannian metric of S+(3), is equal to the squared geodesic distance
between Σ0and Σ1, so that Eq. 13 is true.
Noting that V is, by definition, the logarithm map of Σ1at Σ0logΣ0(Σ1) =˙Σ0, we
10

Page 11

know this is true since one of its properties is:
?˙Σ0,˙Σ0?Σ0= D2(Σ0,Σ1)
Indeed, we have
?V,V?Σ0=1
2tr
??
0Σ1 and Σ−1/2
−Σ−1
0Σ0log(Σ−1
1Σ0)
?2?
=1
2tr
??
log(Σ−1
0Σ1)
?2?
and since the matrices Σ−1
?
within the Riemannian framework, we simply calculate the geodesic distances in
0
Σ1Σ−1/2
0
are similar, we have ?V,V?Σ0=
1
2tr log2?
Σ−1/2
0
Σ1Σ−1/2
0
??
= D2(Σ0,Σ1). Hence, to obtain the gradient magnitude
orthogonal directions following Eq. 13.
Let us now compare the differences between the gradient magnitude in a Euclidean
sense and our definition in the Riemannian framework. For the sake of simplicity,
we are going to study the case of tensors in a two-dimensional neighborhood N
containing a boundary like that shown in Fig. 1, where white dots represent tensors
T1= λ1v1v1T+λ2v2v2TandblackdotsrepresenttensorsT2= k1λ1v1v1T+k2λ2v2v2T,
with λi,(i = 1,2) the eigenvalues of T1, associated to eigenvectors viand constants
ki. Using Eq. 13 and the definition of the geodesic distance in Eq. 4, it is not hard
to see that:
| ∇N |2= D2(T1,T2) =1
2(log2(k1) + log2(k2))(17)
while the magnitude of the Euclidean gradient is:
| ∇N |2
E=? T1− T2?2
F= λ2
1(1 − k1)2+ λ2
2(1 − k2)2
(18)
It is remarkable that the Riemannian gradient magnitude is independent of the
eigenvalues since it only depends on the constant factors, that is to say, the
magnitude is scale-independent. Fig. 2 presents both gradient magnitudes for
different values of k1 = k2 = K and different values of λ1, λ2. While the
11

Page 12

Fig. 1. 3 × 3 Neighborhood N of a tensor field. White dots represent tensors T1whereas
black dots are tensors T2
012345
K
6789 10
0
2
4
6
8
10
12
Gradient Magnitude
Fig. 2. Gradient magnitudes as a function of parameter K. Red Line: Riemannian Gradient
Magnitude. Blue Lines: Euclidean Gradient Magnitude for different eigenvalues Dashed
Line: λ1 = 0.22,λ2 = 0.22. Solid Line: λ1 = 0.94,λ2 = 0.33. Dashed-Dotted Line:
λ1= 1.05,λ2= 0.80.
Riemannian gradient is not affected by different values of λ1, λ2, the slope of the
Euclidean gradient magnitude is determined by the square root of the sum of the
squared eigenvalues, showing that the greater the eigenvalues, the bigger the slope
represented by the blue lines in Fig. 2 Let us now study the different behaviors
when we have, in our neighborhood N, the same tensors but rotated by an angle α.
12

Page 13

The white tensors are now defined as T1= λ1v1v1T+ λ2v2v2Twhile T2= PTT1P,
with P an orthogonal unitary rotation matrix. Again, using Eqs. 13 and 4 it is not
difficult to derive the following expression for the Riemannian gradient magnitude:
| ∇N |2= D2(T1,T2) =1
2(log2(A+) + log2(A−)) (19)
where A+and A−are, respectively
A±=(2λ1λ2+ sin2(α)(λ1− λ2)2± sin(α)(λ1− λ2)
?
sin2(α)(λ1− λ2)2+ 4λ1λ2)
2λ1λ2
(20)
On the other hand, the Euclidean gradient magnitude is:
| ∇N |2
E=? T1− T2?2
F= 2sin2(α)(λ1− λ2)2
(21)
Again, the dependence of the Riemannian gradient magnitude on eigenvalues is
less important than that of the Euclidean gradient magnitude. Fig. 3 shows different
responses of the gradient magnitudes as a function of the angle α ∈ [0,π]. On the
left,weshowthegradientmagnitudesforanisotropictensors,forwhicheigenvalues
ratio is large. In that case, for λ1= 0.99,λ2= 0.2 it can be seen that the response
of the Riemannian gradient (solid red line) is bigger than the response of the
Euclidean gradient. On the right, we show the gradient magnitudes with more
isotropic tensors, with eigenvalues ratio close to one (λ1 ? λ2). In this case, the
Riemannian gradient magnitude provides a smaller response (solid red line) than
the Euclidean gradient magnitude (solid blue line). However, as the Euclidean
gradient magnitude is proportional to the eigenvalues, a problem may arise with
isotropic tensors if the eigenvalues are too large, as the dashed lines show. In that
case, with λ1 = 19,λ2 = 17.5, the Euclidean gradient magnitude is much bigger
than the Riemannian one which, on the contrary, stays almost identical to its value
for small λi.
13

Page 14

00.51 1.522.53 3.5
0
0.5
1
1.5
Angle
Gradient Magnitude
00.51 1.52 2.533.5
0
0.5
1
1.5
2
2.5
Angle
Gradient Magnitude
Fig. 3. Gradient magnitudes as a function of rotation angle α. Red Lines: Riemannian
Gradient Magnitude. Blue Lines: Euclidean Gradient Magnitude. Left Image: Response
for anisotropic tensors. Right Image: Response for more isotropic tensors.
4DT-MRI Anisotropic Filtering
We now make use of the concepts previously presented to develop our Riemannian
anisotropic smoothing algorithm. It is detailed in section 4.1. We also review, in
section 4.2, another filtering algorithm based on the generalization of PDEs to
tensor fields. These methods will be quantitatively compared in the next section.
4.1 Riemannian anisotropic smoothing
In this section we use the mathematical tools presented in section 3 to develop
a boundary preserving smoothing algorithm. In practice, we simply use the step-
forward operator shown in Eq. 12 to compute local weighted averages. The
anisotropic behavior is introduced by weighting each sample, within a local
neighborhood, by a function that depends on the Riemannian gradient magnitude
developed in section 3.3. This function is chosen so that, in homogeneous regions,
the weights are constant and the tensors are isotropically averaged. On the contrary,
when lying on an edge of the image, we would like that only samples on that
14

Page 15

boundary, and not those across, contribute to the local averaging. To achieve
this goal and avoid mixing structures of the image, one possible choice for the
weighting function is ωk= ?+ | ∇Σ(x) |2. A major advantage of this approach is
that a straightforward C++ implementation yields a quite computationally efficient
algorithm since, to regularize a 50×50×50 volume of 3×3 tensors, using a 3×3×3
averaging neighborhood, we obtain an average processing time of 8 minutes on a
1.7 GHz Pentium M CPU with 1 Gb of RAM. Moreover, it is easy to automatically
detect the convergence of the gradient descent, detailed in Eq. 12, by checking the
evolution speed
Σw
l
??N
k=1ωklog(Σ−1
?N
kˆΣw
l)
?
k=1ωk
and stopping whenever a given norm (Frobenius for instance) of this symmetric
matrix has reached a certain threshold (1e-6 in practice). Hence not only do we
ensure the convergence of the weighted mean but we also discard the need for a
parameter such as the number of iterations.
The theoretical framework developed in sections 2 and 3 implies that the tensor
field consists of symmetric positive definite tensors. Depending on the estimation
procedure used to compute those tensors (see [29] and references therein), positive
semidefinite tensors may arise, yielding inaccurate results, since any semidefinite
tensorisataninfinitedistancefromadefiniteone.Tosolvethisissue,itisnecessary
to resort to a dedicated estimation algorithm, such as the one proposed in [19], that
ensures to stay within S+(3).
4.2Nonlinear diffusion of tensor fields
In this section, we focus on nonlinear diffusion filters, as presented in [33] and
studied, in the context of local structure tensor estimation, in [6]. For scalar images,
15