a viewpoint determination system for stenosis diagnosis

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998 121

A Viewpoint Determination System for StenosisDiagnosis and Quantification in Coronary

Angiographic Image AcquisitionYoshinobu Sato,*Member, IEEE, Takahiro Araki, Masayuki Hanayama,

Hiroaki Naito, and Shinichi Tamura,Member, IEEE

Abstract—This paper describes the usefulness of computerassistance in the acquisition of “good” images for stenosis di-agnosis and quantification in coronary angiography. The systemrecommends the optimal viewpoints from which stenotic lesionscan be observed clearly based on images obtained from initialviewpoints. First, the viewpoint dependency of the apparentseverity of a stenotic lesion is experimentally analyzed using soft-ware phantoms in order to show the seriousness of the problem.The implementation of the viewpoint determination system is thendescribed. The system provides good user-interactive tools forthe semiautomated estimation of the orientation and diameter ofstenotic segments and the three-dimensional (3-D) reconstructionof vessel structures. Using these tools, viewpoints that will not giverise to foreshortening and vessel overlap can be efficiently deter-mined. Experiments using real coronary angiograms show thesystem to be capable of the reliable diagnosis and quantificationof stenosis.

Index Terms—Active vision, appearance analysis, computer-assisted diagnosis, coronary angiography, imaging strategy,stenosis diagnosis, stenosis quantification.

I. INTRODUCTION

COMPUTER-ASSISTED diagnosis of stenosis using coro-nary angiograms has been intensively studied. Various

aspects of the image measurement and processing of coronaryangiograms have been investigated, including the quantifica-tion of stenotic lesions [1]–[6], automated extraction of vesselcontours [7]–[9], and three-dimensional (3-D) reconstructionof vessel structures and local shapes [10]–[12]. These studiesare aimed at providing methods that will permit the accuratequantification of stenotic lesions and 3-D reconstruction ofthe surrounding vessel structures. However, such methodsare effective only if sufficient information to enable accuratequantification and reconstruction is included in the images

Manuscript received March 5, 1996; revised December 24, 1997. TheAssociate Editor responsible for coordinating the review of this paper andrecommending its publication was C. Roux.Asterisk indicates correspondingauthor.

*Y. Sato is with the Division of Functional Diagnostic Imaging, BiomedicalResearch Center, Osaka University Medical School, Suita, Osaka, 565, Japan(e-mail: [email protected]).

T. Araki, H. Naito, and S. Tamura are with the Division of FunctionalDiagnostic Imaging, Biomedical Research Center, Osaka University MedicalSchool, Suita, Osaka, 565, Japan.

M. Hanayama is with the Department of Radiology, Osaka UniversityHospital, Suita, Osaka, 565, Japan.

Publisher Item Identifier S 0278-0062(98)03009-2.

to be analyzed, that is, the stenotic lesions can be observedclearly. In the computer vision field, the importance of adap-tive control in image acquisition has recently been recognizedin practical terms and a new field has emerged called “activevision” [13]. In fact, from the clinical aspect as well, it ismost important to acquire “good” images, in which stenoticlesions can be observed clearly. Furthermore, these “good”images need to be acquired within a limited time, with as lowan X-ray dose as possible, and using the minimum amountof contrast material. These requirements suggest the necessityof employing computer-assisted adaptive control in coronaryangiographic image acquisition.

In this paper, we describe a computer-assisted diagnosticsystem [14], [15] that is designed for the acquisition of“good” images rather than only for analyzing given images. Incoronary angiographic image acquisition, images are obtainedinitially from a few standard viewpoints. Using only theseimages, however, it is often difficult to make reliable diagnoseson the basis of the imaged stenoses. There are two maincauses of this difficulty; foreshortening resulting from largeinclination against the viewing direction, and the overlappingof peripheral arteries. In such cases, another image must be ob-tained in which the stenoses are imaged without foreshorteningand overlap. The problem is how to determine the viewpointof this next image. Determination of the optimal viewpoint isoften a difficult task in the clinical situation. Our aim is todemonstrate the usefulness of a system designed to assist insuch a decision.

Stenotic lesions with elliptical cross sections are oftencharacterized as important examples of stenosis involving alarge degree of viewpoint dependency [11]. Although we havenot considered elliptical stenoses in the work reported here,we show the importance of viewpoint determination evenunder the assumption that stenotic lesions have circular crosssections.

The organization of the paper is as follows: In Section II, weexperimentally examine how the apparent severity of stenosisis dependent on the viewpoint, using software phantomsmodeled on typical severe stenoses. In Section III, we describein detail the implementation of a viewpoint determinationsystem. In Section IV, we present experimental results usingreal coronary angiographic images. In Section V, we discussrelated work and present our conclusions.

0278–0062/98$10.00 1998 IEEE

122 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

II. V IEWPOINT DEPENDENCY OFSTENOSIS APPEARANCE

The methods employed to estimate stenosis severity can bebroadly classified into those involving computerized quanti-tative analysis and those relying on human visual analysis.Three types of severity measurement are commonly used incomputerized quantification

-LENGTH stenosis (1)

-AREA stenosis

(2)

-DENSITOMETRIC stenosis

(3)

where and are estimates of the minimal diameter ofthe stenotic part of a vessel and the mean diameter of thenonstenotic part, respectively. and are estimates ofthe cross-sectional area of the lumen at the stenotic part; and

and are estimates of the cross-sectional area of thenonstenotic part. While the areas and in %-AREAstenosis are estimated by assuming a circular cross section,in the cases of and in %-DENSITOMETRIC stenosisthe area of the densitometric profile is used, which should beproportional to the cross-sectional area so as to potentially beable to deal with noncircular cross sections.

In stenosis evaluation by visual analysis, the AmericanHeart Association (AHA) grading scheme [16] is most com-monly used in routine clinical examinations. In the AHAscheme, %-LENGTH stenosis is estimated by visual analysisand classified into one of several categories of severity, asshown in Table I. In practice, experienced radiologists usuallymake severity evaluations under the AHA grading scheme byconsidering various factors, including densitometric findings.

We make a distinction between two kinds of severities;apparent severity and true severity. In the case of %-AREAstenosis, the apparent severity is based on the area estimatesobtained from an image, while the true severity is based onthe actual values measured three-dimensionally. The ordinarydefinition of stenosis severity is the apparent severity fromthe viewpoint where the stenosis is observed most severely.In this section, we examine the viewpoint dependency of theapparent severity measured from images generated using 3-D artery stenosis models with known true severity. In thefollowing, we consider two types of vessel models; a straightvessel without a branch and a branch vessel.

We modeled coronary arteries with stenosis using general-ized cylinders [17], [18] (see Appendix A). Typical severestenoses were modeled under the supervision of a radiol-ogy specialist (one of the authors, H. Naito). Fig. 1 showssimulated X-ray projection images of the 3-D vessel modelsobserved from several different viewpoints. The models weredesigned so as to possess 90%-AREA stenosis in terms of

TABLE IAHA CLASSIFICATION OF STENOSES

Classified Value Severity

100% 100%99% 91% - 99%90% 76% - 90%75% 51% - 75%50% 26% - 50%25% � 25%

true severity and circular cross section.1 Using these models,we experimentally analyzed the viewpoint dependency of thestenosis appearance by generating projection images fromvarious viewpoints. The 3-D vessel models are represented as3-D binary volumes in which the value of voxels representinga vessel region is one and that of the other voxels is zero. Weused an orthographic image projection model. Each gray-levelvalue in a generated image was calculated by integrating the3-D binary volumes of the vessel model along each optical rayto simulate X-ray projection process. The details of the vesselmodels used in the simulation are described in Appendix A.

After applying Gaussian blurring to the generated projectionimages of the vessel models to simulate the effect of spatialand temporal blurring of an imaging system, the vessel profileorthogonal to the vessel direction was used to estimate thediameter and cross-sectional area. The diameter was estimatedbased on the convolution of the normalized second derivativeof the Gaussian function with the profile, which is given by

(4)

where * denotes convolution, is the Gaussian functionwith a standard deviation and is the profile function.The normalization factor is multiplied so that the responsehas properties suitable for diameter estimation. The profile ofangiographic images orthogonal to the vessel direction can betypically modeled by

(5)

which is the ideal profile of the projection of a cylinder witha cross-sectional radius The response in (4) for this profile

has the following properties.Property 1: The response becomes minimum at

and when is fixed. That is,where is a constant.

Property 2: The minimum has the same value for any valueof That is, is constant for any

These properties are satisfied for other line-like profilefunctions which can be written as In the case of(5), and

1%-AREA stenosis is equivalent to %-DENSITOMETRIC stenosis in termsof true severity. %-LENGTH stenosis is somewhat difficult to be defined interms of true severity. When a circular cross section is assumed, however, trueseverity in %-LENGTH stenosis can be based on the diameter of the circularcross section.

SATO et al.: VIEWPOINT DETERMINATION SYSTEM FOR STENOSIS DIAGNOSIS AND QUANTIFICATION 123

(a)

(b)

Fig. 1. Simulated projection images of vessel models. Gaussian blurring was applied to the images to simulate the effect of spatial and temporal blurring ofan imaging system. The standard deviations of Gaussian blurring was 10% of the radius of the nonstenotic part of the vessel. (a) Straight (nonbranching) vesselmodel. The viewpoint of the leftmost image was orthogonal to the vessel direction. From left to right, the deviations from orthogonality to the vesseldirectionare 0�, 10�, 20�, and 30�. As the deviations from orthogonality increase, underestimation of stenosis becomes more evident. (b) Branch vessel model. Theviewpoints of all four images were orthogonal to the vessel direction, but were varied around the vessel axis. From left to right, the angles of rotation aroundthe vessel direction are 0�, 40�, 60�, and 80�. In this case, the problem of vessel overlap becomes serious as the viewpoint rotates around the axis.

We define the estimated diameteras

(6)

should be proportional to the absolute diameter (Property 1).Because the severities are defined based on a ratio,can bedirectly used as the estimates for and in equationsfor %-LENGTH and %-AREA stenoses. The advantage ofthis method is that it does not suffer from the problem ofscale dependence; in contrast, diameter estimation based onthe edges of vessel contours depends on the scale at which firstand/or second derivatives are estimated. To estimate the cross-sectional area in %-DENSITOMETRIC stenosis, we integratethe profile function and use to obtain

andA viewpoint can be represented as a point on a spherical

surface, which can be parameterized using latitudeandlongitude We set the center position of the stenotic partas the origin of the spherical coordinate system, and assignthe direction of the stenotic vessel as to the-axis. Fig. 2(a)shows the viewpoint dependency of the apparent severity inthe straight vessel model, focusing on the problem of fore-shortening. With all three types of severity measurement, theapparent severity was underestimated as the latitude became

higher, that is, when the deviation from orthogonality to thevessel direction was large. Evaluation of the apparent severityin terms of %-DENSITOMETRIC stenosis was especiallysensitive to the viewpoint. Variations in the estimated diameteralong the vessel direction are shown in Fig. 2(b). The plotssuggest that the viewpoint affects the evaluation of the extentof a stenotic lesion as well as its severity.

Fig. 3 shows the viewpoint dependency of the branchmodel, focusing on the problem of vessel overlap. Viewpointsfrom which overlapping occurs are shown. The degree ofoverlap is classified into three types: “nonoverlap,” “partial-overlap,” and “overlap.” When overlapping occurs, but mostof the stenotic part is still visible, the term “partial-overlap”is assigned to the viewpoint; when most of the stenotic part isinvisible, “overlap” is assigned. The instances of nonoverlap,partial-overlap, and overlap in Fig. 3 correspond respectivelyto the rotation angles of 0, (40 , 60 ), and 80 in Fig. 1(b).Fig. 3 shows that the stenotic part was overlapped by theperipheral vessel from a considerable numbers of viewpoints.It is also clear that the nearer a peripheral vessel that hasthe potential to overlap is to the stenotic part of interest, thewider will be the range of viewpoints from which the stenosisis invisible. Therefore, the overlap problem becomes serious


(a)

(b)

Fig. 2. Viewpoint dependency for the straight vessel model shown in Fig. 1(a). (a) Viewpoint dependency of apparent severity using three types of severitymeasurement. The results with two standard deviations of Gaussian blurring are shown, 5% (left) and 10% (right) of the radius of the nonstenotic part of thevessel. The straight vessel model is rotationally symmetric about thez-axis, and mirror symmetric about thexy-plane. Thus, the dependency is a functionof only the latitude and symmetric about the origin. (b) Variations in estimated diameter and area along these vessel direction. The variations in estimateddiameter (left) and area (right) are shown for the cases where the deviations from orthogonality are 0�, 20�, and 30�.

when a stenotic part is located near the root of a branching ves-sel. It can be seen from Fig. 3 that the number of viewpointsorthogonal to the vessel direction (i.e., the latitude is 0) wherethere is no overlap is quite limited. For reliable diagnosis ofsevere stenoses, it is desirable that the stenosis should beobserved from one of this limited number of viewpoints. Inthe branch vessel simulation, the viewpoint is optimal whenit is orthogonal to the direction of the main vessel as well asto that of the stenotic branch vessel. As shown in Fig. 1(b),the angle between the main and branch vessels is closest to aright angle when viewed from this optimal viewpoint.

The above results suggest that cases of severe and steepstenosis can be underestimated if observed from inappropriateviewpoints. Although the accurate estimation of the severityof mild stenoses has been emphasized [3], [4], clinicallyit is more important to avoid underestimating the severityof severe stenoses. In addition, as we have shown, vesseloverlap can be a serious problem, especially if a stenoticlesion is located near a branch root. Guidance on theavoidance of vessel overlap is thus also desirable. To addressthese needs, we have endeavored to devise a system forcomputer-assisted viewpoint determination.

Fig. 3. Viewpoint dependency of apparent severity for the branch vesselmodel shown in Fig. 1(b) with respect to vessel overlap. The viewpointsfrom which vessel overlap occurs are shown. See text for denotations of“nonoverlap,” “partial-overlap,” and “overlap.”


III. V IEWPOINT DETERMINATION SYSTEM

As shown in Section II, a severe case of stenosis can berecognized as a stenotic lesion from a relatively wide range ofviewpoints even though its severity may be underestimated.Even in the branch case, the relatively large extent of the“nonoverlap” and “partial-overlap” viewpoints means that atleast the existence of a stenotic lesion will be apparent in asubstantial number of cases. There is thus a high possibilitythat a stenotic lesion will be imaged in two or more imagesobtained from the standard viewpoints used in a routineexamination. We therefore assume that two images will beavailable in which the same stenotic lesion is observed. Usingthese two images, our system determines the viewpoints fromwhich the stenotic part would be observed most clearly. Thesystem uses the following three criteria

Criterion 1: Viewpoints should be as orthogonal to thedirection of the stenotic segment as possible.

Criterion 2: Viewpoints should be determined so that thestenotic segment is never overlapped by part of other vessels.

Criterion 3: Viewpoints should be determined in a mannerthat does not require large cranial and caudal angles.2

Criterion 1 has been already implemented in several systems[19], [20], [22], [23]. Criterion 2 has received relativelylittle attention and is therefore emphasized in this paper.Criterion 3 originates from imaging system restrictions. Inan oblique projection with large cranial and caudal angles,suitable arrangement of the patient’s body and the imageintensifier becomes difficult. Furthermore, in order to obtain animage of an acceptable quality using large cranial and caudalangles, the X-ray dose has to be increased because the X-rayshave to travel a longer distance.

Our system determines the viewpoints satisfying the abovecriteria by executing the following steps.

Step 1: Select two input images in which a stenosis canbe observed from among images obtained from the initialviewpoints, and manually specify corresponding points in thetwo images that can be used to find translational parametersrelating the two views.

Step 2: Estimate the two-dimensional (2-D) positions andorientations of the stenosis in the two images semiautomati-cally, and then calculate the 3-D position and orientation ofthe stenosis based on the principle of binocular stereo.

Step 3: Specify the vessel of interest, including the stenoticpart, and any peripheral vessel which may overlap the stenoticlesion manually with computer guidance.

Fig. 4 illustrates the range of viewpoints satisfying Criteria1 and 3. In the following, the details of each of the abovesteps are described.

A. Finding Translational Parameters Relating Two Views

With respect to the imaging parameters of an X-ray system,we assume that the source to image intensifier distance (SID),the radius of the image intensifier (II), and the viewpoints

2When a patient’s body is aligned along thez-axis of a spherical coordinatesystem as described in Section II, the variation in the latitude� correspondsto the cranial and caudal angles, while the variation in the longitude�

corresponds to the left- and right-anterior-oblique view (LAO and RAO)angles (see Appendix C).

Fig. 4. Viewpoints satisfying Criteria 1 and 3. The black tubes representvessels. The sphere represents the set of all the viewpoints. The stenotic lesionis located at the center of the disk, which represents the viewpoints satisfyingCriterion 1. The normal vector of the disk is the orientation of the vessel atthe stenotic lesion. The sectors (semitransparent gray regions) represent theviewpoints satisfying both Criteria 1 and 3.

(cranial/caudal and RAO/LAO angles) are known. It is some-what difficult to keep track of information on translationalparameters between a patient’s body and the X-ray system.For instance, the patient’s table is sometimes moved to obtainthe optimal field of view, even during the injection of contrastmaterial.

We assume perspective projection as the image projectionmodel. The imaging parameters of perspective projection in thelocal coordinate system of each view can be obtained using theknown II radius and the SID value. Rotational parameters thatrelate the two local coordinate systems can be obtained usingthe known viewpoint (cranial/caudal and RAO/LAO angles)parameters. Given the parameters of perspective projection ofthe two views and the rotational parameters, we need to findtranslational parameters from the local coordinate system ofone view to that of the other view in order to obtain 3-Dinformation on the vessels from the two angiographic views.Translational parameters can be found by specifying at leastthree pairs of corresponding points in the two images. Branchand stenotic points are typically selected as correspondingpoints. At least three correspondences are necessary to ob-tained the desired parameters. The details of the equations usedto find the translational parameters are given in Appendix B.

B. Estimating Stenosis Orientation and Position

Estimation of the orientation of a stenotic segment isan important step because its accuracy directly affects theestimation of the 3-D orientation of the stenotic segment.Obtaining good viewpoints as specified in Criterion 1 is mostlyreliant on the accuracy of the estimated 3-D orientation of thestenosis. Therefore, to determine the stenosis orientation as


objectively as possible, we use a semiautomated method basedon the Hessian matrix [24], [25]. Given an image intensityfunction the Hessian matrix at each point isgiven by

(7)

The Hessian matrix describes the second-order structure oflocal intensity variations around each point. Let the eigenval-ues of be and and their correspondingeigenvectors be and , respectively.The eigenvector corresponding to the larger (smaller)eigenvalue represents the direction along which thesecond derivative becomes maximum (minimum), andgives the maximum (minimum) second derivative value. Whenthe vessels are imaged as dark (bright) curvilinear structures,we can regard the orientation orthogonal to the vessel directionas By replacing in (7) with

to calculate the normalized second derivative withGaussian blurring, we can make the eigenvaluesandof have properties similar to those of (4) described inSection II.

We use the eigenvalues and eigenvectors of basedon the normalized second derivative with Gaussian blurringat continuous values of at each point. Further, we combinethe gradient with Gaussian blurring at continuous values ofto discriminate between line structures and other structures[25]. If we consider the case where vessels are imagedas dark curvilinear structures, then the maximum of theeigenvalue through at each point is given by

Let the eigenvector corresponding tobe and be . Wecan then regard the direction orthogonal to and

at each point as hypothesized estimates of the vesselorientation and diameter, respectively. We assume that a point

is on the center line of a vessel. In this case,there should be strong edges at both sides along the direction

These edges should occur at a distance proportional to. We define the strengths of these edges as

(8)

and

(9)

where is a gradient vector with Gaussian blurring,and and are scale parameters. (In the experiments, weused and .) If the point is on the centerline of a vessel, both and should be large. Thus,we define a line-ness measureas

(10)

Although can also be used as a line-ness measure,responds to other structures such as step edges as well as linestructures. In contrast to has been shown to discriminate

line structures more strongly [25]. At the point on the centerline, should take a local maximum along the direction

Based on the above considerations, we formulate a methodthat is intended to minimize the amount of user interactionneeded and maximize the objectivity of the estimates of thestenosis orientation and position. The method is as follows.

Step 1: A user specifies a rectangular region of interest(ROI) which encloses the vessels around the stenotic segmentin an image. The orientation as well as the position and sizeof the ROI can be changed interactively. The orientation ofone axis of the ROI is roughly adjusted to the direction of thevessel of interest [Fig. 5(a)].

Step 2: The direction orthogonal to the vesseland the line-ness measure are calculated at each pointwithin the ROI.

Step 3: The candidate points on the center line of the vesselare determined as follows: Let the-axis be the axis of theROI that is roughly adjusted to the vessel direction, and the-axis be the axis orthogonal to the-axis. The candidate pointsare extracted where takes local maximum along the-axis. We then extract the points having a value morethan 80% of the maximum value along the-axis.

Step 4: The vessel diameter is estimated accurately at eachextracted center point of the vessel as follows. The profilealong is evaluated at each point to obtain the estimateddiameter using

(11)

where is on the center line point, and is thenormalized second derivative with Gaussian blurring shownin (4).

Step 5: The estimated diameters (radii) are plotted in the 2-D space whose horizontal axis represents the-coordinate andwhose vertical axis represents[Fig. 5(e)]. Also, the position,diameter, and orientation at each extracted points are displayedby superimposing them onto the image of the ROI [Fig. 5(b)].Finally, the user finds the stenotic point by examining the plotof the estimated diameters and the superimposed image, andspecifies it: the stenotic point is the point with the smallestestimated diameter, and its orientation is given by the directionorthogonal to at its position.

In the above method, user interaction is needed only inspecifying the ROI (Step 1) and in the final selection of thestenotic point (Step 5). Once the 2-D orientations and positionsin the two images are determined, the 3-D orientation andposition of the stenosis are estimated based on the principleof binocular stereo. The viewpoints satisfying Criterion 1 aredetermined using the estimated 3-D orientation of the stenoticsegment (see Appendix C for the method of calculatingthe viewpoints orthogonal to 3-D orientation of a stenoticsegment).

Fig. 5 shows the results of an evaluation of the method usinggenerated projection images of a curved vessel model withGaussian blurring (see Appendix A for details of the model).The standard deviation of the Gaussian blurring was 10% ofthe radius of the nonstenotic parts of the vessel. Fig. 5(c)–(e)shows the estimated center position, orientation, and radius,


(a) (b)

(c) (d)

(e)

Fig. 5. Estimation of vessel center line position, radius, and orientation. (a) Specification of ROI. In this case, the orientation of the ROI was 15� againstthe horizontal axis of the image. (b) Estimated center position, radius, and orientation superimposed on the ROI image. (c) Estimated center position. Theresults are shown for the cases where the ROI orientation was�30�, �15�, 0�, 15�, and 30� The size and center position of the ROI were the same at eachorientation. The estimated center positions were transformed in the coordinates of the original images and plotted. (d) Estimated orientations at extracted centerpositions. The estimated orientations were plotted along the horizontal coordinates of the original images. (e) Estimated radii at extracted center positions.The estimated radii were calculated using�opt=0:57; where 0.57 is the value of constant c in Property 1 described in Section II.

respectively. To assess the sensitivity of the procedure to theROI orientation specified by the user, the ROI was positioned

at several orientations; the results showed the method to bequite insensitive to user-specified orientation of the ROI. The


radius estimates were almost equal to the theoretical values.The orientation estimates included some error, but this wasretained within 5.

C. Avoiding Vessel Overlap

In order to determine viewpoints without vessel overlapfrom among those viewpoints satisfying Criteria 1 and 3,it is necessary to obtain the 3-D structures of vessels thatmight overlap the stenotic lesion. In doing so, one problemto be addressed is how the task should be divided betweena computer and a human operator to obtain a reliable 3-Dreconstruction of the vessel structures within a minimumamount of time. However, even when a good human-computerinteractive system is designed, 3-D reconstruction of an entireset of vessels is not suitable for the use during the examinationdue to the time-consuming nature of the procedures for recon-structing a large number of vessels. Thus, the other importantproblem that needs to be considered is how the vessels to bereconstructed should be selected.

The basic approach is to use geometric constraints toguide the user as much as possible in performing the userspecification tasks. First, we use the constraints given fordetermining the viewpoints satisfying Criteria 1 and 3. Second,we fully utilize the epipolar constraint in human-computerinteraction to find correspondences of the vessels in the twoimages for 3-D reconstruction based on the binocular stereo.The details of the method are as follows.

The set of viewing directions satisfying Criterion 1 can berepresented as a plane in 3-D space which passes through thestenotic point and whose normal vector is the orientation ofthe vessel at the site of the stenosis. Vessels that might overlapthe stenotic lesion should intersect with this plane. The extentof coronary vessels on the plane should be limited. Also, sincethe possibility of overlap is greater for vessels nearer to thestenosis, we can constrain vessels to those on a disk in 3-Dspace whose center is the stenotic point and whose radius isa finite length (Fig. 4). That is, this disk can be regarded asthe area in which there is a high possibility of the existenceof overlapping vessels. Criterion 3 can be incorporated so asto further reduce the possible extent of overlapping vessels,creating two sectors on the disk (Fig. 4). By projecting thesesectors onto the two input image planes, the areas wherevessel overlap is likely to exist are displayed. Fig. 4 shows anexample in which the resultant area with a high possibility ofoverlap is clearly depicted. The projected sectors give guidancefor the performance of the user specification tasks. That is, theusr is advised to pay close attention to the projected sectorsin the knowledge that vessels selected for reconstruction arelikely to be from within these regions.

The 3-D reconstruction of the selected vessels is basedon the correspondence of the user-specified vessels in thetwo images. We use the following interactive method for thereliable and fast determination of correspondences.

Step 1: The user finds vessel segments in the two imagesthat appear likely to correspond one another. These segmentsshould fall within the sectors likely to contain overlappingvessels as displayed by the system for both images.

TABLE IISUMMARY OF DATA SETS OF CORONARY ANGIOGRAMS. SEVERE STENOSES

REPRESENT99–99% STENOSES ANDNONSEVERE(MODERATE OR MILD)STENOSESREPRESENT75% OR LESS. SEVERITY EVALUATIONS WERE

BASED ON A VISUAL ANALYSIS BY A RADIOLOGY SPECIALIST

Image Set # # of viewpoints# of severe

stenoses# of nonsevere

stenoses

1 8 2 02 4 1 03 8 1 14 6 1 0

Step 2: The user specifies the vessel segment in one image(Image 1) as a sequence of 2-D points connected by straightline segments.

Step 3: The system displays the epipolar lines (epi-lines) inthe other image (Image 2) corresponding to the first and lastpoints of the sequence in Image 1.

Step 4: The user specifies the points corresponding to thefirst and last points in Image 1 on the epi-lines in Image 2.Even if a point is not specified exactly on the epi-line, thesystem automatically moves it to the nearest point on theepi-line.

Step 5: The system places the initial points in Image 2corresponding to other points specified in Image 1. Thesepoints are put on the epi-lines so as to be smoothly interpolatedbetween the first and last points.

Step 6: The user modifies the positions of the points in bothImages 1 and 2. During this user modification, the systemalways controls the movement of points so as to satisfy thefollowing epipolar constraints.

• When the user modifies a position in Image 1, the systemmoves the position of the corresponding point in Image2 so that it is always on the epi-line corresponding to themoved point in Image 1.

• When the user modifies a position in Image 2, it can onlybe moved along the epi-line.

Step 7: Step 6 is repeated until satisfactory correspon-dences are determined.

The procedures from Steps 1–7 are executed for eachsegment. The function performed by the system in Step 6 wasfound to be very helpful in allowing correspondences to bedetermined rapidly and reliably.

The system displays, for the user, projection images of thereconstructed 3-D structures of vessels with the stenotic lesionsfrom viewpoints satisfying Criteria 1 and 3, The user can selectthe optimal viewpoints that carry the minimum risk of vesseloverlap.

IV. EXPERIMENTAL RESULTS

In order to evaluate the system, we used four sets ofcoronary angiograms, each of which had been obtained fromfour to eight viewpoints for one patient. Since the system hasnot yet been used clinically, the experiments were performedretrospectively. All the angiograms were taken using a systemcapable of adjusting the RAO/LAO and Cran/Caud angles with1 precision. Table II summarizes the stenotic lesions used inthe experiments and their severity as judged on the basis of a


TABLE IIIVIEWPOINTS OF CORONARY ANGIOGRAMS USED IN EXPERIMENTS

Figure # Image set # Input Image 1 Input Image 2 Resultant image

6 and 8 1 RAO 30� LAO 43�, Cran. 33� RAO 15�, Caud. 30�

10 2 RAO 27�, Caud. 20� RAO 30� RAO 30�, Cran. 21�

11 3 RAO 31� LAO 60�, Caud. 30� LAO 60�

12(a) 1 RAO 30� RAO 15�, Caud. 30� LAO 90�

12(b) 3 RAO 31�, Cran. 32� LAO 59�, Cran. 28� RAO 31�

12(c) 4 RAO 30� LAO 45�, Caud. 34� LAO 45�, Caud. 22�

visual analysis by a radiology specialist. Also, the viewpointsof the images are summarized in Table III.

The experimental method was as follows.Step 1: Select two images in which the same stenotic lesion

can be located.Step 2: Determine the optimal viewpoints based on the two

images using the system.Step 3: From all the available angiograms, select a new

image taken from the viewpoint nearest to the determinedviewpoints.

Step 4: Judge whether or not more reliable diagnosis ispossible by using the new image.

The judgement in the final step was based on both a visualanalysis and computerized quantification. The visual analysiswas performed by a radiology specialist with more than 15years’ experience (one of the authors, H. Naito). Computerizedquantification was performed by the Hessian-based methoddescribed in Section III-B. Since the aim of the experimentwas to confirm whether the system can determine betterviewpoints from images in which stenosis is unclearly imaged,the two images initially selected were not necessarily takenfrom the standard viewpoints normally employed in coronaryangiography. We selected two initial images such that theircardiac phases were almost the same, and adjusted the phasesof the two images to the end-systolic (ES) or the end-diastolicphase (ED) phase.

A. Case of Severe and Steep Stenosis

Fig. 6 shows two input images. The size of input images are640 480 pixels, and Fig. 6 shows the region of 300240(pixels) trimmed from the original input images. The outlinedregion of 75 75 pixels containing the stenosis part alsoshown in enlargement. The severity evaluation by visualanalysis based on the AHA classification [16] (see Table I)was 50% stenosis for the lesion shown in Fig. 6(a) and 25%stenosis for that in Fig. 6(b).

Fig. 7 illustrates the viewpoint determination proceduresperformed by the system. Fig. 7(a) shows the estimated 2-Dposition and orientation of the stenosis in the two imagesobtained by the semiautomated method based on the Hessianmatrix described in Section III-B. Fig. 7(b) shows the possibleareas (sectors) in which overlapping vessels might exist, andthe user-specified vessels obtained by the interactive systemdescribed in Section III-C. The amount of time spent by theuser in providing the required input was typically less than5 min. However, this does not include the time needed tofind the stenotic lesion and the corresponding vessel segments

(a)

(b)

Fig. 6. Input images. The right-hand images are close-ups of the stenoticpart, indicated by arrows. See Table III for the viewpoint of each image. (a)Input image 1. (b) Input image 2.

in the two images, which is highly dependent on the user’sexperience in reading coronary angiograms.

Fig. 7(c) shows the range of viewpoints recommended bythe system, the two viewpoints from which the input imageswere taken, and the nearest viewpoint to the recommededviewpoints among all the coronary angiograms obtained. Be-cause the simulation results presented in Fig. 5(d) indicatesthat we need to assume there may be an error of5 in thestenosis orientation estimation, we also plotted in Fig. 7(c) thevariations in the optimal viewpoints with an error of5 inimage measurement with respect to the stenosis orientation.However, the plots showed that the recommended viewpointswere not very sensitive to error within5 . Fig. 8 shows theimage taken from the nearest viewpoint to those recommendedby the system and the expected vessel image from the sameviewpoint. The expected vessel image confirmed that theviewpoint was not affected by vessel overlap. The severityof the stenosis shown in Fig. 8 was evaluated as 90%–99%by visual analysis. This judgement shows that the image taken


(a)

(b)

(c)

Fig. 7. Viewpoint determination procedures and results. (a) Semiautomatically estimated 2-D position and orientation of stenosis. The 2-D position andorientation of the stenosis are depicted by a line that originates from the 2-D stenosis position. (b) Peripheral vessels specified in the two input images.The areas in which the existence of overlapping vessels is possible are displayed by the system as sectors. (c) Optimal viewpoints determined using thesystem and the viewpoints from which Figs. 6 and 8(a) were taken. The variations in the determined viewpoints with an error of�5� in the estimationof the stenosis orientation are also shown.

TABLE IVSUMMARY OF SEVERITY EVALUATION AND DEVIATION FROM ORTHOGONALITY j90

�� j AT EACH VIEWPOINT

Input Image 1 Input Image 2 Resultant imageFigure # Phase

Severity j90�� j Severity j90

�� j Severity j90

�� j

6 and 8 ED 50% 18.1� 25% 77.9� 90% - 99% 2.0�

10 ED 75% - 90% 55.6� 75% - 90% 36.7� 90% - 99% 18.0�

11EDES

50%50% - 75%

47.6�

65.6�75%75%

20.7�

18.4�99%

overlap1.0�

5.9�

12(a) ES 99% 42.5� 75% - 90% 70.5� 99% 6.5�

12(b) ES 25% - 50% 51.8� 25% - 50% 10.4� 25% - 50% 6.1�

12(c) ES overlap 64.5� 75% 7.8� 90% 7.7�


(a) (b)

Fig. 8. Resultant images recommended by the system. (a) Image taken from the nearest viewpoint to those recommended by the system. See Table III for theviewpoint. The right image is a close-up of the stenotic part. (b) Expected vessel image from the same viewpoint. The stenotic part is shown as an encircled dot.

(a) (b) (c)

Fig. 9. Results of orientation and diameter estimation using the Hessian-based method for the images shown in Figs. 6 and 8. The top row shows thespecified ROI’s. The middle row shows the estimated results superimposed on the image in each ROI. The bottom row shows the plots of the estimateddiameters. The horizontal axis represent the axis along the vessel direction of each ROI. The vertical axis represent the estimated diameters. (a) Results forinput image 1 [Fig. 6(a)]. (b) Results for input image 2 [Fig. 6(b)]. (c) Results for the resultant image [Fig. 8(a)].

from the viewpoint recommended by the system enabled areliable diagnosis to be made, which had proved difficult usingonly the two input images. If we let be the angle betweenthe viewing direction and stenosis orientation estimated by thesystem, because the optimal value ofis 90 , we can definethe deviation from the optimal viewpoint as Ourresults showed that was 18.1 for the viewpointof Fig. 6(a), 77.9 for Fig. 6(b), but only 2.0 for Fig. 8(a).Therefore, it was confirmed that Fig. 8 was acquired fromalmost the optimal viewpoint. Table IV summarizes the resultsof the severity evaluation by visual analysis and the deviationfrom orthogonality, of each viewpoint.

Fig. 9 shows the results of the computerized quantificationas well as the procedure for estimating the stenosis orientation,position, and diameter using the Hessian-method described in

Section III-B. The top row shows the ROI specified in eachimage. The ROI was enlarged four-fold using sinc interpola-tion [26] to make accurate diameter estimation possible. In themiddle row, the results of the estimated position, orientation,and diameter are shown. In the lower row, the estimateddiameters are plotted along the axis corresponding to the vesseldirection in each ROI. Although some false extractions canbe observed in Fig. 9(c), it is easy for the user to removethe outliers and trace the estimated diameter originating fromthe vessel of interest. The method tends to become unstablewhen severe stenosis is observed from a good viewpointbecause densitometric profiles often have insufficient contrastnear the stenotic point. While the visual difference betweenFigs. 6(a) and 8(a) would seem subtle to a nonexpert, thequantification results in Fig. 9 show that the two images differ


(a) (b) (c)

Fig. 10. Results of orientation and diameter estimation using the Hessian-based method for a less-steep stenotic lesion. The top row shows enlarged imagesof the stenotic lesion from each viewpoint. The second row shows the specified ROI’s. The third row shows the estimated results superimposed on theimage in each ROI. The bottom row shows the plots of the estimated diameters. See Table III for the viewpoint of each image. (a) Results for inputimage 1. (b) Results for input image 2. (c) Results for the resultant image.

significantly in both the extent and apparent severity of thestenotic lesion.

B. Case of Less-Steep Stenosis

Fig. 10 shows the results of a case in which the diametervariation along the vessel direction is less steep around astenotic lesion. These depictions make it easier to understandthe viewpoint dependency of a stenotic lesion. In this set ofangiograms, although the deviation from orthogonality in theresultant image was not particularly small (18.0; Table IV),improved visualization was achieved, especially on the extentof the stenotic lesion. The plots of the estimated diametersin Fig. 10 show that the observed length of the stenoticlesion changed significantly depending on the deviation fromorthogonality.

C. Case of Vessel Overlap

Fig. 11 shows a case in which the effect of vessel overlap isserious. In this case, the optimal viewpoints recommended bythe system were quite narrow at the ES phase to satisfy Criteria1–3. Also at the ED phase, the recommended viewpoints com-prised less than half of all the RAO/LAO angles [Fig. 11(c)].Fig. 11(d) and (e) shows the resultant images which wereselected based only on Criteria 1 and 3. The viewpoint was

evaluated as being almost optimal with respect to Criteria 1and 3 (Table IV).

Fig. 11(f) shows the expected image at the ED phase gener-ated by the system. The system suggested a high risk of vesseloverlap from this viewpoint. In the actual image [Fig. 11(d)],however, vessel overlap could be avoided by a small margin.Fig. 11(d) shows the upper-right region (240240 pixels) ofthe original image. The border of the II is also visible inthe upper-right corner, indicating that the stenotic part wasimaged in the peripheral area of the II. However, the expectedimages in Fig. 11(f) were generated assuming that the stenoticpart was imaged at the center of the II. Because we thoughtthis difference in the imaged position in the II may not benegligible,3 we also generated an expected image at a similarposition on the II and from the same viewpoint as the imagein Fig. 11(d) This new expected image [Fig. 11(g)] was moresimilar to the real image [Fig. 11(d)]. The distance in 3-Dspace between the overlapping vessel in Fig. 11(f), and thestenotic part was relatively large. In such a case, only a smallvariation in the viewpoint significantly affects the appearance,and vessel overlap does not occur from a wide range of

3The SID value was about 1 meter,(= 100 cm), and the radius of theimage intensifier was 3.5 in(� 8.9 cm). Thus, the maximum angle betweenthe ray direction and the direction orthogonal to the image intensifier planecould be expected to bearctan(8:9=100) � 5:1�:


(a) (b) (c)

(d) (e)

(f) (g) (h)

Fig. 11. Results in the case of vessel overlap. See Table III for the viewpoints of the input and resultant images. (a) Input image 1 (ED phase). (b) Inputimage 2 (ED phase). (c) Optimal viewpoints determined by the system. (d) Resultant image (ED phase). (e) Resultant image (ES phase). (f) Expected imagefrom the same viewpoint as the resultant image (ED phase). (g) Expected image at a similar position on the II and from the same viewpoint as the imagein (d) (ED phase). (h) Expected image from the same viewpoint as the resultant image (ES phase).

viewpoints. In fact, this overlapping vessel resulted in onlya small number of viewpoints being deleted—at the ED phasenear LAO 60 and Cran/Caud 0in Fig. 11(c). The severityevaluation of the image in Fig. 11(b) by visual analysis was99% (Table IV). In this case, the computerized quantificationfailed because the density at the stenotic part was so weakthat our Hessian-based method could not stably estimate theorientation and diameter around this region. However, it couldbe shown that the system accurately expected the existence ofvessel overlap.

Fig. 11(h) shows the expected image at the ES phasegenerated by the system. The system suggested a high risk

of overlap. In the real image [Fig. 11(e)] as well, the stenoticpart was overlapped by a different peripheral vessel from theone observed at the ED phase. Because this overlapping vesselwas located close to the stenotic part in 3-D space, it causeda considerable number of viewpoints satisfying Criteria 1 and3 to be deleted at both the ED and ES phases.

D. Other Cases

Fig. 12(a) shows the case of long segmental stenosis. Thevisualization of a severe and long segmental stenotic lesionis relatively clear, even in the left-most image of Fig. 12(a),


(a)

(b)

(c)

Fig. 12. Examples of input images (left and middle columns) and resultant images with improved visualization (right column). See Table III for theviewpoint of each image. (a) Long segmental stenosis. (b) Mild stenosis. (c) Branch with complex local structure.

which was taken from an unfavorable viewpoint with a devia-tion from orthogonality of 42.5 (see Table IV). Although theseverity estimation is similar, the stenotic part appears longerand can be seen more clearly from a better viewpoint [theright-most image of Fig. 12(a)]. Fig. 12(b) shows a case ofmild stenosis, in which the severity evaluation depended littleon the viewpoint.

Fig. 12(c) shows a tricky case. Here, a stenotic lesion islocated at the root of a branch. Although the resultant imagegave a better visualization, the deviation from orthogonalitywas almost the same as that of one of the input images (image2). The simulation in Section II using a branch vessel modelshowed that overlapping by the main vessel is a problem evenif the deviation from orthogonality is small. In the simulation[Figs. 1(b) and 3], when viewed from the optimal viewpointamong the viewpoints satisfying Criterion 1, the angle betweenthe main and branch vessels in an image was the closest to theright angle. In Fig. 12(c), however, this angle is narrower inthe resultant image but it gives a giving better visualization.Some of the junction structures between main and branchvessels may not be simple like that of the model shown inFig. 1(b). A complex local structure at a junction will makeit difficult for the system to expect the appearance from inputimages that do not contain sufficient information at the localstructure. This is one of limitations of the current system.

V. DISCUSSION AND CONCLUSION

Several techniques have been developed to minimize fore-shortening [19], [20], [22], [23], one of the two main dif-

ficulties caused by unsatisfactory viewpoints. Wollschlageret al. introduced the concept of triple orthogonality for optimalimaging without foreshortening [19], and different implemen-tations of the concept, as well as experimental results withphantoms, have been reported in [20] and [23]. Further, Dumayet al. elaborated and validated the system in great detail [22].These systems focused on the accurate positioning of thegantry to prevent foreshortening under the assumption thatthe orientation and position of stenosis lesions on projectedimages and complete information on the projection geometryare given. However, the aspect of vessel overlap as well asprocedures for the estimating the orientation and position ofstenosis lesions on the images are not emphasized.

With respect to concerns the problem of vessel overlap,Solzbachet al. reported a system which indicates vessel seg-ments orthogonal to a given viewpoint after the 3-D structuresof whole vessels are reconstructed [21]. Although their systemis intended to avoid vessel overlap, a time-consuming step isneeded for the reconstruction of whole vessels. The favorableviewpoints are not automatically calculated, but interactivelyadjusted by visualizing the reconstructed vessels.

In contrast with previous work, we have elaborated a view-point determination system for angiographic image acquisitionfrom the standpoints of image analysis and computer visionresearch. Our system emphasizes the aspects of automationand vessel overlap, especially with regard to the followingpoints. First, we proposed a new method for the semiautomatedestimation of the position, orientation, and diameter of astenotic part of the vessel using a multiscale Hessian-based


approach. The advantage of our method is that the orientationand diameter are estimated as continuous values directly fromlocal intensity structures, which are viewed as a continuous2-D function. Thus, the intermediate representations needed inconventional methods, such as the left and right edges of vesselborders, are unnecessary. Such intermediate representationshave the drawbacks that they often depend on the scale atwhich first and/or second derivatives are estimated. Further-more, additional user interaction is necessary in order to obtainthe intermediate representations. In our multiscale Hessian-based method, we make positive use of scale dependencyto formulate the diameter estimation as an optimal scaleselection. User interaction is thus minimized, and the methodis insensitive to user-specified information. Second, in orderto minimize the amount of vessel reconstruction needed, oursystem is constrained to the area highly likely to containperipheral vessels that may overlap the stenosis lesion ofinterest. Third, when the 3-D structures of vessels are to bereconstructed, the system assists in obtaining vessel correspon-dences in the two images such that the epipolar constraint isdefinitely satisfied. These second and third features allow thecorrespondences to be determined and the reconstruction tobe completed much more easily and reliably. Based on thereconstructed 3-D vessel structures, our system generates theexpected images when the gantry is rotated around the stenoticvessel axis by computer simulation to find the possibility ofvessel overlap. An alternative way to avoid vessel overlapis to take real angiograms while rotating the gantry aroundthe stenotic vessel axis, keeping Criterion 1 satisfied duringthe rotation. However, this latter method needs an extra X-raydose and the use of more contrast media, which is clinically notdesirable from the standpoint of reducing invasive examinationprocedures to a minimum.

Our method particularly emphasizes the viewpoint depen-dency of the perceived severity of stenosis lesions. Using syn-thesized and real images, we have shown that the severity ofcases of severe stenosis was often drastically underestimated,and that this was dependent on the viewpoints from which theimages were taken. The results of the analyzes not only justifythe necessity of employing a viewpoint determination systembut also suggest that the deviation from orthogonality of theangle between the vessel orientation and the viewing directionshould be incorporated as a kind of confidence measure instenosis quantification and diagnosis.

The performance of our system relies on the accuracy ofthe estimated orientation of a stenotic lesion of interest. Themain error sources that we have not addressed are patientmotion, uncertainty concerning the rotation angles given by thepositioner, and geometric distortion of the image arising fromthe II. Among these factors, the geometric distortion of theII can be effectively reduced by correction using calibrationdata [1].

In addition to the factors considered in this paper, thevisualization of stenotic lesions also depends on various otherconditions, such as the density of the contrast material andmotion blurring. However, the experimental results presentedhere could suggest that viewpoint determination is one of themost important factors affecting stenosis severity evaluation

and quantification, and that computer-assisted viewpoint deter-mination can be especially useful for severe stenotic lesions.In the case shown in Fig. 11, the recommended viewpointsconsiderably depended on the cardiac phase. A recent studyhas shown that more reliable feature detection is possible whentime-varying images of a stenotic vessel are observed under astabilized condition rather than by observing only one image[27]. Therefore, it is important that a stenotic lesion should beable to be observed not at only one phase but during the wholecardiac cycle. As shown in Fig. 11(c), the system presentedhere can be extended so as to deal with the variation of optimalviewpoints during the cardiac cycle based on a combinationof the results at the ES and ED phases.

The following problems remain to be addressed. First, inits present form the system can only deal with stenoseshaving concentric cross sections. It needs to be extended toaccommodate stenoses with elliptic cross sections, which areimportant instances of stenoses entailing a large degree ofviewpoint dependency. Second, the system does not deal withthe time-variation of optimal viewpoints during the cardiaccycle. New criteria should be incorporated so that a stenoticsegment can be observed from a viewpoint as orthogonal tothe direction of the stenotic vessel as possible and withoutincurring overlap, for as long a time as possible during thecardiac cycle. Finally, the system has not yet been used at theclinical stage. Thus far, our experiments have been performedretrospectively. Although we believe that these experimentspresent a meaningful first step in our evaluation of the system,its clinical evaluation is the next important task.

APPENDIX ASTENOSIS MODEL USED IN SIMULATION

The 3-D vessel models were represented as 3-D binaryvolumes in which the value of voxels representing a vesselregion was one and that of the other voxels was zero. Thevolume was 255 255 255 voxels. Its center was regardedas the origin of the coordinate system. The surface model ofcoronary artery with stenosis was defined using a generalizedcylinder. The region inside the surface model was assignedas one and the region outside as zero so as to create the 3-Dbinary volume of a vessel model. Simulated X-ray projectionimages with an area of 255255 pixels were generated usingthe AVS (Advanced Visual Systems Inc., Waltham, MA) tracermodule, which performs ray-traced volume rendering on 3-Dbinary volumes by orthographic projection without scaling. Inthe following, the details of each vessel model are described.The unit of length is the pixel.

1) Stenosis in Straight Vessel Model:A coronary arterywith stenosis was modeled by

(12)

where is a sweep function given by

if

else

(13)


in which and are the diameter of the nonstenotic partand the minimal diameter of the stenotic part, respectively;

and is the range of the stenoticpart. We used and for themodel shown in Fig. 1(a). When andthe %-AREA stenosis is in trueseverity.

2) Stenosis in Branch Vessel Model:Segments of two ves-sels—a main vessel and a branch vessel—were combined. Themain segment is given by

(14)

where is the straight axis of the main segment,and and satisfy

and We usedwhere is the junction point, and is the directionof the main segment. The branch segment is given by

where is a sweepfunction given by

if

if or(15)

We used andfor the model shown in Fig. 1(b).

3) Stenosis in Curved Vessel Model:The curved vesselmodel is given by

where is the curved axis of the vesselgiven by

if

if

if

if(16)

and is a sweep function given by

if

else

(17)

where is the radius of the curve andis the range of thecurve. We used and

for the model shown in Fig. 5(a).

APPENDIX BESTIMATING TRANSLATIONAL PARAMETERS

OF TWO X-RAY COORDINATE SYSTEMS

An X-ray system can be modeled by a cone whose tipis the X-ray source and whose base is the image intensifier.We assume that the direction of the axis, its length, and thediameter of the base of each cone are known. These valuescorrespond to the gantry angle, SID value, and II diameter,respectively. Without loss of generality, we assume that thetip of one cone is at the origin. If we then let be the 3-Dposition of the tip of the other cone, the problem is to findthe position of When correspondences on two imageintensifiers are given, we can obtain

for (18)

where and are unit vectors representing the directions ofthe optical rays of the corresponding points in the two images.

and can be calculated from the image coordinates inthe II and the direction of the cone axis. Thus, the unknownparameters are and In 3-D translation,

includes three unknowns. One pair of correspondencebasically adds three equations as well as two new unknownparameters. However, (18) is essentially underconstrained,even if is sufficiently large. We fixed one value amongand to solve (18) using the least square method. This fixedvalue was selected so as to represent a typical distance froman X-ray source to the arteries of a patient. Therefore, at leastthree correspondences are necessary to solve the problem.

APPENDIX CCALCULATING VIEWPOINTS ORTHOGONAL

TO 3-D ORIENTATION OF STENOTIC SEGMENTS

Let be the 3-D orientationof a stenotic segment, and be the viewpointsorthogonal to the stenosis orientation, whereViewpoint can be represented as a point on a sphericalsurface parameterized using latitudeand longitude whichis given by If we set thedirection of the -axis as being from the feet to the head ofa patient, the -axis from right to left, and the -axis fromback to front, the variation in the latitude corresponds tothe cranial and caudal angles. Similarly, the variationin the longitude corresponds to LAO and RAO (left-and right-anterior-oblique view) angles. Using wehave When isfixed at we obtain

(19)

When we vary from 90 to 90 , we have the at each

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a viewpoint determination system for stenosis diagnosis

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