Flux maximizing geometric flows - Pattern Analysis and ...shape/publications/pami02.pdfMalladi et al. [22], [23], [24] and was also developed independently by Caselles et al. [5]

Flux Maximizing Geometric FlowsAlexander Vasilevskiy, Student Member, IEEE, and Kaleem Siddiqi, Member, IEEE

Abstract—Several geometric active contour models have been proposed for segmentation in computer vision and image analysis.The essential idea is to evolve a curve (in 2D) or a surface (in 3D) under constraints from image forces so that it clings to features ofinterest in an intensity image. Recent variations on this theme take into account properties of enclosed regions and allow for multiplecurves or surfaces to be simultaneously represented. However, it is still unclear how to apply these techniques to images of narrowelongated structures, such as blood vessels, where intensity contrast may be low and reliable region statistics cannot be computed. Toaddress this problem, we derive the gradient flows which maximize the rate of increase of flux of an appropriate vector field through acurve (in 2D) or a surface (in 3D). The key idea is to exploit the direction of the vector field along with its magnitude. The calculationslead to a simple and elegant interpretation which is essentially parameter free and has the same form in both dimensions. We illustrateits advantages with several level-set-based segmentations of 2D and 3D angiography images of blood vessels.

Index Terms—Geometric active contours, gradient flows, shape analysis, divergence and flux, blood vessel segmentation.

æ

1 INTRODUCTION

WHEREAS geometric flows have a long history in theliterature on front propagation, e.g., they figure

prominently in Osher and Sethian’s development of thelevel-set method for hyperbolic conservation laws [29], theywere introduced relatively recently to the computer visionand image analysis communities. Perhaps the first applica-tion of a geometric flow in the latter setting was Kimia et al.reaction-diffusion space for shape analysis [13], [14], whichconsidered curvature dependent motions of the typestudied earlier by Osher and Sethian. The first level-setbased technique for image segmentation was introduced byMalladi et al. [22], [23], [24] and was also developedindependently by Caselles et al. [5]. Here, the essential ideawas to halt an evolving curve in the presence of intensityedges by multiplying the evolution equation with an image-gradient based stopping potential. This led to new activecontour models which, though inspired by and closelyrelated to the parametric snakes introduced by Kass et al.[11], had the advantage that they could handle changes intopology due to the splitting and merging of multiplecontours in a natural way. These geometric flows for shapesegmentation were later given formal motivation as well asunified with the classical energy minimization formulationsthrough several independent investigations [6], [12], [33],[34]. The main idea was to modify the Euclidean arc lengthor the Euclidean area by a scalar function and to then derivethe resulting gradient evolution equations. Mathematically,this amounted to defining a new metric on the plane,tailored to the given image, and then deriving thecorresponding gradient flows. The results generalized tothe case of evolving surfaces in 3D by adding one moredimension to the variational formulation.

Recently there have been other advances in the use ofgeometric flows in computer vision, which have boththeoretical and practical value. First, it has been recognizedthat a practical weakness of most geometric flows withstopping terms based purely on local image gradients is thatthey may “leak” in the presence of weak or low contrastboundaries, are not suitable for segmenting textures, andtypicallyrequire the initialcurveorcurvesto lieentirely insideor outside the regions to be segmented. Thus, a number ofresearchers have sought to derive flows which take intoaccount the statistics of the regions enclosed by the evolvingcurves [31], [41]. Further developments include multiphasemotions,whichallowtriplepoints tobecaptured[7],aswellasthe incorporation of an external force field based on a diffusedgradient of an edge map [40]. Second, most geometric flowsare not able to capture elongated low contrast structures well,such as blood vessels viewed in 2D and 3D angiographyimages. At places where such structures are narrow, edgegradients may be weak due to partial volume effects and it isalso unclear how to robustly measure region statistics.Approaches to regularizing the flow in 3D by introducing aterm proportional to mean curvature have the unfortunateeffect of annihilating such structures. To address this issue,Lorigo et al. have proposed the use of active contours withcodimension two (curves in 3D) [21], [19], [20]. The idea is toregularize the flow by a term proportional to the curvature of a3D curve. The approach is grounded in the level set theory formean curvature evolution of surfaces of arbitrary codimen-sion developed in [3] and has a variational formulation alongwith an energy minimizing interpretation. However, thederived flow is later modified with a (heuristic) multiplicativeterm to tailor it to blood vessel segmentation [19], [20].

In this work, we seek an alternate approach to segment-ing elongated structures that appear as bright regions in anintensity image but may have low contrast. The key idea isto incorporate not only the magnitude but also the directionof an appropriate vector field. The development can bemotivated by the following illustration. Consider a planarclosed curve placed in a dense vector field, as shown inFig. 1a. The inward flux of the vector field through thiscurve provides a measure of how well the curve is alignedwith the direction perpendicular to the vector field. When

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 24, NO. 12, DECEMBER 2002 1565

. A. Vasilevskiy is with the Java JIT Development Group, IBM Canada Ltd.,8200 Warden Avenue, Markham, Ontario L6G 1C7, Canada.E-mail: [email protected].

. K. Siddiqi is with the School of Computer Science & the Center forIntelligent Machines, McGill University, 3480 University Street, Mon-treal, QC AH3A 2A7, Canada. E-mail: [email protected].

Manuscript received 28 Aug. 2001; revised 12 Apr. 2002; accepted 23 Apr. 2002.Recommended for acceptance by S. Dickinson.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number 114861.

0162-8828/02/$17.00 ß 2002 IEEE

the vector field is obtained as the gradient of an imagecontaining bright structures such as blood vessels, one canexpect it to be locally orthogonal to boundaries of interest,in their vicinity. Returning to the example in Fig. 1, anatural principle to use toward the recovery of theseboundaries is to maximize the flux of the vector fieldthrough the evolving curve. The flux maximizing config-uration to which the evolution converges is one where theinward normals to the curve are everywhere aligned withthe vector field, as illustrated in Fig. 1b.

In this paper, we formulate the flux maximizing flowproblem and derive the gradient flow that solves it. Thesolution turns out to be elegant and essentially parameterfree. Furthermore, it maintains the same form whenextended to 3D. The measure of inward flux underlyingthis flow can be of interest in computer vision and imageanalysis problems that involve the detection and tracking ofsingularities of a motion field or a shading field. Theparticular example that we develop in some detail in thispaper is that of blood vessel segmentation. The main idea isto incorporate an adaptive scale at which to compute theinward flux of the gradient vector field which correspondsto an estimate of the local width of a vessel. We illustrate thepotential of this technique with several level-set-basedsimulations on a variety of 2D and 3D angiography data.The method is able to handle regions where structures arenarrow and have low contrast since it incorporates not onlythe magnitude but also the direction of the gradient vectorfield. The constraint of orthogonality between the gradientvector field and the desired boundary continues to hold, nomatter how faint the latter may be, and the flux maximizingflow is designed to exploit it.

The paper is organized as follows: In Section 2, we derivethe gradient flows that determine how to evolve a curve (in2D) or a surface (in 3D) so as to increase the inward flux atthe fastest possible rate. In Section 3, we specialize to thecase of blood vessel segmentation by considering the

gradient of an intensity image as the vector field and by

computing the flux at an adaptive scale which corresponds

to a local estimate of vessel width. We present several

examples illustrating the effectiveness of the flux maximiz-

ing flow when applied to both 2D and 3D angiography data

in Section 4. Finally, we conclude with a discussion of the

results and present directions for future work in Section 5.

2 FLUX MAXIMIZING FLOWS

2.1 The 2D Case

Let C ¼ Cðp; tÞbe a smooth family of closed curves evolving in

the plane. Here, t parametrizes the family and p the given

curve. Without loss of generality, we shall assume that

0 � p � 1, i.e., that Cð0; tÞ ¼ Cð1; tÞ. We shall also assume that

the first derivatives exist and that C0ð0; tÞ ¼ C0ð1; tÞ. The unit

tangent T and the unit inward normalN to C are given by

T ¼

xpyp

� �jjCpjj

¼ xsys

� �;N ¼

ÿypxp

� �jjCpjj

¼ ÿysxs

� �;

where s is the arc length parametrization of the curve. Now,

consider a vector field V ¼ ðV1ðx; yÞ; V2ðx; yÞÞ defined for

each point ðx; yÞ in R2. The total inward flux of the vector

field through the curve is given by the contour integral

FluxðtÞ ¼Z 1

0

hV;Ni jjCpjj dp ¼Z LðtÞ

0

hV;Ni ds; ð1Þ

where LðtÞ is the Euclidean length of the curve. The

circulation of the vector field along the curve is defined in

an analogous fashion as

CircðtÞ ¼Z 1

0

hV; T i jjCpjj dp ¼Z LðtÞ

0

hV; T i ds:

1566 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 24, NO. 12, DECEMBER 2002

Fig. 1. (a) A closed planar curve is placed in a 2D vector field and (b) the curve evolves so as to increase the inward flux through its boundary asfast as possible. The resting flux maximizing configuration is one where the inward normals to the curve are everywhere aligned with the directionof the vector field.

The first technical result of this paper is the followingtheorem:

Theorem 1. The direction in which the inward flux of the vectorfield V through the curve C is increasing most rapidly is givenby @C

@t ¼ divðVÞN .

In other words, the gradient flow which maximizes therate of increase of the total inward flux is obtained bymoving each point of the curve in the direction of theinward normal by an amount proportional to the diver-gence of the vector field. As we shall later see, this result canbe exploited to recover low contrast elongated structuressuch as blood vessels in angiography images.

Proof. Define the perpendicular to a vector W ¼ ða; bÞ asW? ¼ ðÿb; aÞ. The following properties hold:

hU;W?i ¼ ÿhU?;WihU?;W?i ¼ hU;Wi:

ð2Þ

We now compute the first variation of the flux functionalwith respect to t.

Flux0ðtÞ ¼Z 1

0

hVt;Ni jjCpjj dp|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}I1

þZ 1

0

hV; ðN jjCpjjÞti dp|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}I2

:

Switching to parametrization by s for I1 and using

Vt ¼@V1

@t;@V2

@t

� �¼ hrV1; Cti; hrV2; Ctið Þ;

we have

I1 ¼Z LðtÞ

0

hCt; xsrV2 ÿ ysrV1i ds:

With N ¼ ðÿyp; xpÞ=jjCpjj; I2 works out to beZ 1

0

V; ÿyptxpt

� �� dp:

Now, using integration by parts

I2 ¼ V; ÿytxt

� �� #1

0|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}equals 0

ÿZ 1

0

ÿytxt

� �;Vp

� �dp:

Using the properties of scalar products in (2) and the factthat

Vp ¼@V1

@p;@V2

@p

� �¼ rV1; Cp �

; rV2; Cp �ÿ �

;

we can rewrite I2 as follows:

I2 ¼Z 1

0

xt

yt

� �;Vp ?

� �dp

¼Z 1

0

Ct;ÿ rV2; Cp �rV1; Cp � !* +

dp:

Switching to arc length parametrization

I2 ¼Z LðtÞ

0

Ct;ÿ rV2; Th irV1; Th i

� �� ds:

Combining I1 and I2, the first variation of the flux isZ LðtÞ

0

Ct; xsrV2 ÿ ysrV1 þÿ rV2; Th irV1; Th i

� �� ds:

Thus, for the flux to increase as fast as possible, the twovectors should be made parallel:

Ct ¼ xsrV2 ÿ ysrV1 þÿ rV2; Th irV1; Th i

� �:

Decomposing the above three vectors in the Frenet framefT ;Ng, dropping the tangential terms (which affect onlythe parametrization of the curve), and making use of theproperties of scalar products

Ct ¼ xshrV2;Ni ÿ yshrV1;Ni

þÿ rV ?2 ;N �rV ?1 ;N � !

;N* +!

N :

Expanding all terms in the above equation

Ct ¼ xsðÿV2x:ys þ V2y:xsÞ ÿ ysðÿV1x:ys þ V1y:xsÞ

þÿV2y:ys ÿ V2x:xs

V1y:ys þ V1x:xs

� �;N

� �!N

¼ ðÿV2x:xs:ys þ V2y:xs2 þ V1x:ys

2 ÿ V1y:xs:ys

þ V2y:ys2 þ V2x:xs:ys þ V1y:xs:ys þ V1x:xs

2ÞN¼ ðV1xðxs 2 þ ys 2Þ þ V2yðxs 2 þ ys 2ÞÞN¼ ðV1x þ V2yÞN ¼ divðVÞN :

ð3Þ

tu

As a corollary to Theorem 1, we have

Corollary 1. The direction in which the circulation of thevector field V along the curve C is increasing most rapidlyis given by @C

@t ¼ divðV?ÞN .

Proof. Using the properties of scalar products in (2)

CircðtÞ ¼Z LðtÞ

0

hV; T i ds

¼Z LðtÞ

0

hV?; T ?i ds

¼Z LðtÞ

0

hV?;Ni ds:

tu

Hence, the circulation of the vector field V along thecurve is just the inward flux of the vector field V? through itand the result follows from Theorem 1.

2.2 The 3D Case

We now consider the volumetric extension of the fluxmaximizing flow. In order to do this, we will need to set upsome notation. Let S : ½0; 1� � ½0; 1� ! R3 denote a compactembedded surface with (local) coordinates ðu; vÞ. LetN be theinward unit normal. We set

VASILEVSKIY AND SIDDIQI: FLUX MAXIMIZING GEOMETRIC FLOWS 1567

Su :¼ @S@u

; Sv :¼ @S@v

:

Then, the infinitesimal area on S is given by

dS ¼ ðjjSujj2jjSvjj2 ÿ hSu;Svi2Þ1=2dudv ¼ jjSu ^ Svjjdudv:

Let V ¼ ðV1ðx; y; zÞ; V2ðx; y; zÞ; V3ðx; y; zÞÞ be a vector fielddefined for each point ðx; y; zÞ in R3. The total inward fluxof the vector field through the surface is defined by thesurface integral

FluxðtÞ ¼Z AðtÞ

0

hV;Ni dS; ð4Þ

where AðtÞ is the surface area of the evolving surface. Oursecond technical result is that the flux maximizing gradientflow has the same form in 3D.

Theorem 2. The direction in which the inward flux of the vectorfield V through the surface S is increasing most rapidly is givenby @S

@t ¼ divðVÞN .

Proof. The essential idea is to calculate the first variation ofthe flux functional with respect to t, as before, but now tomanipulate properties of cross products. The calculationturns out to be more subtle than in the 2D case.

Flux0ðtÞ ¼Z 1

0

Z 1

0

hVt;Ni jjSu ^ Svjjdudv|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}I1

þZ 1

0

Z 1

0

hV; ðN jjSu ^ SvjjÞti jjdudv|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}I2

:

With S ¼ ðxðu; v; tÞ; yðu; v; tÞ; zðu; v; tÞÞ, the unit normalvector is given by the normalized cross product of twovectors in the tangent plane:

N ¼ Su ^ SvjjSu ^ Svjj

¼ ðN1; N2; N3ÞjjSu ^ Svjj

¼ ðyuzv ÿ yvzuÞ; ðxvzu ÿ xuzvÞ; ðxuyv ÿ xvyuÞjjðyuzv ÿ yvzuÞ; ðxvzu ÿ xuzvÞ; ðxuyv ÿ xvyuÞjj

:

ð5Þ

I1 is then given byZ 1

0

Z 1

0

hSt; ðN1rV1 þN2rV2 þN3rV3Þi dudv;

where the integrand is the inner product ofSt with anothervector. We shall now simplify I2 so that it takes on a similarform. It turns out to be advantageous to express the unitnormal vector in (5) as Su^Sv

jjSu^Svjj and expand it in terms of thepartial derivatives xu; xv; yu; yv; zu; and zv only later. I2 canthen be rewritten asZ 1

0

Z 1

0

hV; ðSu ^ SvÞti dudv:

The trick now, is to exploit the fact that for any vectorsA,B,andC the following properties of inner products and crossproducts hold:

A ^B ¼ ÿB ^AhA; ðB ^ CÞi ¼ hðA ^BÞ; CiðA ^BÞt ¼ ðAt ^BÞ þ ðA ^BtÞ:

Hence, I2 can be written as

I2 ¼Z 1

0

Z 1

0

hV; ðSut ^ Sv þ Su ^ SvtÞi dudv

¼Z 1

0

Z 1

0

hV; ðSut ^ SvÞi dudvþZ 1

0

Z 1

0

hV; ðSu ^ SvtÞi dudv

¼Z 1

0

Z 1

0

ÿhV; ðSv ^ SutÞi dudv

þZ 1

0

Z 1

0

hV; ðSu ^ SvtÞi dudv

¼Z 1

0

Z 1

0

ÿhðV ^ SvÞ;Suti du� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

I3

dv

þZ 1

0

Z 1

0

hðV ^ SuÞ;Svti dv� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

I4

du:

Using integration by parts, I3 works out to be

ÿhðV ^ SvÞ;Sti�10|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}equals 0

þZ 1

0

hSt; ðV ^ SvÞui du:

Similarly, using integration by parts, I4 works out to be

hðV ^ SuÞ;Sti�10|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}equals 0

ÿZ 1

0

hSt; ðV ^ SuÞvi dv:

Combining I3 and I4, I2 works out to beZ 1

0

Z 1

0

hSt; ðV ^ SvÞu ÿ ðV ^ SuÞvi dudv:

It can now be seen that the integrand in I2 has thedesired form of the inner product of St with anothervector. Hence, combining I1 and I2, the first variation ofthe flux isZ 1

0

Z 1

0

hSt; N1rV1 þN2rV2 þN3rV3 þ ðV ^ SvÞuÿ ðV ^ SuÞvi dudv:

Note that

ðV ^ SvÞu ÿ ðV ^ SuÞv ¼ ðVu ^ SvÞ þ ðV ^ SvuÞÿðV ^ SuvÞ ÿ ðVv ^ SuÞ ¼ ðVu ^ SvÞ ÿ ðVv ^ SuÞ:

Hence, the first variation of the flux can be written as the

surface integralZ AðtÞ

0

St;N1rV1 þN2rV2 þN3rV3 þ ðVu ^ SvÞ ÿ ðVv ^ SuÞ

jjSu ^ Svjj

� �dS:

Thus, for the inward flux to increase as fast as possible,the two vectors should be made parallel:

St ¼N1rV1 þN2rV2 þN3rV3 þ ðVu ^ SvÞ ÿ ðVv ^ SuÞ

jjSu ^ Svjj:

ð6Þ


The above expression for the 3D flux maximizinggradient flow can be further simplified by noting thatthe components of the flow in the tangential plane to thesurface S affect only the parametrization of the surface,but not its evolved shape. Hence, they can be dropped.The normal component of the flow can be calculated bytaking the inner product of the right hand side of (6) withthe unit normal vector in (5) to give

St ¼N1rV1 þN2rV2 þN3rV3 þ ðVu ^ SvÞ ÿ ðVv ^ SuÞ

jjSu ^ Svjj;

�Su ^ SvjjSu ^ Svjj

�N :

It is now a straightforward task to expand the terms inthe expression by using (5):

St ¼ðyuzv ÿ yvzuÞjjSu ^ Svjj2

ðV1xðyuzv ÿ yvzuÞ þ V1yðxvzu ÿ xuzvÞ

þ V1zðxuyv ÿ xvyuÞÞ þðxvzu ÿ xuzvÞjjSu ^ Svjj2

ðV2xðyuzv ÿ yvzuÞ

þ V2yðxvzu ÿ xuzvÞ þ V2zðxuyv ÿ xvyuÞÞ þðxuyv ÿ xvyuÞjjSu ^ Svjj2

ðV3xðyuzv ÿ yvzuÞ þ V3yðxvzu ÿ xuzvÞ þ V3zðxuyv ÿ xvyuÞÞ

þ 1

jjSu ^ Svjj2hðVu ^ SvÞ; ðyuzv ÿ yvzu; xvzu ÿ xuzv; xuyv

ÿ xvyuÞi ÿ1

jjSu ^ Svjj2

hðVv ^ SuÞ; ðyuzv ÿ yvzu; xvzu ÿ xuzv; xuyv ÿ xvyuÞi:

Since

Vu ^ Sv ¼ðzvðV2xxu þ V2yyuþ V2zzuÞÿ yvðV3xxu þ V3yyuþ V3zzuÞ;xvðV3xxu þ V3yyu þ V3zzuÞÿ zvðV1xxu þ V1yyuþ V1zzuÞ;yvðV1xxu þ V1yyuþ V1zzuÞÿ xvðV2xxu þ V2yyuþ V2zzuÞÞ;

and

ÿ Vv ^ Su ¼ðÿzuðV2xxv þ V2yyv þ V2zzvÞ þ yuðV3xxv þ V3yyv þ V3zzvÞ;ÿ xuðV3xxv þ V3yyv þ V3zzvÞ þ zuðV1xxv þ V1yyv þ V1zzvÞ;ÿ yuðV1xxv þ V1yyv þ V1zzvÞ þ xuðV2xxv þ V2yyv þ V2zzvÞÞ;

the terms can be grouped and simplified. The curiousresult is that most cancel, leaving the following simpleand elegant form for the 3D flux maximizing flow:

St ¼ ðV1x þ V2y þ V3zÞN ¼ divðVÞN : ð7Þ

tu

2.3 Properties of the Flux Maximizing Flow

We now remark on several interesting properties of the fluxmaximizing flow and demonstrate its connection to relatedwork in the literature.

1. As pointed out by a reviewer, it is possible tosimplify the calculation of the flux maximizing flowsin Sections 2.1 and 2.2 by using the divergencetheorem. This allows the outward flux of a vector

field through a bounding curve or surface to bewritten as the integral of the divergence of thatvector field within the region enclosed.

2. The flows given by (3) and (7) are hyperbolic partialdifferential equations since they depend solely onthe external vector field V and not on properties ofthe evolving curve (2D) or surface (3D). It is easy tosee that these flows will drive toward and thenconverge to a zero level set of the divergence of V.Thus, the existence and uniqueness of a solution tothese flows is guaranteed, unless the vector field hasdivergence that is everywhere positive or every-where negative.

3. The derived flows apply for an arbitrary vector fieldV. However, for the special case that V is thegradient of a potential function such as an intensityimage I, the flux maximizing flow has a veryinteresting connection to Marr and Hildreth’s workon edge detection [25]. Subsequent to our work, butdeveloped independently of it, this idea waspresented in [15]. The intuition can be seen bysubstituting rI into (3):

Ct ¼ divðVÞN ¼ divðrIÞN ¼ �IN :

Thus, the flux maximizing flow moves toward andconverges to zero-crossings of the Laplacian of theimage. Hence, Marr and Hildreth’s proposal foredges as zero-crossings of the Laplacian may beviewed as a solution of the flux maximizing flowassociated with the gradient vector field of theimage. This raises the question of why one wouldwant to implement the flux maximizing flow asopposed to simply finding the zero-crossings of theLaplacian, the latter being a far simpler task. Theanswer, as we shall see in Section 3, is that the flowinterpretation affords the very important advantagethat a subset of relevant zero-crossings for applica-tions of interest such as blood vessel segmentationcan be captured. This can be done by introducing anotion of multiscale flux and by initializing theevolution in regions of high inward flux.

4. For the special case that V is the normalized (unit)

gradient of an intensity image, r Ijj r I jj , the flux

maximizing flow reduces to a form of the well

known geometric heat equation

Ct ¼ divðVÞN ¼ divrI

jjrIjj

� �N ¼ �IN ;

where �I is the Euclidean curvature (in 2D) or meancurvature (in 3D) of the level curve or surface of theimage, respectively, at that point. Thus, the initialcurve or surface is evolving according to the localcurvature of the level sets of the image. The geometricheat equation has been extensively studied in themathematics literature and has been shown to haveremarkable smoothing properties, particularly in 2D[9], [10]. It is also the basis for several nonlineargeometric scale-spaces such as those studied in [2],[1], [13], [14]. It is important to emphasize that despitethis history, the flux maximizing flow is a distinctpartial differential equation since the vector field is notin general a normalized gradient.


5. The use of a gradient vector field as a static externalforce for a parametric snake model has beenproposed in [40]. This gradient vector flow field isderived by minimizing a particular energy func-tional associated with edges in an intensity image.This technique has been shown to be useful inmaking parametric active contours less sensitive toinitialization while being better able to captureboundary concavities. Similar ideas have also re-cently been incorporated in a geometric activecontour framework using level sets [30]. It isimportant to emphasize that although the vectorfield V may be selected as the gradient of anintensity image, the flux maximizing flow is distinctfrom these methods.

6. A number of nonlinear diffusion filters have beenproposed in the literature for image smoothing whilepreserving or enhancing features of interest. Severalof these can be expressed in divergence form via theheat equation

@u

@t¼ divðD:ruÞ:

A very nice overview of these methods, along with adiscussion of their theoretical foundations and theirapplications to image analysis problems appears in[38]. Here, u is the input, typically an intensity image,and D is a diffusion tensor, a positive definitesymmetric matrix. As pointed out by Weickert, thereis a connection between nonlinear diffusion filteringand energy minimization since the basic partialdifferential equation may be viewed as the gradientflow that minimizes the integral of an appropriatepotential function. However, the flux maximizingflow introduced in the present paper is distinct fromthese techniques.

3 BLOOD VESSEL SEGMENTATION

We now tailor the flux maximizing flow to the segmentationof blood vessels in angiography images, an applicationwhich is of great interest in medical imaging. We begin byreviewing a few recent approaches to this problem.

3.1 Background

McInerney and Terzopoulos have extended the classicalparametric snakes by introducing an affine cell decomposi-tion of the underlying space to give them topologicalflexibility [26]. These models have been applied with somesuccess to the segmentation of angiography as well as otherforms of medical data and extensions to 3D have beendeveloped [28], [27]. Wilson and Noble have introduced aGaussian mixture model to characterize the physicalproperties of blood flow [39]. The parameters are estimatedusing the expectation maximization (EM) algorithm andstructural criteria are then used to refine the initialsegmentation. Krissian et al. propose a method whichincorporates a Gaussian model for the intensity distributionas a function of distance from vessel centerlines, andexploits properties of the Hessian to obtain geometricestimates [18]. Koller et al. have introduced a multiscalemethod for the detection of curvilinear structures in 2D and3D data [16] which combines the responses of steerablelinear filters and also exploits the Hessian matrix to obtain

geometric estimates. Bullitt et al. have introduced a methodfor obtaining 3D vascular trees which calculates vesselcenterlines as intensity ridges in the data and estimatesvessel widths via medialness calculations [4]. Several of theabove approaches require second derivative computations,e.g., to compute the Hessian. Numerically accurate esti-mates of principal curvature magnitudes and directions areobtained only when the intensity images have been suitablysmoothed. Approaches to smoothing the data whilepreserving vessel-like structures include [17], [8].

Whereas the potential of several of the above approacheshas been empirically demonstrated, their ability to recoverlow contrast thin vessels remains unclear. A recent frame-work which has been developed with this as one of its goalsis the work of Lorigo et al. [19], [20]. The main idea is toregularize a geometric flow in 3D using the curvature of a3D curve, rather than the classical mean curvature basedregularizations which tend to annihilate thin structures. Thework is grounded in the recent level set theory developedfor mean curvature flows in arbitrary codimension [3]. Thisflow is given by [19], [20]:

t ¼ �ðr ;r2 Þ þ �ðr :rIÞ g0

gr :H rI

jjrIjj :

Here, is an embedding surface whose zero level set is theevolving 3D curve, � is the smaller nonzero eigenvalue of aparticular matrix [3], g is an image-dependent weightingfactor, I is the intensity image, and H is its Hessian. Fornumerical simulations, the evolution of the curve isdepicted by the evolution of an �-level set. It should benoted that, without the multiplicative factor �ðr :rIÞ, theevolution equation is a gradient flow which minimizes aweighted curvature functional. The multiplicative factor is aheuristic which modifies the flow so that normals to the�-level set align themselves (locally) to the direction of imageintensity gradients (the inner product of r and rI is thenmaximized). However, with the introduction of this termthe flow loses its pure energy minimizing interpretation.

3.2 The Flux Maximizing Flow

In order to adapt the flux maximizing flow to blood vesselsegmentation, we shall consider the gradient rI of theoriginal intensity image I to be the vector field V whoseinward flux through the evolving curve (or surface) is to be


Fig. 2. An illustration of the gradient vector field in the vicinity of a bloodvessel in (top) 2D and (bottom) 3D. Assuming a uniform backgroundintensity, at its centerline, at the scale of the vessel’s width, the totaloutward flux is negative. Outside the vessel, at a smaller scale, the totaloutward flux is positive. Thus, when seeds are placed within vessels thesinks will drive them toward boundaries while the sources will preventthem from leaking.

maximized. An important consideration in the implementa-tion of (3) is that, since the divergence of the vector field

needs to be calculated, implicitly second derivatives of I arebeing used. This may be problematic at locations where thegradient vector field is becoming singular, such as at blood

vessels, which are precisely the areas of interest. Ratherthan explicitly calculate the divergence, we shall make thenumerical computation much more robust by exploiting a

consequence of the divergence theorem. The divergence at apoint is defined as the net outward flux per unit area, as the

area about the point shrinks to zero

divðVÞ � lim�R!0

R�R < V;N > ds

�R: ð8Þ

Here, �R is the area of the region R, �R is its bounding

contour,N is the outward normal at each point on the contour,

and ds is an arc length element. Via the divergence theorem,ZR

divðVÞdR �Z�R

< V;N > ds: ð9Þ

In other words, the integral of the divergence over a region

is given by the outward flux through that region’s bounding

contour. The formulation extends to 3D by replacing the

contour integral with a surface integral.For our numerical implementations, we shall use this flux

formulation along the boundaries of discs (in 2D) or spheres

(in 3D). Strictly speaking, the outward flux leads to the

divergence only in the limit as the region shrinks to a point.


Fig. 3. An illustration of the flux maximizing flow in 2D. A cropped portion of a retinal angiography image is shown on the top left with the multiscale

outward flux of the gradient vector field on its right. Bright regions correspond to negative flux. The other images show the evolution of a few isolated

seeds to reconstruct the vessel boundaries. Observe that the very low contrast vessel at the top is successfully reconstructed.

However, an important advantage of the integral form is thatthe measure allows for the incorporation of an appropriatelocal scale for the computation, which corresponds to thewidth of a vessel, if one is present. The idea is as follows: Ateach location consider discs (in 2D) or spheres (in 3D) ofincreasing radii, where the radii cover a range between theminimum and the maximum expected blood vessel radii.Compute the outward flux over all such discs (in 2D) orspheres (in 3D) by discretizing the right-hand side of (9) anddividing by the number of entries in the discrete sum. Now ateach location select the flux value with the largest magnitude,over the range of radii considered.

In our experiments, we have found this to be an effectivemeans of numerically estimating the flux by which to drivethe flow for blood vessel segmentation in angiographyimages. In contrast to other multiscale approaches wherecombining information across width scales is nontrivial[18], normalization across scales is quite straightforward.Locations where the total outward flux is negative may beviewed as generalized sinks and locations where the totaloutward flux is positive may be viewed as generalizedsources, as illustrated in Fig. 2. Hence, this implementationof the flux maximizing flow has the desirable effect thatwhen seeds are placed within blood vessels the sourcesoutside boundaries will prevent the flow from leaking.

A final important consideration in the application of theflow is the regularization of the vector field V whose flux is

being maximized. We have assumed that the vector field hasbeen smoothed prior to the application of the flow. This is areasonable assumption because the nature of regularizationmust depend on the particular application. A few iterations ofthe geometric heat equation applied to the original dataworks very well for blood vessel segmentation in angiogra-phy images. In other applications, there is a growing interestin the development of regularization methods for particulartypes of vector valued data. Two very interesting recentdevelopments in the computer vision literature are thosedescribed in [36], [37]. Whether or not the regularization of thevector field can be included in the derivation of a flow relatedto the one we have presented remains an interesting (butnontrivial) subject to investigate. It is nontrivial because thecalculation would have to combine a maximization of a fluxwith a minimization of an energy functional related to a normof the vector field. Furthermore, even if such a flow could bederived, the existence and uniqueness of a solution would bein question.

3.3 Level Set Implementation

In order to implement the flow, we use the level setrepresentation for curves flowing according to functions ofcurvature [29]. This is now a standard approach forimplementing partial differential equations of this type inthe literature, since it allows for topological changes to


Fig. 4. An illustration of the flux maximizing flow in 2D. A different cropped portion of the retinal angiography image is shown on the top left. The multi-

scale outward flux of the gradient vector field is shown on its right. Bright regions correspond to negative flux. The other images show the evolution of

a single isolated seed to reconstruct the vessel boundaries.

occur without any additional computational complexity.Further details are presented in [29], [32].

Let Cðp; tÞ : S1 � ½0; �Þ ! R2 be a family of curvessatisfying the curve evolution equation

Ct ¼ FN ;

where F is an arbitrary (local) scalar speed function. Then,it can be shown that, if Cðp; tÞ is represented by the zerolevel set of a smooth and Lipschitz continuous function : R2 � ½0; �Þ ! R, the evolving surface satisfies

t ¼ F jjrjj:

This last equation is solved using a combination of straight-forward discretization and numerical techniques derivedfrom hyperbolic conservation laws [29]. For hyperbolic terms,care must be taken to implement derivatives with upwindingin the proper direction. The evolving curve C is then obtained

as the zero level set of . The formulation is analogous for thecase of surfaces evolving in 3D.

4 EXAMPLES

We now illustrate the flux maximizing flow with several 2Dand 3D simulations. For the 2D examples, the seeds wereplaced manually in order to illustrate the properties of theflow. For the 3D examples, the volumetric data was sampleduniformly and seeds were placed at locations of high inwardflux, where the inward flux was calculated at an appropriatelocal scale, as described in Section 3.2. Our experimentsindicate that the technique is not sensitive to the precisethreshold used to locate seeds. In most cases, complexvasculature can be recovered from a very sparse initialization(corresponding to a conservative threshold), provided thatconnected paths to the desired vessels exist.


Fig. 5. An illustration of the flux maximizing flow for a portion of a 171� 256� 256 3D MRA image of blood vessels in the head. Two distinct views of

the same evolution are shown (top two rows and bottom two rows). For each view, a maximum-intensity projection of the cropped portion is shown

on the top left and the other images depict the evolution of a few isolated blobs according to the flux maximizing flow. The full reconstruction is shown

in Fig. 6.

Fig. 3 and Fig. 4 show the flow on two different portions

of a 2D retinal angiogram.1 Notice how a few seeds evolve

along the direction of shading (orthogonal to the image

intensity gradient direction) to reconstruct thin or low

contrast structures, e.g., the top portion of Fig. 3. Most other

flows, particularly ones with a constant inflation term,

would leak through such boundaries. The introduction of acurvature-based regularization term may prevent leaking toan extent, but the flow would then be halted at narrowregions as well, since the curvature term would dominateand would push the evolving curve back.

Fig. 5 illustrates the 3D flow on a portion of a 171�256� 256 magnetic resonance angiography (MRA) image ofblood vessels in the head. A maximal intensity projection ofthe data is shown on the top left, followed by the evolutionof a few seeds initialized uniformly in regions of highinward flux, which is similar to the idea of using “bubbles”


Fig. 6. An illustration of the flux maximizing flow for the full 171� 256� 256 3D MRA image, of which a portion was shown in Fig. 5. A maximum-

intensity projection of the data is shown on the top left and the other images depict the evolution of a few isolated seeds. The main vessels, which

have higher inward flux, are the first to be reconstructed.

Fig. 7. An illustration of the flux maximizing flow for a 60� 256� 256 MRA data set obtained under the same imaging conditions as those used in

[19], [20]. A maximum-intensity projection of the data is shown on left and the reconstructed blood vessels are shown on the right.

1. These were cropped from a gray-level image which was obtained froma Web page. A considerable amount of numerical precision was lost sincethe saved image had only 256 gray levels. Hence, these serve as good testcases for the flow.

[35]. Notice how the seeds elongate in the direction of bloodvessels, which is once again the evolution we expect since itmaximizes the rate of increase of inward flux through them.The effectiveness of the flow in reconstructing the full dataset is illustrated in Fig. 6. The main blood vessels, whichhave the higher inward flux, are the first to be captured.

Fig. 7 illustrates the reconstruction of blood vessels on a60� 256� 256 MRA image that is very similar to those usedin [19], [20]. It was taken under the same imaging conditionsbut the data corresponds to a different patient. The resultsillustrate the ability of the flow to recover several thin lowcontrast vessels along with the main structures.

We conclude with experiments on a 360� 330� 420computed rotational angiography (CRA) data set of thehead, from which we have selected four distinct regionscontaining vascular networks of varying complexity. Theevolution results for one of these regions is presented inFig. 8. The initial spheres were placed automatically butsparsely in regions of high inward flux. Once again, themain blood vessels which have the higher inward flux arethe first to be captured. The evolution has the intuitivebehavior that it follows the direction of blood flow toreconstruct the blood vessel boundaries.

Although CRA data is of higher resolution than MRA,the vessel structures exhibit a wider range of intensities and


Fig. 8. An illustration of the flux maximizing flow for a portion of a 360� 330� 420 3D CRA image of blood vesels in the head. A maximum-intensity

projection of the region being viewed is shown on the top left. The other images depict the evolution of a few isolated spheres. Notice how the

evolution follows the direction of blood flow to reconstruct the blood vessel boundaries.

there are also a number of other structures whose intensitiesoverlap with those of the thin vessels. Thus, simplethresholding of the intensity data generally gives poorresults, although this is a commonly used initialization stepin many algorithms including the approach of [19], [20].This point is illustrated in Fig. 9. The first row shows theresults of a high threshold on the four regions. As onewould expect, many thin low contrast vessels are notcaptured. When the threshold is decreased, more thinvessels begin to be captured but many voxels are alsoincorrectly labeled as vessels (Fig. 9, second row). Thesegmentation results obtained by the flux maximizing floware presented in the last row. The arrows point to some of

the thin low contrast vessels that are successfully captured,but are not seen even in the low threshold case. Our ownexperience with several of the related geometric flows in theliterature is that many would fail in low contrast regions orwould not be able to capture the thinner vessels.

5 CONCLUSION AND DISCUSSION

The main contribution of this paper is the formulation andderivation of a flux maximizing geometric flow. Thecalculation leads to the simple interpretation that in orderto increase the inward flux as fast as possible, each point ona curve (2D) or surface (3D) should move in the inward


Fig. 9. A comparison of the segmentation results obtained by the flux maximizing flow with simple thresholding on four different regions of a

360� 330� 420 CRA image. First Row: A conservative high threshold fails to capture many thin low contrast vessels. Second Row: A lower

threshold captures some of the thinner vessels but also incorrectly labels many voxels. Third Row: The segmentation results obtained by the flux

maximizing flow, with arrows pointing to some of the thin low contrast vessels that are captured.

normal direction, by an amount proportional to thedivergence of the vector field. When combined with anotion of multiscale flux, this result can be exploited todevelop an algorithm for recovering the boundaries ofblood vessels from an angiography image. The idea is toapproximate the divergence by measuring the outward fluxat the local scale which best corresponds to the width of avessel. Several numerical experiments implemented in alevel set framework have been presented to demonstrate thepotential of this technique.

We chose to initialize the flow by placing seeds in regionsof high inward flux. However, one might consider thresh-olding the flux to obtain an initialization which has alreadyreconstructed large portions of blood vessels. In practice, wehave found that this performs better than simple thresholdingof the original data as in [39], [19], [20]. We also chose not tointroduce a regularization term in the variational formula-tion. Whether this can be incorporated in the derivationremains to be investigated. One choice, which has a verysimilar behavior to the curvature of a 3D curve used in [3],[19], [20], is the minimum principal curvature of the surface.Intuitively, as the cross-sectional width of a tubular structureshrinks to zero, the minimum principal curvature approachesthe curvature of the 3D curve obtained in the limit. Anotheruseful direction for future work is the validation of the fluxmaximizing flow for vessel segmentation against otherapproaches in the literature as well as manual segmentationsby experts. We hope to gain access to the data sets used in [19],[20] in order to be able to carry out such quantitativecomparisons. Our preliminary experiments on an MRA dataset obtained under the same imaging conditions indicate thatthe flux maximizing flow is able to recover many thin lowcontrast vessels.

ACKNOWLEDGMENTS

This work was supported by grants from the CanadianFoundation for Innovation, FCAR Quebec, and the NaturalSciences and Engineering Research Council of Canada. Theauthors are grateful to Vincent Hayward, Terry Peters,Bruce Pike, David Holdsworth, Ron Kikinis, and Carl-Fredrik Westin for the MRA and CRA data. This article isbased on papers presented at the International Conferenceon Computer Vision, 2001, and the International Workshopon Engergy Minimization Methods in Computer Vision andPattern Recognition, 2001.

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Alexander Vasilevskiy received the BA degreein mathematics and computer science in 1999,and the MSc degree in computer science in2001, all with honors from McGill University.Prior to that, he studied mathematics andpsychology at Odessa State University, Ukraine.He is currently a software developer with theIBM Toronto Research Laboratory, Canada. Hisresearch interests are in computer vision,

medical image analysis, and optimizing compilers. He is a studentmember of the IEEE.

Kaleem Siddiqi received the BS degree fromLafayette College in 1988, and the MS and PhDdegrees from Brown University in 1990 and1995, respectively, all in the field of electricalengineering. He is currently an associate pro-fessor in the School of Computer Science atMcGill University and is a member of McGill’sCenter for Intelligent Machines. Before movingto McGill in 1998, he was a postdoctoralassociate in the Department of Computer

Science at Yale University (1996-1998) and held a visiting position inthe Department of Electrical Engineering at McGill University (1995-1996). His research interests are in computer vision, medical imageanalysis, and human psychophysics. He is a member of Phi Beta Kappa,Tau Beta Pi, and Eta Kappa Nu, and is a member of the IEEE and theIEEE Computer Society.

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Flux maximizing geometric flows - Pattern Analysis and ...shape/publications/pami02.pdfMalladi et al. [22], [23], [24] and was also developed independently by Caselles et al. [5]