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Digital Object Identifier (DOI) 10.1007/s00791-004-0129-0 Comput Visual Sci 6: 197–209 (2004) Computing and Visualization in Science Regular article Morphological image sequence processing Karol Mikula 1 , Tobias Preußer 2 , Martin Rumpf 2 1 Department of Mathematics, Slovak University of Technology, Bratislava, Slovakia (e-mail: [email protected]) 2 Fakulty 4, Institute for Mathematics, Duisburg University, 47057 Duisburg, Germany (e-mail: [email protected], [email protected]) Received: 1 December 2002 / Accepted: 15 April 2003 Published online: 24 February 2004 – Springer-Verlag 2004 Communicated by: J. Kaˇ cur Abstract. We present a morphological multi-scale method for image sequence processing, which results in a truly coupled spatio-temporal anisotropic diffusion. The aim of the method is not to smooth the level-sets of single frames but to de- noise the whole sequence while retaining geometric features such as spatial edges and highly accelerated motions. This is obtained by an anisotropic spatio-temporal level-set evo- lution, where the additional artificial time variable serves as the multi-scale parameter. The diffusion tensor of the evolu- tion depends on the morphology of the sequence, given by spatial curvatures of the level-sets and the curvature of trajec- tories (= acceleration) in sequence-time. We discuss different regularization techniques and describe an operator splitting technique for solving the problem. Finally we compare the new method with existing multi-scale image sequence pro- cessing methodologies. 1 Introduction During the last decade scale-space methods have proven to be useful in image processing, including image denoising, edge enhancement and shape recovery from noisy data [1, 25, 33, 38]. A given image is thereby considered as initial data to some suitable evolution problem. The artificial time param- eter acts as the scale parameter, which guides the user from noisy fine scale representations to enhanced and coarse scale representations of the original image. Within many applications not only single images but whole image sequences are of particular interest. The ob- served time period thereby ranges from a few seconds to days, months and years. In medical image processing recent acquisition hardware such as ultrasound (US), magnetic res- onance imaging (MRI) and computed tomography imaging (CT) enable for an observation of e.g. the human heart during a cardiac cycle, the flow of a tracer through blood vessels, or the growth of tumors. These image sequences and especially ultrasound data are characterized by high frequent noise typ- ically due to measurement errors of the underlying imaging device. The particular interest in medical applications is un- derstanding of growth and flow phenomena of tissue and the quantitative volume change in time (e.g. blood volume in the heart). Thus one often is interested in the extraction of certain level-surfaces from the data which bound volumes or separate regions of interest. Moreover the extraction of the velocities describing the motion of the level-sets in the sequence, the so called optical flow, is desired. The aim of this paper is to discuss a new anisotropic morphological method for the denoising of image sequences. The presented model takes the curvature of level-sets in space into account as presented in an previous paper [28], and hence is capable of preserving edges and corners on the level-sets. Moreover the method takes into account the velocity in whose direction the level-sets move within the image sequence and finally the acceleration of the level- sets which characterizes this motion in sequence-time. Let us emphasize that the resulting evolution is a truly coupled anisotropic spatio-temporal smoothing process which treats the image sequence as a unit and not as a compilation of sin- gle frames. The paper is organized as follows: First, in Sect. 2 we discuss some background work on image processing, image sequence processing and the optical flow problem. In Sect. 3 we review an anisotropic level-set diffusion model for the processing of single frames. This further motivates the mod- eling of the final evolution. Before we give a detailed descrip- tion of the new model in Sect. 5, we will have to discuss the extraction of motion velocities from given image sequences in Sect. 4. In Sect. 6 we discuss the robust evaluation of cur- vatures on level-sets and the discretization by finite elements. Before we draw conclusions in Sect. 8, we would like to com- pare the new method with existing image sequence process- ing methodology in Sect. 7. In the Appendix we give further details on the spatio-temporal discretization. 2 Related work Scale Space methods in image processing define an evolution operator E(t) which acts on initial data u 0 and delivers a scale
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Page 1: Morphological image sequence processing - Math · Morphological image sequence processing 199 3.1 The shape operator Since our goal is a morphological multi-scale model, we need a

Digital Object Identifier (DOI) 10.1007/s00791-004-0129-0Comput Visual Sci 6: 197–209 (2004) Computing and

Visualization in Science

Regular article

Morphological image sequence processing

Karol Mikula1, Tobias Preußer2, Martin Rumpf2

1 Department of Mathematics, Slovak University of Technology, Bratislava, Slovakia(e-mail: [email protected])

2 Fakulty 4, Institute for Mathematics, Duisburg University, 47057 Duisburg, Germany(e-mail: [email protected], [email protected])

Received: 1 December 2002 / Accepted: 15 April 2003Published online: 24 February 2004 – Springer-Verlag 2004

Communicated by: J. Kacur

Abstract. We present a morphological multi-scale method forimage sequence processing, which results in a truly coupledspatio-temporal anisotropic diffusion. The aim of the methodis not to smooth the level-sets of single frames but to de-noise the whole sequence while retaining geometric featuressuch as spatial edges and highly accelerated motions. Thisis obtained by an anisotropic spatio-temporal level-set evo-lution, where the additional artificial time variable serves asthe multi-scale parameter. The diffusion tensor of the evolu-tion depends on the morphology of the sequence, given byspatial curvatures of the level-sets and the curvature of trajec-tories (= acceleration) in sequence-time. We discuss differentregularization techniques and describe an operator splittingtechnique for solving the problem. Finally we compare thenew method with existing multi-scale image sequence pro-cessing methodologies.

1 Introduction

During the last decade scale-space methods have proven to beuseful in image processing, including image denoising, edgeenhancement and shape recovery from noisy data [1, 25, 33,38]. A given image is thereby considered as initial data tosome suitable evolution problem. The artificial time param-eter acts as the scale parameter, which guides the user fromnoisy fine scale representations to enhanced and coarse scalerepresentations of the original image.

Within many applications not only single images butwhole image sequences are of particular interest. The ob-served time period thereby ranges from a few seconds todays, months and years. In medical image processing recentacquisition hardware such as ultrasound (US), magnetic res-onance imaging (MRI) and computed tomography imaging(CT) enable for an observation of e.g. the human heart duringa cardiac cycle, the flow of a tracer through blood vessels, orthe growth of tumors. These image sequences and especiallyultrasound data are characterized by high frequent noise typ-ically due to measurement errors of the underlying imaging

device. The particular interest in medical applications is un-derstanding of growth and flow phenomena of tissue and thequantitative volume change in time (e.g. blood volume in theheart). Thus one often is interested in the extraction of certainlevel-surfaces from the data which bound volumes or separateregions of interest. Moreover the extraction of the velocitiesdescribing the motion of the level-sets in the sequence, the socalled optical flow, is desired.

The aim of this paper is to discuss a new anisotropicmorphological method for the denoising of image sequences.The presented model takes the curvature of level-sets inspace into account as presented in an previous paper [28],and hence is capable of preserving edges and corners onthe level-sets. Moreover the method takes into account thevelocity in whose direction the level-sets move within theimage sequence and finally the acceleration of the level-sets which characterizes this motion in sequence-time. Letus emphasize that the resulting evolution is a truly coupledanisotropic spatio-temporal smoothing process which treatsthe image sequence as a unit and not as a compilation of sin-gle frames.

The paper is organized as follows: First, in Sect. 2 wediscuss some background work on image processing, imagesequence processing and the optical flow problem. In Sect. 3we review an anisotropic level-set diffusion model for theprocessing of single frames. This further motivates the mod-eling of the final evolution. Before we give a detailed descrip-tion of the new model in Sect. 5, we will have to discuss theextraction of motion velocities from given image sequencesin Sect. 4. In Sect. 6 we discuss the robust evaluation of cur-vatures on level-sets and the discretization by finite elements.Before we draw conclusions in Sect. 8, we would like to com-pare the new method with existing image sequence process-ing methodology in Sect. 7. In the Appendix we give furtherdetails on the spatio-temporal discretization.

2 Related work

Scale Space methods in image processing define an evolutionoperator E(t) which acts on initial data u0 and delivers a scale

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198 K. Mikula et. al.

of representations E(t)u0t≥0. The time parameter t servesas the scale parameter that guides from fine scales on the ini-tial data (t = 0) to successively coarser and smoother scales.Throughout this paper we will always denote the multi-scaleparameter by t whereas – to avoid any confusion – for thesequence-time parameter we will use s, which represents timein the image-sequence data.

The simplest linear image processing model given by theheat equation ∂tu −∆u = 0 with the noisy image u0 as ini-tial data leads to smooth images but also destroys edges in theimage, indicated by high gradients. The proposal of Peronaand Malik [24] and the modification of Cattè et al. [7] avoidsthis drawback considering an evolution problem

∂tu −div (G(|∇uσ |)∇u)= 0 ,

where the diffusion coefficient depends on the magnitude ofthe gradient of a (regularized) version of the actual image u.Here, uσ = Kσ ∗ u is the convolution of the image witha Gaussian kernel Kσ of variance σ > 0. In contrast to theoriginal Perona/Malik model (σ = 0) the regularization turnsthis model into a mathematically well posed problem andmoreover it avoids the detection and accentuation of artificialedges which are due to noise. A suitable choice for the dif-fusion coefficient is G(s) = (1+ s2/λ2)−1 for some λ > 0. Atleast formally, decreasing the diffusion coefficient in areas ofhigh gradients then results in a backward diffusion and thusan enhancement of edges, whereas areas of low gradients aresmoothed in an isotropic way. The method was improved byWeickert [37] who took anisotropic diffusion into account.Thereby the diffusion is of original Perona/Malik respec-tively Cattè et al. type in directions of the image gradient(i.e. orthogonal to level-sets) and of linear type in directionstangential to level-sets. This leads to an additional smooth-ing tangential smoothing of level-sets and enables to amplifyintensities or correlations along level-sets. In [26] Preusserand Rumpf applied this type of anisotropic diffusion to visu-alize arbitrary vector fields. Convergence of a finite elementmethod and finite volume methods were shown by Kacur andMikula [18] and Mikula and Ramarosy [22]. Furthermoreadaptivity was considered in [5, 19, 27].

In the axiomatic work of Alvarez et al. [1] general non-linear evolution equations were derived from a set of axioms.Including the axiom of gray value invariance (i.e. the model issupposed to be invariant under monotone transformations ofthe gray value) lead to a curvature evolution model. Curvaturemotion has been studied intensively in geometry and physics,where interfaces are driven by surface tension [4, 34]. Al-ready in the basic model for mean curvature motion

∂tu −|∇u|div (∇u/|∇u|)= 0 ,

singularities in the evolution may occur. In this setting exis-tence of viscosity solutions has been shown independently byEvans and Spruck [13] and Chen et al. [8]. Anisotropic cur-vature motion has for instance been studied by Belletini andPaolini [6]. Moreover Sapiro proposed a modification of themean curvature motion model which takes into account theimage gradient magnitude [31].

The detection of motion in image sequences, also knownas the optical flow problem, is one of the fundamental tasksin computer vision and image processing. For two dimen-

sional (2D) images it has been studied extensively in thepast [2, 3, 12, 23, 30]. The velocity of a level-set splits up intoa component normal to the level-set and a component tangen-tial to it. The extraction of the tangential velocity is in generalnot well posed [30]. Thus, one has to restrict the set of pos-sible solution velocities and instead work with the apparentvelocity [15], which arises from locally constant translationsin space. As an alternative one might ask for regularizations interms of elastic stresses or viscous fluid effects [9–11, 14, 17,20, 35], which is computationally expensive and mostly paysoff in cases of large deformations in between frames of thesequence, which we rule out in our applications consideredhere.

The image processing models discussed above do not im-mediately apply to image sequence processing. Since thereis no coupling between successive frames of the sequencein any of the approaches, it is only possible to process thesequence as a collection of steady-images. Still this lacksa correlation of the smoothed versions of the single frames.Therefore modifications of the standard image processingmethods have to be taken into account, which introducea coupling between the frames of the sequence in terms ofthe velocity or acceleration of the sequence. In the 2D moviemulti-scale analysis [1, 15] an evolution equation was derivedfrom a set of axioms, which depends on the curvature (givenin terms of the eigenvalues and eigenvectors of the shape-operator S, cf. Sect. 3) of level-sets and the acceleration of themotion:

∂tu −|∇u|F(t, S, accel) .

This forms the base for the approaches presented by Sartiet al. in [32] and Mikula et al. in [21]. In Sect. 7 the lat-ter will be compared to the method being presented in thispaper.

3 Review of anisotropic level-set diffusion insteady-image processing

In this section we will briefly review an anisotropic level-setdiffusion model which was originally presented in [28]. Ascommon for level-set models, we deal simultaneously withall level-sets. Although in certain applications our interest isfocused on one specific implicit surface, possibly in advanceconverted from a parametric to an implicit representation.

Let us denote by u0 : Ω → R the gray value function ofthe initial image with inscribed level-sets

Mc0 := x ∈ Ω | u0(x) = c .

We assume u0 and the set of corresponding implicit sur-faces Mc

0c to be noisy and ask for a family of successivelysmoothed images u(t, ·) | t ∈R+

0 where u(t, ·) : Ω →R andu(0, ·) = u0(·). Throughout this paper Ω will always be theunit square or cube [0, 1]d, d = 2, 3. The variable t serves asthe scale parameter. Thereby, for each gray value c a familyof surfaces Mc

t t∈R+0

is generated, with Mc(t=0) = Mc

0. Herewe assume u(·, ·) to be sufficiently smooth and ∇u(t, x) = 0for all (t, x) ∈ R+

0 ×Ω. Indeed, due to the implicit functiontheorem the corresponding sets Mc

t then are actually smoothsurfaces.

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Morphological image sequence processing 199

3.1 The shape operator

Since our goal is a morphological multi-scale model, weneed a characterization of the level-set geometry on im-ages. To this end let us consider the normal to a level-setN(x) := ∇u

|∇u| of some image u. We denote the tangent spaceby TxM := (spanN(x))⊥. We compute the Jacobian of thenormal

DN = (Id− N ⊗ N)D2u

|∇u|on R3 and consider the restriction

S := DN(Id − N ⊗ N)

on the tangent space TxM. S is a symmetric mapping andon the tangent space TxM it coincides with the Shape Op-erator STxM. Therefore S is characterized by the eigenval-ues κ1, κ2, 0 and the eigenvectors v1, v2, N. The eigenval-ues κi correspond to the principal curvatures of the level-setand the eigenvectors vi are the principal directions of curva-ture. Thus, the geometry of the level-set is determined by Svia its eigenvalues and eigenvectors.

3.2 The anisotropic level-set model

We consider the following type of nonlinear boundaryand initial value problem on Ω: Given an initial imageu0 : Ω → R find a family of images u(t, ·) : Ω →Rt∈R+which obey the following anisotropic evolution equation

∂tu −|∇u|div(

aσTxM

∇u

|∇u|)

= 0 in R+×Ω ,

aσTxM

∂u

∂ν= 0 on R+×∂Ω ,

u(0, ·) = u0(·) in Ω , (1)

where ν denotes the outer normal to Ω. The anisotropic geo-metric level-set diffusion model should depend on the geom-etry of the level-sets. Thus it is natural to base the definitionof the diffusion coefficient aσ

TxM on a regularized version Sσ

of the shape operator S. We assume this regularized versiondiagonalizes with respect to the basis v1,σ , v2,σ , Nσ havingeigenvalues κ1,σ , κ2,σ , 0. We then consider the scalar func-tion G(s) := (1 + s2/λ2)−1 from the basic image processingmodels now acting on Sσ . In matrix representation we thusobtain

aσTxM := aσ

TxM(Sσ ) = BTσ

G(κ1,σ )

G(κ2,σ )0

Bσ ,

where Bσ = (v1,σ , v2,σ , Nσ )T , i.e. the basis transformationfrom the regularized frame of principal directions and the nor-mal v1,σ , v2,σ , Nσ onto the canonical basis e1, e2, e3.

Let us recall that in the function G the parameter λ actsas a steering parameter for the detection of edges. For largervalues of λ, more features on a level-set will be regarded asedges. In the standard Perona Malik model the value λ is ex-actly the switch between forward and backward diffusion.

Remark 1. Although we have based this short review on 3Dimages and therefore level-sets which are 2D-surfaces, wewill present examples of 2D-image-sequences in later sec-tions. The definition of the diffusion tensor of the anisotropicdiffusion tensor for 2D images then obviously has the formBT

σ diagG(κσ), 0Bσ , where κσ is the regularized curvatureof the level-lines.

4 Extracting motion velocities from image sequences

Let us from now on assume, we are concerned with an imagesequence. At first, we consider a continuous family of im-ages on some time interval [0, T ] each image again defined onΩ = [0, 1]d, d = 2, 3, which we will denote by

u : Q → R , (s, x) → u(s, x) ,

- where Q denotes the sequence-time/space cylinder Q :=[0, T ]×Ω. Here and in the following s always denotes thesequence-time parameter and x as before spatial coordinates.Again the perspective of level-sets will play a central role inour model. As before we denote

Mc(s) = x ∈ Ω | u(s, x) = c,

N(x, s) = ∇u(s, x)

|∇u(s, x)| if |∇u(s, x)| = 0 ,

the level-set of u(s, x) to level-value c ∈ R respectively thenormal to this level-set, which now depend on the sequence-time s. Hence we have families of level-sets Mc(s)c∈Rwhich change in sequence-time. Assuming there is some cor-respondence between consecutive images in the sequence(i.e. the sequence is continuous in sequence-time), it willbe an essential part of the new model, to extract the under-lying motion, which influences the observed image inten-sity. Before proceeding to the description of the new time-space coupled smoothing model, we therefore will brieflyfocus on the extraction of these motion-velocities from theimage-sequence. A more detailed discussion can be foundin [29].

Suppose

v : Q →Rd , (s, x) → v(s, x)

is the velocity field generating the motion in space and time.Therefore a single motion trajectory is described by x(s)with

∂sx(s) = v(s, x(s)) .

It is obvious that this optical flow problem – the extraction ofv from the image data – is an ill posed problem: Any tangen-tial motion, that only moves one level-set within itself cannotbe captured by the process. Nevertheless following two as-sumptions will allow us to derive a formula for the so calledapparent velocity:

(A1) Intensities are preserved along motion trajectories:

u(s0, x(s0)) = u(s0 + τ, x(s0 + τ))

− s0 ≤ τ ≤ T − s0 .

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200 K. Mikula et. al.

This assumption is reasonable since it is related to theinvariance of the image acquisition device, which usu-ally measures physical quantities. If this physical quan-tity moves in time, so does the corresponding imageintensity.

(A2) Locally the underlying motion is a translation:

N(s0, x(s0)) = N(s0 + τ, x(s0 + τ))

− s0 ≤ τ ≤ T − s0 .

This assumption is of course fulfilled, assuming thescenery consists of solid objects moving in space.

Differentiating these assumptions with respect to τ and eval-uating at τ = 0, we get the following two expressions forv = vn N +vtg

vn = v · N = − ∂su

|∇u| if |∇u| = 0 , (2)

vtg = −S−1∂s N . (3)

Equation (2) is an expression for the normal componentvn N = v · NN of the velocity. For equation (3) we rememberthat the Shape Operator S operates on the tangent space TxMand ∂s N ∈ TxM. Adding the two parts we obtain the apparentvelocity

vapp := vn +vtg = −(

∂su

|∇u| N + S−1∂s N

). (4)

In 2D this formula was already obtained by Guichard [15, 16]although he did not explicitly express it in terms of the in-trinsic Shape Operator. From (3) we again see the limitationsof the tangential motion capturing, because it involves the in-verse of the Shape Operator, which of course may not exist.

Fig. 1. From an image sequence, taken by an ultrasound device, and showing the left ventricle of the human heart during one cardiac cycle we have extractedthe velocities of the underlying motion. From top left to bottom right for successive frames of the sequence one iso-surface of the muscle of the heart isdepicted. The coloring codes the normal velocity going inward (red) or outward (blue). Since the tissue of the heart’s muscle does not allow for tangentialmovements it is sufficient to consider the normal velocity in this application

Clearly our assumption impose restrictions on the types ofmotion fields, which can be extracted. Especially in case oflarge deformations the optical flow field can be very compli-cated. We restrict our motion analysis to small deformationsbetween the successive frames of the sequence. For these con-figurations our assumptions enable to approximate the motionfields. Moreover, in many physical applications it will be suf-ficient to have the normal velocity vn, if the observed processgives reason that vtg = 0. For example in porous mediumflow we already know from the physical model, that the flowwill be in direction of the pressure gradient, which in sim-ple settings will be the normal to the level-sets. Also in thesituation depicted in Fig. 1 we conclude that the normal vel-ocity is sufficient to characterize the motion since the tissue ofthe human heart will not allow for tangential motions. Therewe have depicted the extraction of motion velocities from animage sequence showing one ventricle of the human heartduring a cardiac cycle. Moreover Fig. 2 shows the extractionof the velocity from an artificial data set, in which ellipsoidallevel-sets change their half-axes in time.

Given the apparent velocity we can furthermore computethe acceleration of the motion, which is equivalent to the cur-vature of the apparent trajectory, resulting from the apparentvelocity (cf. [15, 16])

accel(s, x) := ∂τvapp(s + τ, x(s + τ))

∣∣∣τ=0

= ∂svapp + (∇vapp) vapp . (5)

In Fig. 1 we have depicted the extraction of motion vel-ocities from an image sequence showing one ventricle of thehuman heart during a cardiac cycle.

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Morphological image sequence processing 201

Fig. 2. As a test case we extract the motion from the evolution of ellipsoidal level-sets with oscillating half axes. I.e. we consider the image sequenceφ(s, x1, x2, x3) := x2

1/a(s)+ x22/b(s)+ x2

3 , where a(s) := 4s − (1− s), b(s) := s −4(1− s) for s ∈ [0, 1]. We have depicted the results of the velocity com-putation on the same level-set (iso-surface) in different frames of the sequence. In the upper row a color ramp from blue (moving inward) to red (movingoutward) indicates the normal component of the velocity. In the lower row the color ramp from blue to red indicates the absolute value of the tangentialcomponent of the velocity

5 Coupled spatio-temporal anisotropic level-set diffusionin image sequence processing

We are now equipped to formulate the new coupled spatio-temporal anisotropic level-set diffusion model. We would liketo combine the good edge and corner preserving behavior ofthe model reviewed in Sect. 3 with an anisotropic smooth-ing in sequence-time in direction of the apparent velocity.To this end let us denote the sequence-time/space gradi-ent by ∇(s,x) := (∂s,∇) and the corresponding divergence bydiv(s,x) := ∂s +∇.

Given a noisy image sequence u0 : Q → R, we writedown the following spatio-temporal level-set problem:

Find u : R+ × Q → R such that in R+ × Q:

∂tu −|∇(s,x)u|div(s,x)

(Aσ ∇(s,x)u

|∇(s,x)u|)

= 0 . (6)

We impose the initial condition

u(0, ·, ·) = u0(·, ·) in Q →R ,

and furthermore one of the following boundary conditions

∇(s,x)u · ν(s,x) = 0 on R+ × ∂Q , (BC1)

∇u(t, s, ·) · ν = 0 on ∂Ω

u(·, 0, ·) = u(·, T, ·) in R+ and Ω ,

(BC2)

where ν(s,x) denotes the outer normal to the sequence-time/space cube Q and ν denotes the outer normal to ∂Ω.The two different boundary conditions have the followingmeaning. In (BC1) we prescribe generally natural boundaryconditions to the whole sequence, i.e. we have no flux across

the spatial boundary of the single frames and moreover noflux at the beginning and the end of the sequence. It maybe more convenient to impose natural boundary conditionsin space and periodicity in sequence-time which is stated in(BC2).

Again the variable t in the problem acts as the scale pa-rameter and we again emphasize that we make a distinctionbetween t and s; s denoting the sequence-time parameter. Thedefinition of the problem indeed increased the dimension ofthe data by one, which results in 4D respectively 5D problemsfor 2D respectively 3D image data. In the following sectionswe will describe how to solve these 4D respectively 5D prob-lems with moderate effort.

It remains to define the diffusion tensor Aσ for the newmodel. Denoting the tensor product by v⊗w := (viwj)ij , weconsider the normalized sequence-time/space velocity vec-tors V σ := (1, vσ

app)/|(1, vσapp)| based on regularized apparent

velocities vσapp, and the diffusion coefficient already known

from the steady image model to build

Aσ = aσv V σ ⊗ V σ +

(0 0

0 aσTxM(Sσ )

),

with aσv = G(|accelσ |). The function G(s) = (1 + s2/λ2)

again is the well known function from image processing(cf. Sect. 3). With this definition of the diffusion tensor weindeed prescribe a behavior of the evolution that is edge pre-serving in space but also smoothing the sequence nonlinearlyin direction of V σ . If the acceleration is high the diffusionwill be decreased via the function G. This leads to a goodpreservation of highly curved motion trajectories (i.e. highlyaccelerated motion) as shown in Fig. 3.

In general the decomposition in the definition of Aσ isnot orthogonal. Only if the complete apparent velocity vapp

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202 K. Mikula et. al.

Fig. 3. We show the evolution of single trajectories under the new coupledanisotropic evolution. From a noisy image sequence showing a ball bounc-ing at some invisible boundary, marked by the red line in the image (topleft) we have extracted two motion trajectories (top right). Clearly the vel-ocity is disturbed much and does not at all reflect the underlying motion.In the bottom row we see the same extracted trajectories from the fifth andninth scale step of the evolution. Clearly the non-accelerated motion hasbeen smoothed much, whereas the rapid velocity change in the middle ofthe sequence has been preserved very well

Fig. 4. From a sequence whose frames are piece-wise constant in space,we show the evolution of one single frame. The sequence shows a squarebouncing at some invisible object (top left) and we have depicted the third(top right) and the sixth (bottom left) scale step of the evolution. Fromthe magnified section around the square (bottom right) we clearly see, thatthe coupled diffusion smooths the data across successive slices in directionof the velocity (here diagonally from bottom left to top right). Althoughthe single frames therefore may loose sharpness of edges perpendicular tothe velocity, the diffusion makes the whole movie smoother. Obviously, theeffect is weaker if the sequence-step-width ∆s is smaller since then thepixel/voxel offset between slices is smaller

Fig. 5. From a sample data set showing an object bouncing at some solidobject, which here is depicted with a red line, we show the coupled multi-scale evolution (cf. Fig. 3). The image data is the continuous functionu0(s) = |x − d(s)|1 to which noise was added, where d(s) is the movingcenter of the object. The right column shows successive frames of thenoisy sequence, whereas the left column shows the same frames after thethird scale step of the evolution. We have extracted the sets u(s, x) ≤ 0.2and drawn them in black color. The computations were performed ona 129×129 grid

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Morphological image sequence processing 203

Fig. 6. From the 3D ultrasound image-sequence (cf. Fig. 1) we have ex-tracted a 2D image sequence showing a slice through the three-dimensionalvolume. From top to bottom successive frames of the sequence are depicted.The left column shows the original noisy data, whereas the right columnshows the result of the new coupled diffusion model after the third scalestep

vanishes, the diffusion tensor reduces to a diagonal ma-trix. Therefore in general we actually have a coupled dif-fusion, with a mixed spatio-temporal diffusion componentG(|accelσ |)V σ ⊗ V σ . This can be observed from the exampleshown in Fig. 4, where we see a diffusion across the sharpedge of the square in direction of the underlying velocity.

Figure 5 shows the evolution of a noisy sample data setunder the coupled diffusion model. The image-sequence con-sists of a continuous function whose level-sets were disturbedrandomly in normal direction. The application to real data isshown in Fig. 6 where we have extracted one slice of the 3Dechocardiographical heart image sequence (cf. Fig. 1).

Remark 2. Here and in the sequel we have denoted regular-ized quantities (like Sσ , vσ

app, accelσ ) with a superscript σ . Weemphasize that we do not distinguish between regularized ge-ometrical quantities and quantities based on regularized data,although they in general do not coincide. In the next sectionwe will focus on the type of regularization we choose in ourapplications.

6 Discretization and numerical solution

Up to now we have considered image-sequences to be suffi-ciently smooth in space and time Q. Since in the applicationsimage-sequences arise as a finite sequence of single images(the frames) consisting of arrays of pixels or voxels, we willdiscretize the model in an appropriate way. For each singleframe, we interpret the pixel/voxel values as nodal valueson a uniform quadrilateral respectively hexahedral mesh Ccovering the whole spatial domain Ω. Moreover since typ-ically the time offset ∆s between successive frames is con-stant in image sequences, we introduce an equidistant latticein the sequence-time direction. In any coordinate direction,we consider the data to be piece-wise multi-linear, meaningpiece-wise linear in sequence-time and piece-wise bilinear re-spectively trilinear in space. To simplify the notation, we willalways denote discrete quantities by upper case letters to dis-tinguish them from their continuous correspondence in lowercase letters.

6.1 Shape operator and apparent velocity on discrete data

The model presented above makes extensive use of regular-ized geometric quantities such as the shape operator Sσ andthe apparent velocity vσ

app. It is obvious that on noisy image-sequence data a regularization is necessary, but also the defin-ition of these quantities involving higher order derivatives onimages which are usually piece-wise constant or rarely givenas bilinear respectively trilinear data is not clear. We willtherefore in the following focus on these regularized geomet-ric quantities.

For the regularization of the underlying images we havedifferent methods at hand:– The simplest non-morphological regularization method,

which is quite standard in image processing is the convo-lution of the image with a Gaussian kernel. Thereby onesolves a short time step of the heat equation

∂tφ−∆φ = 0 on Q ,

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204 K. Mikula et. al.

with the given image u as initial value to the problem.– The morphological analogue of the Gaussian convolution

is the mean curvature evolution, which lets all level-setssimultaneously evolve in direction of their normal witha speed according to their mean curvature. The corres-ponding level-set evolution would be

∂tφ−|∇φ|div( ∇φ

|∇φ|)

= 0 on Q ,

again with the given image u as initial value for thisparabolic problem.

Both approaches implemented numerically regularize thedata, but we are still left with the problem of defining higherorder derivatives on piece-wise multi-linear image data.A better approach would be the one used in [28]:– For each (s, x) ∈ Ω take Bσ (s, x) to be some neighbor-

hood of (s, x). Furthermore let P be some polynomialspace of degree greater than one. Now compute the localL2 projection φ of the image data onto P , i.e. solve∫

Bσ (s,x)

(u(r, y)−φ(r, y)) q(r, y) dr dy = 0 ∀q ∈ P .

Finally the Shape operator can be computed from thederivatives of the projection.

This last approach is consistent, but also computational veryexpensive, since a lot of integrations and inversions of smalllinear systems have to be performed. In the sequel we willtherefore describe a third regularization variant (cf. [22])which is based on convolutions with symmetric smoothingkernels Kσ ∈ C∞, but additionally uses the convolution prop-erty for any derivative Dα

Dα (Kσ ∗u) (s, x) = Dα

∫Kσ (r, y)u(s − r, x − y) dr dy

= Kσ ∗ Dαu(s, x) = Dαx Kσ ∗u(s, x) .

Since for computations it is essential to have kernels withcompact support, we choose

Kσ (s, x) = 1

Ze(|x|2+s2

)/(|x|2−s2−σ2

)

having support in Bσ (s, x) around (s, x). The constant Z ischosen such that

∫Kσ = 1. We replace the convolution as

usual with a weighted summation

(Dα Kσ ∗u)(s, x) =∫

Dα Kσ (r, y)u(s − r, x − y) dr dy

=∑

E⊂Bσ

u(sE)

∫E

Dα Kσ (r, y) dy

over the values of u at the center sE of the involved elem-ents E. The weights are thus obtained by integrating thederivatives of Kσ over the elements E ⊂ Bσ (s, x) and there-fore can be precomputed in advance. Thus we are nowequipped with weights for the computation of the derivatives∂s,∇, D2, on discrete data represented by piece-wise multi-linear functions on the elements of the sequence-time/spacegrid.

In the above formulas for the computation of vapp we haveassumed that ∇u = 0 and moreover we have made use of theinverse of the shape operator (Sσ )−1. In general we cannotguarantee that ∇u = 0 during the evolution even if the initialdata fulfills this assumption (cf. [13]). We therefore have tofurther regularize the problem by substituting

| · | | · |ε :=√

| · |2 + ε2

as proposed by Evans and Spruck in [13]. Moreover in areaswhere the image is flat at least in one direction (i.e. κi,σ = 0for some i), we replace the inverse (Sσ )−1 by the pseudo-inverse (Sσ )†, by inverting Sσ only on spanvi,σ |κi,σ = 0 andextending it trivially again to TxM. These are the areas, wherewe cannot expect the tangent part vtg of the velocity to containany information, since the image is flat.

Finally, we obtain the following formulas for the Shapeoperator Sσ , the apparent velocity vσ

app and the accelerationaccelσ :

Nσ (s, x) = ∇Kσ ∗u

|∇Kσ ∗u|ε (s, x) ,

Sσ (s, x) = 1

|∇Kσ ∗u|ε(

D2 Kσ ∗u −

Nσ ⊗ (D2 Kσ ∗u)Nσ)(s, x) ,

vσapp(s, x) = −

(∂s Kσ ∗u

|∇Kσ ∗u|ε Nσ +

(Sσ )†(∂s Kσ ∗ Nσ )

)(s, x) ,

accelσ (s, x) =((∂s Kσ ∗vapp)+

(∇Kσ ∗vapp)vapp

)(s, x) .

6.2 An operator splitting scheme

The coupled problem (6) is a 4D respectively 5D problem for2D respectively 3D image sequences. We will in the sequelpresent an operator splitting like scheme which uses appro-priate quadrature rules to simplify the solution approach. Weare in favour of using finite elements, since they are known toresolve anisotropic smoothing directions in a better way thanfinite differences. Especially in image processing the use ofquadrilateral or hexahedral elements is as efficient as the usedifference schemes, since the elements and the data structuresare completely aligned with the pixel/voxel structure. For thehigh dimensional problem we are considering here, we firstarrange space-time finite elements, but simplify the temporalpart in form of a block solver. Therefore, we retain the goodanisotropy resolving behavior of the spatial finite elements,while having a finite difference like scheme for the temporalparts. So in the sequel we will present an operator splittinglike scheme which uses appropriate quadrature rules to sim-plify the solution approach.

We start with the weak formulation of the coupled prob-lem. To this end we discretize in time by a semi-implicitbackward Euler scheme, denoting the scale step by ∆t andwriting un(s, x) = u(n∆t, s, x). We furthermore test the prob-lem with a function ψ ∈ C∞(Q) and integrate by parts over Qto obtain the time-discrete problem:

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Morphological image sequence processing 205

Find a family (un)n>0, un : Q → R such that:

(un+1 −un

∆t|∇(s,x)un| , ψ)

Q

−(

Aσ,n ∇(s,x)un+1

|∇(s,x)un| ,∇(s,x)ψ

)Q

= 0

where (·, ·)Q denotes the L2 scalar product on Q and

u0(·, ·) = u0(·, ·) in Q .

The semi-implicity of this scheme results in the evaluation ofthe nonlinearities at the old time step, i.e. at scale step n +1we take into account Aσ,n and |∇(s,x)un|.Remark 3. With the definition of Aσ,n we have for two func-tions φ,ψ

Aσ,n∇(s,x)φ ·∇(s,x)ψ

= aσ,nv (V σ ·∇(s,x)φ)(V σ ·∇(s,x)ψ)+

aσ,nTxM∇φ ·∇ψ

= aσ,nv (∂sφ+vσ

app ·∇φ)(∂sψ +vσapp ·∇ψ)+

aσ,nTxM∇φ ·∇ψ

=: aσ,nv

D

∂sφ

D

∂sψ +aσ,n

TxM∇φ ·∇ψ

where D∂s = ∂s +vσ

app ·∇ = V σ ·∇(s,x) is the material derivativealong the apparent motion trajectories.

We proceed with the discretization by formally introduc-ing multi-linear tensor product finite elements on the discretedomain [0, T ]×Ω. To this end let us denote the nodes in se-quence time direction s by Latin indices i, j ∈ 0, . . . , M andthe spatial degrees of freedom by Greek multi-indices α, β ∈0, . . . , N3. We then have the notation Un = (Un

0 , . . . , UnM)

Fig. 7. The matrix of the system splits up into various parts. The sequence-time/space matrix (upper row, left term) and the spatial stiffness matrix (upperrow, right term) lead to diagonal blocks in the resulting scheme, whereas the coupled diffusion terms (upper row, middle term) futher split up (inner box, cf.Appendix A)

Table 1. Super- and subscripts in the discretized coupled diffusion problem

t ∈ R+0 Scale

s ∈ I Sequence time coordinatex ∈ Ω Spatial coordinate

n ∈N Scale stepi, j = 0, . . . , M Temporal node index (sequence frame)α, β = 0, . . . , N Spatial node index (image voxel)

∆t Scale-step-width∆s Temporal grid width (sequence-step-width)∆x Spatial grid width

for the nodal values of the nth scale step, where Uni = Un

i,ααis the discrete image at sequence time i and scale step n. Thus,the identification u(n∆t, i∆s, α∆x) = Un

i,α holds. In Table 1we have collected all the indices which are now in use.

A basis for the multi-linear finite element space is given by

φi(s) ψα(x)

where φi are simple hat functions on the sequence-time latticeand ψα also hat functions but on the space-discrete quadtreerespectively octtree. We have the following basis decompos-ition for the nth scale

un(s, x) =∑

i

∑α

Uni,αφi(s)ψα(x) .

From this we derive the standard discrete formulation of theproblem, testing with each basis function. Using the last re-mark, the components of the corresponding matrix system(see Fig. 7) in scale step n +1 will be given by(

φiψα

|∇(s,x)un|ε , φjψβ

)Q

+ (MM)

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206 K. Mikula et. al.

Fig. 8. We compare existing image processing methodology applied to an image-sequence, which shows noisy spherical level-sets bouncing at some solidobject (cf. Fig. 5). In the left pictures one iso-surface from the 3D representation of the image-sequence smoothed at scale 3 is displayed. The leftmostimage shows the result of the model (7) which smooths out the highly accelerated motion of the level-sets. In contrast to that, the anisotropic geometricmodel (middle left) preserves this behavior quite well. The right images show magnified sections from the iso-surface representations (middle right: model(7), right: anisotropic geometric model)

∆t

(aσ,n

v

|∇(s,x)un|εD

∂s(φiψα),

D

∂s(φjψβ)

)Q

+ (CP)

∆t

(aσ,n

TxM

|∇(s,x)un|ε ∇(φiψα),∇(φjψβ)

)Q

, (MS)

where the integration on Q reduces to the support of the basisfunctions. The terms correspond to the sequence-time/spacemass matrix (MM), a coupled part (CP) consisting of a mix-ing between sequence-time and space derivatives and the spa-tial stiffness matrix (MS) multiplied by the sequence-timemass matrix.

The key to simplify these entries is the application of masslumping in sequence-time [36] and a midpoint quadraturerule in space. The mass lumping results in a diagonalizationof the terms (MM) and (MS) in sequence-time.

Furthermore we evaluate the denominator |∇(s,x)un |ε ats = i∆s always by a central difference D±

(s,x)uni in time and

therefore can completely split off the sequence-time integra-tion. We obtain

(MM)(α,i),(β, j) =(i+1)∆s∫

(i−1)∆s

∫Ω

φiψα φiψβ

|∇(s,x)un|ε ds dx

≈ ∆s

(ψα

|D±(s,x)u

ni |ε

, ψβ

.

Since we have given the nonlinearity aσ,nTxM of (MS) on the

sequence-time nodes, we can handle this part in a similar wayto obtain

(MS)(α,i),(β, j)

= ∆t

(i+1)∆s∫(i−1)∆s

∫Ω

aσ,nTxM

φi∇ψα φi∇ψβ

|∇(s,x)un|ε ds dx

≈ ∆t∆s

((aσ,n

TxM)i∇ψα

|D±(s,x)u

ni |ε

,∇ψβ

,

which evaluates aσ,nTxM only at the frame i of the sequence.

The remaining term (CP) does not diagonalize in sequence-time. If we split this term up into its parts (according to thelast remark), then for each part split off the sequence-timeintegral by using Fubini’s theorem and midpoint integration

on the involved sequence-time intervals, we obtain a scheme,that has 3-band block-structure in sequence-time (cf. Ap-pendix A). The blocks again correspond to the frames in thesequence and moreover the off-diagonal blocks reflect thesequence-time derivatives similar to a difference scheme withstencil [−1, 2,−1] in the sequence-time direction.

The numerical integration of the sequence-time parts con-sidered here separates the sequence-temporal operations fromthe spatial ones and thus simplifies the resulting matrix.

6.3 A block solver

The complete matrix of our system now has a 3-band block-structure and therefore we can formally rewrite the discreteproblem as (cf. Fig. 9):

For each scale n > 0 find frames uni such that

〈Fni (un

i−1, uni , un

i+1), ψα〉 = 0 ∀α ,

where the 〈Fni (. . . ), ψα〉 corresponds to the row i of the above

derived matrix structure. To solve the system of equations,we use a symmetric block Gauß–Seidel solver which can besketched as follows:

If we would set kmax = 1 the solution strategy would cor-respond to an explicit scheme in sequence-time. This is notdesirable since the strength of the approach is its nonlin-ear sequence-time behavior. We therefore fix a small value

Fig. 9. The block solver considers in each step of the inner frame loop al-ways only three successive images of the sequence. This corresponds to thefact that the resulting system matrix has a 3-band block-structure

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Morphological image sequence processing 207

greater than one for kmax to obtain a better approximation.In our computations we always chose kmax = 3. The solu-tion of the subsystems Fn

i is done by a conjugate gradient(CG) method preconditioned by diagonal scaling. In a typ-ical example with a 129 × 129 image sequence consistingof 60 frames and with a stopping criterion of 10−10 for theCG solver, one scale-step takes about 600 seconds computingtime on an Intel Itanium II processor with 1.0 GHz.

For each scale n = 0, 1, . . . do

For each frame i = 0, 1, . . . , M set un+1,0i = un

i .For k = 1, 2, . . . , kmax do

Sweep from left to right: For each framei = 0, 1, . . . , M solve for un+1,k

i the system

〈Fni (un+1,k

i−1 , un+1,ki , un+1,k−1

i+1 ), ψα〉 = 0 .

Sweep from right to left: For each framei = M, M −1, . . . , 0 solve for un+1,k

i the system

〈Fni (un+1,k−1

i−1 , un+1,ki , un+1,k

i+1 ), ψα〉 = 0 .

For each frame i = 0, 1, . . . , M set un+1i = un+1,kmax

i .

7 Comparison to other methods

In this section we would like to compare some of the image-sequence processing models mentioned in Sect. 2 with thenew model. We will not discuss any steady-image method-ology which may be applied to the single frames of the se-quence, since a model taking into account velocity and accel-eration of an image sequence clearly gives better correlationbetween successive frames of the sequence.

In [32] Sarti et al. have presented a model for nonlinearimage sequence smoothing, which is based on the method-ology derived by Guichard in [15]. They took into accountthe apparent acceleration and the apparent velocity in termsof the curvature of Lambertian trajectories (clt) and used thefollowing model:

∂tu − clt(u)div(G(|∇uσ |)∇u) = 0 . (7)

The model treats the frames of the sequence separately, buta coupling is given by the modulation of the speed of dif-fusion via the clt term in front of the divergence. Since thecurvature of the trajectories is proportional to the accelerationwe conclude that the diffusion will be larger where high ac-celeration is detected, whereas for non-accelerated motion theequation degenerates to the identity ∂tu = 0.

A similar approach was taken in [21] where the non-morphological Perona–Malik like behavior was replaced bythe anisotropic level-set smoothing, which was describedin Sect. 3. Again the frames are treated separately and thecoupling is done via the clt term steering the speed of diffu-sion:

∂tu − clt(u)div

(aσ

TxM(Sσ )∇u

|∇u|)

= 0 .

We can characterize the time-smoothing character of bothapproaches in the following way: The resulting images willbe smoother in regions of high acceleration, whereas nosmoothing is applied to regions, that move uniformly, non-accelerated. In this sense the behavior of the model presentedin this paper is contrary to the clt approaches. First, wehave a coupled smoothing, where the sequence is treated asa whole, second, the sequence-time diffusion is decreased ifthe trajectory has high curvature. On one hand, this resultsin a smoothing of the images even if the motion is non-accelerated. The important features are then spatial cornersand edges. On the other hand accelerated motions especiallywith nearly discontinuous velocities will be preserved muchbetter.

Since the comparison of the different image sequencemodels is very difficult, when only static frames of the se-quence are shown, we are going to identify a 2D image se-quence width 3D space in the following: We consider theimage sequence

u0(s, x) = |x −d(s)+ rand(N) N| for N = x

|x| ,

where d(s) models the bouncing at some solid object as be-fore and rand is a random factor in [−5h, 5h]. Thus, theimages consist of spherical level-sets which are disturbedin normal direction and which bounce at some solid object.We run the desired diffusion, which creates a scale u : R+ ×[0, T ]×Ω of smoothed image sequences. From each scalewe create the 3D representation U(x, y, z) by identifying thesequence-time s with the third spatial coordinate z, i. e. fort ∈ R+

U(x, y, z) = u(t, z, w) with w = (x, y) ∈ Ω .

From this 3D image we then draw iso-surfaces which showthe movement of level-sets in the space-time cube. Slicesthrough this cube correspond to the frames of the smoothedsequence.

In Fig. 8 we compare the behavior of the clt-model (7)with the new anisotropic model by looking at iso-surfaces ofthe space-time image U for a fixed scale. We clearly see, thatthe anisotropic model preserves the high acceleration at thepoint where the motion changes its direction much better thanthe clt-model does. This verifies our expectations from above.

8 Conclusions

We have presented a new morphological anisotropic smooth-ing approach for image sequences, which takes into accounttemporal and spatial curvature information. The multi-scalediffusion thereby is truly coupled in sequence-time and spaceand the anisotropy directions correspond to the apparent di-rection of motion in sequence-time and to principal directionsof curvature in space. The diffusivity is decreased in areas ofhigh curvature which results in a good preservation of spa-tial corners and edges as well as highly accelerated motionsin sequence-time.

The discretization takes into account a mass lumping insequence-time and a suitable mid-point integration rule inthe corresponding sequence-time intervals. Therefore the ma-trix scheme resulting from a tensor product multi-linear finite

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208 K. Mikula et. al.

element approach in sequence-time and space has a 3-blockstructure, where the single blocks correspond to the framesin the sequence. This makes a treatment with moderate effortfeasible even for 3D image sequences, which would result ina 5D problem.

On the web site

http://www.numerik.math.uni-duisburg.de/exports/anisoseq/

more examples and movies showing image sequences de-noised by the new model are available.

Acknowledgements. We thank G. Dziuk and J. Weickert for inspiring discus-sions on the topic. We furthermore acknowledge C. Lamberti from DEIS,Bologna University and TomTec Imaging Systems for providing the ultra-sound image data shown in Fig. 1.

Appendix. Operator splitting in the coupled sequence-time/space problem

Within our explanations in Sect. 6.2 we have claimed that theresulting matrix scheme has a 3-band block structure. So farthis has only been shown for a part of the resulting matrix.Let us recall that the matrix entries of the resulting system aregiven by(

φiψα

|∇(s,x)un |ε , φjψβ

)Q

+ (MM)

∆t

(aσ,n

v

|∇(s,x)un|εD

∂s(φiψα),

D

∂s(φjψβ)

)Q

+ (CP)

∆t

(aσ,n

TxM

|∇(s,x)un|ε ∇(φiψα),∇(φjψβ)

)Q

. (MS)

In Sect. 6.2 it has become clear that together with mass lump-ing in sequence time, the terms (MM) and (MS) only lead todiagonal block-entries in the resulting system. In the follow-ing we will show that (CP) leads to two off-diagonal blocks,such that the final matrix has a 3-band block structure. Ac-cording to the Remark 3 we split up the term (CP) to

(CP)(i,α),( j,β) (A.1)

= ∆t

(aσ,n

v

|∇(s,x)un|ε ∂s(φiψα), ∂s(φjψβ)

)Q

(CP1)

+∆t

(aσ,n

v

|∇(s,x)un|ε ∂s(φiψα), v·∇(φjψα)

)Q

(CP2)

+∆t

(aσ,n

v

|∇(s,x)un|ε v ·∇(φiψα), ∂s(φjψβ)

)Q

(CP3)

+∆t

(aσ,n

v

|∇(s,x)un|ε v·∇(φiψα), v·∇(φjψα)

)Q

. (CP4)

Again we inspect these terms separately. The first component(CP1) is the elliptic sequence-time term. Let us perform theintegration:

(CP1)(i,α),( j,β)

= ∆t

T∫0

∫Ω

aσ,nv

|∇(s,x)un|ε ∂s(φiψα)∂s(φjψβ) dx ds

≈ ∆t∑

k

(k+1)∆s∫k∆s

∂sφi ∂sφj ds∫Ω

(aσ,nv )k

|D+(s,x)u

nk |ε

ψα ψβ dx,

where we assume aσ,nv to be constant over each sequence-time

interval [k∆s, (k +1)∆s] and denoted by (aσ,nv )k, and D+

(s,x)unk

is the spatio-temporal-gradient evaluated on such an interval.Since the support of ∂sφi and ∂sφj only overlap in the case|i − j| ≤ 1 we conclude that the resulting matrix may onlyhave 3-band structure, where the entries in row i are approxi-mated by−∆t

∆s

Ω

(aσ,nv )i−1ψα ψβ

|D+(s,x)u

ni−1|ε

α,β

,

∆t

∆s

Ω

[(aσ,nv )i−1 + (aσ,n

v )i]ψα ψβ

|D±(s,x)u

ni |ε

α,β

,

−∆t

∆s

Ω

(aσ,nv )iψα ψβ

|D+(s,x)u

ni |ε

α,β

,

where we again have used the central derivative D±(s,x)u

ni in the

diagonal element.The second and third term (CP2) and (CP3) consist of

mixed derivatives in sequence-time and space. If the tempo-ral diffusion coefficient aσ,n

v was constant, these terms wouldvanish for symmetry reasons. But since we have built the newmodel upon nonlinear temporal diffusion, we have to takeinto account these terms. Again due to symmetry reasons, weonly have to take into account the temporal diagonal

(CP2 +CP3)(i,α),( j,β)

= ∆t

T∫0

∫Ω

aσ,nv ∂s(φiψα)

|∇(s,x)un|ε v ·∇(φiψβ)

+ aσ,nv ∂s(φiψβ)

|∇(s,x)un|ε v ·∇(φiψα) dx ds

≈ ∆t∆s∫Ω

[(aσ,nv )i−1 − (aσ,n

v )i]|D±

(s,x)uni |(

ψα v ·∇ψβ +v ·ψα ψβ

)dx .

Finally we have the spatial anisotropic elliptic term (CP4)which can be handled in exactly the same way as before(MM) and (MS) to obtain

(CP4)(i,α),( j,β)

= ∆t∆s∫Ω

[(aσ,nv )i−1 + (aσ,n

v )i]|D±

(s,x)uni |

v·∇ψα v·∇ψβ dx .

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Morphological image sequence processing 209

Adding up all the terms (MM)+ ∑4i=1 (CPi) + (MS), we

obtain the stiffness matrix which is then treated by the blocksolver as shown in Sect. 6.3.

References

1. Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fun-damental equations of image processing. Arch. Ration. Mech. Anal.123(3), 199–257 (1993)

2. Alvarez, L., Weickert, J., Sanchez, J.: A scale-space approach to nonlo-cal optival flow calculations. In: Nielsen, M., Johansen, P., Olsen, O.F.,Weickert, J. (eds.), Scale-Space Theories in Computer Vision. SecondInternational Conference, Scale-Space 1999, Corfu, Greece, Septem-ber 1999, Lecture Notes in Computer Science 1682, pp. 235–246,Springer, 1999

3. Alvarez, L., Weickert, J., Sanchez, J.: Reliable estimation of dense op-timal flow fields with large displacements. Int. J. of Computer Vision39(1), 41–56 (2000)

4. Angenent, S.B., Gurtin, M.E.: Multiphase thermomechanics with inter-facial structure 2, evolution of an is othermal interface. Arch. RationalMech. Anal. 108, 323–391 (1989)

5. Bänsch, E., Mikula, K.: A coarsening finite element strategy in imageselective smoothing. Comput. Visual. Sci. 1, 53–63 (1997)

6. Bellettini, G., Paolini, M.: Anisotropic motion by mean curvature inthe context of finsler geometry. Hokkaido Math. J. 25, 537–566(1996)

7. Catte, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smooth-ing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal.29(1), 182–193 (1992)

8. Chen, Y.-G., Giga, Y., Goto, S.: Uniqueness and existence of viscositysolutions of generalized mean curvature flow equations. J. Diff. Geom.33(3), 749–786 (1991)

9. Christensen, G.E., Joshi, S.C., Miller, M.I.: Volumetric transformationsof brain anatomy. IEEE Trans. Medical Imaging 16(6), 864–877(1997)

10. Christensen, G.E., Rabbitt, R.D., Miller, M.I.: Deformable templatesusing large deformation kinematics. IEEE Trans. Medical Imaging5(10), 1435–1447 (1996)

11. Davatzikos, C.A., Bryan, R.N., Prince, J.L.: Image registration basedon boundary mapping. IEEE Trans. Medical Imaging 15(1), 112–115(1996)

12. Deriche, R., Kornprobst, P., Aubert, G.: Optical–flow estimation whilepreserving its discontinuities: A variational approach. In Proc. SecondAsian Conf. Computer Vision (ACCV ’95, Singapore, December 5–8,1995), Vol. 2, pp. 290–295, 1995

13. Evans, L., Spruck, J.: Motion of level sets by mean curvature I. J. Diff.Geom. 33(3), 635–681 (1991)

14. Grenander, U., Miller, M.I.: Computational anatomy: An emerging dis-cipline. Quarterly Appl. Math. LVI, 617–694 (1998)

15. Guichard, F.: Axiomatisation des analyses multi-echelles dimages etde films. PhD thesis, University Paris IX Dauphine, 1994

16. Guichard, F.: A morphological, affine, and galilean invariant scale–space for movies. IEEE Trans. on Image Processing 7(3), 444–456(1998)

17. Joshi, S.C., Miller, M.I.: Landmark matching via large deformationdiffeomorphisms. IEEE Trans. Medical Imaging 9(8), 1357–1370(2000)

18. Kacur, J., Mikula, K.: Solution of nonlinear diffusion appearing inimage smoothing and edge detection. Appl. Numer. Math. 17(1),47–59 (1995)

19. Kriva, Z., Mikula, K.: An adaptive finite volume scheme for solvingnonlinear diffusion equations in image processing. J. Vis. Comm. andImage Repres. 13, 22–35 (2002)

20. Maes, F., Collignon, A., Vandermeulen, D., Marchal, G., Suetens, P.:Multi–modal volume registration by maximization of mutual informa-tion. IEEE Trans. Medical Imaging 16(7), 187–198 (1997)

21. Mikula, K., Preußer, T., Rumpf, M., Sgallari, F.: On anisotropic geo-metric diffusion in 3D image processing and image sequence analysis.In Trends in Nonlinear Analysis, pp. 305–319, Springer, 2002

22. Mikula, K., Ramarosy, N.: Semi–implicit finite volume scheme forsolving nonlinear diffusion equations in image processing. Nu-merische Mathematik, 2001

23. Nagel, H.H., Enkelmann, W.: An investigation of smoothness con-straints for the estimation of displacement vector fields from imagessequences. IEEE Trans. Pattern Anal. Mach. Intell. 8, 565–593 (1986)

24. Perona, P., Malik, J.: Scale space and edge detection using anisotropicdiffusion. In IEEE Computer Society Workshop on Computer Vision,1987

25. Perona, P., Malik, J.: Scale space and edge detection using anisotropicdiffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

26. Preußer, T., Rumpf, M.: Anisotropic nonlinear diffusion in flow visu-alization. In Proceedings Visualization 1999, pp. 325–332, 1999

27. Preußer, T., Rumpf, M.: An adaptive finite element method for largescale image processing. J. Vis. Comm. and Image Repres. 11, 183–195(2000)

28. Preußer, T., Rumpf, M.: A level set method for anisotropic diffusion in3D image processing. SIAM J. Appl. Math. 62(5), 1772–1793 (2001)

29. Preußer, T., Rumpf, M.: Extracting motion velocities from 3D imagesequences and coupled spatio-temporal smoothing. In Proceedings Vi-sual Data Analysis, pp. 181–192, 2003

30. Radmoser, E., Scherzer, O., Weickert, J.: Scale-space properties of reg-ularization methods. In: Nielsen, M., Johansen, P., Olsen, O.F., We-ickert, J. (eds.), Scale-Space Theories in Computer Vision. SecondInternational Conference, Scale-Space ’99, Corfu, Greece, Septem-ber 1999, Lecture Notes in Computer Science 1682, pp. 211–220,Springer, 1999

31. Sapiro, G.: Vector (self) snakes: A geometric framework for color, tex-ture, and multiscale image segmentation. In Proc. IEEE InternationalConference on Image Processing, Lausanne, September 1996

32. Sarti, A., Mikula, K., Sgallari, F.: Nonlinear multiscale analysis of 3Dechocardiography sequences. IEEE Trans. Medical Imaging 18(6),453–466 (1999)

33. Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cam-bridge University Press, 1999

34. Taylor, J.E., Cahn, J.W., Handwerker, C.A.: Geometric models of crys-tal growth. Acta metall. mater. 40, 1443–1474 (1992)

35. Thirion, J.P.: Image matching as a diffusion process: An analogy withmaxwell’s demon. Medical Imag. Analysis 2, pp. 243–260, 1998

36. Thomee, V.: Galerkin – Finite Element Methods for Parabolic Prob-lems. Springer, 1984

37. Weickert, J.: Anisotropic diffusion filters for image processing basedquality control. In: Fasano, A., Primicerio, M. (eds.), Proc. SeventhEuropean Conf. on Mathematics in Industry, pp. 355–362, Teubner,1994

38. Weickert, J.: Anisotropic diffusion in image processing. Teubner, 1998