Model Based Detection of Tubular Structures in 3D Imageslibvolume3.xyz/.../tubularstructures/tubularstructurestutorial1.pdf · Model Based Detection of Tubular Structures in 3D Images

Model Based Detection of Tubular Structures in 3DImages

Karl Krissian�

, Grégoire Malandain, Nicholas AyacheEpidaure project INRIA - 2004, route des Lucioles

06902 Sophia Antipolis France;

and

Régis Vaillant, Yves Trousset

General Electric Medical Systems 1 BP 3478533 Buc Cedex, France

E-mail:�

[email protected]

Detection of tubular structures in 3D images is an important issue for vascular

medical imaging. We present in this paper a new approach for centerline detection

and reconstruction of 3D tubular structures. Several models of vessels are introduced

for estimating the sensitivity of the image second order derivatives according to

elliptical cross-section, to curvature of the axis, or to partial volume effects. Our

approach uses a multiscale analysis for extracting vessels of different sizes according

to the scale. For a given model of vessel, we derive an analytic expression of the

relationship between the radius of the structure and the scale at which it is detected.

The algorithm gives both centerline extraction and radius estimation of the vessels

allowing their reconstruction. The method has been tested on synthetic images, an

image of a phantom, and real images, with encouraging results.

Key Words: filtering, vessel detection, multiscale analysis, segmentation

CONTENTS1. Introduction.2. Study of second order derivatives on several models.3. The method.4. Experiments and Results.5. Conclusion and Discussion.

�

Supported in part by General Electric Medical Systems

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1. INTRODUCTION

1.1. MotivationIn this paper, we present a new method for segmentation and detection of tubular struc-

tures in 3D images. We propose a general method for extracting centerlines of tubularstructures independently of the modality of acquisition of the image. Thus, we focus oursegmentation on a geometric approach where the first and second order derivatives of theimage are obtained using a linear multiscale analysis. Although the proposed method canbe applied to any kind of 3D image, we focus here on the extraction vascular networksfrom medical images.

An accurate detection of the vascular network from various organs (liver, lungs, brain)is a very important issue in medical image analysis. Indeed, problems like aneurysm orstenosis can occur in a vessel, and the physicians need tools to help them in interpreting andquantifying the images for evaluating the pathology, for proposing a therapy or a surgicaloperation, for planning minimally invasive treatment.

Working on 2D projection can easily mislead the comprehension and the interpretation ofthe structures as shown in [36] in the case of stenosis quantification. Different acquisitiontechniques allow to obtain 3D images of the vessels network, we experimented our methodon 3D reconstructed images from 2D X-ray subtracted angiographies and on phase contrastMagnetic Resonance Angiographies (MRA).

In order to make a decision on a case, the physicians need tools both for visualizing andfor quantizing the vessels. The visualization of the whole volume data can be obtained byprojecting a selected information of the image according to a given point of view. We candistinguish three methods that allow this kind of visualization:

� Maximum Intensity Projection (MIP) displays at each point the maximum of theintensities of the voxels that project on this point. This technique is efficient when theintensity of the vessels is higher that the intensity of the other structures in the studiedimages, which is the case in the studied images. If it allows a visualization of the wholeimage, it nevertheless has the drawback of giving no information about the relative positionof vessels along the projection axis. Working directly on this projection leads also to vesselwidth underestimation and a decreased contrast-to-noise ratio [5].

� Surface rendered view gives better information of the relative position of the vessels,allowing rendering effects like lighting and depth-cueing. It requires a binary segmentationof the image for selecting the viewed surface. In the case of an isosurface view, thissegmentation is simply done by choosing a threshold on the image intensity, and a smoothedsurface can be obtained using the Marching Cubes algorithm [25].

� Volume Rendering is more flexible as it allows to view a projection of the wholevolume of data by allocating to each voxel a measure of probable occupancy, typicallybased on the image intensity. It also allows rendering effects that give an idea of therelative position of vessels. Nevertheless, it requires more interactivity for choosing theprobability map.

If the visualization allows to find a region of interest that contains the pathology and toplan a minimal invasive treatment, the physicians needs also a quantification of the degreeof pathology to make a decision. In order to help both visualization and quantification of thevessels, we propose to represent the vessels network as a set of centerlines and to associate,at each point of a centerline, a circular cross-section whose radius is a function of the

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curvilinear coordinate. This representation allows both to visualize the set of centerlinessuperimposed on the initial image and to view a reconstructed surface of the detectedvessels. We use a linear multiscale analysis for extracting the vessels centerlines, with aresponse function that is locally maximal at the center of the vessel and at a scale that is afunction of its size.

1.2. Previous workA challenge in processing of vascular images is to detect vessels of different sizes. A

way to take into account the varying size of vessels in the image is to apply a multiscaleanalysis. Multiscale analysis allows to detect structures of various sizes according to thescale at which they give a maximal response. We successively present the notions of linearscale-space, of medialness and ridge, and previous work dedicated to vessel detection.

1.2.1. Linear Scale-Space

When applying a multiscale analysis to an image, the use of the convolution productwith a Gaussian kernel and its linear partial derivatives has been shown to be the only wayto ensure the following properties: - linearity, - invariance under translation (spatial shiftinvariance), - invariance under rotation (isotropy), - invariance under rescaling [17, 23, 10].Florack et al. [10] show that the evolution through scales can be written using twodimensionless variables

��and � �� by the means of the Pi-theorem which states that a

function that relates physical observables must be independent of the choice of dimensionalunits.

�denotes the standard deviation of the Gaussian kernel,

��is the response obtained

from the initial image and�

is the response obtained at a scale � � � .In his works on scale-space theory [23, 24], Lindeberg shows the necessity of normalizing

the derivatives of the image in the multiscale analysis. He introduces the notion of � -normalized derivatives :

��

(1)

When the parameter � equals one, the normalization ensures invariance under imagerescaling, which is compatible with the dimensionless variable � !� �"� :

�$#� �

�$#� ��

� �$#� �However, for certain specific task (extraction of 2D blob, of edges, of 2D ridges),

Lindeberg studied on analytical models the relationship between the scale at which anobject is detected (gives the maximal response), the normalization parameter � , and theobject size, which can lead to choose other values for � .

In the following, we will implicitly suppose that the scale-space used is linear andobtained from Gaussian convolution of the image and its derivatives.

1.2.2. Medialness

Pizer et al. [34] uses the notion introduced by Blum [3, 4] in order to characterizethe shape of an object by the means of medial axes containing width information. In 2Dimages, Blum defined the medial axis as the locus of centers of disks of maximal fit withinan object. Making use of the boundariness which measures the presence of contours, Pizeret al. define the medial axis, and then the multiscale medial axis (MMA) which defines boththe central axis and the width of objects. Medialness at a given point and scale %'&(�*)�+ � )-,


measures the degree of belonging of the point � ) to the medial axis of the object. In [34],it is defined as the integration over space, scale and direction of a weighted boundariness� & � ) +�� + � ) + � � +��,�� &(�� + � � +�� , , where the weight

�is maximum when - �� is

at a distance from ��) proportional to� � with a constant of proportionality � , -

� ) isproportional to

� � with a constant of proportionality , - � � has the same orientation anddirection as � �� ) .

In a more recent work [35], he generalized this notion. The medialness can be definedas a convolution product of the initial image with a kernel � &(�*+ � , :

%'& � + � , # & ��,� �� & � + � ,To ensure the properties of invariance under rotation, translation, and rescaling, � is basedon normalized Gaussian derivatives of intensity, computed at a distance from � proportionalto�

and at positions that are rotationally invariant relative to � .He classifies medialness function in two ways: first, central or offset medialness; second,

linear or adaptive medialness. On one hand, central medialness is obtained by localinformation, using spatial derivatives of the image at a point � and a scale

�. Offset

medialness uses the localization of boundaries by averaging spatial information about �over some region whose average radius is proportional to

�. On the other hand, medialness

is said to be linear when � is radially symmetric and data-independent; and adaptive when� is data-dependent.

We give below examples of the different possibilities of 3D medialness, according to theprevious classification:

� the normalized Laplacian operator that uses convolution with the second order deriva-tives of the Gaussian defines a central linear medialness;

� the sum (in absolute value) of the two largest magnitude eigenvalues of the Hessian ma-trix, where the components of the Hessian matrix are obtained by convolution of the imagewith the second order derivatives of the Gaussian defines a central adaptive medialness;

� the integration of the boundary information around a sphere whose radius is propor-tional to scale defines an offset linear medialness; where the boundary information can bedefined as the norm of the gradient operator for example;

� the integration of the boundary information around a circle located in the plan givenby the current point and two eigenvectors of the Hessian matrix, corresponding to the twolargest magnitude eigenvalues, defines an offset adaptive medialness.

1.2.3. Ridges of medialness

The different definitions of ridges and their invariance properties where reviewed byEberly et al. [8]. They also propose an extension of the concept of ridges of dimension �in � dimensional images:�� "!��"!

If# &$#�$, is a real-valued function defined for #�&%(' � , and ) &*#� ,

is the Hessian matrix of#

at #� . Assume that the eigenvalues of ) &$#��, are ordered as+�,.-0/1/2/3-(+ �with associated eigenvectors &54�6 , 687:9 , � �<; , and assume that = - � - � :

#� is a ridge point of type � � if and only if > 4 , /2/1/ 4@?2ACBED # &*#� , (F and+ ?HGIF .


In the context of multiscale analysis, ridges can be extracted in a space including thespatial and scale dimensions. The Multiscale Medial Axis [34, 28, 13] or also called coreis an example. Extraction of such ridges requires specific algorithms [24, 12, 14, 35] as forexample the so-called Marching Lines [43, 44] derived from the Marching cubes [25] andapplied for multiscale crest lines extraction in medical images [9].

1.2.4. Works dedicated to vessel detection

We concentrate here on works that use a linear multiscale analysis for vessel detection,especially in 3D, and that propose different response functions (or medialness). Otherworks that do not use this kind of analysis can be found, for example Verdonck [45] fitsa model of the vessels contours to the image, Szekely et al. [41] propose a symbolicdescription of the vessels network and show results on MRA, Summers et al. [40] use amultiresolution data structure to extract vascular morphology and local flow parametersfrom phase contrast magnetic resonance angiograms.

In the following, we will refer to linear structures for structures that can be locallyapproximated as a line at every point of the structure, this definition applies for 2D or 3Dimages. A structure can be approximated as a line if it is elongated in one direction of thespace, and it has a low thickness in any other orthogonal orientation. They are in generalvessels in the studied images. We will refer to planar structures in 3D images for structuresthat can be locally appromixated as a plan at every point of the structure. A structure canbe approximated as a plan if it has a low thickness in a given orientation of space, and it iselongated in any other orthogonal orientation. An example of a planar structure is the skinin 3D images of the brain.

The work of Koller et al. [18] propose a multiscale response in order to detect linearstructures in 2D images. The response function uses eigenvectors of the Hessian matrixof the image to define at each point � an orientation orthogonal noted � to the axis of apotential vessel that goes through � . From this direction, the two points located at equaldistance of � in the direction � are noted � , and � � . The response at � is defined bythe minimum of � �� 6 / D # &�� 6 ,�� *3��6 � for � % ��=�+�� .

The authors put the emphasis on the discrimination between contours and vessels centers.They propose also an extension to 3D, but they use only four points equidistant to � . Wewill propose instead, for symmetry reason and for more robustness, to use the set of pointslocated on a circle centered on � and in the plan of the cross-section that contains thepoint � .

Following this work, Lorenz et al. [26] decided to use further information from theHessian matrix: its eigenvalues. Indeed, after a Taylor expansion to second order of theimage intensity Eq. (2), the eigenvalues of the Hessian matrix, when the gradient is weak,express the local variation of the intensity in the direction of the associated eigenvectors.

# &�� ,- # &�� ,��HD # �� B ) & # ,�� &�� , (2)

In this way, for white structures on dark background, a linear structure has two negativeand high eigenvalues and a third one which is low in absolute value, and a planar structurehas only one negative and high eigenvalue and two other low eigenvalues. This notingleads them to define a response function which depends on the eigenvalues of the Hessianmatrix. However, the authors show only a single result on a three dimensional image whichcontains only two tubular structures.


A more recent work done by Sato et al. [38, 37] also proposes to choose a responsefunction based exclusively on the eigenvalues of the Hessian matrix. The choice of theresponse function which combines the three eigenvalues is heuristic and is based on anexperimental study on various cases (curved vessels, junctions of vessels).

The interest of their work is to show that a single method can given results on severalmodalities: MRA, CT and still describing different anatomical structures: vessels in brain,bronchi or liver. Their approach is to provide a visual help in the interpretation of the imageafter filtering. However, the images used in their experiments seem to have a higher spatialresolution than usual images used in clinical practice, and their algorithm, which uses veryfew discrete scales, doesn’t detect vessel axes and doesn’t seem suitable for an accurateestimation of vessel size. In the same state of mind, Frangi et al. [11] propose anotherresponse function by interpreting geometrically the eigenvalues of the Hessian matrix.

Using the classification of Pizer et al. described in section 1.2.2, all those works presentdifferent choices of medialness that are adaptive because they depend on the Hessian matrixin a non-symmetric way, and are either central [26, 38, 11] or offset [18].

1.3. Contributions and organization of the articleThe contributions of this paper, based on previous works [21, 22, 20], are twofold.First, we propose a new adaptive medialness measure for detection of tubular structures

in 3D images. The adaptive property of the medialness is based on the characteristics of theHessian matrix of the image, its eigenvectors and eigenvalues. We present an analytic studyof those characteristics on different models of vessels including elliptical cross-sections andvessel axis curvature. This study shows that eigenvalues are sensitive to the vessel curvatureand to elliptical cross-section whereas the estimation of plan of the cross-section based ontwo eigenvectors is more stable. Thus a response function based on both eigenvectorsand gradient information must be more robust than a response function based only oneigenvalues. This leads us to choose an offset rather than a central medialness response.

Second, we use a simple model of cylindrical vessel with circular Gaussian cross-sectionto guide our detection. The analytic computation allows a scale-selection in the same wayas Lindeberg [24]. We then express the relationship between the parameters � , the selectedscale, and the structure width and choose those parameters according to the model. Fromthis relationship, we make a full reconstruction of the vessels network. Other works do notuse an explicit model to derive optimal parameters of their response function. This modelis suitable for extracting the centerlines because the cross-sections converge to this modelwhen convolved with a Gaussian with increasing standard deviation. However, we willshow in the experimental study that the size estimation based on the multiscale analysis isdependant on the intensity profile of the cross-section. For this reason, we also propose amore realistic model with a bar-like convolved cross-section and propose a correction ofthe radius estimation based on this new model.

The first section describes a first cylindrical circular model and two derived models thatare curved and with non-circular but elliptical cross-section. Those theoretical modelsare used to compute eigenvalues and eigenvectors of the Hessian matrix and to interprettheir values and their sensitivity with respect to the position of the current point, the imageintensity, the radius of the structure, the vessel curvature, the non-circular cross section.The second section describes the proposed measure of medialness, and the automatic scale-selection based on the cylindrical circular model. It also gives the relationship betweenthe size of the structure and its selected scale. Eventually, it explains the extraction of the


local extrema and the reconstruction stages. Experiments and results are detailed in a thirdsection. Synthetic images are used to validate the analytical study and to show the behaviorof the algorithm under different kinds of tubular structures. Experiments on a phantomimage are done to validate our radius estimation. And an application on real images, thatare 3D reconstruction of the brain vessels from 2D X-ray angiographies and a brain phasecontrast MRA, is also presented, where the vessels network reconstruction is compared tousual isosurfaces rendering or MIP views.

2. STUDY OF SECOND ORDER DERIVATIVES ON SEVERAL MODELS

Following the work of [26], several articles have been dedicated to the visualization ofvessels after a multiscale filtering, whose response is exclusively based on the eigenvaluesof the image Hessian Matrix [38]. In order to understand the link between the eigenvaluesand the local structure of the image, we evaluate in this section the analytic expression ofthese eigenvalues for several theoretical models derived from a simple cylindrical circularmodel. The cross-section in each model is either a circular or an elliptical Gaussian blob.In this section, we use the following notations:-# �

is the initial image,- � denotes the studied point of coordinates &(�*+��$+��, in the definition domain of the image,-� �

denotes the radius of the initial vessel model which is also the standard deviation of aGaussian,- �� is a Gaussian function with standard deviation

�,

- ) is the Hessian Matrix of the image, )�� is a simplified matrix proportional to ) ,-+�, + + � + + � are eigenvalues of the Hessian matrix with � + , � � + � �� + � � ,

- �� + �� + �� are the associated eigenvectors.

2.1. First model: Cylindrical circular model with Gaussian cross-section

FIG. 1. Initial model of a vessel. On the left, representation of the model in the basis �� . On theright, representation of the Gaussian-like intensity profile of the cross-section.

The first vessel model that we introduce is cylindrical where &�� , is the vessel axis andthe vessel section is a Gaussian blob:

# � & �*+!�$+!��, #"$� �&% &(�*+�� , "�(' � � �*)

�,+!-!.0/1--�2 % - / (3)


where " is a function of� �

and �� % - represents the intensity at the center of the vessel(Fig. 1). " depends on the size of the vessel, this dependence is due to partial volumeeffect that decreases the small vessels intensity. The partial volume effect correspondsto an average of the intensities at the frontier between different tissues, according to theproportion of volume occupied by each tissue in the voxel.

The model properties are:

� the frontier of the vessel is considered to be at the points where the first derivative inthe gradient direction is maximum, i.e., for the points which verify � � �$� � � � � , thusthe vessel radius is

� �. In this paper, we will further use the word frontier to express the

vessel boundaries.� if the model is convolved with a Gaussian kernel of standard deviation

�, the resulting

image is another vessel which matches our model but with a radius� � �� . This result

can be directly deduced from the semi-group property.

2.1.1. Expression of eigenvalues and eigenvectors of the Hessian matrix

The Hessian matrix can be expressed as ) � %� %�� ) � where

) � �� *� F� � � � � � � FF F F

�and the eigenvalues and eigenvectors of ) are given in Table 1, where

# � # � & � +��$+!��, isthe initial image, and

� �is the radius of the initial image.

TABLE 1

Eigenvalues and eigenvectors for the cylindrical circular model.�� %� -%�� -%�� -�!#"�-�$� -% % �

&�� %� -%'(*) � � � � �,+!� '(*- � ��(� � � '(/. � � �� This means that our model has the following properties:

� Inside the vessel ( � � � � � G � � � ) we have two negative eigenvalues with eigenvectorsin the plane orthogonal to the axis of the vessel.

� The third eigenvalue is null and the associated eigenvector is in the direction of theaxis.

� The eigenvalues+�,

and+ � are maxima in absolute value when � � F and are

equal to � %10 �32 � � 412 �65�&% - . And

+ � increases faster than+ ,

as a function of the distance to thecenter and becomes positive outside the vessel where

� �� & � � � � � , is negative.

For the multiscale process, the model is convolved with a Gaussian kernel of standarddeviation

�and the results are still valid due to the semi-group property but

� �has to be

replaced by� � �� . In order to better take into account the reality of the vessels, we

will study two variations of this model. The first one is a toroidal circular vessel whichallows us to introduce a curvature of the vessel. The second one is a cylindrical vessel withan elliptical cross-section which introduces a variation in the circular shape of the vessel.


2.2. Toroidal circular model

FIG. 2. Toroidal model of a vessel.

In this case, the eigenvalues and eigenvectors are similar except that the third eigenvalueis not zero everywhere but only at the center of the vessel (see appendix A).

We model the vessel with a torus, the big circle parallel to the plane XY and with a radius�and the small circle with a radius equal to � , as shown in Fig. 2.The intensity function of the model is given by the expression:

# � &(�*+!��+��, #" )� &�� + - .0/ - , - .� -- 2 -% /

From the circular symmetry around the &�� , axis, we can choose � F and �� F .Then the Hessian matrix H can be expressed as:

) # � & �*+!�$+!��,� �� ) �

where

) � �� & � ��, � � �� F ��& � ��,F 0�� $� 5 � -%� F ��& � �$, F � � � ��

� �The eigenvectors and eigenvalues of ) are expressed in Table 2 (the notations are il-

lustrated in Fig. 2), where � is the current point of coordinates & � +��$+!��, and " � is itsdistance to the torus axis.

In order to interpret the value of+� depending on the curvature of the vessel � ,� ,

we center the reference to the center C of the torus in the plane (Ox,Oy). With the newcoordinate � �� , we have

+�

# ��

� � �� # �� =�� /

When the curvature is null,+� F as in the cylindrical case. Nevertheless, this result

shows that a vessel curvature can lead to positive ( �� ) or negative ( � G �

) values of


TABLE 2

Eigenvalues and eigenvectors for the toroidal circular model.�� %� -%�� 3�� %� -% � � -% �� -� -% % �

�� %� -%'(*) � � � + � � � '(*- �� '( . � �� TABLE 3

Sign of the eigenvalues at the point M for the elliptical model.

(1) � �� + � �� "� / � � � + � �M is inside the ellipse,

� � � � � � �

(2) � �� + � � � "� / � � � + �M is on the ellipse,

� � � � � ��

(3) � �� + � �� "� / � � � + � �M is outside the ellipse,

� � � � � � �

+� in the vicinity of the vessel center. This mean that the absolute value of

+� may not

be negligible compared to the absolute value of the two other eigenvalues when the vesselcurvature is high.

2.3. Cylindrical elliptical modelThe elliptical section is defined by one standard deviation along the x axis,

� �, and one

standard deviation along the y axis,� 4

. The model is thus defined by:

# � & �*+!�$+!��, " )��- � & +2 + , -�� /2 /�� - % /

The Hessian matrix can be expressed as

) # � & �*+!�$+!��,� �� 4 ) �

where

) � �� 4 �� = � � � � 4 �� F� � � 4 �� = � F

F F F

�

�� F� � FF F F

�with

� �� + and

� 4� / .

The eigenvalues are (see appendix B for more details):

+�, # �

� � �� 4 & � � � � & � � � , � ��& � � � � ,�,+ �

# �� 4 & � � � � � & � � � , � ��& � � � � ,�,

As� � � � � �� 4 � & �� + , � � & 4

� / , � = % , we can distinguish three cases depending on

the position of � & � +�� , , summarized in Table 3. We can also study the eigenvalues along

the � and � axis. In both cases, if we choose� � � 4

, �� &8F +1= + F�, and �� &E=�+ F + F�, ,


TABLE 4

Expression of eigenvalues along � and � axis for the elliptical model.

x axis (y = 0)

� �� %� -/ � + � " -� -/�� &� � %� -+

y axis (x = 0)

� �� %� -/ � &� � %� -+ � + � � -� -+ �

center (x = y = 0)

� �� %� -/ � &� � %� -+

Table 4 gives the analytic expression of the eigenvalues. At the center of the elliptical

vessel, one interesting property is that the ratio of the two main eigenvalues is equal to theinverse of the square of the ratio of the respective radii:

+�,+ � �� 4��

This means that a given variation on the radii ratio will lead to a higher variation of theeigenvalues ratio. Fig. 3 shows the section of an elliptical Gaussian vessel with

� � and� 4 �� , and the representation of the eigenvalues+ , & � +�� , and

+ � & � +�� , .Eigenvectors are given in Eq. (B.2) in appendix. Fig. 4 shows the curves which are

tangent to the eigenvectors in the section of the vessel.

2.3.1. Conclusion about the eigenvalues of the Hessian matrix

The three studied models have in common the following properties: at the vessel centerone eigenvalue is null with the corresponding eigenvector in the direction of the vesselaxis, and the two other eigenvalues are negative and equal if the section is circular, orapproximatively equal if the section is an ellipse nearly circular. The three followingconditions express these properties:

+�, & � � � � �,� � B # � = (4)

+�, � + � = (5)

� +�, � � � + � � / (6)

Eq. (4) comes from the relation+ , # � �� in the cylindrical circular model. It

expresses the relationship between the highest negative eigenvalue+ ,

, the intensity of thevessel at the current scale and the vessel radius.

When applying the multiscale analysis, the computation of the derivatives at a scale is equivalent to the computation the derivatives of the image after a convolution with aGaussian kernel of standard deviation � . We already noticed in section 2.1 that the radiusof the cylindrical circular model evolves according to the relation ��&(�, � � �� where� & �, is the radius of the structure as a function of the scale . Thus, applying the relation+ , # � �"� �� for the model at a scale leads to

�� 0��&% - � B 5�� % = where the initial radius� &8F�,� � � is replaced by ��&(�,� � � �� and the intensity# �

is replaced by the intensityof the convolved image � � B # � .


However, as the value of� �

is not known, this relation is difficult to exploit and thesimple criterion

+ � GIF which implies that+�,

is also negative is used.Two main difficulties arise for using the previous criteria. The first one is the discrete

representation of the image combined with the small size of vessels. Actually, vesselssizes are sometimes thinner than two or three voxels and the eigenvalues are not computedat the real vessel center but at the center of a voxel. The second one is the non-circularcross-section that increases the uncertainty on criterion (5).

Thus, the vessel models presented in this section emphasize the difficulty in relying oneigenvalues of the Hessian matrix for an accurate detection of vessel center and size. Forthis reason, we propose to use eigenvalues of the Hessian matrix for the discrimination ofvessel-like structures from other structures, and to use a gradient-based response functionfor the extraction of the vessel centerlines. This extraction is explained in the followingsection.

3. THE METHOD

Our approach can be split into three steps. We first compute the multiscale response fromresponses at a discrete set of scales, we then extract the local maxima in this multiscaleresponse in order to estimate the vessels centerlines. Vessels are then reconstructed usingboth the centerlines and the size information. In the first step, we use a model of the vesselsboth for interpreting the eigenvalues and the eigenvectors of the Hessian matrix and forchoosing a good normalization parameter.

Computation of the single scale response requires different steps. First, a numberof points are pre-selected using the eigenvalues of the Hessian matrix. These pointsare expected to be near a vessel axis. Then, for each pre-selected point, the responseis computed at the current scale. The response function uses information from botheigenvectors of the Hessian matrix and gradient vectors located on a circle centered onthe current point. Finally, this response is normalized in order to give a multiscale scaleresponse that combines interesting features of each single scale response. These steps aredetailed in the following paragraphs.

The following notations are used:- denotes the current scale,- � &$#�$, denotes a point in the definition domain of the image

# �, #� & � +��$+!��,�% ' � ,

-�B &$#��, is the response for a scale and at a given location #� ,

-� �B is the normalized response for a scale ,

- � is a normalization parameter,- �� is the scale at which the normalized response is maximal,-� &*#� +��,- # � &*#� , � � B is the image at a scale t.

3.1. Pre-selection of candidates using eigenvalues of the Hessian matrixIn order to compute the response

�B at one scale , we pre-select points that are expected

to be near the vessel axis. This pre-selection is both a discrimination of non-tubularstructures and a way to save computation time. In the first section, we studied properties ofthe eigenvectors and the eigenvalues of the Hessian matrix on different models of vessels.For this pre-selection, we use a weak version of the criteria given in Eq. (4,5,6) where weonly test that

+�,and+ � are negative.


3.2. Computation of the response��

at one scale tA first choice for

�B can be a 3D extension of the 2D response proposed by Koller et

al. [18]. For a point #� , the response is set to the minimum of the absolute value of theintensity’s first order derivative computed at 4 points equidistant from #� . An advantageof this choice is to ensure that a high response results in a high probability of being at avessel’s center, but this medialness response is too sensitive to noise.

The choice of taking the minimal response among four distant points has been exper-imentally found to give incomplete results. In fact, this choice can lead to a very lowresponse for voxels that are actually located at a vessel center. This can be explained bythe following statement: if only one of the considered four distant points has a very lowboundary information, the response at the corresponding voxel will be low. The possibilitythat one of the considered point has a low boundary information is not negligible and canbe due to local variation of the image intensity, noise, or non-circular cross-section.

It seems more natural to use information from the first derivative at every point of acircle than just four points. This circle " &$#� +�� , is centered at the current point #� , hasa radius � � , and is the plane defined by #� and the two eigenvectors �� and �� . Theproportionality constant � will be chosen according to the model. This constant is theinverse of the constant � ,� already introduced in [35]. We will see in section 3.3.3 howthis constant can be chosen to optimize the response at the maximal scale.

We propose to use the following medialness response:

�B &$#�$, =

�('� � �� 2 � D B # � � #� �� / �� (7)

with �� &� , �� -&� , �� . This response is the mean of first order derivativeinformation taken at the circle " &*#� +�� , . �� is the radial direction and D B

# �is the

gradient vector of the initial image, computed at the scale . To ensure a positive responsefor white structure on dark background, we take the negative of the scalar product betweenthe gradient and the radial direction.

In practice, we must compute this response for a discrete image. Thus, we use � � � &��(' � � = , points along the circle " &�� + �, , where ��% ' + � � &5�*, is the integer partof � , it leads to the discrete formulation:

�B &*#�$, =��

� ,�6 2 � D B # � � #� �� / �� + with ��' � � � / (8)

The value of the gradient vector at the point #�� is obtained by trilinear interpolationof the gradient vector, allowing a better boundary estimation.

3.3. Computation of the multiscale response3.3.1. Response normalization through scales

One difficulty with a multiscale approach is that we want to compare the result of aresponse function at different scales while the intensity and its derivatives are decreasingfunctions of scale. Lindeberg [23, 24] introduced the notion of normalized derivatives inorder to deal with this problem. If the scale is defined as � � where

�is the standard

deviation of the Gaussian, the � –normalized derivative� �

was already defined by Eq. 1in section 1.2.1.


At a scale , the cylindrical circular model with a constant " F leads to

� &$#� + �,- # � &*#� , � � B &*#�$, " � � � -% � B &*#� , /Appendix C gives some details about the calculation of the maximum of the normalized

response� �B : � �

B #" � �� . �-��' & � � �� , ��)

�� - �-�� . 2 -%�� (9)

We find a relation of proportionality between the scale � � � that gives a maximal responseand the initial radius of the vessel

� �:

� � � � & � +�� , � �� (10)

where � is a function of the normalization parameter � and the proportionality constant � :

�*& � +�� , � � � �"� � � �� & � � , with

� � � � &�� , � � �(= &�� , � (11)

Usually, � is chosen to allow the response�B to be maximal for a scale corresponding to

the size of the structure we want to detect. To keep generality, we establish the relationshipbetween the scale � � � where the maximal response is reached and the initial radius

� �of the vessel. This relationship depends on both the normalization parameter � and thedistance coefficient � . However, this relation still depends on the choice of the structuremodel, we will see in the experimental study (section 4.1.2) a comparison of this relationbetween the Gaussian cross-section model and other models.

3.3.2. Zoom-invariant criterion

We define the zoom or rescaling operation of an image as a transformation � with a realpositive parameter � and the following property:

� # % � + � &(�*+!��+��, %�� & # , +�� & # , &�� *+ � ��+ �&��,- # &(�*+��$+��, /where � is the set of all images and � & # , is the definition domain of the image

#.

A transformation � is zoom-invariant if it commutes with � :

� # %�� +�� &�� & # , ,-�� &�� & # ,�, (12)

A criterion for normalization is to choose � in order to ensure a zoom-invariant responsefunction. This choice will avoid privileging vessels of certain radii in the multiscaleintegration, because the maximal response at the center of a tubular structure will notdepend on its size. It will also help to handle interaction between vessels of different sizes.For example, if the maximum response at the center of a big vessel is higher than themaximum response at the center of a small vessel, and the two vessels are neighbors, thebig vessel may create side effects which will disturb the extraction of the small vessel inthe multiscale integration. Moreover, finding a single threshold to extract the centerlinesof all vessels in the final image will be more difficult. Thus, we choose the value 1 for � .


This value is the only one that ensures a zoom-invariant property of our response functionas shown in [23].

3.3.3. Choice of the constant �The purpose of introducing the parameter � is to compute the boundariness at a distance

which is equal to the frontier of the vessel at the maximal scale � � � . This can be achievedfor our vessel model. At a given scale , the frontier of the vessel is at a distance

� � �� from the vessel center. As the response uses the gradient at a distance �� , we would liketo have the following relation

� � � � � � � ��

Introducing Eq. (10) and having set � to the value 1, we find a solution to this relation givenby � � � .

Once we have chosen the two parameters � and � , two numerical relations can be deducedfor our vessel model. The first one is proportional relation between the size of the vesseland the scale at which it is detected � � � � & � +�� , � �� F / � �� . The second one is thevalue of the maximal response

� �B�� + :

� �B�� +

� "�(' � �� ,� � � & � +�� , � . �-

&��*& � +�� , �(= , � )� � -��

� �-�� . � � 0F / � � � � "�(' � �� (13)

This value is proportional to the intensity of the vessel center when the background intensityis null.

� �B �� + is approximatively equal to F / � � � times the intensity at the vessel center�� -% .

3.3.4. Multiscale response

The scales are discretized from �� to � using a logarithmic scale in order to have moreprecision at low scales.

Looking a little ahead to Figures 20-22, Fig. 21 shows Maximum intensity Projections(MIP views) of the normalized responses computed on image 1 of Fig. 20. Six scales wereused for radii of vessels ranging from 1.0 to 3.5: � 1.0, 1.28, 1.65, 2.12, 2.72, 3.5 � . If � isthe radius of the vessel we want to detect, the associated scale chosen for the detection is� & � , � � .

A MIP view of the initial image is shown at the top left of Fig. 22, and a MIP view ofthe multiscale response which is the maximum response across the set of scales is shownbeside.

3.4. Extraction of local maximaOur definition of local extrema is a special case of the Height ridge definition [14]. Some

recent work [15, 24, 9] in ridge extraction use the “Marching Lines” algorithm [25, 44].For each spatial point � &(�*+!��+��, % ' � , we associate the scale-space point � B & � +��$+!��,

of ' � � ' � . We also define �� as a vector in the scale direction.We define a local maximum in the scale-space normalized response as a point &*#��, %' �� ' � which is locally maximal in the directions of �� &$#�� , , �� &*#��, and �� . We caneasily state that for every central point � of a vessel, its associated scale-space point � B �� +


is locally maximal in the scale-space response. If the converse is true, i.e. all the localmaxima in the scale-space response are located at central points of a vessel, then detectingthe centerlines is equivalent to extracting those local maxima. This assumption can beverified in the vicinity of the central points. This vicinity is obtained by pre-selection ofcandidates (paragraph 3.1).

In practice, we use Eq. (13) as a characterization of the local maxima.

&*#��E6 , is locally maximal �� B�� &*#� ,

� �B�� &$#�� , and

� �B�� &*#� ,

� �B�� &*#�� ,

and� �B � &*#��,

� �B �� &$#�$, (14)

The image of the local maxima has a zero value at the voxels that are not detected aslocal maxima, and contains the value of the response function at the maximal scale at thevoxels that are detected as local maxima. Fig. 22 shows at the bottom left a MIP view ofthe local maxima extracted from the upper left image.

3.5. Reconstruction and visualizationIt is not an easy task to visualize the local extrema image in order to improve the

interpretation of the original data image. For that purpose, we propose to extract someinformation from the local extrema image, to superimpose it into some 3D representationof the original data image (volume or surface rendering) or to use it for a vessel networkreconstruction.

Line extraction. The image of local maxima contains enhanced information on thevessels centerlines. However, all the extracted maxima are not centers of vessels, some ofthem with lower value are detected due to noise or to the irregularity of the vessels frontiers.

In order to obtain a representation of the centerlines from the image of the local maxima,we apply the following treatment:

� we first binarize the local extrema image by applying a hysteresis thresholding.� Second, we thin this result to obtain a skeleton-like representation of the vessels.

Thinning is achieved by deleting the simple points. These points are the ones whoseremoval does not change the topology of the image. More details of the skeletonizationalgorithm can be found in [2]. The resulting skeleton is composed of pieces of curves, eachof them representing a piece of vessel.

� Third, the skeleton is simplified by removing small pieces of curves.� For a better visualization, the remaining curves are smoothed using an energy mini-

mization including data attachment. The smoothing method is derived from [6] and doesn’tmodify the localization of the extremities of each line.

The result obtained is an image of the vessel axes.Reconstruction. The centerline image also contains information about the size of the

vessel, which is proportional to the scale at which the current point has been extracted. Therelation between a vessel size (

� �) and the scale at which it is detected ( �� ) is given by

Eq. (10), where the two parameters � and � are set respectively to = and � � . The bottomright image of Fig. 22 represents a MIP view of the centerlines obtained, where centralpoints are colored according to the scale at which they have been extracted, six colors areused ranging from blue to red for the six single-scale responses shown in Fig. 21.


Each piece of vessel is described by a sequence of points � 6 � , each of them being asso-ciated with an estimated radius � 6 . We reconstruct each segment > 16�+ $6 � , A independently.If the orthogonal projection of a point � on the line 6 6 � , is into the segment > 6 + 6 � , A ,we estimate the radius in , and deduce from it the intensity in � with Eq. (3). This way,we reconstruct a grey-level image and we visualize easily all the reconstructed vessels withan isosurface.

Visualization. The usual means of visualizing the vessels network, that are MIP viewsand isosurface rendering, give both a partial representation of the vessel tree.

On the one hand, MIP views can mislead the physicians because they don’t containinformation about the relative position of the vessels in depth. One can add depth-cueingto them but a high intensity vessel located behind a low intensity vessel may still appear tobe in front of it, or hide it.

On the other hand, an isosurface of the initial image can account for the the relative posi-tion of the vessels, but it contains partial information about the image which is insufficient.With a low threshold an isosurface contains the small vessels but they are hidden by the bigvessels. With a high threshold, it contains only the thickest vessels as shown in Fig. 20.

In both cases, MIP view or isosurface, the superimposition of the detected 3D center-lines can help the interpretation of the real vessels network. Moreover, an isosurface ofthe reconstructed vessels network have the advantages of an initial image isosurface with-out having its drawbacks, because all vessels are reconstructed with the same centerlineintensity. Thus, it can help to understand the local structure of the vessels network.

4. EXPERIMENTS AND RESULTS4.1. Experimental study on synthetic images

In this section, we present some experiments made on synthetic images. The purpose isto estimate the sensitivity and to understand the limits of our method on several criteria:normalization, radius estimation, curvature, tangency of vessels, junctions. The createdimages have a Gaussian blob cross-section and their difference from the theoretical modelslies in their discrete representation. This choice allows to check the expected results foundby the analytic study. However, we also compare the response profile obtained for bar-likeand Gaussian-like cross-sections on a cylindrical circular model.

This study on synthetic images is not exhaustive, but we hope that it leads to a betterunderstanding of problems arising in vessels segmentation. In the ideal case, the spiritof the work on synthetic images is to first find all the difficulties; second create syntheticimages that isolate each difficulty, understand the behavior of the method on this problemand try to improve it; third test the accuracy and the robustness on phantom images; andfinally make experiments on a database of real images. When dealing with problems onsynthetic images, we make the assumption that those problems are independent, and thatan algorithm which handles each of these difficulties separately will have a good behavioron real images.”

4.1.1. Cylindrical circular vessels with Gaussian cross-section

Response profile.The response profile is the evolution of the medialness response as a function of scale,

here taken at the vessel center. Fig. 5 shows a comparison between the theoretical andthe obtained profiles. The synthetic image contains a circular cylinder with Gaussian blobcross-section, radius 3 voxels and intensity equal to 100 at the center. The theoretical


response profile is given by Eq. (9) where� � � , � = , � � � and " ��' � �� =1F�F .

The experimental response profile is obtained from twenty scales ranging from 0.7 to 3.5.This comparison shows that the two profiles match, and that the experimental profile isslightly lower than the theoretical one near the maximal scale.

Normalization.The relationship between the vessel radius and the optimal scale is � � � � & � +�� , � �� F / � �� where � � is the radius of the vessel with Gaussian-like cross-section. The responseat the optimal scale and at the vessel center should be zoom invariant and equal to

� �B �� +

times the intensity at the vessel center (see Eq. (13)). The initial image of Fig. 6 containsfour vessels with Gaussian blob cross-section and respective radii: 1.25, 1.75, 2.5, 3.5.

After applying the multiscale analysis on this image with 20 scales for vessels radiiranging from 1 to 4 voxels, the second row of Table 5 presents the maximum intensityobtained at the center of each vessel. The difference between the obtained maximalresponse and the theoretical expected value 23.3 is stronger for small vessels. As we usea high number of scales, the main difference between the theoretical value and the valueobtained on synthetic images is the discrete representation of the image. Thus, this erroris due to the trilinear interpolation of the gradient when computing the response function.However, this difference remains small , below 11%, which confirms the zoom invariantproperty of the normalization, and will allow an easy threshold of the local extrema image(Fig. 6).

Radius estimation. Rows 4 and 5 of Table 5 show radius estimation for the same image.Except for the vessel of size 1.25, the maximum response is obtained at the nearest scaleassociated to the size of the vessel. The error in size estimation is below 0.3 voxel andimproves when the vessel size increases. This result shows that, due to discretization,we cannot hope to get an accurate sub-voxel estimation of the size of small vessels, i.e.vessels of radius below 1.5 voxels. However, the detection of vessels of 1.5 voxels is stillan acceptable limit relative to other published works.

4.1.2. Other cross-section models

These first tests set the problem of sensitivity to the cross-section model. In real images,there should not be high intensity variations inside the vessel. Two main reasons of intensityvariation can be noise and partial volume effect. Concerning noise, the multiscale processthat uses Gaussian kernel convolutions tends to reduce it, but depending on the acquisitionmodality, one can apply a pre-filtering technique like anisotropic diffusion. The partialvolume effect disturbs the detection of small vessels and also reduce their intensity. In fact,big vessels can be considered as having a bar-like cross-section whereas small one have aGaussian-like cross-section and a lower intensity caused by partial volume effect. Thus,the extension of our work to bar-like cross-sections is very important for many applicationswhere one is interested in large vessels, as for example for endovascular treatments.

Definition of a bar-like convolved cross-section.We are currently working on a vessel model of a bar-like cross-section convolved with

Gaussian kernel with a constant and small standard deviation. In this cross-section model,the Gaussian kernel convolution acts like a partial volume effect and can lower the intensityof small vessel: big vessels are bar-like and small ones are Gaussian-like. Using this kindof model closer to real images, size estimation can be considerably improved.


An analytic expression of the model is given by:

# � & � +��$+!��, �� &(�*+�� , � �� & � +�� , (15)

where � is the radius of the vessel,��

is a constant, and � is defined by

� &(�*+�� , = if� � � � � � - �

F otherwise.

Fig. 7 and Fig. 8 give an illustration of this model for 1D and 2D cross-sections respectively,and with different values of the radius parameter � , the convolution parameter

��being

fixed.

Comparison of the response profiles. Fig. 9 shows response profiles for different cross-sections on a cylindrical circular vessel of radius 3. The profile for a Gaussian-likecross-section represented in plain line is the same profile as in Fig. 5. There are importantvariations between this profile and the bar-like cross-section profile: the bar-like cross-section has its maximum with a higher response value and at a lower scale. This resultshows that our vessel size estimation can not be accurate without having a good modelof the vessel cross-section. Fig. 9 shows also the response profile obtained for a bar-likecross-section of radius three and convolved with a Gaussian kernel of standard deviation 1,and the profile using a standard deviation 3 (equal to the vessel radius).

Radius estimation for the convolved bar-like model. Using Maple, which is a programfor symbolic calculus, and the expressions of our response for the bar-like convolved modelgiven by Eq. D.1 in appendix D, we computed the curve that represents the maximal scale� � � � � � � � as a function of the vessel radius. The result is represented in Fig 10.Although the size estimation requires the use of more complex cross-section models, thereis a increasing function � that links the maximal scale and the radius of a vessel. Simulationallows to estimate this function on a bar-like cross-section model convolved with a constantGaussian. A vessel of a given radius is detected at a lower scale for a bar-like convolvedmodel, thus, for a given maximal scale, the radius of the detected vessel is bigger for abar-like convolved model than for a Gaussian-like model. Using this simulation, a betterradius estimation can be achieved, and this result will be used in the next section on aphantom image.

4.1.3. One vessel with varying width

Fig. 11 and Fig. 12 show experiments made on vessels with varying width. The vesselsize of the images is a periodic sinusoid and the radius varies from 2 to 4 voxels with aperiod of � � � � ) � � voxels � %�> =�+�� + � A and � � � � ) = F@F , along z axis. The equation of thevessel radius for � = +�� + � is :� &��,- � �� ('��

� � � � ) �The local extrema in Fig. 11 shows that the vessel center has been well detected and alsothat some extrema were detected near the vessel frontiers when the radius goes though amaximum. In this case, there are two negative eigenvalues in the plane tangent to the vesselcontour, and it is normal to obtain local extrema. Nevertheless, the response obtained at


the vessel center is higher and the false responses can be removed either by thresholdingof the image of local extrema or by removal of small connected components.

Fig. 12 shows the estimated radius along z axis compared to the real radius profile of thevessel. For smooth variations, on the left, the size is well estimated, but for fast variationsof radius, on the right, in regions of maximum radius the size is under-estimated due to theGaussian convolution at high scales that decreases the intensity near those frontiers, fasterthan in the cylindrical case.

4.1.4. Curved vessels

For a single torus with a Gaussian cross-section, the local extrema gives high responseat the torus center where the intensity is higher than 18.00 and some response near theoutside frontier of the torus. This second type of response is explained by the negativevalue of the third eigenvalue that becomes higher in absolute value than the second one(see section 2.2). However, it has an intensity lower than 9.0 and can easily be threshold.Fig. 13 right shows the threshold extrema superimposed on the initial image. The locationof the vessel center doesn’t have a sub-voxel precision, but the voxels found for the vesselcenter contain the real vessel center independently of the curvature (bottom row of Fig. 13).

4.1.5. Tangent vessels

We say that two vessels are tangent when their frontiers are enough near to disturb theestimation of the gradient. Generally, the tangency concerns two vessels but in some casesmore vessels can be involved, or a vessel can be tangent to a non-vessel structure. Werestrict the study to the case of two vessels.

The tangency can be characterize by three parameters: 1) the minimal distance � betweenthe two vessels frontiers compared to the size of the vessels; 2) the ratio between the twovessels radii; 3) the angle % > F +�' � � A between the two vessels axis at the tangent locus.

In our experiments, we set the ratio of the two vessels to 1 (their radius is three voxels)and tested the cases (F in Fig. 14 and ' � � in Fig. 15.

Fig. 14 shows results on the first case ! F where the distance � is equal to zero onthe right and to the vessel radius i.e. three voxels on the left. When � &F , a third line isdetected between the two vessels and at a higher scale (bottom right), while the continuitiesof the two vessels centerlines are preserved. As the detected lines are not connected, itis possible to remove the “wrong” line by removing small connected components, but notby thresholding the local extrema image. On the bottom left image, the distance betweenthe two vessels is equal to their radius and a thresholding of the local extrema image issufficient for removing the “wrong” detected local extrema.

Fig. 15 shows results for ' � � , where the distance � decreases from 4 voxels to0. In this case, there is no third line created by the tangency, but the disturbance on thecenterline position is more important. This important displacement of the vessel center fora vessel denoted 4 , at the right of Fig. 15 can be explain by the low curvature of the tangentsurface of the other vessel 4 � in the direction orthogonal to 4 , . This low curvature, equalto zero here, disturb the medialness response which integrate boundariness along a circleorthogonal to 4 , axis.

In the same way, when a small vessel is tangent to a “bigger” one, we can expectdisturbance in the small vessel axis detection due to the low curvature of the big vessel,even when vessels are parallel i.e. for low values of .


As a conclusion, when the boundaries of tangent vessels are not in contact, one canexpect to keep the continuity of the vessels centers. Nevertheless, tangency of vesselshave the following negative effects: - it decreases the response function and makes thethresholding more difficult, - it increases the estimated size of the vessel near the tangentarea, - it changes the location of centerlines. One way to improve the detection of tangentvessels can be to make an iterative process. The information of the detected vessels can beused to localize the region of tangency and to discard the information of gradient in thoseregions for the next iteration.

4.1.6. Junctions

A junction in a vessels network is a branching of vessel, where one vessel divides intoseveral branches, in general two. We restrict this experimental study to the case of twobranches.

Although the modelization does not include any vessels junction, some of them may beproperly detected. Fig. 16 shows experiments made on three synthetic junction images.

The centerlines detection, obtained from the extraction of local maxima of the multiscaleresponse, does not ensure the continuity of the junction detection. In the top image, themain vessel divides into two branches of the same size and the continuity is preserved,but in the middle and in the bottom image, the two branches don’t have the same size andthe junction continuity is not preserved by the centerlines. This discontinuity can still bepresent after the reconstruction (middle image).

Disconnection disturbs the interpretation of the reconstruction and the characterization ofthe vascular tree topology. To solve this problem, the junctions can be connected using thecenterlines and the radii information. Assuming that the bigger vessel keeps its continuity,a junction is restored when the distance between extremity

�of a vessel 4 , and the axis of

second vessel 4 � is lower than the radius of 4 � :��& � +�4 � ,�G � &54 � , �� (16)

where stands for the error in radius estimation and � is the sum of the error in the locationof 4 � axis and in the location of the extremity

�of 4 , . To perform the junction connection,

each extremity�

of a vessel 4 , is projected on every segment of every vessel differentfrom 4 , and located in the vicinity of

�, and � is the distance to the nearest projected point�

. We set = / � and � � voxels. Fig. 16 shows the restored centerlines and thereconstruction from those centerlines for two junction images.

4.2. Image of a phantom4.2.1. Characteristics of the phantom image

We have a phantom image created by General Electric Medical Systems, on which thesizes of the different structures is known before the acquisition. This image is coded on 2bytes and its size is 3= � � := � � 3= � , the voxels are isotropic with a size of 0.267 mm.Fig. 17 shows two Maximum Intensity Projections of this image compared with its mapcontaining real sizes of some structures. This phantom image allows us to estimate theerror made in the estimation of the vessel size. The Gaussian blob is not a good modelfor the section of the vessel and using this model leads to under-estimation of the size ofstructures. Thus, in the following experiments, we use the relation between the maximalscale and the radius of the structure for the bar-like convolved cross-section model presented


in paragraph 4.1.2. This relation is was given in Fig. 10. The standard deviation of theGaussian convolution is set to

� � = / voxels.

4.2.2. Extraction of tubular structures and size estimation

cylinder �� .The size of

��is 2 mm, i.e. 3.744 voxels. From our multiscale analysis with 20

scales for radii ranging from 2 to 6 voxels, we extracted the whole centerline at a scalecorresponding to a radius of 3.778 voxels. The corresponding estimated diameter is 2.018mm.

cylinder � .The cylinder named

�in Fig. 17 has a size of 4mm. From our multiscale analysis

with 20 scales for radii ranging from 6 to 9 voxels, we extracted the whole centerline atscales corresponding to radii of 7.270 or 7.427 voxels. In fact, the responses for those twosuccessive scales are very close. The corresponding estimated diameters in are 3.884 mmand 3.968 mm.

cylinders �� and �� .Fig. 19 shows the results on the structure with varying width:

� � =4mm and �� =2mm.The multiscale analysis was run on 20 scales ranging from sizes of 3 voxels to 9 voxels.The maximum radius found for

� � is 3.6 mm and the minimum radius found for �� is2.14 mm.

4.3. Real Images4.3.1. Brain Vessels from X-ray images

Image Acquisition.Our algorithm was tested on a set of images produced by General Electric Medical

Systems, Buc, France. They are obtained by 3D reconstruction of the vessels from 2DX-ray subtracted angiographies. Details of the reconstruction scheme can be found in[31, 32]. Compared to the other 3D acquisition modalities which are Magnetic ResonanceAngiography and Computerized Tomography Angiography, this 3D reconstruction givesa high isotropic resolution over the whole reconstructed volume. However, it requires agood opacification of the vascular network obtained with an intra-arterial injection. Theleft images in Fig. 20 are MIP views of a typical sub-images centered on an aneurysm.They contain different artefacts: noise, partial volume effect, consequences of the patientmotion between different acquisitions and 3D reconstruction artefacts which lead to a non-homogeneity of the intensity of the vessels for different sizes of the vessel. The two rightcolumns of Fig. 20 show isosurfaces of the images, where small vessels are only visiblewith a low threshold (surface holes in black are due to the image boundaries).

Choice of parameters. We tested our algorithm on ten images = �� = �� = �� ofvarying complexity, that are ten sub-images from ten separate patient scans. Because smallvessels have a lower intensity than bigger ones, we used a parameter � lower than 1 forthe normalization. Decreasing the value of � has the effect of enhancing small vesselscompared to big ones, and helps to compensate for intensity variations. We used the value0.75 for � . The minimum and maximum scales are chosen according to the radii of thethinnest and the thickest vessels in the initial image. The algorithm was run with thenscales that detect vessels of radii ranging from 0.4 to 6 voxels. The computation time for


obtained the local maxima image is approximately 15 minutes on a Dec Alpha station 500,running at 400 MHz. Then, after manual thresholding, the centerlines extraction takes afew seconds.

Results.Results on the three images of Fig. 20 are shown in Fig. 23. On the left, the detected

centerlines are represented with an isosurface of the initial image and using transparency.On the right, a surface of the reconstructed network is represented, the reconstructionis based on the previous centerlines and the estimated radii. The following point areemphasized: First image, a separate detection of two tangent vessels. Second image, onthe left, a continuous detection of a vessel with high curvature; on the right, the detectionof a vessel in the vicinity of the aneurysm, the location of this vessel was difficult tounderstand in both MIPs and Isosurfaces. Third image, the detection of a vessel with lowand decreasing intensity and several well-detected junctions.

However, we can also note a few limitations of the current algorithm. For example, weobserve an over-estimation of the size of the vessels near junctions where the model ofthe vessels network is more complex. Another limitation is the lack of global treatmentto improve the connectivity of the whole vessels network. To face the first limitation, onecould envisage a post-treatment that uses more complex models of vessels on a selectedregion of interest of the image [16, 27], or that uses a deformable model of the vesselscontours with our reconstruction as an initialization of the model [39]. For the connectivityof the whole vessels network, one could envisage some heuristic based on a distancebetween detected structures to allow their connection or to exclude a detected centerlinefrom the vessels network.

4.3.2. Brain MRA

Magnetic Resonance Angiography allows to measure the magnetic properties of theatoms, and to obtain 3D volume data. The angiography is obtained by subtraction that canbe done either on phase (phase contrast MRA) or on time of flight (time of flight MRA).Time of flight MRA measure the inflow of relatively unsaturated blood into a surround ofhighly saturated tissue [29] whereas phase contrast MRA produces a voxel by voxel mapof velocity field [7]. Fig. 24 shows a brain phase contrast MRA := � � 3= � � �F withvoxel dimensions F / � � � �� F / � � � �� F / � �� . From this image we extracteda sub-volume = � � = � � � @F shown at the top left of Fig. 25. Compared to previousX-ray images, vessel detection is more difficult because the intensity inside the vessels isless homogeneous, the vessels are not isotropic and noise is more important. A solutionto reduce noise and inhomogeneous intensity inside vessels is to apply a pre-filteringmethod like anisotropic diffusion. Several works have been done on nonlinear diffusionand anisotropic filtering [33, 1, 42], and also on its application to vessel enhancement[19, 30].

Top right image of Fig. 25 shows the image after anisotropic diffusion where a thresholdon the gradient norm is used to control the diffusion.

To deal with non-isotropic voxels, the program adapts the sizes of the Gaussian kernelconvolution along x,y and z axis. Voxel dimensions are also used to estimate first and secondderivatives in each direction and to convert spatial coordinates into image coordinates whencomputing the response function.


The bottom left image represents the local extrema extracted from the filtered image andthe bottom right image represents a surface rendering of the vessels reconstruction.

5. CONCLUSION AND DISCUSSION

We presented a multiscale approach for tubular structures detection in 3D images. Ourapproach uses gradient information at a given distance of the vessels centers. Based ondifferent models of vessels, we expressed eigenvectors and eigenvalues of the Hessianmatrix and showed their sensitivity to elliptical cross-section, to vessel curvature and tothe distance of the point to the real vessel center. Using a cylindrical circular modelwith Gaussian blob cross-section, we found the optimal distance for computing gradientinformation at a given scale, and we also expressed the vessel radius as a function of themaximal scale.

Then, we proposed a method for extracting centerlines and reconstructing the wholevessels network. Experiments on synthetic images show the limits and the robustness ofthe algorithm according to radius variations, to curvature, and to junctions for vessels withGaussian cross-sections. A bar-like convolved cross-section model was also introducedand we derived a new size estimation for this model with formal calculus simulations.Robustness of the size estimation was tested on a phantom image and results were presentedon real X-ray and MRA images of brain vessels.

Several conclusions and research directions follow from this work. The first point is theimportance of extracting the vessels centerlines, and ensuring their continuity to understandthe topology of the vessels network. For this purpose, the variable intensity at the centerof vessels of different sizes is still an important issue and has to be taken into accountin the vessel model. This can lead for example to a specific response normalization. Asecond important issue is the discrimination between vessel and non-vessel structures. Thisdiscrimination is present in our method at the pre-selection stage based on Hessian matrixeigenvalues and also in the response function that enhances vessel centers. However,another discrimination of the local maxima may be necessary when the image containsnon-vessel structures with high gradients or to remove wrong extrema obtained near thevessels frontiers. Finally, once a good detection of the centerlines is obtained, a second andmore precise detection of the vessels contours may be done based on this information andwithout assuming a circular cross-section profile.


FIG. 3. Left to right, surface of the Gaussian ellipse with � � ��and �

" ��, surface � �

� � ��&� , andsurface � � �� .

FIG. 4. Representation of the curves which are tangent to the eigenvectors for the cylindrical ellipticalmodel.


FIG. 5. Response obtained at the center of the vessel for different scales. The obtained profile (linewith dots) is superimposed on the theoretical profile. The vertical line shows the theoretical scale for which theresponse is maximal.

initial image local extrema

scale� � / F�� scale

� � / =�� scale� &= / scale

� = / = FFIG. 6. Cylinder with circular Gaussian cross-section. Responses obtained for the optimal scales.

TABLE 5Intensity obtained at the center of the vessels for a range of 10 scales, estimated

sizes and error in the estimation.

Real vessel radius 1.25 1.75 2.5 3.5

Maximal reponse � 20.8085 22.3763 22.7294 22.9052Difference to the expected value 10.8667% 4.1510% 2.6385% 1.8855%

Estimated radius 1.55 1.79 2.58 3.46Error in voxels 0.3 0.04 0.08 0.04

� The theoretical expected maximal response is 23.3.


0

0.2

0.4

0.6

0.8

1

1.2

-6 -4 -2 0 2 4 6

BarGaussian

Convolved bar

0

0.2

0.4

0.6

0.8

1

1.2

-6 -4 -2 0 2 4 6

BarGaussian

Convolved bar

0

0.2

0.4

0.6

0.8

1

1.2

-6 -4 -2 0 2 4 6

BarGaussian

Convolved bar

FIG. 7. Illustration of the 1D bar-like convolved model. From left to right: models for a 1D vessel withradius 1,2 and 3. Superimposition of a bar-like model, a Gaussian-like model, and the convolution of a bar-likemodel with a Gaussian of standard deviation 0.7.

FIG. 8. Illustration of the cross-section of the 3D bar-like convolved model. At the top, representation ofthe bar-like cross-section with radii equal to 1, 2 and 3 from left to right. At the bottom, the same cross-sectionsafter convolution with a 2D Gaussian of standard deviation 0.7.


0

5

10

15

20

25

30

35

40

0.5 1 1.5 2 2.5 3 3.5 4 4.5

GaussianBar

Convolved bar 1Convolved bar 3

FIG. 9. Response profiles obtained for a Gaussian-like cross-section, for a bar-like cross-section, fora bar-like cross-section convolved with a Gaussian �� + , and for a bar-like cross-section convolved with aGaussian ��

.

FIG. 10. Computation of the maximal scale �� as a function of the vessel radius. Comparisonbetween the Gaussian cross-section model where � ��

�� and the bar-like model convolved with aGaussian of standard deviation ��

, represented by the thick line.


synthetic images local extrema

FIG. 11. Tests on an Gaussian cross-section vessel with varying radius.

n=1

n=2

n=4

FIG. 12. Comparison of the real and the detected radii along z axis.


r=3 R=15

r=3 R=10

r=3 R=5

r=1.5 R=3

FIG. 13. Detection of torus with Gaussian cross-section and different curvatures. At the top, MIPs of theinitial images; At the bottom, superimposition of the real torus center (in grey) on the local maxima image.


� � voxels � (F

FIG. 14. Tangent vessel, tangency parallel to the vessel axis ( � �

).

� � voxels

� � voxels

� F voxel

FIG. 15. Tangent vessels, tangency orthogonal to the vessel axis ( � ��

).


FIG. 16. Centerlines detection and reconstruction on three synthetic junction images. Left, initial image andthe detected centerlines. Middle, reconstruction before junction connection. Right, centerlines and reconstructionafter junction connection.


� � � � �� ,� � ��

FIG. 17. Phantom image. Left, map of the phantom with the known structures radii. Right, MIP view ofthe phantom image.


obtained centerline. estimated contours of the vessel.

FIG. 18. Results on a sub-image of the phantom centered on the diameter � (see Fig. 17).

sub-image extracted = � � � � � � @F contours of the reconstructed image

FIG. 19. Results on a sub-image of the phantom for the diameter � � and �� .


image 1 threshold=871 threshold=500



FIG. 20. Top, MIP view and isosurfaces of three X-ray 3D images.


scale 1.0 scale 1.28



FIG. 21. MIP views (Maximum Intensity Projections) of the responses obtained for 6 differents scales.


initial image maximum response across scales

local extrema from the center of the vessels with colorsmaximum response depending on the detected scale

FIG. 22. MIP views at different stages of the multiscale analysis.


FIG. 23. Results on the images represented in Fig. 20. Left, detected centerlines superimposed on anisosurface of the initial image. Right, reconstruction of the vessels network from centerlines and radii estimation.


FIG. 24. Initial MRA Image.

FIG. 25. MIP of a sub-image on the top left and the resulting image after anisotropic filtering on the topright. Bottom left, image of the local extrema; and bottom right, vessels reconstruction.


APPENDIX A

Eigenvalues for a toroidal model with circular sectionIf we modelize the vessel with a torus, the big circle parallel to the plane XY and with a

radius�

and the small circle with a radius equal to � .The square distance from a given point � & � +��$+!��, to the axis of the torus is given by:� � � � � � � � � � � � �

derived from the cylindrical model, we can take the following function of intensity:

# � & �*+!�$+!��, " )� &�� + - .�/ - , - .�� --�2 -%

The first and second derivatives are:� � %� � � % �� -% � �� - � 4 - = �� - � %� � � 4 � % �34� -%

� ,� -% � �� - � 4 - = � � �0 � - � 4 - 5�� -�� - � %� � - � % � -� �% � �� - � 4 - = � � � � %� -% � � 4 -0 � - � 4 - 5��- = �� - � %�� - � %� �% &�� ,� - � %� � �� % � �� % � �� - � 4 - = �

From the circular symmetry around Oz axis we can choose � F and � F , then theHessian can be expressed as:

) # � & �*+!�$+!��,� �� ) �

where

) � �� & � ��, � � �� F ��& � ��,F 0�� $� 5 � -%� F ��& � �$, F � � � ��

� �The discriminant of )�� + # � is:

� ) -� & � ��, � ��

� + � � + � & � ��, � � � � � �� + � � �� and the eigenvalues of )�� are:

+� 0 � �� 5 � -%� �� &8F +2= + F�,+ � & � ��, � � � � � �� & � � + F +!��,+�, � �� &�� + F + � ��,


and the eigenvalues of ) :

+�

# ��

+ � # ��

�= � " ��

+ , # ��

APPENDIX B

Eigenvalues for a cylindrical model with elliptical sectionThe elliptical section is defined by one standard deviation along the x axis:

� �and one

standard deviation along the y axis:� 4

. The model is then defined by:

# � & � +��$+!��, #" )��- � & +2 + , - �� /2 /�� - %

The Hessian matrix can be expressed as

) # � & �*+!�$+!��,� �� 4 ) �

where ) � is expressed as:�� 4 �� = � � � � 4 �� F� � � 4 �� = � FF F F

�

�� F� � FF F F

�with

� �� + and

� 4� / .

The discriminant of )�� + # � is��

� �� "�� " �� " �! � � � � � � � �

Let� & + , + � + & � � � , � � � � � ,

the discriminant of�

is

� & � � � , � ��& � � � � , & � � , � �� F if

� �#" � 4+�, =� &

� � � � � , + � =� &� � � � � � , (B.1)


The expression of the eigenvalues is,� & � � � �

� & � � � , � ��& � � � � ,�, where

� � � � � �� 4 �& �� , � � & �� 4 , � = � /

We can then distinguish three cases depending on the position of � & �*+!� , , expressed inTable 3. We can also study the eigenvalues along the � and � axis. In both cases,

� Fand ) � is diagonal so �� &�= + F + F�, and �� &8F +1= + F�, , the result is given in Table 4.

�� 6 �� + 6 ��F

��

�� + 6 �F

��for � % ��=�+�� / (B.2)

APPENDIX C

Expression of the maximal scale depending on �We want to detect the axis of the vessel which is defined by � � F . The response at

a point � & F + F +!��, is given by:�B &*#�$, =

� � ' � �� 7 � � � � � D � &*#� +��, / �� E� #�

which corresponds to the mean of the vector product of the gradient with the unit radialvector along the circle of center &8F + F +��, and of radius � � . The gradient and the normalvector �� have the following expressions:

D � &$#� + �,- � &*#� +��, =�� F

��

�� =� � � � � �� F

��with � � � � � � �

then

� D � &$#� + �, / �� " � � ��' & � � �� , ��)

� � - �-�� . 2 -% �This last expression is no longer a function of #� , then the mean of this expression along thecircle is straightforward, and�

B " � � ��' & � � �� , � )

� � - �- � � . 2 -% �The normalized response

� �B is defined by

� �B

�� B and its partial derivative on t is:

� � �B� " � � �-��' & �� , � ) � � - �-�� . 2 -% �


with & � ��, � � &�� , � �� &E=�� , � �� /We are looking for the value of � which gives a maximum for the function

� �B at � � �� .

Thus we want� �� B to have a positive root which corresponds to a maximum.

The sign of� �� B is the same as the sign of A, and the expression , when � G � and the

discriminant�

is also positive, has only one positive root which corresponds to a maximumfor

� �B :

� & � +�� ,- � � �� & � � ,

with� � � � &�� , � � � = &�� , � /

APPENDIX D

Convolved bar-like modelThe analytic definition of the model was given by Eq. (15), and is

# & � +��$+!��, � � & � +�� , � �� & � +�� , � � � & �*+!� , � � � &(��, � � � � & � ,

The convolution product can be replaced by integrals, then the integral through x axis isexpressed as following:

� � & �*+!� , � � � &(��, � B 2 � �B2 � � � � &(� +�� ,1� � � & �,��

� B 2 �� B2 �$��$� � � � & � +�� ,�� & �,��

where � , � � � � �� &(�*+!� ,� � � � & �$,

� B 2 �$� � � �B2 �$��$� � � � � &(�,E��

� � � & �� , � � � & � � � � � � , � �� & � � � � � � � , � �� & � � � � � � ,

where� � � is a primitive of � �� chosen to be an odd function.

� � � =� ) � � � �� where the function ) � � is defined by:

) � � &(��, �� '� �B2 � ) � B - �


# �� & � +��$+!��, � � & � +�� , � �� & �*+!� , � � � &(�*+!� ,� � � � &(��, � � � � &�� ,

� � �� &(�� , � � � & � � � � � � , % � � � &�� $,��

The derivative of the intensity along x axis can also be expressed analytically, and thenthe expression of the response at the vessel center and for a scale � � � is deduced:

�$# �� &(�*+!��+��, � � ��$� � � � � &(� � � � � � � , � � � &(� � � � � � , % � � � &�� $,��

� � �$# �� &�� + F +��,

� � �� & � � � � � � � � , � � & � � � � � � � , % � � & �$,�� (D.1)

with � � �� .ACKNOWLEDGMENT

We authors thank Marc Thiriet for providing the Magnetic Resonance Angiography of the brain.

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Model Based Detection of Tubular Structures in 3D Imageslibvolume3.xyz/.../tubularstructures/tubularstructurestutorial1.pdf · Model Based Detection of Tubular Structures in 3D Images