With the introduction of diffusion tensor magnetic resonance imaging (DT-MRI) 
, there has been an ever increasing demand on rigorous, reliable and robust methods for the processing of tensor-valued data such as the estimation, filtering, regularization and segmentation. Many well established PDE-based methods used for the processing of scalar-valued data have been extended in various ways to the processing of multi-valued data such as vector-valued data and smoothly constrained data[5, 11, 20, 21, 22, 23]. Recently, some efforts have been directed toward the extension of these methods to tensor fields [3, 8, 7, 13, 16, 24, 25]
. The generalization of the methods used for scalar- and vector-valued data to tensor-valued data is being pursued with mainly three formalisms: the use of geometric invariants of tensors like eigenvalues, determinant, trace; the generalization of Di Zenzo’s concept of a structure tensor for vector-valued images to tensor-valued data; and recently, differential-geometric methods.
The aim of the present paper is to generalize the total variation (TV) flow, mean curvature motion (MCM), modified mean curvature flow and self snakes to tensor-valued data such as DT-MRI. The key ingredient for these generalizations is the use of the Riemannian geometry of the space of symmetric positive-definite (SPD) matrices. The remainder of this paper is organized as follows. In Section 2 we give a compilation of results that gives the differential geometry of the Riemannian manifold of symmetric positive-definite matrices. In Section 3 we fix notation and recall some facts about immersions between Riemannian manifolds and their mean curvature. We explain in Section 4 how to describe a DT-MR image by differential-geometric concepts. Section 5 is the key of our paper in which we extend several mean curvature-based flows for the denoising and segmentation from the scalar and vector setting to the tensor one. In Section 6 we present some numerical results.
2 Differential Geometry of
Positive-definite matrices are omnipresent in many engineering and physical contexts. They play important roles in various disciplines such as control theory, continuum mechanics, numerical analysis, covariance analysis, signal processing, etc. Recently, they gained an increasing attention within the diffusion tensor magnetic resonance imaging (DT-MRI) community as they are used as an encoding for the principal diffusion directions ans strengths in biological tissues.
We here recall some differential-geometric facts about the space of symmetric positive-definite matrices that have been recently published by the authors. We denote by the vector space of symmetric matrices. A matrix is said to be positive semidefinite if for all , and positive definite if in addition is invertible. The space of all symmetric, positive-definite matrices will be denoted by . We note that the set of positive-semidefinite matrices is a pointed convex cone in the linear space of matrices, and that is the interior of this cone. It is a differentiable manifold endowed with a Riemannian structure. The tangent space to at any of its points is the space , which for simplicity is identified with . On each tangent space we introduce the base point-dependent inner product defined by .
This inner product leads to a natural Riemannian metric on the manifold that is given at each by the differential
where is the symmetric matrix with elements . We note that the metric (1) is invariant under congruent transformations: and under inversion .
For an matrix we denote by the -column vector that is obtained by stacking the columns of . If is symmetric, then then elements of are redundant. We will denote by the -vector that is obtained from by eliminating the redundant elements, e.g., all supradiagonal elements of . We note that there are several ways to arrange the independent elements of into . In any case, there exists a unique matrix, called the duplication matrix and denoted by , that by duplicating certain elements, reconstructs from , i.e., is the matrix such that
The duplication matrix , which has been studied extensively by Henderson and Searle , and by Magnus and Neudecker , has full column rank . Hence, is non-singular and it follows that the duplication matrix has a Moore-Penrose inverse denoted by and is given by
It follows from (2) that
For differential-geometric operators on it is important to obtain the expression of the inverse of the metric and that of its determinant. The matrix of components of the inverse metric tensor is given by
and the determinant of is
In the coordinate system , the Christoffel symbols are given by 
where is the dual basis associated with the local coordinates . As the elements of and are either 0, 1, or , it follows from the above theorem that each non-vanishing Christoffel symbol is given by an element of or half of it.
Let be an element of and let be a (symmetric) infinitesimal variation of it
Hence, the complete and reduced vector forms of are respectively,
The components of the inverse metric tensor and the Christoffel symbols are given explicitly in the appendix.
3 Immersions and Mean Curvature
Let and be two connected Riemanian manifolds of dimensions and , respectively. We consider a map that is of class , i.e., . Let be a local coordinate system of in a neighborhood of a point and let be a local coordinate system of in a neighborhood of .
The mapping induces a metric on defined by
This metric is called the pull-back metric induced by , as it maps the metric in the opposite direction of the mapping .
An isometry is a diffeomorphism that preserves the Riemannian metric, i.e., if and are the metrics for and , respectively, then . It follows that an isometry preserves the length of curves, i.e., if is a smooth curve on , then the curve is a curve of the same length on . Also, the image of a geodesic under an isometry is again a geodesic.
A mapping is called an immersion if is injective for every point in . We say that is immersed in by or that is an immersed submanifold of . When an immersion is injective, it is called an embedding of into . We then say that is an embedded submanifold, or simply, a submanifold of .
Now let be an immersion of a manifold into a Riemannian manifold with metric . The first fundamental form associated with the immersion is . Its components are where . The total covariant derivative is called the second fundamental form of and is denoted by . The second fundamental form takes values in the normal bundle of . The mean curvature vector of an isometric immersion is defined as the trace of the second fundamental form divided by 
In local coordinates, we have 
where are the Christoffel symbols of and is the Laplace-Beltrami operator on given by
4 Diffusion-Tensor MRI Data as Isometric Immersions
A volumetric tensor-valued image can be described mathematically as an isometric immersion of a three-dimensional domain in the fiber bundle , which is a nine-dimensional manifold. We denote by the image manifold and its metric and by the target manifold and its metric. Here and . Consequently, a tensor-valued image is a section of this fiber bundle. The metric of is given by
where . The corresponding metric tensor is
where is the metric tensor of as defined in Section 2.
Since the image is an isometric immersion, we have . Therefore
We note that is the codimension of . In compact form, we have
where is given by (4). (We take for a slice and for a volumetric DT-MRI image.)
5 Geometric Curvature-Driven Flows for Tensor-Valued Data
The basic concept in which geometric curvature-driven flows are based is the mean curvature of a submanifold embedded in a higher dimensional manifold. Here we generalize the scalar mean curvature flow to mean curvature flow in the space-feature manifold . For this, we embed the Euclidean image space into the Riemannian manifold , and use some classical results from differential geometry to derive the Riemannan Mean Curvature (RMC). We then use the RMC to generalize mean curvature flow to the tensor-valued data. Given the expression of the mean curvature vector , we can establish some PDEs based tensor-image filtering. Especially, we are interested of the so called level-set methods, which relay on PDEs that modify the shape of level sets in an image.
5.1 Riemannian Total Variation Flow
The total variation norm (TV) method introduced in  and its reconstructions have been successfully used in reducing noise and blurs without smearing sharp edges in grey-level, color and other vector-valued images [6, 12, 17, 20, 21, 22]. It is then natural to look for the extension of the TV norm to tensor-valued images.
The TV norm method is obtained as a gradient-decent flow associated with the -norm of the tensor field. This yields the following PDE that express the motion by the mean curvature vector
This flow can be considered as a deformation of the tensor field toward minimal immersion. Indeed, it derives from variational setting that minimize the volume of the embedded image manifold in the space-feature manifold.
5.2 Riemannian Mean Curvature Flow
The following flow was proposed for the processing of scalar-valued images
where is the grey level of the image to be processed, is its smoothed version that depends on the scale parameter .
The “philosophy” of this flow is that the term represents a degenerate diffusion term which diffuses in the direction orthogonal to its gradient and does not diffuse at all in the direction of .
The natural generalization of this flow to tensor-valued data is
We note that several authors have tried to generalize curvature-driven flows for tensor-valued data in different ways. We think that the the use of differential-geometric tools and concepts yield the correct generalization.
5.3 Modified Riemannian Mean Curvature Flow
where is a smoothing kernel (a Gaussian for example), is therefore a local estimate of for noise elimination, and is a nonincreasing real function which tends to zero as . We note that for the numerical experiments we have used .
The generalization of the modified mean curvature flow to tensor-field processing is
The role of is to reduce the magnitude of smoothing near edges. In the scalar case, this equation does not have the same action as the Perona-Malik equation of enhancing edges. Indeed, Perona-Malik equation has variable diffusivity function and has been shown to selectively produce a “negative diffusion” which can increase the contrast of edges. Equation of he form (17) have always positive or forward diffusion, and the term merely reduces the magnitude of that smoothing. To correct this situation, Sapiro have proposed the self-snakes formalism , which we present in the next subsection and generalize to the matrix-valued data setting.
5.4 Riemannan Self-Snakes
The method of Sapiro, which he names self-snakes introduces an edge-stopping function into mean curvature flow
Comparing equation (19) to (17), we observe that the term is missing in the old model. This is due to the fact that the Sapiro model takes into account the image structure. Indeed, equation (19) can be re-written as
The term is as in the anisotropic flow proposed in . The second term in (20), i.e., , increases the attraction of the deforming contour toward the boundary of “objects” acting as the shock-filter introduced in  for deblurring. Therefore, the flow is a shock filter acting like the backward diffusion in the Perona-Malik equation, which is responsible for the edge-enhancing properties of self snakes. See  for detailed discussion on this topic.
We are now interested in generalizing Self-Snakes method for the case of tensor-valued data. We will start the generalization from equation (20) in the following manner
This decomposition is not artificial, since the covariant derivative on follow the same chain rule as the Euclidean directional derivative: leta vector field on which components are , and let a scalar function. From the classic differential geometry we have
and in compact form
6 Numerical Experiments
In Fig. 1 (left), we give a slice of a 3D tensor field defined over a square in . We note that a symmetric positive-definite matrix
is represented graphically by an ellipsoid whose principal directions are parallel to the eigenvectors ofand whose axes are proportional to the eigenvalues of . Figure 1 (right) shows this tensor field after the addition of noise. The resultant tensor field
is used as an initial condition for the partial differential equations (21) which we solve by a finite difference scheme with Neumann boundary conditions. We used 50 time steps of 0.01.s. Figure 2 represents the tensor smoothed by (21).
In this paper we generalized several curvature-driven flows of scalar- and vector-valued data to tensor-valued data. The use of the differential-geometric tools and concepts yields a natural extension of these well-known scalar-valued data processing methods to tensor-valued data processing.
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We give here the explicit form of the inverse metric tensor and Christoffel symbols for the Riemannian metric on . The components of the inverse metric tensor are given by
The determinant of is . Let , where is the adjoint matrix of .
The Christoffel symbols are arranged in the following six symmetric matrices: