Left-invariant Diffusions on R3 S2 and their Application to Crossing ...

Left-invariant Diffusions on R3 o S 2 and their Application to Crossing-preserving Smoothing on HARDI-images. Remco Duits1,2 and Erik Franken2 Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands. Department of Mathematics and Computer Science1 , CASA Applied analysis, Department of Biomedical Engineering2 , BMIA Biomedical image analysis. e-mail: [email protected], [email protected]

November 10, 2009

Abstract HARDI (High Angular Resolution Diffusion Imaging ) is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. In this article we study left-invariant diffusion on the group of 3D-rigid body movements (i.e. 3D Euclidean motion group) SE(3) and its application to crossing-preserving smoothing of HARDI images. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space of positions and orientations in 3D embedded in SE(3) and can be solved by R3 o S 2 -convolution with the corresponding Green’s functions. We provide analytic approximation formulas and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (non-linear) convection-diffusions on HARDI data we explain the underlying differential geometry on SE(3). We write our left-invariant diffusions in covariant derivatives on SE(3) using the Cartan connection. This Cartan connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI data containing crossing-fibers. Keywords: High Angular Resolution Diffusion Imaging (HARDI), Scale spaces, Lie groups, Partial differential equations.

1

Introduction

High Angular Resolution Diffusion Imaging (HARDI) is a recent magnetic resonance imaging technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. HARDI provides for each position in 3-space (i.e. R3 ) and for each orientation (antipodal pairs on the 2-sphere S 2 ) an MRI signal attenuation profile, which can be related to the local diffusivity of water molecules in the corresponding direction. It is generally believed that such profiles provide rich information in fibrous tissues. DTI (Diffusion Tensor Imaging) is a related technique, producing a positive symmetric rank-2 tensor field. A DTI tensor (at each position in 3-space) can also be related to a distribution on the 2-sphere, albeit with limited angular resolution. DTI is incapable of representing areas with complex multimodal diffusivity profiles, such as induced by crossing, “kissing”, or bifurcating fibres. HARDI, on the other hand, does not suffer from this problem, because it is not restricted to functions on the 2-sphere induced by a quadratic form. See Figure 1 where we used glyph visualizations as defined below. 1

Definition 1 A glyph of a distribution U : R3 × S 2 → R+ on positions and orientations is a surface Sµ (U )(x) = {x + µ U (x, n) n | n ∈ S 2 } ⊂ R3 for some x ∈ R3 and µ > 0. A glyph visualization of the distribution U : R3 × S 2 → R+ is a visualization of a field x 7→ Sµ (U )(x) of glyphs, where µ > 0 is a suitable constant. For the purpose of tractography (detection of biological fibers) and visualization, DTI and HARDI data should be enhanced such that fiber junctions are maintained, while reducing high frequency noise in the joined domain of positions and orientations.

fibertracking

fibertracking

DTI

HARDI

Figure 1: This figure shows glyph visualizations of HARDI and DTI-images of a 2D-slice in the brain where neural fibers in the corona radiata cross with neural fibers in the corpus callosum. In HARDI and DTI socalled “glyphs” (i.e. angular diffusivity profiles) reflect, per position, the local diffusivity of water in all directions. More, precisely, to a DTI-tensor field x 7→ D(x) we associate a distribution on positions and orientations (x, n) 7→ nT D(x)n of which a glyph visualization (according to Definition 1) is depicted on the left. The rank-2 limitation of a DTI-tensor constrains the corresponding glyph to be ellipsoidal, whereas no such constraint applies to HARDI.

Promising research has been done on constructing diffusion (or similar regularization) processes on the 2-sphere defined at each spatial locus separately [14, 24, 25, 46] as an essential pre-processing step for robust fiber tracking. In these approaches position and orientation space are decoupled, and diffusion is only performed over the angular part, disregarding spatial context. Consequently, these methods are inadequate for spatial denoising and enhancement, and tend to fail precisely at the interesting locations where fibres cross or bifurcate. 2

Original

+Noise Original

CED-OS t+Noise = 10

CEDCED-OS t = 10 t = 10

CED t = 10

Therefore in this article we extend recent work on enhancement of elongated structures in 2D greyscale images [2, 27, 26, 23, 22, 18, 17, 21, 20] to the genuinely 3D case of HARDI/DTI, since this approach has proven to be capable of handling all aforementioned problems in various feasibility studies. See Figure 2. original 2D-image

CED: standard approach

CED-OS: our approach

Fig. 5. Shows the typical different behavior of5. CED-OS compared toFig. CED. In CED-OS Fig. Shows the typical different behavior CED-OS compared to CED. In CED-OS 5. Shows theoftypical different behavior of CED-OS compared to CED. In CED-OS crossing structures are better preserved. crossing structures are better preserved. crossing structures are better preserved. Original

Original

CED-OS t = 2

CED-OS Originalt = 30

+Noise Original

CED-OS t = 10 Original +Noise

CEDOriginal CED-OS t = 230

CED-OS t = 30t = 2 CED-OS

CED tt = +Noise CED-OS = 10 10

CED-OS t = 10 CED t = 10

CED t = 30 t = 30 CED-OS

CED t = 30

CED t = 10

Fig. 6. Shows results on antwo image constructed two ofrotated 2-photon images of Fig. 6. Shows results on an image constructed rotated 2-photonfrom images Fig. 6. Shows results on an image constructed from two rotated 2-photon images of from collagen tissue the heart. t = 2 we obtain nice enhancement of the image. collagen in enhancement the heart. Atoft = 2 image. weinobtain a niceAtenhancement of thea image. collagen tissue in the heart. At t = 2 we obtaintissue a nice the Comparing with t = 30 a nonlinear behavior can be seen. For comparison, Comparing with can t = 30 nonlinear scale-space behavior can bescale-space seen. For comparison, Comparing with t = 30 a nonlinear scale-space behavior be aseen. For comparison, right column shows the behavior of CED. the right column shows thethe behavior of CED. the right column shows the behavior of CED.

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9

Conclusions

Conclusions

9

Conclusions

2 Figure 2: Left-invariant diffusion viaon diffusion on In SE(2) =R o introduced S 1 is on thethe right approach to generically deal with this paper we nonlinear diffusion on the Euclidean motion group. Indiffusion this paper we nonlinear diffusion Euclidean motion group. In this paper we introduced nonlinear the introduced Euclidean motion group. Starting from a 2Da image, constructed a three-dimensional orientation score Starting fromLeft a 2Dcolumn: image, we constructed three-dimensional orientation score coherence crossings and bifurcations in practice. original images. Middlewecolumn: result of standard Starting from a 2D image, we constructed a three-dimensional orientation score using rotated versions afilter. directional quadrature considered the orirotated versions a directional quadrature We considered thefilter. ori- We using rotated versionsdiffusion of a directional quadrature filter. Weofconsidered ori- cf. of[48]. enhancing appliedusing directly in the image domain R2the (CED), Right column: coherence enhancing entation score as a in function ongroup the Euclidean motion group and entation score as a function on the Euclidean motion andmotion defined the SE(2), entation score as a via function on the Euclidean motion group and defined the diffusion the corresponding invertible orientation score (CED-OS) the 2D-Euclidean group cf. defined the left-invariant diffusion equation. We showed how one can use normal left-invariant diffusion equation. We showed how one can use normal Gaussian left-invariant diffusion equation. We showed how one can use normal Gaussian [21, 27]. Top row: 2-photon microscopy image of bone tissue. Second row : collagen fibers of the heart. Third row: Gaussian derivatives to noncalculateinregularized derivatives in the orientation score. The nonderivatives to regularized derivatives the orientation score. The nonderivativesartificial to calculate derivatives thecalculate orientation score. The noisyregularized interference pattern. in Typically, these 2D-applications clearly show that coherence enhancing diffusion linear diffusion isoriented steered by estimates for oriented feature strength and curvature linear diffusion is steered by estimates for feature strength and curvature linear diffusion is steered by estimates for oriented feature strength and curvature Figure 15: Shows the typical different behavior of CED-OS compared to CED. In CED-OS crossing Figure 15: Showsto typical different behavior of CED-OS compared to CED. whereas In CED-OS(CED) crossingproduces Figure 15: Shows on the invertible typical different behavior of CED-OS compared In CED-OS crossing orientation scores (CED-OS) istheCED. capable of handling crossings and bifurcations, that are obtained Gaussian derivatives. Furthermore, we proposed to use that are Furthermore, obtained Gaussian derivatives. Furthermore, we proposed to use that are arebetter obtained from Gaussian derivatives. we proposed to use from structures are better preserved. structures are betterfrom preserved. structures preserved. 3 2 spurious artifacts at such junctions. Now in the genuinely 3D-case of HARDI images U : R o S → wederivative do not operators. finite differences that approximate left-invariance ofR, the differencesofthat the left-invariance of the the derivative operators. finite differences that approximate thefinite left-invariance theapproximate derivative operators. Original +Noise CED-OS t==30 30 CED t = 30 and about invertibility of the transform between a grey-value image its orientation score as the input- patterns Original +Noise CED-OS t = 30 CED t = 30 Original have to bother +Noise CED-OS t = 30 CED t Theweexperiments showtothat we areelongated indeed able to enhance elongated The able experiments show that are indeed able enhance patterns The experiments show that we are indeed to enhance elongated patterns data itself already gives rise to a function on the 3D Euclidean motion group SE(3). This is now simply achieved by large curin images and that including curvature helps to enhance lines with in images and that including curvature helps to enhance lines with large curin images and that including curvature helps to enhance lines with large cur-

˜ (x, R) = U (x, Rez ), R ∈ SO(3), x ∈ R3 , ez = (0, 0, 1)T and the challenge rises to generalize our previous setting U work on crossing preserving diffusion to 3D and to apply the left-invariant diffusion directly on the HARDI images.

In contrast to the previous works on diffusion of DTI/HARDI images [14, 24, 25, 46, 40], we consider both the spatial and the orientational part to be included in the domain, so a HARDI dataset is considered as a function U : R3 × S 2 → Figureand 16:CED Result CED-OS and CED on microscopy images bone Additional Gaussian Figure 16: ofResult of CED-OS and of CED ontissue. microscopy images of bone tissue. Additional Gaussian Figure 16: ResultR. of CED-OS on of microscopy images bone tissue. Additional Gaussian Furthermore, explicitly employ proper underlying group structure, that naturally arises by embedding the noise is addedwe to noisy verify the noise behaviour onthe noisy images. is added to verify the behaviour on noisy images. noise is added to verify the behaviour on images. coupled space of positions and orientations into the group SE(3) of 3D-rigid motions. The relevance of group theory in DTI/HARDI imaging has also ofbeen stressed in promising and well-founded recent line works [30, 31, 32]. However these Figure 15 shows the effect CED-OS compared to CED on artificial images with crossing Figure shows the effect of CED-OS compared to CED on artificial images with crossing line Figure 15 shows the effect of CED-OS compared to CED on15artificial images with crossing line structures. The upper image shows anThe additive of additive two images with concentric worksimage rely shows on bi-invariant Riemannian metrics on compact groups (such as SO(3)) our case group SE(3) structures. image showsconcentric an superimposition ofand twoinimages with the concentric structures. The upper an additive superimposition of upper two superimposition images with circles. Our methoddoes is ableitcircles. to preserve this structure, while CED can not. The same holds forcan not.ofThe Our method is able to same preserve this structure, while CED holds for is neither compact permit aCED bi-invariant metric [4, 21]. general the advantage oursame approach on SE(3) circles. Our method is able to preserve nor this structure, while can not. The holds forIn the lower image with crossing straight lines,with where it should be noted that our method leads to that our method leads to the lower image crossing straight lines, be noted the lower image with crossing straight lines, where it should be noted that our method leads towhere it should is that we can enhance the original HARDI/DTI data using simultaneously orientational and spatial neighborhood amplification of the crossings, which is because the lines in the original image are not superimposed amplification of theimage crossings, which is because the lines in the original image are not superimposed amplification of the crossings, which is because the lines in the original are not superimposed linearly. which In this potentially experiment, no deviation from horizontality was taken into account,algorithms. and thetaken Figure information, leads towas improved enhancement and detection 3 shows linearly. In this experiment, no deviation from horizontality was into account, and an theexample linearly. In this experiment, no deviation from horizontality taken into account, and the numerical scheme of of Section 7.2 is used. Theofnon-linear diffusion parameters for CED-OS are:parameters for CED-OS are: numerical scheme Section 7.2 is used. The non-linear diffusion clarifying the structure a HARDI image. numerical scheme of Section 7.2 is used. The non-linear diffusion parameters for CED-OS are:

n = 32, ts = 12, ρs = 0, β = = 0.08. parameters that=we used forparameters CED are (see nθ0.058, = 32, and ts =c12, ρs we = The 0,used β= 0.058, 0.08. The that we used for CED are (see nθ = 32, ts = 12, ρs = 0, βθ = 0.058, and c = 0.08. The parameters that for CEDand arec (see [40]): σ = 1, ρ = 1, C = 1, and ασ == 0.001. The C images and haveαa=size of 56The × 56images pixels.have a size of 56 × 56 pixels. [40]): ρ= 0.001. [40]): σ = 1, ρ = 1, C = 1, and α = 0.001. The images have a 1, size of 1, 56 ×=561,pixels. Figure 1 at the beginning ofFigure the paper shows the results on an image of collagen fibres obtained This paper is organized as follows. In Section 2 we will start with the introduction the group 1 atimage the beginning of fibres the paper shows the results on an image of of collagen fibresstructure obtained on the Figure 1 at the beginning of the paper shows the results on an of collagen obtained using 2-photon microscopy. These kind ofwill images are acquired in tissue engineering research, where domain of a HARDI image. Here we explain that the domain of a HARDI image of positions and orientations using 2-photon microscopy. These kind of images are acquired in tissue engineering research, where using 2-photon microscopy. These kind of images are acquired in tissue engineering research, where the goal is to create artificial valves. All parameters during theseAll experiments were set the theheart goalduring is to create artificial heart valves. the goal is to create artificial heart valves. All parameters these experiments were set the parameters during these experiments were set the same as the artificial imagessame mentioned above except for CED parameter ρ = 6. The image size the parameter artificial images same as the artificial images mentioned above except for as CED ρ = 6.mentioned The imageabove size except for CED parameter ρ = 6. The image size is 160 × 160 pixels. is 160 × 160 pixels. is 160 × 160 pixels. 3 of the method Figures 16 and 17 show examples of16 the method on examples other microscopy data. The same microscopy paramFigures and 17 data. show on other data. The same paramFigures 16 and 17 show examples method onfor other microscopy The same parameters are usedofasthe above except ts = 25 as in Figure 17. Clearly, the curve enhancement and noise eters are used above except for t = 25 in Figure 17. Clearly, the curve enhancement and noise s eters are used as above except for t = 25 in Figure 17. Clearly, the curve enhancement and noise suppressions of the crossing suppression curves is good in our method, while standard coherence enhancing of standard the crossing curves is good in our method, while standard coherence enhancing suppression of the crossing curvestends is good in our method, coherence enhancing diffusion to destruct crossings while and create artificial oriented structures. diffusion tendsstructures. to destruct crossings and create artificial oriented structures. diffusion tends to destruct crossings and create artificial oriented Figure 18 demonstrates the advantage of including curvature. Again, the same parameters and Figure 18 demonstrates theparameters advantage of including curvature. Again, the same parameters and Figure 18 demonstrates the advantage of including curvature. Again, the same and

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Figure 3: Visualization of a simple HARDI image (x, y, z, n(β, γ)) 7→ U (x, y, z, n(β, γ)) containing two crossing straight lines, visualized using Q-ball glyphs in the DTI tool (see http://www.bmia.bmt.tue.nl/software/dtitool/) from two different viewpoints. At each spatial position x a glyph (cf. Fig.1 and Definition 1) is displayed. carries a semi-direct product structure rather than a direct Cartesian product structure reflecting a natural coupling between position and orientation. We embed the space of positions and orientations into the group of positions and rotations in R3 , which is commonly denoted by SE(3) = R3 o SO(3). As a result we must write R3 o S 2 := R3 o SO(3)/({0} × SO(2)) rather than R3 × S 2 for the domain of a HARDI image. In Section 3 we will discuss basic tools from group theory, which serve as the key ingredients in our diffusions on HARDI images later on. Within this section we also provide an example to embed a recent paper [8] by Barmpoutis et al. on smoothing of DTI/HARDI data in our group theoretical framework. We show that their kernel operator indeed is a correct left-invariant group convolution on R3 o S 2 . However their practically intuitive kernel does not satisfy the semigroup property and does not relate to diffusion or Tikhononov energy minimization on R3 o S 2 . Subsequently, in Section 4 we will derive all linear left-invariant convection-diffusion equations on SE(3) and R3 oS 2 (the actual domain of HARDI images) and show that the solutions of these convection-diffusion equations are given by group-convolution with the corresponding Green’s functions, which we explicitly approximate later. Furthermore, in Subsection 4.2, we put an explicit connection with probability theory and random walks in the space of orientations and positions. This connection is established by the fact that the convection-diffusion equations are Fokker-Planck (i.e. forward Kolmogorov) equations of stochastic processes (random walks) on the space of orientations and positions. This in turn brings a connection to the actual measurements of water-molecules in oriented fibrous tissues. Symmetry requirements for the linear diffusions on R3 o S 2 yields the following four cases: 1. the natural 3D-generalizations of Mumford’s direction process on R2 o S 1 [38, 23], which is a contour completion process in the group SE(2) = R2 o S 1 ≡ R2 o SO(2) of 2D-positions and orientations. 2. the natural 3D-generalizations of a (horizontal) random walk on R2 o S 1 , cf. [20], corresponding to the diffusions proposed by Citti and Sarti [13], which is a contour enhancement process in the group SE(2) = R2 oS 1 ≡ R2 o SO(2) of 2D-positions and orientations, 3. Gaussian scale space [35, 36, 3, 16] over position space, i.e. spatial linear diffusion, 4. Gaussian scale space over angular space (2-sphere), [14, 40, 24, 25, 46], i.e. angular linear diffusion, or combinations of these four types of convection-diffusions. Previous approaches of HARDI-diffusions [14, 40, 24] fit in our framework (third and fourth item), but it is rather the first two cases that are challenging as they involve a natural coupling between position and orientation space and thereby allow appropriate treatment of crossing fibers.

4

In Section 5 we will explore the underlying differential geometry of our diffusions on HARDI-orientation scores. By means of the Cartan-connection on SE(3) we put a useful relation to rigid body mechanics expressed in moving frames of reference, providing geometrical intuition behind our left-invariant (convection-)diffusions on HARDI data. Furthermore, we show that our (convection-)diffusion may be expressed in covariant derivatives and we show that both convection and diffusion locally take place along the exponential curves in SE(3), that are explicitly derived in subsection 5.1. In Section 6 we will derive suitable formulae and Gaussian estimates for the Green’s functions of our linear left-invariant convection-diffusions on HARDI images. These formulas are used in the subsequent section in our numerical convolution-schemes solving the left-invariant diffusions on HARDI images. Section 7 explains the basic numerics of our left-invariant PDE- and/or convolution schemes, which we use in the subsequent experimental section, Section 8. Section 8 contains preliminary results of linear left-invariant diffusion on artificial HARDI datasets over the joined coupled domain of positions and orientations (i.e. over R3 o S 2 ). The final section of this paper provides the theory for non-linear adaptive diffusion on HARDI images, which is a generalization of our non-linear adaptive diffusion schemes on the 2D-Euclidean motion group [27, 21].

2

The Group Structure on the Domain of a HARDI Image: The Embedding of R3 × S 2 into SE(3)

In order to generalize our previous work on line/contour-enhancement via left-invariant diffusions on invertible orientation scores of 2D-images we first investigate the group structure on the domain of an HARDI image. Just like orientation scores are scalar-valued functions on the coupled space of 2D-positions and orientations, i.e. the 2D-Euclidean motion group, HARDI images are scalar-valued functions on the coupled space of 3D-positions and orientations. This generalization involves some technicalities since the 2-sphere S 2 = {x ∈ R3 | kxk = 1} is not a Lie-group proper1 in contrast to the 1-sphere S 1 = {x ∈ R2 | kxk = 1}. To overcome this problem we will embed R3 × S 2 into SE(3) which is the group of 3D-rotations and translations (i.e. the group of 3D-rigid motions). As a concatenation of two rigid body-movements is again a rigid body movement, the product on SE(3) is given by (x, R) (x0 , R) = (Rx0 + x, RR0 ),

R, R0 ∈ SO(3), x, x0 ∈ R3 .

The group SE(3) is a semi-direct product of the translation group R3 and the rotation group SO(3), since it uses an isomorphism R 7→ (x 7→ Rx) from the rotation group onto the automorphisms on R3 . Therefore we write R3 oSO(3) rather than R3 × SO(3) which would yield a direct product. The groups SE(3) and SO(3) are not commutative. Throughout this article we will use Euler-angle parametrization for SO(3), i.e. we write every rotation as a product of a rotation around the z-axis, a rotation around the y-axis and a rotation around the z-axis again. R = Rez ,γ Rey ,β Rez ,α ,

(1)

where all rotations are counter-clockwise, for explicit formulas for matrices Rez ,γ , Rey ,β , Rez ,α , see for example [26, ch:7.3.1]. The advantage of the Euler angle parametrization is that it directly parameterizes SO(3)/SO(2) ≡ S 2 as well. Here we recall that SO(3)/SO(2) denotes the partition of all left cosets which are equivalence classes [g] = {h ∈ SO(3) | h ∼ g} = g SO(2) under the equivalence relation g1 ∼ g2 ⇔ g1−1 g2 ∈ SO(2) where we identified SO(2) with rotations around the z-axis and we have SO(3)/SO(2) 3 [Rez ,γ Rey ,β ] = {Rez ,γ Rey ,β Rez ,α | α ∈ [0, 2π)} ↔ n(β, γ) := (cos γ sin β, sin γ sin β, cos β) = Rez ,γ Rey ,β Rez ,α ez ∈ S 2 .

(2)

1 If S 2 were a Lie-group then its left-invariant vector fields would be non-zero everywhere, contradicting Poincar´ e’s “hairy ball theorem” (proven by Brouwer in 1912), or more generally the Poincaré-Hopf theorem (the Euler-characteristic of an even dimensional sphere S 2n is 2).

5

Like all parameterizations of SO(3)/SO(2), the Euler angle parametrization suffers from the problem that there does not exists a global diffeomorphism from a sphere to a plane. In the Euler-angle parametrization the ambiguity arises at the north and south-pole: Rez ,γ Rey ,β=0 Rez ,α = Rez ,γ−δ Rey ,β=0 Rez ,α+δ , and Rez ,γ Rey ,β=π Rez ,α = Rez ,γ+δ Rey ,β=0 Rez ,α+δ , for all δ ∈ [0, 2π) . (3)

Consequently, we occasionally need a second chart to cover SO(3); R = Rex ,˜γ Rey ,β˜ Rez ,α˜ ,

(4)

which again implicitly parameterizes SO(3)/SO(2) ≡ S 2 using different ball-coordinates β˜ ∈ [−π, π), γ˜ ∈ (− π2 , π2 ), ˜ γ˜ ) = Re ,˜γ R ˜ ez = (sin β, ˜ − cos β˜ sin γ˜ , cos β˜ cos γ˜ )T , ˜(β, n x ey ,β

(5)

but which has ambiguities at the intersection of the equator with the x-axis , for all δ ∈ [0, 2π) . Rex ,˜γ Rey ,β=± π Rez ,α±δ π R ez , α ˜ ˜ ˜ = Rex ,˜ γ ∓δ Rey ,β=± ˜ 2

2

(6)

See Figure 4. Away from the intersection of the z and x-axis with the sphere one can accomplish conversion between ˜ γ˜ ) or (α, β, γ) in Re ,˜γ R ˜ Re ,α˜ = Re ,γ Re ,β Re ,α . the two charts by solving for for either (˜ α, β, x z z y z ey ,β Now that we have explained the isomorphism n = Rez ∈ S 2 ↔ SO(3)/SO(2) 3 [R] explicitly in charts, we return to the domain of HARDI images. Considered as a set this domain equals the space of 3D-positions and orientations R3 × S 2 . However, in order to stress the fundamental embedding of the HARDI-domain in SE(3) and the thereby induced (quotient) group-structure we write R3 o S 2 , which is given by the following Lie-group quotient: R3 o S 2 := (R3 o SO(3))/({0} × SO(2)). Here the equivalence relation on the group of rigid-motions SE(3) = R3 o SO(3) equals (x, R) ∼ (x0 , R0 ) ⇔ x = x0 and R−1 R0 is a rotation around z-axis and set of equivalence classes within SE(3) under this equivalence relation (i.e. left cosets) equals the space of coupled orientations and positions and is denoted by R3 o S 2 .

3

Tools From Group Theory

In this article we will consider convection-diffusion operators on the space of HARDI images. We shall model the space of HARDI images by the space of quadratic integrable functions on the coupled space of positions and orientations, i.e. L2 (R3 o S 2 ). We will first show that such operators should be left-invariant with respect to the left-action of SE(3) onto the space of HARDI images. This left-action of the group SE(3) onto R3 o S 2 is given by g · (y, n) = (Ry + x, Rn),

g = (x, R) ∈ SE(3), x, y ∈ R3 , n ∈ S 2 , R ∈ SO(3)

and it induces the so-called left-regular action of the same group on the space of HARDI images similar to the leftregular action on 3D-images (for example orientation-marginals of HARDI images): Definition 2 The left-regular actions of SE(3) onto L2 (R3 o S 2 ) respectively L2 (R3 ) are given by (Lg=(x,R) U )(y, n) = U (g −1 · (y, n)) = U (R−1 (y − x), R−1 n), x, y ∈ R3 , n ∈ S 2 , U ∈ L2 (R3 o S 2 ), (Ug=(x,R) f )(y) = f (R−1 (y − x)) , R ∈ SO(3), x, y ∈ R3 , f ∈ L2 (R3 ). Intuitively, Ug=(x,R) represents a rigid motion operator on images, whereas Lg=(x,R) represents a rigid motion on HARDI images. 6

z

z Į

α ˜

β˜

ȕ Ȗ

γ˜

y

x

⊗

y

x⊗ x

Figure 4: The two charts which together appropriately parameterize the sphere S 2 ≡ SO(3)/SO(2) where the rotation-parameters α and α ˜ are free. The first chart (left-image) is the common Euler-angle parametrization (1), the second chart is given by (4). The first chart has singularities at north and south-pole (inducing ill-defined parametrization of the left-invariant vector fields (26) at the unity element) whereas the second chart has singularities at (±1, 0, 0). In order to explain the importance of left-invariance of processing HARDI images in general we need to define the following operator. Definition 3 We define the operator M which maps a HARDI image U : R3 o S 2 → R+ to its orientation marginal MU : R3 → R+ as follows (where σ denotes the usual surface measure on S 2 ): Z (MU )(y) = U (y, n)dσ(n). S2

If U : R3 o S 2 → R+ is a probability density on positions and orientations then MU : R3 → R+ denotes the corresponding probability density on position space only. The marginal gives us an ordinary 3D image that is a “simplified” version of the HARDI image, containing less information on the orientational structure. This is analogue to taking the trace of a DTI-image. The following theorem tells us that we get a Euclidean invariant operator on the marginal of HARDI images if the operator on the HARDI image is left-invariant. This motivates our restriction to left-invariant operators, akin to our framework of invertible orientation scores [2, 27, 26, 23, 22, 18, 17, 21, 20]. Theorem 1 Suppose Φ is an operator on the space of HARDI images to itself. Then the corresponding operator Y on the orientation marginals given by Y(M(U )) = M(Φ(U )) is Euclidean invariant if operator Φ is left-invariant, i.e. (Φ ◦ Lg = Lg ◦ Φ, for all g ∈ SE(3)) ⇒ Ug ◦ Y = Y ◦ Ug , for all g ∈ SE(3) . Proof The result follows directly by the intertwining relation Ug ◦ M = M ◦ Lg for all g ∈ SE(3). Now regardless of the fact if Φ is bounded or unbounded, linear or non-linear, we have under assumption of left-invariance of Φ that Y ◦ Ug ◦ M = Y ◦ M ◦ Lg = M ◦ Φ ◦ Lg = M ◦ Lg ◦ Φ = Ug ◦ M ◦ Φ = Ug ◦ Y ◦ M . All useful linear operators in image processing can be written as kernel operators. Therefore, we classify all leftinvariant kernel operators K on HARDI images in the next subsection and we will provide important probabilistic interpretation of these left-invariant kernel operators. 7

3 2 Lemma 1 Let K be a bounded operator from L2 (R3 o S 2 ) into there exists an integrable kernel R L∞ (R o S 0) then 3 2 3 2 2 0 2 0 k : R o S × R o S → C such that kKk = sup |k(y, n ; y , n )| dy dσ(n0 ) and we have 3 2 R oS (y,n)∈R3 oS 2

Z

k(y, n ; y0 , n0 )U (y0 , n0 )dy0 dσ(n0 ) ,

(KU )(y, n) =

(7)

R3 oS 2

for almost every (y, n) ∈ R3 o S 2 and all U ∈ L2 (R3 o S 2 ). Now Kk := K is left-invariant iff k is left-invariant, i.e. Lg ◦ Kk = Kk ◦ Lg ⇔ ∀g∈SE(3) ∀y,y0 ∈R3 ∀n,n0 ∈S 2 : k(g · (y, n) ; g · (y0 , n0 )) = k(y, n ; y0 , n0 ).

(8)

Proof The first part of the Theorem follows by the general Dunford-Pettis Theorem [11, p.113-114]. With respect to the left-invariance we note that on the one hand we have R R k(y, n ; y00 , n00 ) U (R−1 (y00 − x, R−1 n00 ) dy00 dσ(n00 ) (Kk Lg U )(y, n) = SR2 RR3 k(y, n ; Ry0 + x, Rn0 ) U (y0 , n0 ) dy0 dσ(n0 ) = SR2 RR3 = k(y, n ; g · (y0 , n0 )) U (y0 , n0 ) dy0 dσ(n0 ) S 2 R3

whereas on the other hand (Lg Kk U )(y, n) =

k(g −1 (y, n) ; y0 , n0 ) U (y0 , n0 ) dy0 dσ(n0 ) , for all g ∈ SE(3),

R R S 2 R3

U ∈ L2 (R3 o S 2 ), (x, n) ∈ R3 o S 2 . Now SE(3) acts transitively on R3 o S 2 from which the result follows.

From the invariance property (8) we deduce that k(y, n ; y0 , n0 ) = k((Rez ,γ 0 Rey ,β 0 )T (y − y0 ), (Rez ,γ 0 Rey ,β 0 )T n(β, γ), 0, ez ) , k(Rez ,α y, Rez ,α n ; 0, ez ) = k(y, n, 0, ez ), and consequently we obtain the following result : Corollary 1 If we use the well-known Euler-angle parametrization of SO(3), we have SO(3)/SO(2) ≡ S 2 with T the isomorphism [Rez ,γ Rey ,β ] = {Rez ,γ Rey ,β Rez ,α | α ∈ [0, 2π)} ↔ n(β, γ) = (sin β cos R γ,Rcos β sin γ, cos β) = 3 2 3 2 + Rez ,γ Rey ,β ez . Then to each positive left-invariant kernel k : R oS ×R oS → R with S 2 R3 k(0, ez ; y, n)dydσ(n) = 1 we can associate a unique probability density p : R3 o S 2 → R+ with the invariance property p(y, n) = p(Rez ,α y, Rez ,α n),

for all α ∈ [0, 2π),

(9)

such that k(y, n(β, γ) ; y0 , n(β 0 , γ 0 )) = p((Rez ,γ 0 Rey ,β 0 )T (y − y0 ), (Rez ,γ 0 Rey ,β 0 )T n(β, γ)) with p(y, n) = k(y, n ; 0, ez ). We can briefly rewrite [26, eq. 7.59] and (7), coordinate-independently, as Z Z Kk U (y, n) = (p ∗R3 oS 2 U )(y, n) = p(RnT0 (y − y0 ), RnT0 n) U (y0 , n0 )dσ(n0 )dy0 ,

(10)

R2 S 2

where σ denotes the surface measure on the sphere and where Rn0 is any rotation such that n0 = Rn0 ez . By the invariance property (9), the convolution (10) on R3 o S 2 may be written as a (full) SE(3)-convolution. An ˜ : SE(3) → R is given by: SE(3) convolution [12] of two functions p˜ : SE(3) → R, U Z ˜ ˜ (h)dµSE(3) (h) , p˜ ∗SE(3) U (g) = p˜(h−1 g)U (11) SE(3)

8

where Haar-measure dµSE(3) (x, R) = dx dµSO(3) (R) with dµSO(3) (Rez ,γ Rey ,β Rez ,α ) = sin βdαdβdγ. It is easily ˜ (x, R) := U (x, Rez ) the following identity holds: verified that (9) implies that if we set p˜(x, R) := p(x, Rez ) and U ˜ )(x, R) = 2π (p ∗R3 oS 2 U )(x, Rez ) . (˜ p ∗SE(3) U Later on in this article (in Subsection 4.2 and Subsection 4.3) we will relate scale spaces on HARDI data and first order Tikhonov regularization on HARDI data to Markov processes. But in order to provide a road map of how the groupconvolutions will appear in the more technical remainder of this article we provide some preliminary explanations on probabilistic interpretation of R3 o S 2 -convolutions. In particular we will restrict ourselves to conditional probabilities where p(y, n) = pt (y, n) represents the probability density of finding an oriented random walker at position y with orientation n at time t > 0, given that it started at (0, ez ) at time t = 0. In such a case the probabilistic interpretation of the kernel operator is as follows. The function (y, n) 7→ (Kkt U )(y, n) = (pt ∗R3 oS 2 U )(y, n) represents the probability density of finding some oriented particle, starting from the initial distribution U : R3 o S 2 → R+ at time t = 0, at location y ∈ R3 with orientation n ∈ S 2 at time t > 0. Furthermore, in a Markov process traveling time is memoryless, so in such process traveling time is negatively exponentially distributed P (T = t) = λe−λt with expectation E(T ) = λ−1 . Consequently, the probability density pλ of finding an oriented random walker starting from (0, ez ) at time t = 0, regardless its traveling time equals pλ (y, n) =

Z∞

Z∞ pt (y, n) P (T = t)dt = λ

0

pt (y, n)e−λt dt .

(12)

0

Summarizing, we can always apply Laplace-transform with respect to time to map transition densities pt (g) given a traveling time t > 0 to unconditional probability densities pλ (g). The same holds for the probability density P λ (y, n) of finding an oriented random walker at location y ∈ R3 with orientation n ∈ S 2 starting from initial distribution U (i.e. the HARDI data) regardless the traveling time, since  ∞   Z∞ Z PUλ (y, n) = λ e−λt (pt ∗R3 oS 2 U )(y, n)dt = λ e−λt pt  ∗R3 oS 2 U  (y, n) = (pλ ∗R3 oS 2 U )(y, n). (13) 0

3.1

0

Relation of the Method Proposed by Barmpoutis et al. to R3 o S 2 -convolution

In [8] the authors propose2 the following practical decomposition for the kernel k : 1 1 t κ κ 0 kdist (ky − y0 k)korient (n · n0 )kfiber n · (−(y − y )) , k t,κ (y, n ; y0 , n0 ) = 4π ky − y0 k t with kdist (ky − y0 k) =

1 3 (4πt) 2

e−

ky−y0 k2 4t

κ κ (cos φ) = and korient (cos φ) = kfiber

eκ cos(φ) 2πJ0 (iκ)

(14)

with φ ∈ (−π, π] angle between

respectively n and n0 and between n and y − y0 , which denotes the von Mises distribution on the circle, which is R π eκ cos(φ) indeed positive and −π 2πJ dφ = 1. The decomposition (14) automatically implies that the corresponding kernel 0 (iκ) κ κ since operator Kk is left-invariant, regardless the choice of kdist , korient , kfiber s κ κ −1 0 1 kdist (kR−1 (y − x) − R−1 (y0 − x)k)korient (R−1 n · R−1 n0 )kfiber ( kR−1 (y−x)−R n ) · R−1 (y − x − (y0 − x))) −1 (y0 −x)k (R t κ κ = kdist (ky − y0 k)korient (n · n0 )kfiber

1 n ky−y0 k

kt,κ (g −1 (y, n) ; g −1 (y0 , n0 )) = kt,κ (y, n ; y0 , n0 ), 2 We

· (y − y0 )

⇔ for all g = (R, x) ∈ SE(3).

used slightly different conventions as in the original paper to ensure L1 -normalizations in (14).

9

The corresponding probability kernel (which does satisfy (9)) reads p(t,κ) (y, n) =

1 t κ κ k (kyk)korient (ez · n)kfiber (−kyk−1 n · y), 4π dist

y 6= 0.

(15)

For a simple probabilistic interpretation we apply a spatial reflection3 and define p+ (t,κ) (y, n) = p(t,κ) (−y, n). Now + p(t,κ) should be interpreted as a probability density of finding an oriented particle at position y ∈ R3 with orientation n ∈ S 2 given that it started at position 0 with orientation ez . The practical rationale behind the decomposition (14), is that two neighboring local orientations (y, n) ∈ R3 o S 2 and (y, n0 ) ∈ R3 o S 2 are supposed to strengthen each other if the distance between y and y0 is close (represented by the first kernel kdist ), if moreover the orientations are similar κ (represented by korient ), and finally if local orientation (y, n) is nicely aligned according to some a priori fibre model with the local orientation (y0 , n0 ), i.e. if the orientation of ky − y0 k−1 (y − y0 ) is close to the orientation n (represented κ by kfiber ). Furthermore the decomposition allows us to reduce computation time by: = (pt,κ ∗R3 oS 2 U )(y, n)

(Kkt,κ U )(y, n)

=

1 4π

R

t kdist (ky

−y

0

κ k)kfiber (ky

0 −1

−y k

0

n · (y − y))

R3

R

! 0

U (y , n

0

κ )korient (n

0

0

· n )dσ(n ) dy0

(16)

S2

Despite the fact that the practical kernel (14) gives rise to a reasonable connectivity measure between two local orientations (y, n) and (y0 , n0 ) ∈ R3 o S 2 and that the associated kernel operator has the right covariance properties, the associated kernel operator is not related to left-invariant diffusion and/or Tikhonov regularization on R3 o S 2 , as was aimed for in the paper [8]. In this inspiring pioneering paper the authors consider a position dependent energy and deal with the Euler-Lagrange equations in an unusual way (in particular [8, eq. 7]). The kernel (15) involves two separate time parameters t, κ and the probability kernels (14) are not related to Brownian motions and/or Markovprocesses on R3 o S 2 , since they do not satisfy the semi-group property. A disadvantage as we will explain next, however, is that the kernel is not entirely suited for iteration unless combined with non-linear operators such as nonlinear grey-value transformations. The function y 7→ kyk−1 y · n within (15) is discontinuous at the origin. If the origin is approached by a straight-line along n the limit-value is 1 and this seems to be a reasonable choice for evaluating the kernel at y = 0. The finite maximum of the kernel is now obtained at y = 0 (and n = ez ). Since the kernel is single-sided and does not have a singularity at the origin convolution with itself will allow the maximum of the effective kernel to run away4 from its center, similar to the following periodic convolution on a finite grid [0, 0, 0, 0, 1, α, 0, 0] ∗ [0, 0, 0, 0, 1, α, 0, 0] = [0, 0, 0, 0, 1, 2α, α2 , 0] with α ≤ 1. See Figure 5, where we numerically R3 o S 2 -convolved the kernel (14) with itself. However, if the kernel would have satisfied the semigroup-property such artifacts would not have occurred. For example the single-sided exact Green’s function of Mumford’s direction process [23] (and its approximations [45, 18, 23]) on SE(2) = R2 o S 1 has a natural singularity at the origin. Before we consider scale spaces on HARDI data whose solutions are given by R3 o S 2 -convolution (10) with the corresponding Green’s functions (which do satisfy the semigroup-property) we provide, for the sake of clarity, a quick review on scale spaces of periodic signals from a group theoretical PDE-point of view.

3.2

Introductory Example: Reviewing Scale Spaces/Tikhonov regularization on the Circle

The Gaussian scale space equation and corresponding resolvent equation (i.e. the solution of Tikhonov regularization) 0 0 on a circle T = {eiθ | θ ∈ [0, 2π)} ≡ S 1 with group product eiθ eiθ = ei(θ+θ ) , read

∂t u(θ, s) = D11 ∂θ2 u(θ, t), u(0, t) = u(2π, t) and u(θ, 0) = f (θ)

3 Later

and pγ (θ) = γ(D11 ∂θ2 − γI)−1 f (θ),

(17)

on in Subsection 8.2.1 we will return to the important practical consequences of this spatial reflection in full detail. 1 [. . . , 0, 1, α, 0, . . .], then for every n ∈ N the sequence an := a ∗(n−1) a ∈ `1 (Z) has n + 1 non-zero coefficients: a := 1+α k −n k an = (1 + α) α n , k = 0, . . . , n. So the position of the maximum of an increases with n (if α = 1 it takes place at k = b n c). k 2 4 Set

10

3 2 + Figure 5: Left: Glyph visualization (recall Definition 1) of the kernel p+ (t,κ) : R o S → R (15) as proposed in [8], plotted in perspective with respect to indicated horizon (dashed line) and vanishing point. Right: Glyph visualization + 3 2 of p+ (t,κ) ∗R oS p(t,κ) , i.e. the kernel numerically convolved with itself by a convolution algorithm that we will explain later in subsection 8.2 (kernels are sampled on 3 × 3 × 3-grid with 162-orientations). Parameter settings are (t = 21 , κ = 4). The maximum moves away from the origin by iteration: in the right image the second glyph on the z-axis has a larger radius than the glyph at 0. The effective shape of the convolution kernel is destroyed by iteration, as the kernel (15) does not satisfy the semi-group property. This motivates our quest (in Section 6) for appropriate diffusion kernels (related to Brownian motion on R3 o S 2 ) on R3 o S 2 that do satisfy the semigroup D D property pD t ∗R3 oS 2 ps = ps+t , involving only one time parameter.

with θ ∈ [0, 2π) and D11 > 0 fixed, where we note that the function θ 7→ pγ (θ) = γ minimizer of the Tikhonov-energy Z E(pγ ) :=

R∞

u(θ, t)e−γt dt is the

0

2π

γ|pγ (θ) − f (θ)|2 + D11 |p0γ (θ)|2 dθ

0

under the periodicity condition pγ (0) = pγ (2π). By left-invariance the solutions are given by T-convolution (or 11 “periodic convolution”) with their Green’s function (or “impuls-response”), say GD : T → R+ and RγD11 : T → R+ . t Recall that the relation between Tikhonov regularization and scale space theory is given by Laplace-transform with respect to time and thereby we have the following relation between the two regularizations: Z ∞ 11 −γt u(·, t) = Gt ∗T f and pγ = RγD11 ∗T f , with RγD11 = γ GD e dt, (18) t 0

Rπ

where the T convolution is given by f ∗T g(eiθ ) =

−π

0

0

f (ei(θ−θ ) )g(eiθ )dθ0 . Now orthogonal eigenfunctions of the

diffusion process correspond to eigenfunctions of the generator D11 (∂θ )2 and they are given by ηn (θ) = u(θ, t) = pγ (θ) =

P

2

(ηn , f )L2 (T) ηn (θ)e−n

11 , GD (θ) = t

tD11

n∈Z P

(ηn , f )L2 (T) ηn (θ) D11 nγ 2 +γ

, RγD11 (θ) =

n∈Z

P n∈Z P n∈Z

2

ηn (θ)ηn (0)e−n

tD11

einθ √ , 2π

so that

, (19)

ηn (θ)ηn (0) D11 nγ 2 +γ .

A drawback is that the series do not converge quickly if t > 0 resp. γ > 0 are small. Therefore one prefers a spatial implementation over a Fourier implementation, i.e. unfold the circle and calculate modulo 2π-shifts afterwards: 11 11 u(θ, t) = (GD ∗ f )(θ) , where GD (θ) = t t

P

11 ,∞ GD (θ − 2πn) t

(20)

n∈Z θ2

11 ,∞ where GD (θ) = (4πt)−1/2 e− 4t and RγD11 ,∞ (θ) = t

√ γ − γ|θ| . 2e

Again the latter formula follows by Laplace

11 transform of the first. The sums in (20) can be computed explicitly, yielding GD (θ) = t

11

1 ϑ 2π 3

√θ , e−t 2 D11

, where ϑ3

is a theta-function of the 3rd kind. The unique solution u(θ, t) can (at least formally, via Fourier transform on T) be written as 11 u(θ, t) = (et∆T f )(θ) = (GD ∗T f )(θ) with ∆T = (∂θ )2 t and by es ∆T et ∆T = e(s+t) ∆T the heat-kernel on T satisfies the (for iterations) important semi-group property: 11 11 11 GD ∗T G D = GD t s s+t , for all s, t > 0.

In this basic example the generator of a Gaussian scale space on the torus is given by D11 ∂θ2 . Just like the solution operator (D11 ∂θ2 − λI)−1 of Tikhonov regularization, it is left-invariant. This means that these operators commute 0 0 0 with the left-regular representation on T given by Leiθ f (eiθ ) = f ((eiθ )−1 eiθ ) = f (ei(θ −θ) ), for all f ∈ L2 (T), i.e. an ordinary (right) shift of complex-valued functions on T, since T is commutative. Due to this left-invariance, the solutions (18) of a Gaussian Scale Space and Tikhonov regularization are given by T-convolution. This (periodic) convolution is naturally related to our convolution operators (11), as the only difference is the replacement of the group product and the left-invariant Haar measure. Now in order to generalize scale space representations of functions on a torus to scale space representations of HARDI data, i.e. functions on R3 o S 2 embedded in SE(3) = R3 o SO(3), we simply have to replace the left-invariant vector field ∂θ on T by the left-invariant vector fields on SE(3) (or rather R3 oS 2 ) in the quadratic form which generates the scale space on the group, [17]. In the next section we will compute the left-invariant vector fields on SE(3).

3.3

Left-invariant Vector Fields on SE(3) and their Dual Elements

We will use the following basis for the tangent space Te (SE(3)) at the unity element e = (0, I) ∈ SE(3): A1 = ∂x , A2 = ∂y , A3 = ∂z , A4 = ∂γ˜ , A5 = ∂β˜ , A6 = ∂α˜ , where we stress that at the unity element R = I, we have β = 0 and here the tangent vectors ∂β and ∂γ are not defined, which requires a description of the tangent vectors on the SO(3)-part by means of the second chart. The tangent space at the unity element e = (0, 0, 0, R = I), Te (SE(3)), is a 3D Lie algebra equipped with Lie product [A, B] = lim t−2 a(t)b(t)(a(t))−1 (b(t))−1 − e , (21) t↓0

where t 7→ a(t) resp. t 7→ b(t) are any smooth curves in G with a(0) = b(0) = e and a0 (0) = A and b0 (0) = B, for explanation on the formula (21) which holds for general matrix Lie groups, see [19, App.G]. Define {A1 , A2 , A3 } := {eθ , ex , ey }. Then {A1 , A2 , A3 } form a basis of Te (SE(2)) and their Lie-products are [Ai , Aj ] =

6 X

ckij Ak ,

(22)

k=1

where the non-zero structure constants for all three isomorphic Lie-algebras are given by sgn perm{i − 3, j − 3, k − 3} if i, j, k ≥ 4, i 6= j 6= k, k k −cji = cij = sgn perm{i, j − 3, k} if i ≤ 3, j ≥ 4, i 6= j 6= k,

(23)

The corresponding left-invariant vector fields {Ai }6i=1 are obtained by the push-forward of the left-multiplication Lg h = gh by Ai |g φ = (Lg )∗ Ai φ = Ai (φ ◦ Lg ) (for all smooth φ : Ωg → R which are locally defined on some neighborhood Ωg of g) and they can be obtained by the derivative of the right-regular representation: Ai |g φ = dR(Ai )φ(g) = lim φ(g e h↓0

with Rg φ(h) = φ(hg). 12

hAi )−φ(g)

h

,

(24)

Expressed in the first coordinate chart (1) this renders for the left-invariant derivatives at position g = (x, y, z, Rez ,γ Rey ,β , Rez ,α ) ∈ SE(3) (see also [12, Section 9.10]) A1 A2 A3 A4 A5 A6

(cos α cos β cos γ − sin α sin γ) ∂x (− sin α cos β cos γ − cos α sin γ) ∂x sin β cos γ ∂x cos α cot β ∂α − sin α cot β ∂α ∂α

= = = = = =

+ + + + +

(sin α cos γ + cos α cos β sin γ) ∂y (cos α cos γ − sin α cos β sin γ) ∂y sin β sin γ ∂y sin α ∂β cos α ∂ β

− + + − +

cos α sin β ∂z , sin α sin β ∂z , cos β ∂z , cos α ∂ γ, sin β sin α ∂γ , sin β . (25)

for β 6= 0 and β 6= π. The explicit formulae of the left-invariant vector fields (which are well-defined in north- and south-pole) in the second chart (4) are : A1 A2 A3

= cos α ˜ cos β˜ ∂x + (cos γ˜ sin α ˜ + cos α ˜ sin β˜ sin γ˜ ) ∂y ˜ ∂z , +(sin α ˜ sin γ˜ − cos α ˜ cos γ˜ sin β) = − sin α ˜ cos β˜ ∂x + (cos α ˜ cos γ˜ − sin α ˜ sin β˜ sin γ˜ ) ∂y +(sin α ˜ sin β˜ cos γ˜ + cos α ˜ sin γ˜ ) ∂z , = sin β˜ ∂x − cos β˜ sin γ˜ ∂y + cos β˜ cos γ˜ ∂z ,

for β˜ 6=

π 2

A4

= − cos α ˜ tan β˜ ∂α˜ + sin α ˜ ∂β˜ +

A5

= sin α ˜ tan β˜ ∂α˜ + cos α ˜ ∂β˜ −

A6

= ∂α˜ ,

cos α ˜ ˜ cos β

sin α ˜ ˜ cos β

∂γ˜ , (26)

∂γ˜ ,

and β˜ 6= − π2 . Note that dR is a Lie-algebra isomorphism, i.e.

[Ai , Aj ] =

6 X

ckij Ak ⇔ [dR(Ai ), dR(Aj )] =

k=1

6 X

ckij dR(Ak ) ⇔ [Ai , Aj ] = Ai Aj − Aj Ai =

k=1

6 X

ckij Ak .

k=1

These vector fields form a local moving coordinate frame of reference on SE(3), the corresponding dual frame {dA1 , . . . , dA6 } ∈ (T (SE(3)))∗ is defined by hdAi , Aj i := dAi (Aj ) = δji , i, j = 1, . . . , 6, where δji = 1 if i = j and zero else. A brief computation yields the following dual frame (in both coordinate charts): 0 B B B B B B @

dA1 dA2 dA3 dA4 dA5 dA6

1

0

C 0 C (Rez ,γ Rey ,β Rez ,α )T C C=@ C C 0 A

0 Mβ,α

1B B B AB B B @

dx dy dz dα dβ dγ

1

0

C 0 C (Rex ,˜γ Rey ,β˜ Rez ,α˜ )T C C=@ C C 0 A

0 ˜˜ M β,α ˜

dx dy dz dα dβ dγ

1B B B AB B B @

1 C C C C C C A

(27)

˜ ˜ are given by where the 3 × 3-zero matrix is denoted by 0 and where the 3 × 3-matrices Mβ,γ , M β,α ˜ 0

Mβ,α

0 =@ 0 1

sin α cos α 0

0

1

− cos α sin β sin α sin β A , cos β

˜˜ M β,α ˜

− cos α ˜ tan β˜ B = @ sin α ˜ tan β˜ 1

sin α ˜ cos α ˜ 0

cos α ˜ ˜ cos β sin α − cos β˜˜

1−T C A

Finally, we note that by linearity the i-th dual vector filters out the i-th component of a vector field hdAi ,

6 X

v j Aj i = v i ,

.

0

P6

j=1

v j Aj

for all i, j = 1, . . . , 6.

j=1

4

Left-Invariant Diffusions on SE(3) = R3 o SO(3) and R3 o S 2

In order to apply our general theory on diffusions on Lie groups, [17], to suitable (convection-)diffusions on HARDI ˜ : R2 o SO(3) → R+ in the natural way images, we first extend all functions U : R3 o S 2 → R+ to functions U ˜ (x, R) = U (x, Rez ) or in Euler angles: U ˜ (x, Re ,γ Re ,β , Re ,α ) = U (x, n(β, γ)). U (28) z

13

y

z

˜ : R3 oSO(3) → R, given by (28), the HARDI-orientation score corresponding to HARDI Definition 4 We will call U 3 2 image U : R o S → R. ˜ in general is not equal to the wavelet transform of some image f : R2 → R, Here we note that the function U in contrast to our previous works on invertible orientations of 2D-images, [26], [2], [23], [20], [21] and invertible orientation scores of 3D-images, [22]. Then we follow our general construction of scale space representations W of functions U (could be an image, or a score/wavelet transform of an image) defined on Lie groups, [17], where we consider the special case G = SE(3): ( ∂t W (g, t) = QD,a (A1 , A2 , . . . , A6 ) W (g, t) , (29) ˜ (g) . lim W (g, t) = U t↓0

which is generated by a quadratic form on the left-invariant vector fields: QD,a (A1 , A2 , . . . , An ) =

6 X i=1

ai Ai +

6 X

Ai Dij Aj

(30)

j=1

Now the Hörmander requirement, [34], on the symmetric D = [Dij ] ∈ R6×6 , D ≥ 0 and a, which guarantees smooth non-singular scale spaces, for SE(3) tells us that D need not be strictly positive definite. The Hörmander requirement is that all included generators together with their commutators should span the full tangent space. To this end for diagonal D one should consider the set S = {i ∈ {1, . . . , 6} | Dii 6= 0 ∨ ai 6= 0} , now if for example 1 is not in here then 3 and 5 must be in S, or if 4 is not in S then 5 and 6 should be in S. Following the general theory [17] we note that iff the Hörmander condition is satisfied the solutions of the linear diffusions (i.e. D, a are constant) are given by SE(3)-convolution with a smooth probability kernel pD,a : SE(3) → R+ such that t R −1 ˜ )(g) = ˜ (h)dµSE(3) (h), pD,a g)U W (g, t) = (pD,a ∗SE(3) U t (h t SE(3) R ˜ =U ˜, lim pD,a ∗SE(3) U with pD,a > 0 and SE(3) pD,a t t t (g)dµSE(3) (g) = 1. t↓0

where the limit is taken in L2 (SE(3))-sense. The left-invariant diffusions on the group SE(3) also give rise to left-invariant scale spaces on the homogeneous space R3 oS 2 ≡ SE(3)/({0}×SO(2)) within the group. There are however, two important issues to be taken into account: 1. If we apply the diffusions directly to HARDI-orientation scores we can as well delete the last direction in our ˜ = 0. diffusions because clearly A6 = ∂α vanishes on functions which are not dependent on α, i.e. ∂α U 2. In order to naturally relate the (convection-)diffusions on HARDI-orientation scores, to (convection-)diffusions on HARDI images we have to make sure that the evolution equations are well defined on the cosets SO(3)/SO(2), meaning that they do not depend on the choice of representant in the classes. Next we formalize the second condition on diffusions on HARDI-orientation scores more explicitly. A movement along the equivalence classes SO(3)/SO(2) is done by right multiplication with the subgroup Stab(ez ) ≡ SO(2), with Stab(ez ) = {A ∈ SO(3) | Aez = ez }. Therefore our diffusion operator Φt which is the transform that maps ˜ : R3 o SO(3) → R+ to a diffused HARDI-orientation score Φt (U ˜ ) = etQD,a U ˜ , with the HARDI-orientation score U stopping time t > 0, should satisfy ˜ ) = Φt (U ˜) (Φt ◦ Rh )(U

for all h ∈ Stab(ez ) ≡ SO(2), 14

(31)

˜ (g) = U ˜ (gh). Now (31) is satisfied iff where Rh U QD,a (A1 , . . . , A6 ) ◦ R0,Rez ,α = QD,a (A1 , . . . , A6 ) ,

(32)

since AR0,Rez ,α = Zα A, where A = (A1 , . . . , A6 )T and where 0 B B B Zα = B B B @

cos α sin α 0 0 0 0

− sin α cos α 0 0 0 0

0 0 1 0 0 0

0 0 0 cos α sin α 0

0 0 0 − sin α cos α 0

0 0 0 0 0 1

1 C C C C = Re ,α ⊕ Re ,α , Zα ∈ SO(6), Re ,α ∈ SO(3). z z z C C A

(33)

Now for constant D and a (i.e. linear diffusion on the HARDI data) the requirement (32) simply reads QD,a (A) = QD,a (Zα A) = QD,a (Zα A) = Q(Zα )

T

D Zα ,Zα a

(A) ⇔ a = Zα a and D = ZαT DZα ,

(34)

which by Schur’s lemma is the case iff a1 = a2 = a4 = a5 = 0 and D = diag{D11 , D11 , D33 , D44 , D44 }.

(35)

˜, Analogously, for adaptive non-linear diffusions, that is D and a not constant but depending on the initial condition U ˜ ) : SE(3) → R3×3 , with (D(U ˜ ))T = D(U ˜ ) > 0 and a(U ˜ ) the requirement (32) simply reads i.e. D(U ˜ ) = Z T (a(U ˜ )) and D(R0,R U ˜ ) = Zα D(U ˜ )Z T . a(R0,Rez ,α U α α ez ,α

(36)

Summarizing all these results we conclude on HARDI data whose domain equals the homogeneous space R3 o S 2 one has the following scale space representations:

∂t W (y, n, t) = QD(U ),a(U ) (A1 , A2 , . . . , A5 ) W (y, n, t) , W (y, n, 0) = U (y, n) .

(37)

P5 P5 with QD(U ),a(U ) (A1 , A2 , . . . , An ) = a A + A D (U )A , where from now on we assume that i i i ij j i=1 j=1 D(U ) and a(U ) satisfy (36). Again in the linear case where D(U ) = D, a(U ) = a this means that we shall automatically assume (35). In this case the solutions of (37) are given by the following kernel operators on R3 o S 2 : W (y, n, t) = (pD,a ∗R3 oS 2 U )(y, n) t Rπ 2π R R D,a = pt ((Rez ,γ 0 Rey ,β 0 )T (y − y0 ), (Rez ,γ 0 Rey ,β 0 )T n)) U (y0 , n(β 0 , γ 0 )) dy0 dσ(n(β 0 , γ 0 )),

(38)

0 0 R3

˜ γ˜ )) = | cos β| ˜ dβd˜ ˜ γ. where the surface measure on the sphere is given by dσ(n(β 0 , γ 0 )) = sin β 0 dγ 0 dβ 0 ≡ dσ(˜ n(β, 3 Now in particular in the linear case, since (R , I) and (0, SO(3)) are subgroups of SE(3), we obtain the LaplaceBeltrami operators on these subgroups by means of: ∆S 2 = QD=diag{0,0,0,1,1,1},a=0 = (A4 )2 + (A5 )2 + (A6 )2 = (∂β )2 + cot(β)∂β + sin−2 (β)(∂γ )2 , ∆R3 = QD=diag{1,1,1,0,0,0},a=0 = (A1 )2 + (A2 )2 + (A3 )2 = (∂x )2 + (∂y )2 + (∂z )2 . Remark: Recall that in the linear case we assumed (35) to ensure (32) so that (31) holds. It is not difficult to show, [26, p.170], that this implies the required symmetry (9) on the convolution kernel.

15

4.1

Special Cases of Linear Left-invariant Diffusion on R3 o S 2

If we consider the singular case D = diag{1, 1, 1, 0, 0, 0}, a = 0 (not satisfying the Hörmander condition) we get the usual scale space in the position part only −tkω k W (y, n, t) = (et∆ U (·, n))(y) = FR−1 FR3 f (ω)](y) = (Gt ∗ f )(y), with Gt (y) = (4πt)− 2 e− 3 [ω 7→ e 2

3

kyk2 4t

and consequently on R3 o S 2 we have the singular distributional kernel pD,a t (y, n) = Gt (y)δez (n), in (38). If we consider the singular case D = diag{0, 0, 0, 1, 1, 1}, a = 0 we get the usual scale space on the sphere: W (y, n(β, γ), t)

= (et∆S2 U (y, ·)(x) = et∆S2 =

∞ l P P

∞ l P P

(Ylm , U )Ylm (β, γ) =

l=0 m=−l

∞ l P P

(Ylm , U )et∆S2 Ylm (β, γ)

l=0 m=−l

(Ylm , U )e−t(l(l+1)) Ylm (β, γ).

l=0 m=−l

2 where we note that the well-known spherical harmonics {Ylm }m=−l,...,l l=0,...,∞ from an orthonormal basis of L2 (S ) and ∆S 2 Ylm = −l(l + 1)Ylm . Recall s (2l + 1)(l − |m|)! m m Pl (cos β)eimγ l ∈ N , m = −l, . . . , l. (39) Yl (β, γ) = 4π(l + |m|)!

Consequently, on R3 o S 2 we have the singular distributional kernel pD,a t (y, n) = gt (n)δ0 (y), in (38), where gt (n(β, γ)) =

∞ X

Ylm (β, γ)Ylm (β, γ)e−tl(l+1) =

l=0

∞ X l=0

(Plm (cos β))2

(2l + 1)(l − |m|)! −tl(l+1) e . 4π(l + |m|)!

Note that in the two cases mentioned above diffusion takes place either only along the spatial part or only along the angular part, which is not desirable as one wants to include line-models which exploit a natural coupling between position and orientation. Such a coupling is naturally included in a smooth way as long as the Hormander’s condition is satisfied. In the two previous examples, the Hörmander condition is violated since both the span of {A1 , A2 , A3 } and the span of {A4 , A5 , A6 } are closed Lie-algebra’s, i.e. all commutators are again contained in the same 3dimensional subspace of the 6-dimensional tangent space. Therefore we will consider more elaborate simple leftinvariant convection, diffusions on SE(3) with natural coupling between position and orientation. To explain what we mean with natural coupling we shall need the next definitions. Definition 5 A curve γ : R+ → R3 o S 2 given by s 7→ γ(s) = (y(s), n(s)) is called horizontal if n(s) = k˙y(s)k−1 y˙ (s). A tangent vector to a horizontal curve is called a horizontal tangent vector. A vector field A on R3 o S 2 is horizontal if for all (y, n) ∈ R3 o S 2 the tangent vector A(y,n) is horizontal. The horizontal part Hg of each tangent space is the vector-subspace of Tg (SE(3)) consisting of horizontal vector fields. Horizontal diffusion is diffusion which only takes place along horizontal curves. It is not difficult to see that the horizontal part Hg of each tangent space Tg (SE(3)) is spanned by {A3 , A4 , A5 }. So all horizontal left-invariant convection diffusions are given by (37) where one must set a1 = a2 = 0, D2j = D2j = D1j = Dj1 = 0 for all j = 1, 2, . . . , 5. Now on a commutative group like R6 with commutative Lie-algebra {∂x1 , . . . , ∂x6 } omitting 3-directions (say ∂x1 , ∂x2 , ∂x6 ) from each tangent space in the diffusion would be a disaster, since this would imply no indirect smoothing would take place along the global x1 , x2 , x6 -axes. In SE(3) it is different since the commutators take care of indirect smoothing in the omitted directions {A1 , A2 , A6 }, since span {A3 , A4 , A5 , [A3 , A5 ] = A2 , [A4 , A5 ] = A6 , [A5 , A3 ] = A1 } = T (SE(3)) For example we consider the SE(3)-analogues of the Forward-Kolmogorov (or Fokker-Planck) equations of the direction process for contour-completion and the stochastic process for contour enhancement which we considered in 16

our previous works, [20], on SE(2). Here we provide the resulting PDE’s first and explain the underlying stochastic processes later in subsection 4.2. The Fokker-Planck equation for (horizontal) contour completion on SE(3) is ( ∂t W (y, n, t) = (A3 + D((A4 )2 + (A5 )2 )) W (y, n, t) = (A3 + D ∆S 2 ) W (y, n, t) , D = 21 σ 2 > 0. (40) lim W (y, n, t) = U (y, n) . t↓0

where we note that (A6 )2 (W (y, n(β, γ), s)) = 0. This equation arises from (37) by setting D44 = D55 = D and a3 = 1 and all other parameters to zero. The Fokker-Planck equation for (horizontal) contour enhancement is ( ∂s W (y, n, t) = (D33 (A3 )2 + D44 ((A4 )2 + (A5 )2 ) ) W (y, n, t) = ((A3 )2 + D ∆S 2 ) W (y, n, t) , (41) lim W (y, n, t) = U (y, n) . t↓0

The solutions of the left-invariant diffusions on R3 o S 2 given by (37) (with in particular (40) and (41)) are again given by convolution product (38) with a probability kernel pD,a on R3 o S 2 . t

4.2

Brownian Motions on SE(3) = R3 o SO(3) and on R3 o S 2

Next we formulate a left-invariant discrete Brownian motion on SE(3) (expressed in the moving frame of reference). The left-invariant vector fields {A1 , . . . , A6 } form a moving frame of reference to the group. Here we note that there are two ways of considering vector fields. Either one considers them as differential operators on smooth locally defined functions, or one considers them as tangent vectors to equivalent classes of curves. These two viewpoints are equivalent, for formal proof see [5, Prop. 2.4]. Throughout this article we mainly use the first way of considering vector fields, but in this section we prefer to use the second way. We will write {e1 (g), . . . , e6 (g)} for the left-invariant vector fields (as tangent vectors to equivalence classes of curves) rather than the differential operators { A1 |g , . . . , A6 |g }. We obtain the tangent vector ei from Ai by replacing ∂x ↔ (1, 0, 0, 0, 0, 0), ∂y ↔ (0, 1, 0, 0, 0, 0), ∂z ↔ (0, 0, 1, 0, 0, 0),

∂β ↔ (0, 0, 0, α cos β cos γ, α cos β sin γ, −α sin β), ∂γ ↔ (0, 0, 0, α cos γ, α sin γ, 0), ∂α ↔ (0, 0, 0, cos γ sin β, sin γ sin β, cos β),

(42)

where we identified SO(3) with a ball with radius 2π whose outer-sphere is identified with the origin, using Euler angles Rez ,γ Rey ,β Rez ,α ↔ αn(β, γ) ∈ B0,2π . Next we formulate left-invariant discrete random walks on SE(3) expressed in the moving frame of reference {ei }6i=1 given by (26) and (42): (Yn+1 , Nn+1 ) = (Yn , Nn ) + ∆s

5 P i=1

ai ei |(Yn ,Nn ) +

√

∆s

5 P i=1

εi,n+1

5 P j=1

σji ej |(Yn ,nn ) for all n = 0, . . . , N − 1,

(Y0 , n0 ) ∼ U D , with random variable (Y0 , n0 ) is distributed by U D , where U D are the discretely sampled HARDI data (equidistant sampling in position and second order tessalation of the sphere) and where the random variables (Yn , Nn ) are re−1 cursively determined using the independently normally distributed random variables {εi,n+1 }n=1,...,N , εi,n+1 ∼ i=1,...,5 P 5 s N (0, 1) and where the stepsize equals ∆s = N and where a := i=1 ai ei denotes an apriori spatial velocity vector having constant coefficients ai with respect to the moving frame of reference {ei }5i=1 (just like in (30)). Now if we apply recursion and let N → ∞ we get the following continuous Brownian motion processes on SE(3): ! 3 3 Rt P P 1 Y (t) = Y (0) + ai ei |(Y (τ ),N (τ )) + 21 τ − 2 εi σji ej |(Y (τ ),N (τ )) dτ , i=1 j=1 0 ! (43) 5 5 Rt P P 1 − 21 ai ei |(Y (τ ),N (τ )) + 2 τ εi σji ej |(Y (τ ),N (τ )) dτ , N (t) = N (0) + 0

j=4

i=4

17

with εi ∼ N (0, 1) and (X(0), N (0)) ∼ U and where σ =

√

√ 1 2D ∈ R6×6 , σ > 0. Note that d τ = 12 τ − 2 dτ .

Now if we set U = δ0,ez (i.e. at time zero ) then suitable averaging of infinitely many random walks of this process yields the transition probability (y, n) 7→ pD,a t (y, n) which is the Green’s function of the left-invariant evolution equations (37) on R3 o S 2 . In general the PDE’s (37) are the Forward Kolmogorov equation of the general stochastic process (43). This follows by Ito-calculus and in particular Ito’s formula for formulas on a stochastic process, for details see [2, app.A] where one should consistently replace the left-invariant vector fields of Rn by the left-invariant vector fields on R3 o S 2 . In particular we have now formulated the direction process for contour completion in R3 oS 2 (i.e. non-zero parameters in (43) are D44 = D55 > 0, a3 > 0 with Fokker-Planck equation (40)) and the (horizontal) Brownian motion for contour-enhancement in R3 o S 2 (i.e. non-zero parameters in (43) are D33 > 0, D44 = D55 > 0 with Fokker-Planck equation (41)).

4.3

Tikhonov-Regularization of HARDI Images

In the previous subsection we have formulated the Brownian-motions (43) underlying all linear left-invariant convectiondiffusion equations on HARDI data, with in particular the direction process for contour completion and (horizontal) Brownian motion for contour-enhancement. However, we only considered the time dependent stochastic processes and as mentioned before in Markov-processes traveling time is memoryless and thereby negatively exponentially distributed T ∼ N E(λ), i.e. P (T = t) = λe−λt with expectation E(T ) = λ−1 , for some λ > 0. Now recall our observations (12) and (13) and thereby by means of Laplace-transform with respect to time we relate the time-dependent Fokker-Planck equations to their resolvent equations, as at least formally we have Z ∞ D,a t(QD,a (A)) U )(y, n) and Pγ (y, n, t) = λ e−tλ (et(Q (A)) U )(y, n) = λ(λI−QD,a (A))−1 U (y, n), W (y, n, t) = (e 0

for t, λ > 0 and all y ∈ R3 , n ∈ S 2 , where the negative definite generator QD,a is given by (30) and again with AU = (A1 U, . . . , A6 U )T . This is similar to our introductory example on the torus in Subsection 3.2. The resolvent operator λ(λI − QD=diag(Dii ),a=0 (A))−1 occurs in a first order Tikhonov regularization as we show in the next theorem. Theorem 2 Let U ∈ L2 (R3 o S 2 ) and λ, D33 > 0, D44 = D55 > 0. Then the unique solution of the variational problem Z

arg min

P ∈H1 (R3 oS 2 ) R3 oS 2 )

X λ (P (y, n) − U (y, n))2 + Dkk |Ak P (y, n)|2 dydσ(n) 2 5

(44)

k=3

is given by PUλ (y, n) = (RλD ∗R3 oS 2 U )(y, n), where the Green’s function RλD : R3 o S 2 → R+ is the LaplaceR∞ transform of the heat-kernel with respect to time: RλD (y, n) = λ pD,a=0 (y, n)e−tλ dt with D = diag{D11 , . . . , D55 }. t 0

PUλ (y, n) equals the probability of finding a random walker in R3 o S 2 regardless its traveling time at position y ∈ R3 with orientation n ∈ S 2 starting from initial distribution U at time t = 0. Proof By convexity of the energy (44) (together with hypo-ellipticity of the operator variational problem is unique. Along the minimizer W = Pλ we have lim h↓0

E(Pλ + hδ) − E(Pλ ) =0 h

18

P5

2 k=3 (Ak ) )

the solution of this

for all pertubations δ ∈ H1 (R3 o S 2 ). So by integration by parts we find ! 5 X λ(W − U ) − (Ak )2 W, δ k=3

=0

L2 (R3 oS 2 )

P5 for all δ ∈ H1 (R3 o S 2 ). Now H1 (R3 o S 2 ) is dense in L2 (R3 o S 2 ) and therefore λ U = λI − k=3 (Ak )2 W , −1 P5 so W = λ λI − ( k=3 (Ak )2 ) U and by left-invariance and linearity this resolvent equation is solved by a R3 o S 2 -convolution with the smooth Green’s function Rλ,D=diag{Dii } : R3 o S 2 \{e} → R+ . By (13) this Green’s D=diag{Dii } function follows by the smooth Green’s function Gs by Laplace tranform with respect to time.

5

Differential Geometry: The underlying Cartan-Connection on SE(3) and the Auto-Parallels in SE(3)

Now that we have constructed all left-invariant scale space representations on HARDI images, generated by means of a quadratic form (30) on the left-invariant vector fields on SE(3). The question rises what is the underlying differential geometry for these evolutions ? For example, as the left-invariant vector fields clearly vary per position in the group yielding a moving frame of reference attached to luminosity particles (random walkers in R3 o S 2 embedded in SE(3)) with both a position and an orientation, the question rises along which trajectories in R3 o S 2 do these particles move ? Furthermore, as the left-invariant vector fields are obtained by the push-forward of the left-multiplication on the group, Ag = (Lg )∗ Ae , i.e. Ag φ˜ = Ae (φ˜ ◦ Lg ), where Lg h = gh, g, h ∈ SE(3), φ˜ : SE(3) → R smooth , the question rises whether this defines a connection between all tangent spaces, such that these trajectories are autoparallel with respect to this connection ? Finally, we need a connection to rigid body mechanics described in a moving frame of reference, to get some physical intuition in the choice of the fundamental constants5 {ai }6i=1 and {Dij }6i,j=1 within our generators (30). In order to get some first physical intuition on analysis and differential geometry along the moving frame {A1 , . . . , A6 } and its dual frame {dA1 , . . . , dA6 }, we will make some preliminary remarks on the well-known theory of rigid body movements described in moving coordinate systems. Imagine a curve in R3 described in the moving frame of reference (embedded in the spatial part of the group SE(3)), describing a rigid body movement with constant spatial velocity ˆc(1) and constant angular velocity ˆc(2) and parameterized by arc-length s > 0. Suppose the curve is given by y(s) =

3 X

αi (s) Ai |y(s) where αi ∈ C 2 ([0, L], R),

i=1

such that ˆc(1) =

P3

i=1

α˙ i (s) Ai |y(s) for all s > 0. Now if we differentiate twice with respect to the arc-length

parameter and keep in mind that

d ds

Ai |y(s) = ˆc(2) × Ai |y(s) , we get

¨ y(s) = 0 + 2ˆc(2) × ˆc(1) + ˆc(2) × (ˆc(2) × y(s)) . In words: The absolute acceleration equals the relative acceleration (which is zero, since ˆc(1) is constant) plus the Coriolis acceleration 2ˆc(2) × ˆc(1) and the centrifugal acceleration ˆc(2) × (ˆc(2) × y(s)). Now in case of uniform circular 5 Or

later in Subsection 9 to get some intuition in the choice of functions {ai }6i=1 and {Dij }6i,j=1 .

19

motion the speed is constant but the velocity is always tangent to the orbit of acceleration and the acceleration has constant magnitude and always points to the center of rotation. In this case, the total sum of Coriolis acceleration and centrifugal acceleration add up to the well-known centripetal acceleration, ¨ y(s) = 2ˆc(2) × (−ˆc(2) × Rr(s)) + ˆc2 × (ˆc(2) × Rr(s)) = −kˆc(2) k2 Rr(s) = − where R is the radius of the circular orbit y(s) = m + R r(s), the Coriolis acceleration, i.e. ¨ y(s) = ˆc(2) × ˆc(1) .

kˆc1 k2 r(s), R

kr(s)k = 1). The centripetal acceleration equals half

In our previous work [21] on contour-enhancement and completion via left-invariant diffusions on invertible orientation scores (complex-valued functions on SE(2)) we have put a lot of emphasis on the underlying differential geometry in SE(2). All results straightforwardly generalize to the case of HARDI images, which can be considered as functions on R3 o S 2 embedded in SE(3). These rather technical results are summarized in Theorem 3, which answers all questions raised in the beginning of this section. Unfortunately, this theorem requires general differential geometrical concepts such as principal fiber bundles, associated vector bundles, tangent bundles, frame-bundles and the Cartan-Ehresmann connection defined on them. These concepts are explained in full detail in [43] (with a very nice overview on p.386 ). The reader who is not familiar with these technicalities from differential geometry can skip the first part of the theorem while accepting the formula of the covariant derivatives (49), where the anti-symmetric Christoffel-symbols are equal to minus the structure constants ckij = −ckji (recall (23)) of the Lie-algebra. Here we stress that we follow the Cartan view-point on differential geometry, where connections are expressed in moving coordinate frames (we use the frame of left-invariant vector fields {A1 , . . . , A6 } derived in Subsection 3.3 for this purpose) and thereby we have non-vanishing torsion.6 This is different from the Levi-Civita connection for differential geometry on Riemannian manifolds, which is much more common in image analysis. The Levi-Civita connection is the unique torsion free connection on a Riemannian manifold and because of this vanishing torsion of the Levi-Civita connection ∇ there is a 1-to-1 relation7 to the Christoffel symbols (required for covariant derivatives ∇i v j = ∂i v j + Γkij ∂k v j ) and the derivatives of the metric tensor. In the more general Cartan-connection outlined below, however, one has non-vanishing torsion and the Christoffels are not necessarily related to a metric tensor, nor need they be symmetric. Theorem 3 The Maurer-Cartan form ω on SE(3) is given by ωg (Xg ) =

6 X

h dAi g , Xg iAi ,

Xg ∈ Tg (SE(2)),

(45)

i=1

where the dual vectors {dAi }3i=1 are given by (27) and Ai = Ai |e . It is a Cartan Ehresmann connection form on the principal fiber bundle P = (SE(3), e, SE(3), L(SE(3))), where π(g) = e, Rg u = ug, u, g ∈ SE(3). Let Ad denote the adjoint action of SE(3) on its own Lie-algebra Te (SE(3)), i.e. Ad(g) = (Rg−1 Lg )∗ , i.e. the push-forward of conjugation. Then the adjoint representation of SE(3) on the vector space L(SE(3)) of left-invariant vector fields is given by f Ad(g) = dR ◦ Ad(g) ◦ ω. (46) This adjoint representation gives rise to the associated vector bundle SE(3) ×Ad e L(SE(3)). The corresponding connection form on this vector bundle is given by ω ˜=

6 X j=1

e j ) ⊗ dAj = ad(A

6 X

ckij Ak ⊗ dAi ⊗ dAj ,

(47)

i,j,k=1

6 The torsion-tensor T of a connection ∇ is given by T [X, Y ] = ∇ Y − ∇ X − [X, Y ]. The torsion-tensor T of a Levi-Civita connection ∇ X Y ∇ vanishes, whereas the torsion-tensor of our Cartan-connection ∇ on SE(3) is given by T∇ = 3ckij dAi ⊗ dAj . 7 In a Levi-Cevita connection one has Γi = Γi = 1 g im (g mk,l + gml,k − gkl,m ) with respect to a holonomic basis. kl lk 2

20

Then ω ˜ yields the following 6 × 6-matrix valued matrix 1-form ω ˜ jk (·) := −˜ ω (dAk , ·, Aj )

k, j = 1, 2, 3.

(48)

on the frame bundle, [43, p.353,p.359], where the sections are moving frames [43, p.354]. Let {µk }3k=1 denote the sections in the tangent bundle E := (SE(3), T (SE(3))) which coincide with the left-invariant vector fields {Ak }3k=1 . Then the matrix-valued 1-form (48) yields the Cartan connection given by the covariant derivatives D X|γ(t) (µ(γ(t)))

:= Dµ(γ(t))( X|γ(t) ) 6 6 6 P P P ω ˜ kj ( X|γ(t) ) µj (γ(t)) ak (γ(t)) a˙ k (t)µk (γ(t)) + = =

k=1 6 P

a˙ k (t)µk (γ(t)) +

i,j=1

k=1

with a˙ k (t) =

6 P i=1

k=1 6 P

j=1

γ˙ i (t) ak (γ(t)) Γjik µj (γ(t))

γ˙ i (t) ( Ai |γ(t) ak ), for all tangent vectors X|γ(t) =

SE(2) and all sections µ(γ(t)) =

6 P k=1

(49)

6 P i=1

γ˙ i (t) Ai |γ(t) along a curve t 7→ γ(t) ∈

ak (γ(t)) µk (γ(t)). The Christoffel symbols in (49) are constant Γjik = −cjik ,

with cjik the structure constants of Lie-algebra Te (SE(3)). Consequently, the connection D has constant curvature and constant torsion and the left-invariant evolution equations (29) can be rewritten in covariant derivatives (using short notation ∇j := DAj ): 8 6 6 P P > > ∂s W (g, s) = −ai (W )Ai W (g, s) + Ai ( (Dij (W ))(g, s)Aj W )(g, s) > > < i=1 i,j=1 6 6 P P = −ai (W )∇i W (g, s) + ∇i ((Dij (W ))(g, s)∇j W )(g, s) > > > i=1 i,j=1 > :

˜ (g) , W (g, 0) = U

(50)

for all g ∈ SE(3), s > 0.

Both convection and diffusion in the left-invariant evolution equations (29) take place along the exponential curves 6 P t ci Ai γc,g (t) = g · e i=1 in SE(3) which are the covariantly constant curves (i.e. auto-parallels) with respect to the Cartan connection. In particular, if ai (W ) = ci constant and if Dij (W ) = 0 (convection case) then the solutions are −s

˜ (g · e W (g, s) = U

6 P i=1

ci Ai

).

(51)

The spatial projections PR3 γ of these of the auto-parallel/exponential curves γ are circular spirals with constant curvature and constant torsion. The curvature magnitude equals kˆc(1) k−1 kˆc(2) × ˆc(1) k and the curvature vector equals ! 1 sin(t kˆc(2) k) (2) (2) (2) (1) (2) (1) ˆc × ˆc × ˆc κ(t) = (1) , (52) cos(t kˆc k) ˆc × ˆc + kˆc k kˆc(2) k where c = (c1 , c2 , c3 c4 , c5 , c6 ) = (ˆc(1) ; ˆc(2) ). The torsion vector equals τ (t) = |ˆc1 · ˆc2 | κ(t). Proof The proof is a straightforward generalization from our previous results [21, Thm 3.8 and Thm 3.9] on the SE(2)-case to the case SE(3). The formulas of the constant torsion and curvature of the spatial part of the autoparallel curves (which are the exponential curves) follow by the formula (55) p for (the spatial part x(s) of) the exponential curves, which we will derive in Section 5.1. Here we stress that s(t) = t (c1 )2 + (c2 )2 + (c3 )2 is the arc-length d ˙ of the spatial part of the exponential curve and where we recall that κ(s) = ¨x(s) and τ (s) = ds (x(s) × ¨x(s)). Note that both the formula (55) for the exponential curves in the next section and the formulas for torsion and curvature are simplifications of our earlier formulas [26, p.175-177]. In the special case of only convection the solution (51) follows ˜ (g) = ResA U ˜ (g), with A = − P6 ci Ai and dR(A) = − P6 ci Ai with Ai = dR(Ai ). by esdR(A) U i=1 i=1 21

5.1

The Exponential Curves and the Logarithmic Map explicitly in Euler Angles

Next we compute the exponential curves in SE(3) by an isomorphism of the group SE(3) to matrix group SE(3) Rγ,β,α x SE(3) 3 (x, Rγ,β,α ) ↔ ∈ SE(3) with Rγ,β,α = Rez ,γ Rey ,β Rez ,α , 0 1 This isomorphism induces the following isomorphism between the respective Lie-algebra’s 6 X

ci Ai ∈ L(SE(3)) ↔ Te (SE(3)) 3

i=1

6 X

ci Ai ↔

i=1

6 X

ci Xi ∈ R3×3 .

i=1

where {ci }6i=1 ∈ R6 and with matrices {Xi }6i=1 ∈ R3×3 are given by 0

0 B0 B X1 = @ 0 00 0 B0 B X4 = @ 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 −1 0 0

1

1 0C C, 0A 0 1 0 0C C, 0A 0

0

0 B0 B X2 = @ 0 00 0 B 0 B X5 = @ −1 0

0 0 0 0

1

0 0 0 0 0 0 0 0

1 0 0 0

0 1C C, 0A 0 1 0 0C C, 0A 0

0

0 B0 B X3 = @ 0 00 0 B1 B X6 = @ 0 0

0 0 0 0 −1 0 0 0

1

0 0 0 0 0 0 0 0

0 0C C, 1A 0 1 0 0C C. 0A 0

(53)

Note that Ai ↔ Ai ↔ Xi ⇒ [Ai , Aj ] ↔ [Ai , Aj ] ↔ [Xi , Xj ] and indeed direct computation yields: 0

0 B 0 B 6 6 X X B 0 ckij Ak = [Ai , Aj ] ↔ [Xi , Xj ] = ckij Xk with commutator table B B 0 B k=1 k=1 @−X3 X2

0 0 0 X3 0 −X1

0 0 0 −X2 X1 0

0 −X3 X2 0 −X6 X5

X3 0 −X1 X6 0 −X4

1

−X2 X1 C C 0 C C, −X5 C C X4 A 0 (54)

where i enumerates vertically and j horizontally and [Ai , Aj ] = Ai Aj − Aj Ai and [Xi , Xj ] = Xi Xj − Xj Xi . Each element in the Lie-algebra of the matrix group SE(3) can be written ! 6 0 −c6 c5 X Ω ˆc(1) i 6 4 ˆc(2) = (c1 , c2 , c3 ) ∈ R3 , c 0 −c ∈ so(3), A= c Xi = ,Ω = 0 0 −c5 c4 0 i=1

3×3 with | AT = −A}. Note that exp(so(3)) = SO(3) and Ωx = ˆc(2) × x and set q˜ = kˆc(2) k = p so(3) = {A ∈ R (c4 )2 + (c5 )2 + (c6 )2 so that Ω3 = −˜ q 2 Ω and therefore   1 R tsΩ (1) ∞ k tΩ k k−1 ˆ(1) P ˆ R x e t e ds c Ω Ω c t k k tA   A = = ∈ SE(3), ⇒e = 0 k! A = 0 1 0 0 k=1 0 1 ∞ R 1 tsΩ −1 P (tsΩ)k q t) 1 1−cos(˜ with e ds = Ω + q˜−t q˜3sin(˜qt) Ω2 , (k+1)! = I + t q˜2 0

and

k=0 sin(˜ q t) q˜ Ω

R = etΩ = I +

+

1−cos(˜ q t) 2 Ω . q˜2

so that the exponential curves are given by  6  t P ci Ai q t) i=1 e = (tˆc(1) + 1−cos(˜ Ωˆc(1) + (t˜ q −2 − γc (t) = q˜2  (c1 t, c2 t, c3 t , I)

22

sin(˜ q t) q˜3 )

Ω2ˆc(1) , I +

sin(˜ q t) q˜ Ω

+

(1−cos(˜ q t)) ) q˜2

if ˆc(2) 6= 0 , if ˆc(2) = 0 . (55)

As the exponential map is surjective we are also interested in the logarithmic map. This means we have to solve for ˆc(1) ∈ R3 and Ω ∈ so(3), given a group element g = (x, Rγ,β,α ) ∈ SE(3). Note that ΩT = −Ω, (Ω2 )T = Ω2 so that q t) R − RT = 2 sin(˜ q˜ Ω from which the logarithmic map Ω = logSO(3) R, R = Rγ,β,α follows explicitly: c4 = c4γ,β,α := 6

c =

c6γ,β,α

:=

q˜ 2 sin q˜ q˜ 2 sin q˜

sin β(sin α − sin γ) , c5 = c5γ,β,α := (2 cos2 β2 sin(α + γ)) .

p and thereby q˜ = (c4 )2 + (c5 )2 + (c6 )2 = q˜γ,β,α = arcsin

r cos2

α+γ 2

q˜ 2 sin q˜

sin β(cos α − cos γ) ,

sin2 β + cos4

(56)

β 2

sin2 (α + γ). So it

remains to express ˆc(1) = (c1 , c2 , c3 )T in Euler angles (γ, β, α). Now Ω3 = −˜ q 2 Ω implies that I + q˜−2 (1 − cos q˜)Ω + q˜−3 (˜ q − sin q˜)Ω2 ˆc(1) = x ⇔ q˜ q˜ −2 1 ˆc(1) = ˆc(1) )(Ωγ,β,α )2 x. ˜γ,β,α cot γ,β,α (1 − γ,β,α x,γ,β,α := I − 2 Ωγ,β,α + q 2 2

(57)

Now equality (56) and (57) provide the explicit logarithmic mapping on SE(3): logSE(3) (x, Rγ,β,α ) =

3 X

cix,γ,β,α Ai +

i=1

6 X

ciγ,β,α Ai .

(58)

i=4

p d Remark: It can be shown that k dt PR3 γc (t)k = (c1 )2 + (c2 )2 + (c3 )2 . Consequently the arc-length parameter p s > 0 is expressed in t by means of s(t) = t (c1 )2 + (c2 )2 + (c3 )2 . If we want to imposepspatial arc-length parameterizations of curves in SE(3) we must rescale all ci → √ 1 2 ci2 2 3 2 so that kˆc(1) k = (c1 )2 + (c2 )2 + (c3 )2 = 1. (c ) +(c ) +(c )

Remark: The group SE(3) is isomorphic to the group of rigid motions in R3 well-known in mechanics. The vector ˆc(1) denotes constant velocity in the moving coordinate frame {Ai }3i=1 whereas ˆc(2) denotes constant angular velocity with respect to the same moving coordinate frame attached to a particle on a moving rigid body in R3 . Note that κ(0) equals the centripetal acceleration at the moving frame of reference {A1 , A2 , A3 }|γc (0)=e = {A1 , A2 , A3 }, whereas κ(s) equals the centripetal acceleration at the moving frame of reference {A1 , A2 , A3 }|γc (s) , but again expressed in the global coordinate system {A1 , A2 , A3 } = {∂x , ∂y , ∂z } of the spatial part ≡ R3 of the group, which is for s > 0 no longer aligned with the moving frame of reference. In order to re-express κ(s) in {A1 , A2 , A3 } one must rotate κ(0) over an angle of skˆc(2) k around the angular velocity axis ˆc2 , which explains (52).

6

Analysis of the Convolution Kernels of Scale Spaces on HARDI images

It is notorious problem to find explicit formulas for the exact Green’s functions pD,a : R3 o S 2 of the left-invariant t 3 2 diffusions (37) on R o S . Explicit, tangible and exact formulas for heat-kernels on SE(3) do not seem to exist in literature. Nevertheless, there does exist a nice general theory overlapping the fields of functional analysis and group theory, see for example [44, 39], which at least provides Gaussian estimates for Green’s functions of left-invariant diffusions on Lie groups, generated by subcoercive operators. In the remainder of this section we will employ this general theory to our special case where R3 o S 2 is embedded into SE(3) and we will derive new explicit and useful approximation formulas for these Green’s functions. Within this section and in Appendix A we always use the second coordinate chart (4), as it is highly preferable over the more common Euler angle parametrization (1) because of the much more suitable singularity locations on the sphere (we rather avoid singularities at the unity element of SE(3)). Furthermore, in Appendix A we provide an alternative approach to derive the Green’s functions for the direction process 40 (a contour-completion process) in R3 o S 2 . Here we follow a similar approach as in [23], where we managed to derive the exact Green’s functions of the direction process in SE(2). However, unlike the SE(2)-case, we 23

do have to apply a reasonable approximation in the generator in order to get explicit formulas. This approximation is valid for 4sD44 small and is nearly exact in a sharp cone around the z-axis where the Green’s function is concentrated. Moreover, in Appendix A, a highly remarkable geometric connection arises between this approximation of Green’s function of the direction process in SE(3) and the exact Green’s functions for the direction process in SE(2). We shall first carry out the method of contraction. This method typically relates the group of positions and rotations to a (nilpotent) group positions and velocities and serves as an essential pre-requisite for our Gaussian estimates and approximation kernels later on. The reader who is not so much interested in the detailed analysis can skip this section and continue with the numerics explained in Chapter 7.

6.1

Local Approximation of SE(3) by a Nilpotent Group via Contraction

The group SE(3) is not nilpotent. This makes it hard to get tangible explicit formulae for the heat-kernels. Therefore we shall generalize our Heisenberg approximations of the Green’s functions on SE(2), [23], [45],[2], to the case SE(3). Again we will follow the general work by ter Elst and Robinson [44] on semigroups on Lie groups generated by weighted subcoercive operators. In their general work we consider a particular case by setting the Hilbert space L2 (SE(3)), the group SE(3) and the right-regular representation R. Furthermore we consider the algebraic basis {A3 , A4 , A5 } leading to the following filtration of the Lie algebra g1 = span{A3 , A4 , A5 } ⊂ g2 = span{A1 , A2 , A3 , A4 , A5 , A6 } = L(SE(3)) .

(59)

Now that we have this filtration we have to assign weights to the generators w3 = w4 = w5 = 1 and w1 = w2 = w6 = 2.

(60)

For example w3 = 1 since A3 already occurs in g1 , w6 = 2 since A6 is within in g2 and not in g1 . Now that we have these weights we define the following dilations on the Lie-algebra Te (SE(3)) (recall Ai = Ai |e ): 6 P

ci Ai ) =

6 P

q wi ci Ai , for all ci ∈ R, i=1 i=1 y z x γ˜q (x, y, z, Rγ˜ ,β, ˜α ˜) = q w1 , q w2 , q w3 , R γ˜ , γq (

q w4

˜ β q w5

α ˜ , qw 6

, q > 0,

and for 0 < q ≤ 1 we define the Lie product [A, B]q = γq−1 [γq (A), γq (B)]. Now let (SE(3))q be the simply connected Lie group generated by the Lie algebra (Te (SE(3)), [·, ·]q ). This Lie group is isomorphic to the matrix group with group product: 0 (x, Rγ˜ β˜α˜ ) ·q (x0 , Rγ˜ 0 β˜0 α˜ 0 ) = ( x + Sq · Rγ˜ q,βq, ˜ αq ˜ 2 · Sq −1 x , Rγ ˜ β˜α ˜ · Rγ ˜ 0 β˜0 α ˜0 )

(61)

where the diagonal 3×3-matrix is defined by Sq := diag{1, 1, q} and we used short-notation Rγ˜ β˜α˜ = Rex ,˜γ Rey ,β˜ Rez ,α˜ , i.e. our elements of SO(3) are expressed in the second coordinate chart (4). Now the left-invariant vector fields on the group (SE(3))q are given by Aqi |g = (˜ γq−1 ◦ Lg ◦ γ˜q )∗ Ai , i = 1, . . . , 6. Straightforward (but intense) calculations yield (for each g = (x, Rγ˜ β˜α˜ ) ∈ (SE(3))q ):

Aq4 |g

˜ ∂x + (cos(˜ ˜ sin(˜ γ q) sin(αq ˜ 2 ) + cos(αq ˜ 2 ) sin(βq) γ q)) ∂y + = cos(q 2 α) ˜ cos(q β) 2 2 ˜ +q(sin(αq ˜ ) sin(˜ γ q) − cos(αq ˜ ) cos(˜ γ q) sin(βq)) ∂z ˜ ∂x + (cos(q 2 α) ˜ sin(˜ = − sin(αq ˜ 2 ) cos(βq) ˜ cos(˜ γ q) − sin(αq ˜ 2 ) sin(βq) γ q)) ∂y + 2 2 ˜ ˜ ˜ ) sin(˜ γ q)) ∂z +q(sin(αq ˜ ) sin(βq) cos(βq) + cos(αq ˜ ∂x − q −1 cos(βq) ˜ sin(˜ ˜ cos(˜ = q −1 sin(βq) γ q) ∂y + cos(βq) γ q) ∂z 2 cos( αq ˜ ) −1 2 ˜ ∂α˜ + sin(αq) = −q cos(αq ˜ ) tan(βq) ˜ ∂˜ + ∂γ˜

Aq5 |g Aq6 |g

˜ ∂α˜ + cos(αq = q −1 sin(αq ˜ 2 ) tan(βq) ˜ 2 ) ∂β˜ − = ∂α˜ .

Aq1 |g Aq2 |g Aq3 |g

β

24

˜ cos(βq) sin(q 2 α) ˜ ˜ ˜ ∂γ cos(q β)

Now note that [Ai , Aj ]q = γq−1 [γq (Ai ), γq (Aj )] = γq−1 q wi +wj [Ai , Aj ] =

6 P

q wi +wj −wk ckij Ak and thereby we have

k=1

[A4 , A5 ]q = A6 , [A4 , A2 ]q = q 2 A3 ,

[A4 , A6 ]q = −q 2 A5 , [A5 , A1 ]q = −q 2 A3 ,

[A5 , A6 ]q = q 2 A4 , [A5 , A3 ]q = A1 ,

[A4 , A3 ]q = −A2 , [A6 , A1 ]q = q 2 A2 and [A6 , A2 ]q = −q 2 A1 .

(62)

Analogously to the case q = 1, (SE(3))q=1 = SE(3) we have an isomorphism of the common Lie-algebra at the unity element Te (SE(3)) = Te ((SE(3))q ) and left-invariant vector fields on the group (SE(3))q : (Ai ↔ Aqi and Aj ↔ Aqj ) ⇒ [Ai , Aj ]q ↔ [Aqi , Aqj ] . It can be verified that the left-invariant vector fields Aqi satisfy the same commutation relations (62). Now let us consider the case q ↓ 0, then we get a nilpotent-group (SE(3))0 with left-invariant vector fields ˜ x − γ˜ ∂y + ∂z , A0 = −β∂ ˜ α˜ + ∂γ˜ , A0 = ∂ ˜ , A0 = ∂α˜ . A01 = ∂x , A02 = ∂y , A03 = β∂ 4 5 6 β

6.1.1

(63)

The Heisenberg-approximation of the Contour Completion Kernel

Recall that the generator of contour completion diffusion equals A3 + D44 ((A4 )2 + (A5 )2 ). So let us replace the true left-invariant vector fields {Ai }5i=3 on SE(3) = (SE(3))q=1 by their Heisenberg-approximations {A0i }5i=3 that are given by (63) and compute the Green’s function pat 3 =1,D44 =D55 on (SE(3))0 (i.e. the convolution kernel which yields the solutions of contour completion on (SE(3))0 by group convolution on (SE(3))0 ). For 0 < D44 1

(91)

We did not take spatial second order B-spline interpolations into account11 that we used to compute (88). An increase of spatial diffusivity and angular diffusivity yields a decrease in the maximum time step, whereas an increase of the regularity parameter treg allows a larger time step. We also recognize a turning point if treg · (L(L + 1)) = 1. The multiplier l(j)(l(j) + 1)e−treg l(j)(l(j)+1) attains its maximum at j < nSH if treg > (L(L + 1))−1 . This is desirable since the multiplier is supposed to vanish at j = nSH ≈ No . In the experiment of Figure 10, we have set h = 10−1 , no = 162, L = 18 (restricting l to even order), treg = 0.01, D33 = 1, D11 = 10−2 , D44 = 10−4 and ∆t = 0.005.

7.3

Convolution Schemes for Linear R3 o S 2 -Diffusion

Instead of a finite difference scheme one can use the theoretical fact that the solutions of the linear diffusions (37) are given by R3 oS 2 -convolution (38) with the corresponding (smooth) Green’s function pD,a that we derived analytically t 11 The interpolation has a minor effect on the stability bounds. The radius of the Gerschgorin circles associated to the spatial increments functions √ get multiplied with 2 which is the upper-bound for the cardinal B-spline, [26, p.146].

32

in Section 6. The convolution scheme is a relatively straightforward discretization of W (y, n, t) = (pD,a ∗R3 oS 2 U )(y) t given by (10), where the integrals are usually replaced by sums using the mid-point rule (unless one has to deal with the singularity at the origin of the contour-completion kernel). This approach has computational advantages (in contrast to the SE(2)-convolutions) over the rather technical steerable SE(3)-convolutions [2, p.174-179]. We will return to specific practical implementation issues, later in subsection 8.2. In this subsection we restrict ourselves to an overview of options for the computation of the Green’s functions. We propose the following options to evaluate the Green’s function for contour enhancement (i.e. non-zero parameters are D33 > 0, D44 = D55 > 0 and D11 = D22 ≥ 0) in the integration (38): 1. Use the finite difference scheme to numerically compute the Green’s function. Disadvantage: This requires interpolation. Advantage: If the time step ∆t 0, but here we use µ > 0 to avoid confusion with P P Euler-angle β > 0 in SO(3). The left-invariant metric on SE(3) is given by the first fundamental form 3i=1 µ2 dAi ⊗dAi + 6j=4 dAj ⊗dAj .

41

10

Conclusion

For the purpose of tractography (detection of biological fibers) and visualization, DTI and HARDI data should be enhanced such that fiber junctions are maintained, while reducing high frequency noise in the joined domain R3 o S 2 of positions and orientations. Therefore we have considered diffusions on HARDI and DTI induced by fundamental stochastic processes on R3 o S 2 embedded in the group manifold SE(3) of 3D-rigid body motions. We have shown that the processing must be left-invariant and we have classified all linear left-invariant diffusions on HARDI images. We presented two novel diffusion approaches which take place simultaneously over both positions and orientations. These two approaches do allow enhancement of fibres while preserving crossings and/or bifurcations. These two diffusions are Fokker-Planck equations of stochastic processes (random walks) for respectively contour enhancement and contour completion. In a contour completion process a random walker always proceeds forward in space along its prescribed random direction, whereas in a contour enhancement process the random walker randomly moves forward and backward in its prescribed random orientation. As a result the contour completion process is generated by −A3 + D44 ∆S 2 whereas the contour enhancement process is generated by the sub-Laplacian +D33 (A3 )2 + D44 ∆S 2 , with D33 , D44 > 0 and ∆S 2 |H = A24 + A25 the Laplace-Beltrami operator on the sphere and where Ai -denotes the i-th left-invariant vector field on R3 o S 2 . Consequently, the contour enhancement process preserves the angular reflection symmetry of HARDI data, whereas the contour completion process allows a choice between attraction or continuation of glyphs. As the solutions of linear left-invariant diffusion equations are given by R3 o S 2 -convolution with their Green’s functions, we arrive at two types of implementations: Convolution schemes and finite difference schemes. Practical advantages of convolution schemes over finite difference schemes for linear diffusions are: they are unconditionally stable and do not involve the typical numerical blurring of a finite difference scheme. However, the finite difference schemes with sufficiently small time steps do provide crossing preserving diffusion as well, and they are preferable for our extensions to non-linear adaptive diffusions proposed in Section 9. The crucial theoretical observations in our framework lie in the fact that the left-invariant evolution equations are expressed by a quadratic form in the left-invariant vector fields {Ai }6i=1 on R3 o S 2 embedded in SE(3), which form a moving frame of reference consisting of a spatial velocity part {A1 , A2 , A3 } and an angular velocity part {A4 , A5 , A6 }. This moving frame of reference requires the Cartan connection viewpoint13 on the underlying differential geometry and by expressing the left-invariant diffusions in covariant derivatives we see that even the adaptive non-linear left-invariant evolutions locally take place along the covariantly constant curves, which coincide with the exponential curves in SE(3). The spatial part of the exponential curves are circular spirals, i.e. curves in R3 with constant curvature and constant torsion. As a result our non-linear adaptive diffusion schemes allow adaptation for curvature and torsion. The general theory and results in this article apply to any engineering application where enhancement of (possibly crossing and/or bifurcating) fibers is required given an initial probability distribution on 3D-positions and orientations.

Acknowledgements. Vesna Prˇckovska en Paulo Rodriguez, biomedical image analysis group Eindhoven University of Technology are gratefully acknowledged for providing the artificial HARDI test images and the DTI tool supporting HARDI glyphs. Special thanks to Mark Bruurmijn and Paulo Rodriguez, Eindhoven University of Technology, for their support on the visualization and implementation of R3 o S 2 -convolutions. The Netherlands Organization for Scientific Research (NWO) is gratefully acknowledged for financial support. 13 Rather

than the torsion free Levi-Cevita connection on the Riemannian manifold (SE(3), (diag(µ2 , µ2 , µ2 , 1, 1, 1))ij dAi ⊗ dAj ).

42

A

Towards an Exact Analytic Formula for the Direction Process in 3D

Recall from Section 4.1 that the Fokker-Planck equation of the direction process (for contour completion) in R3 o S 2 is given by (40), which is a special case of (37) with D = diag{0, 0, 0, D44 , D55 = D44 } and a = (0, 0, 1, 0, 0). Now in order to derive its Green’s function (i.e. “impuls response”) we must set the initial condition as a single Dirac-delta at the unity element (0, ez ) ∈ R3 o S 2 . So we must derive ps (y, n, s) = et((A4 )

2

+(A5 )2 −A3 )

δ0 ⊗ δez

explicitly. To this end we should get a grip on the spectrum of the generator A24 + A25 − A3 , which is an unbounded operator on the space of HARDI images L2 (R3 o S 2 ). Analogously to our derivation of the exact Green’s function for contour completion in SE(2) we apply a spatial Fourier transform. So we would like to find the exact solutions of ˆ ˆ FR3 (D44 ((A4 )2 + (A5 )2 ) − A3 )FR−1 3 U = λU ,

(103)

ˆ (ω, n) = [FR3 Q(·, n)](ω). We use the following parametrization for the variable in Fourier domain where U π π ω = (ωx , ωy , ωz ) = ρ(sin β, − cos β sin γ, cos β cos γ) , with β ∈ [−π, π), γ ∈ [− , ], ρ = kωk > 0, 2 2 ˜ γ˜ ) ∈ S 2 given by (5). Now note that FA3 F −1 = iω · n and ˜(β, since it is consistent with our parametrization n = n thereby we can rewrite (103) as ˜ γ˜ )) = λ U ˜ γ˜ )) , where the generator in the spatial Fourier domain equals ˆ )(ω, n ˆ (ω, n ˜(β, ˜(β, (Bω U ˜ −γ D44 2 2 2 γ ˜ ˜ Bω = cos2 (β) ˜ ) − iρ cos(β − β) − 2 cos β cos β sin ˜ ) + D44 (∂β ˜ (∂γ 2

(104)

which is rather difficult to solve. Therefore, we apply the approximation cos β˜ = 1 + O(β˜2 ) ≈ 1 while maintaining ˜ This is somewhat similar to the Heisenberg-approximations, [23, ch:4.3] but this time all other dependencies on β. without destroying the periodicity. So we must solve

ˇ + D44 (∂ ˜ )2 − iρ cos(γ − γˇ ) − iρ cos β U ˜ γ˜ )) = λ U ˜ γ˜ )) , âpprox (ω, n âpprox (ω, n ˜(β, ˜(β, D44 (∂γ˜ )2 − iρ cos(β − β) β (105)

so for each ω ∈ R3 fixed, the operator on the left is a sum of two commuting Mathieu-operators, so we can apply the method of separation. To this end we write ˜ γ˜ )) = B(β)C(˜ ˜ âpprox (ω, n ˜(β, U γ) , So by the periodicity constraints we find ! ˜ B 00 (β) C 00 (˜ γ) ˜ + cos(γ − γ˜ )) = λ1 + λ2 = λ + iρ cos β , D44 + − iρ(cos(β − β) ˜ C(˜ γ) B(β) where the eigen-value λ of the generator equals the sum of −iρ cos β and the separations constants λ1 , λ2 . So we have the following Sturm-Liouville-type of system for C: 00 C (˜ γ ) − i Dρ44 cos(˜ γ − γ)C(˜ γ ) = λ2 C(−π) = C(π) where we evaluate C : [−π, π] → R later only from [− π2 , π2 ] whose the countable eigen-solutions are given by 1 γ˜ − γ 2ρ i 2ρ i n Cn (˜ γ) = cen , with eigenvalue λ1 = λ1 = an , n = 0, 1, 2, 3, . . . , 2π 2 D44 D44

43

where the function q 7→ an (q) denotes the Mathieu-characteristics, [42, 1] and where cen (·, q) denotes the cosine elliptic Mathieu-function (with Floquet exponent n), [42, 1], which is a π-periodic solution of Mathieu’s equation y 00 (z) + 2 q cos(2z)y(z) = an (q)y(z). Similarly, the eigensystem of 00 ˜ = λ1 ˜ − i ρ cos(β˜ − γ)B(β) B (β) D44 B(−π) = B(π) ˜ = is given by Bm (β)

1 cem 2π

˜ β−β 2ρ i ,D 2 44

with eigenvalue λm 1 = am

2ρ i D44

, m = 0, 1, 2, 3, . . . ,. Consequently, a com-

14

plete set of bi-orthogonal eigenfunctions (with eigen values λmn ) of the generator Bω is given by 2ρ i β−β 2ρ i ˜ 2ρ 2ρ am D cen γ˜ −γ +an D 2 , D44 cen 2 , D44 44 44 ˜ ˆ ˜(β, γ˜ )) = Umn (ω, n and λmn (ω) = −iρ cos β − < 0. 4π 2 4 44 =D55 ,a3 =1 44 =D55 ,a3 =1 Consequently, we have the following approximation formula for pˆD = FR3 pD : s s

F R3

e

t((A4 )2 +(A5 )2 −A3 )|

˜ cos β=1

∞ X ∞ X

˜ γ˜ )) = ˜(β, (ω, n

δ0 ⊗ δez

˜ γ˜ ))U ˜ γ˜ )). ˆmn (ω, n ˆmn (0, n ˜(β, ˜(β, et λmn (ω ) U

(106)

m=0 n=0

SE(2),D

44 Now this expression is quite similar to one of our exact formulas for the (spatial) Fourier transform FR2 pt SE(2),D44 : (SE(2)\{e}) → R+ of the direction process in SE(2). In fact we have the of the Green’s function pt following intriguing geometrical relation

FR3 et((A4 )

2

+(A5 )2 −A3 ) SE(2),D44

= e−iRt (FR2 pt

with ρ =

)

˜ γ˜ )) ≈ FR3 ˜(β, δ0 ⊗ δez (ω, n iˇ γ

ω ρ ωz ρ , − Ry , e R

e

SE(2),D44

(FR2 pt

t((A4 )2 +(A5 )2 −A3 )|

˜ cos β=1

ˇ

δ0 ⊗ δez

˜ γ˜ )) = ˜(β, (ω, n (107)

) R, ωx , eiβ ,

q q ωx2 + ωy2 + ωz2 and R := ωy2 + ωz2 , which allows us to use efficient algorithms [23, 6] for the compu-

tation of the Green’s function of the contour completion process in SE(2) for the computation of the approximation (107) of the Green’s function of the contour completion process in SE(3).

B

Stability and Step Size of the Finite Difference Scheme

In this section we will derive the upper bound for the step size (90) that guarantees the stability of the finite difference scheme explained in Subsection 7.2. We shall use (88) and (89) as starting point for our Euler-forward discretization of evolution (87). For each q ∈ N∪{0}, q we define the vector wq = (wy,k )k=1...No ,y∈I , where k enumerates the number of samples on the sphere, where y enumerates the samples on the discrete spatial grid I = {1, . . . , N } × {1, . . . , N } × {1, . . . , N } and where q enumerates the discrete time frames. Rewrite (88) and (89) in vector form wq+1 = (I − ∆t(JS2 + JR3 ))wq , where the angular increments block-matrix equals JS2 = 0 ⊕ D44 Λ−1 M T QM Λ ∈ R(3N +No )×(3N +No ) , with Q = diag{Qjj } = diag{l(j)(l(j)+1)e−treg l(j)(l(j)+1) },

14 Here

bi-orthogonality means that the conjugation is omitted in the L2 inner-product, i.e.

ˆm0 n0 , U ˆmn U

L2 ([−π,π]×[0,2π])

R π R 2π −π

0

˜ γ˜ )U ˜ γ˜ )d˜ ˆm0 n0 (β, ˆmn (β, U γ dβ˜ =

= δmm0 δnn0 , which follows from the fact that (Bω )∗ φ = Bω φ for all φ ∈ H2 ([−π, π] × [0, 2π]).

44

according to the second approach in (86) and where the spatial increments block matrix equals JR3 = h−2 (D11 B1 ⊕ D11 B2 ⊕ D33 B3 ) ⊕ 0 ∈ R(3N +No )×(3N +No ) with spatial stepsize h and Bi ∈ RN ×N standard sparse 3-band matrices associated to stencil (−1, 2, −1) with cyclic boundary conditions, according to (88). Both JS 2 and JR3 are positive semi-definite. We have guaranteed stability iff kI − ∆t(JS 2 + JR3 )k = supkwk=1 k(I − ∆t(JS 2 + JR3 ))wk < 1, which is by I − ∆t(JS 2 + JR3 ) = (νI − ∆tJR3 ) + ((1 − ν)I − ∆tJS 2 ), with ν ∈ (0, 1), the case if kI −

∆t ∆t JR3 k < 1 and kI − JS 2 k < 1 ⇒ k(νI − ∆tJR3 ) + ((1 − ν)I − ∆tJS 2 )k < ν + (1 − ν) = 1. (108) ν 1−ν

Sufficient (and sharp) conditions for the first inequality are obtained by means of the Gerschgorin Theorem [29] : ∆t ≤

νh2 . 4D11 + 2D33

(109)

1 ∆tJS 2 ) Sufficient and sharp conditions for the second equality are obtained by considering the spectrum σ(I − 1−ν 1 † † of symmetric matrix I − 1−ν ∆tJS 2 and by M M ≈ I (recall Figure 6) and M M → I as nSH → ∞ we have 1 1 ∆tJS 2 ) = σ(I − 1−ν D44 ∆t Λ−1 M T QM Λ) σ(I − 1−ν 1 → σ(I − 1−ν D44 ∆t Q) as nSH → ∞.

Now in general σ(I − ∆t ≤

2(1−ν) D44 max(Qjj )

1 2 1−ν ∆tJS )

≈ σ(I −

1 1−ν D44 ∆t Q)

(110)

and thereby the spectrum is contained in (−1, 1) iff

where the maximum of the diagonal elements Qjj equals max(Qjj ) =

L(L + 1)e−treg L(L+1) (e treg )−1

since 0 < x 7→ xe−treg x attains its only maximum at x =

1 treg .

if treg · L(L + 1) < 1 else

(111)

Now (108), (109), (110) and (111) imply result (90).

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