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Simultaneous Correspondence and Non-Rigid 3D Reconstruction of the

Coronary Tree from Single X-ray Images

Eduard Serradell1

Adriana Romero2,4

Rub´ en Leta3

Carlo Gatta2,4

Francesc Moreno-Noguer1

1Institut de Rob` otica i Inform` atica Industrial, CSIC-UPC, Barcelona, Spain

2Centre de Visi´ o per Computador, UAB, Barcelona, Spain

3Hospital de la Santa Creu i Sant Pau, Barcelona, Spain

4Departament de Matem` atica Aplicada i An` alisi, UB, Barcelona, Spain

{eserradell,fmoreno}@iri.upc.edu, aromero@cvc.uab.es, rleta@hsp.santpau.es, carlo.gatta@ub.edu

Abstract

We present a novel approach to simultaneously recon-

struct the 3D structure of a non-rigid coronary tree and es-

timate point correspondences between an input X-ray im-

age and a reference 3D shape. At the core of our approach

lies an optimization scheme that iteratively fits a generative

3D model of increasing complexity and guides the matching

process. As a result, and in contrast to existing approaches

that assume rigidity or quasi-rigidity of the structure, our

method is able to retrieve large non-linear deformations

even when the input data is corrupted by the presence of

noise and partial occlusions. We extensively evaluate our

approach under synthetic and real data and demonstrate a

remarkable improvement compared to state-of-the-art.

1. Introduction

Recovering the 3D structure of a non-rigid coronary tree

from single X-ray images is a highly ambiguous problem

since many different 3D configurations can virtually have

the same projection. As shown in Fig. 1 the problem be-

comes even more challenging because X-ray images are of-

ten affected by a series of artifacts such as noise, blurring,

partial occlusions and vessel discontinuity. Thus, solving

this problem requires from prior knowledge about the type

of deformations the structure can undergo.

Standard approaches within medical imaging assume a

reference 3D scan of the tree is known and that defor-

mations in the input image are negligible. This reduces

the shape recovering task to a rigid 3D-to-2D registra-

tion [9, 11, 16]. There exist a recent attempt of addressing

the non-rigidity nature of the problem, although it has only

been shown effective for relatively small deformations [8].

We may find other related areas in computer vision that

essentially solve the same problem but in a different con-

text, for instance, the techniques for 3D non-rigid surface

Figure 1. Recovering the structure of a non-rigid coronary tree.

Top: Given an input X-ray image (left), and a reference struc-

ture (shown in red at the top-right image) we are able to retrieve

the 3D configuration of the coronary tree in the input image (yel-

low). Bottom: Results on synthetic data, for which we know the

ground truth and allows us to evaluate the method under noise, oc-

clusions (blue dots), and different levels of deformation. The left

figure depicts the 2D results and the figure on the right represents

the ground truth (black), the prior (red) and our solution (yellow).

Note that even when the prior significantly differs from the ground

truth, our approach yields very accurate results.

reconstruction [5, 14, 17] and articulated human pose esti-

mation from monocular images [1, 18, 21, 27]. In these ap-

proaches, though, it is often easy to obtain large amounts of

training data and build detailed parametric models for spe-

cific deformations or 2D-to-3D mappings that directly link

2D observations with 3D configurations.

producing these detailed models and mappings is beyond

our possibilities, because X-ray images are harmful for the

patient and, besides one single reference 3D scan of the

coronary tree, no further prior knowledge can be used.

We therefore propose a novel approach that, given solely

Unfortunately,

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one single X-ray image and a reference 3D configuration,

simultaneously recovers the 3D structure of the coronary

tree in the input image and establishes matches with the ref-

erence shape. As shown in Fig. 1, our method can recover

the 3D structure in the input image even when it highly dif-

fers from that of the reference configuration. In addition we

can handle large amounts of noise and occlusions.

The key contributions that make this possible are

twofold. First we use a generative model that progressively

increases its complexity and allows a coarse to fine fitting

while 3D-to-2D matches are estimated. Second, we take

advantage of a recursive parameterization of the coronary

tree that introduces dependencies between all the nodes of

the tree, and diffuses the local constraints to the whole

structure. Both the parameterization and the generative

model are then integrated within a Kalman-based optimiza-

tion framework. In the results section we will show that the

overall methodology has significant advantages when com-

pared to state-of-the art approaches.

2. Related Work

Recovering the 3D structure of the coronary tree from

single vascular images involves dealing with many differ-

ent issues. Besides the inherent ambiguity of the monocular

non-rigid reconstruction, the problem is further accentuated

due to the presence of noise in the images and partial oc-

clusions between different branches of the tree. This com-

plexity has been traditionally alleviated by considering the

vascular system as a rigid structure [9, 11, 16] and using

multiple views [26, 29]. To the best of our knowledge, [8]

is the only approach in the medical imaging literature that

considers the non-rigid nature of the problem. They intro-

duce 3D priors and inextensibility constraints into a steepest

descent scheme to solve for the shape. Yet, their optimiza-

tion procedure is only effective under relatively simple de-

formations as those occurring in the liver artery.

On the other hand, our approach has similarities with

the techniques to reconstruct non-rigid 3D surfaces and es-

timate articulated pose from monocular images. Among the

former, it has been shown that 3D shape can be retrieved by

imposing local inextensibility and constraints introduced by

a set of 3D-to-2D correspondences between the input im-

age and a reference shape [5, 14, 17, 19]. In essence we

will also use the same kind of assumptions, although we

will need from additional constraints since in our context

the 3D-to-2D correspondences are unknown and have to be

resolved simultaneously with the shape. Moreover, local

distance constraints are much less restrictive when dealing

with points linked through a tree-like structure than when

dealing with neighboring points on a surface. In addition,

many of these approaches impose strong shape priors based

on previously acquired training data [17, 19] while in our

approach accurate training data is hard to obtain and we

have to rely on very weak shape priors.

Since the coronary tree may be regarded as an articu-

lated structure, one might think in applying the techniques

of articulated pose estimation to our problem [1, 21, 27].

These approaches rely on large amounts of data for learn-

ingamappingfrom2Dimageobservationsto3Dposes, and

have the advantage of not requiring to solve the 2D-to-3D

correspondence problem. Yet, as said above, while obtain-

ing sufficient training data is feasible for applications such

as human pose estimation [2, 23], it becomes prohibitive in

our framework, as the number of X-ray images that may be

captured for each patient is limited.

Recent works suggest introducing similar constraints as

those used for non-rigid shape recovery into the formula-

tion of articulated pose estimation problems [18, 22, 24].

This allows fitting more detailed parametric 3D models [22]

and reducing the dependency of articulated pose estimation

techniques on the training data [18]. However, reducing the

dependency on training data has the drawback of increasing

the sensitivity to artifacts into the input data.

Drawing particular inspiration on these approaches, our

method also combines tools from the techniques for articu-

latedpose estimationand shape recovery. However, inorder

totackleproblemswithmuchlargeramountsofimagenoise

and occlusions, we propose using a generative 3D model

that progressively increases its complexity and adaptability

while establishing correspondences and detecting and re-

jecting outlier points. In addition, we represent the articu-

lated structure using a recursive parametrization that, as we

will show in the results section, yields remarkable improved

results when compared against [18].

3. Algorithm Overview

The focus of this paper is on retrieving the 3D structure

of the coronary tree while establishing 3D-to-2D point cor-

respondences between an input X-ray image and a reference

3D scan. However, the overall algorithm requires additional

tasks as we next detail:

1. Feature extraction: Our input data is an X-ray im-

age and a volumetric 3D Computed Tomography (CT)

scan. In a preprocessing step we segment the vessel re-

gions on both sets of data and extract points of interest.

2. Generative model for the coronary tree: We repre-

sent the 3D feature points as a tree, parameterized

by the joint angles and distances between consecutive

points. Since we cannot explicitly compute deforma-

tion modes from training data, we estimate them by

performing a Probabilistic Principal Component Anal-

ysis (PPCA) over a set of synthetic samples for which

we randomized the values of the joint angles. The flex-

ibility of the model will be controlled by the number of

components of the PPCA and by the magnitude of the

noise used to generate the deformed samples.

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(a) (b)(c) (d)

(e)(f)(g) (h)

Figure 2. Feature extraction. Top: Extraction of the 3D skeleton from the 3D Computed Tomography. (a) Two levels of thresholding.

(b) Vessel segmentation. (c) 3D skeletonization. (d) Feature point orientations. Bottom: 2D Feature extraction from the X-ray images. (e)

Original image. (f) Vesselness segmentation. (g) Extracted feature points (h) Estimated feature orientations.

3. Non-rigid reconstruction and matching:

Kalman-based approach, we iteratively solve for the

3D-to-2D correspondences and progressively fit the

deformation modes onto the 3D points.

Using a

In the following sections we discuss each one of these

constituent pieces.

4. Feature Extraction

We next briefly describe the steps we perform to seg-

ment the vessel regions and extract the 3D and 2D features

from the raw 3D scans and from the X-ray images respec-

tively. Note that this task is by itself an active area of re-

search within medical imaging [4, 7, 20]. Yet, we design

relatively simple solutions, partially based on the popular

vessel enhancement filter proposed in [6], which although

not being free of error, demonstrate the robustness of our

algorithm for 3D non-rigid reconstruction.

4.1. 3D Features

The raw data of the volumetric 3D CT scanner is com-

posed of a stack of grayscale image slices. The CT volume

is initially segmented using the fuzzy connectedness tree al-

gorithm [28], and as shown in Fig. 2a, a number of regions

corresponding to different physiological structures are gen-

erated. Fig. 2b shows the vessel-like formations. The ves-

sel centerlines are then accurately detected based on a local

steepest gradient ascent of the vesselness measure [6]. Fi-

nally, we homogeneously distribute an arbitrary number of

nodes along the skeleton, and enforce connection using a

minimum spanning tree approach [15](Fig. 2c).

In order to perform the 3D-to-2D matching in a subse-

quent step, besides using the 3D position, we describe each

node based on a unitary vector that indicates the local artery

orientation, which is computed from the eigenvectors of the

Hessian matrix at the appropriate scale. Although we could

have used more elaborated descriptors, we found the po-

sition and orientation to be discriminative enough in our

context of 3D heart scans, where despite non-rigid defor-

mations, the global orientation of the heart remains quite

stable throughout the cardiac cycle. In Fig. 2d we depict the

orientation assigned to each node of the tree.

4.2. 2D Features

A reliable and robust detection of the vessel centerlines

in the 2D X-ray images is more challenging than in the 3D

case. Occlusions produced by vessel crossings, reduced lo-

cal contrast in the diaphragm area, discontinuities due to

lack of the contrast-agent (or stenosis), are some of the ar-

tifacts we may find in these images. Even though a Digi-

tal Subtraction Angiography technique can help to reduce

some of these problems, its practical use is limited to se-

quences in which the relative pose of the X-ray equipment

does not change with respect to the patient, which is not our

case. In addition, while in practice the 3D feature extrac-

tion can be performed off-line in a pre-operative stage, 2D

features must be computed in real-time and without manual

intervention. As a consequence, we will only extract points

of interest and will not attempt to solve the connectivity tree

of the vascular structure.

The extraction of points of interest is performed by ap-

plying the multi-scale vesselness filtering proposed in [6].

We then perform a non-maxima suppression on the vessel-

ness map, followed by a thresholding. For each detected

location, a ridge traversal algorithm, similar to the one pro-

posed in [3] is applied, providing the centerlines of arteries.

In order to select a specific number of keypoints and ensure

they are homogeneously distributed, we further apply a k-

means clustering algorithm, were we set k to the number of

desired keypoints. Finally, each of the keypoints is again

described by the vessel orientation at the point, computed

from the eigenvectors of the Hessian matrix. Fig. 2(g,h)

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show the final keypoints and their respective orientations.

5. Non-rigid Reconstruction Algorithm

In this section we discuss the core elements of our ap-

proach. After formalizing the problem, we introduce the re-

cursive model we use to represent the deformable coronary

tree, as its particular representation will play a decisive role

for being robust to high levels of occlusions and allowing

to retrieve large deformations. We then describe the weak

priors we use to build a probabilistic generative model of

the tree, and finally we describe the iterative algorithm we

propose that combines all the previous ingredients to solve

for the shape and assign correspondences.

5.1. Problem formulation

Let Mref= {xref

we have extracted from the 3D reference scan and U =

{u1,...,unf} the nffeature points extracted from the 2D

X-ray image, corresponding to a projection of M, a non-

rigidly deformed version of Mref. Our goal is to retrieve

both as many 3D-to-2D correspondences {xref,u} as pos-

sible, and the 3D configuration of the deformed structure

M = {x1,...,xnm}. Note that since in practice the key-

points are obtained from uniformly sampling the segmented

3D scans and X-ray images, it may not exist a perfect one-

to-one match between the sets Mrefand U. However, in

the synthetic results section we will show that our algorithm

tolerateslargeamounts ofnoiseandocclusions, whichcom-

pensates for all these inaccuracies in the matches.

1

,...,xref

nm} be the nmmodel points

5.2. Recursive 3D Model Parameterization

Since we know the links between the 3D points of M,

we can represent the structure of the deformable model by

the vector:

m = [x⊤

1,ρ⊤

2,...,ρ⊤

nm]⊤

(1)

where x1are the 3D coordinates of the root node, and ρj=

[rij,θij,φij]⊤are the spherical coordinates of the vector

joining the i-th and j-th nodes. Thus, the 3D position of a

node xkmay be recursively written as:

xk= x1+

?

i,j∈Ak

rijcosθijsinφij

rijsinθijsinφij

rijcosφij

where Akcontains all the ancestors of the k-th node. Ob-

serve that using this formulation, when the 3D position of

a node is updated, the location of all its ancestors is also

updated. This is a remarkable novel contribution of our for-

mulation, as it naturally introduces constraints that go be-

yond local neighborhoods. In addition, the propagation of

the error using this recursive parameterization is very well

suitedtodealwithstructureslikethecoronarytree, inwhich

the root node remains almost at constant position while the

terminal nodes are usually highly deformed.

Figure 3. Building weak deformation priors. Since we do not

explicitly use training data, we build a weak prior of the tree struc-

ture by assigning gaussian noise to its nodes (left). Note that since

our parameterization is recursive, we assign larger noise values to

the terminal nodes of the tree. Using this prior, we then produce

multiple random shape samples (right) and fit a PPCA model.

5.3. Introducing Synthetic Deformation Priors

Although we do not explicitly use training data besides

the reference 3D scan, we synthetically define weak priors

on the feasible deformations, which will be used within the

optimization scheme to progressively fit the 3D model to

the input data.

Given the 3D parameterization mrefof the reference

shape, we compute the weak priors by first generat-

ing multiple samples {si} from the normal distribution

N(mref,Σm), where Σmis a 3nm× 3nmdiagonal co-

variance matrix generated by concatenating the covariances

of x1and the vectors {ρj}j=2,...,nm. As shown in Fig. 3

these covariances are set to relatively large values in order

to deal with different types of deformations. Yet, in order to

avoid completely random shapes with no physical meaning

we slightly smooth the resulting sampled shapes. In addi-

tion, the variance associated to the inter-node lengths rij

is set to a very small value, as we assume the vessels do

not stretch. Note that although real vessels may stretch, we

represent the tree by a sufficiently dense set of points that

makes the inextensibility assumption locally correct.

We then learn a low-dimensional deformation model of

the coronary tree, by applying Probabilistic Principal Com-

ponent Analysis [25] over the set of deformed samples {si}.

By doing this, we can approximate the pose parameters of

Eq. 1 as a weighted sum of a mean structure m0and nq

deformation modes Q = [q1,...qnq] :

m = m0+

nq

?

i=1

αiqi= m0+ Qα

(2)

where α = [α1,...,αnq]⊤are unknown modal weights

that define the current structure, and whose covariance is

defined by a nq× nqmatrix Σα.

5.4. Iterative Fitting and 3D-to-2D Matching

Representing the coronary tree by means of the modal

weights of Eq. 2 allows rewriting the problem as that of

estimating the parameters α∗such that the following repro-

jection error is minimized

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InitializationIteration #1Iteration #2Iteration #3 Rigid Registration

Figure 4. Iterative Fitting and 3D-to-2D Matching. We plot the 3D results (top) and 2D results (bottom) of the fitting process at different

iterations and the matches we establish. The left-most figures show the initial prior (red) and the ground truth structure. The next 3 images

show the fitting process of our approach (yellow) for different iterations. The figures on the right show what might be obtained using just

a rigid registration, which is a standard solution to this kind of problems. Obviously the rigid approach yields large errors, specially on the

terminal nodes of the coronary structure.

α∗= argmin

α

nm

?

i=1

?Proj(xi;α) − Match(xi,U)?

(3)

where Match(xi,U) returns the match uj ∈ U for a 3D

point xi, and Proj(xi;α) is the perspective projection of

the 3D point xigiven the modal weights α.

Note that we assume we know the projection matrix of

the imaging system, including both the pose of the patient

with respect to the CT and X-ray equipment, and the intrin-

sic parameters of the detector. In practice, though, there is

always some translation in the position of the patient with

respect to the global reference system. In addition, the heart

deformation during the cardiac cycle and the global heart

shift produced by the breathing introduce additional trans-

lation and small rotation effects. In order to address these

issues, we have considered the translations and small rota-

tions as part of the non-rigid model.

We next turn to the algorithmic steps to minimize Eq. 3.

We achieve this by alternatively solving correspondences

and fitting the 3D model, which we have initialized with the

reference 3D model provided by the 3D scanner.

Establishing 3D-to-2D Correspondences. Let us assume

that as input of an iteration we have α and Σα, the modal

weights and their covariance matrix estimated at the pre-

vious iteration, the set M of 3D model points deformed

according the weights α, and the set U of 2D features ex-

tracted from the X-ray image. We then compute 3D-to-2D

matches with the following steps:

1. We project the 3D model points onto the image, con-

sidering the current configuration of modal weights.

We denote these projections by V = {v1,...,vnm}

where vi= Proj(xi;α).

2. For each vi we establish an uncertainty region sur-

rounding it, by propagating the covariance Σαof the

modal weights to the image plane. This region will be

a Gaussian centered at each viand with covariance:

Σi

v= J(xi)ΣαJ(xi)⊤

where J(xi) is the 2 × nqJacobian of the projection

equationProj(xi;α)withrespecttothemodalweights

α, evaluated at the 3D point xi.

3. Given the set {vi,Σi

points uj∈ U as a standard Optimal Assignment Prob-

lem using the Hungarian algorithm [12]. For this pur-

pose, for each potential match {vi,uj} we introduce

a cost defined as a linear combination of the Maha-

lanobis distance between the two points and their sim-

ilarity in the orientation computed in Section 4:

v} we solve the matching with the

Cij= λ1Mah(vi,uj) + λ2Angle(vi,uj)

where Mah(vi,uj) = (vi− uj)(Σi

and Angle(vi,uj) is the difference in the orientations

between the points of ujand xi, the 3D model point

projected on vi. Note that since the orientation vector

of xiis originally computed in the 3D space, we need

to project it on the image plane to compare it with the

orientation of vi. The terms λ1and λ2are constant

scale factors used to give similar orders of magnitude

toeach ofthecomponents ofthecostfunction. Inprac-

tice, although the Hungarian algorithm is already an

efficient technique for determining the optimal match-

ing, we further reduce its complexity by only consid-

ering those costs Cijwhich are below a certain thresh-

old. This also prevents from fitting the shape to outlier

correspondences or mismatches.

v)−1(vi− uj)⊤,

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Deformation Level [σθ,σφ]Image Noise [σn]Occlusions (%)

RMS Error 3D [voxels]

RMS Error 2D [pix]

Deformation Samples

σθ = σφ= 0.1

Error 3D: 5.8 [vox]

σθ = σφ= 0.3

Error 3D: 14.7 [vox]

σθ = σφ= 0.5

Error 3D: 28.8 [vox]

Figure 5. Synthetic experiments. RMS error distribution for 3D reconstruction (top) and 2D reprojection (bottom) for each experiment.

The whiskers denote min. and max. errors, the box spans from first to third quartile and the inter-box lines show the mean RMS error.

Updating Modal Weights. Given the set {xi,ui}i=1,...,nc

of estimated correspondences we then use Kalman filter

equations to update the modal weights α and its covari-

ance matrix Σα. Since we simultaneously use all of the

estimated matches, we define an extended 3ncvector of 3D

points ˆ x = [x⊤

locations ˆ u = [u⊤

Jacobian matrixˆJ = [J(x1)⊤,...,J(xnc)⊤]⊤. We then

update α and Σαas

α+

=

α + K(ˆ u − Proj(ˆ x;α))

(I − KˆJ)Σα

1,...,x⊤

1,...,u⊤

nc]⊤, an extended 2ncvector of 2D

nc]⊤, and an extended 2nc× nq

Σα+

=

where K is the Kalman gain and I the identity matrix.

Iterating and Increasing Flexibility. The matching and

modal weight updating processes are iteratively repeated

until the convergence of Eq. 3.

In order to adapt the reference model to highly deformed

structures, such as those shown in Fig. 1, the number of

modes we use is increased at each iteration. This allows to

progressively fit the structure, starting from the most rigid

parts up to the more deformed ones. In addition, using

more rigid structures at early stages yields robustness to

mismatches, preventing to adapt the model towards outlier

2D features. In practice, for a coronary tree with nmpoints,

and thus with a maximum number of 3nmmodes, at itera-

tion number niterwe used nq= 3nm/(10 − niter) modes.

We found the number of iterations to converge to be always

≤ 5. For instance, Fig. 4 shows an example in which con-

vergence was achieved after 3 iterations.

6. Results

We now present the results on both synthetic and real

data. In the synthetic results we compare our approach (de-

noted ArtDeform) to [18]1, which is a representative exam-

ple of the state-of-the-art in articulated pose recovery that

has been shown successful in recovering human and hand

pose. We also compare it to the solution obtained from

a 3D-to-3D rigid transformation between the reference 3D

shape and the ground truth shape associated to the input im-

age, which would represent the best solution that a rigid

2D-to-3D registration algorithm could obtain. For this pur-

pose, we use a standard technique for absolute orientation

estimation [10]. Note that both in [18] and [10], the corre-

spondencesareassumedtobeknown, whileinourapproach

we simultaneously estimate them with the shape.

6.1. Synthetic Experiments

In this section we extensively evaluate the performance

of each algorithm against noise in the correspondences, dif-

ferent levels of deformation and partial occlusions.

generated random shapes of 50 nodes within a volume of

300×300×300 voxels, such as the tree-like structure shown

in Fig. 1, and simulated the deformations undergone in

the coronary tree by applying increasing levels of noise

{σθ,σφ} to the joint angles. We then projected each 3D

shape on a 512×512 image and added gaussian noise of

standard deviation σnto the 2D correspondences. In addi-

tion, a percentage poof the projected points was randomly

removed in order to simulate partial occlusions. Given the

original reference shape and the set of projected points of

the deformed shape, we then performed the reconstruction

with each of the algorithms.

Three different types of experiments were performed.

We initially evaluated the amount of deformation each al-

gorithm was able to recover by sweeping the variances of

We

1We thank Dr. Mathieu Salzmann for kindly testing the data of our

experiments on the algorithm proposed in [18].

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Left Coronary ArteryRight Coronary Artery

2D Reprojection Error

Rigid Reg.ArtDeform

5

10

15

20

25

30

Reprojection Error [pix]

Figure 6. Reconstruction results on real vessel structures. Left: Several samples of the LCA and RCA reconstructions. Upper row:

initial model (red) and the final non-rigid tree (yellow) overlaid on the input X-ray images. We also plot the results of the 2D segmentation

(blue), which, as can be seen, contains certain amounts of false positives. Bottom row: Initial and retrieved 3D shapes. Note that despite

the segmentation suffers from occlusions and false positive regions our approach yields an accurate solution. Right: 2D reprojection error,

which is used to quantitatively assess the quality of the real results, because the 3D ground truth is not available.

the joint angles within the range [σθ,σφ] ∈ [0 − 0.5] rad,

and randomly setting σn ∈ [1 − 3] and po ∈ [5 − 20]%.

To give significance to the levels of deformation and recon-

struction errors, Fig. 5-right depicts different deformations

of the model corresponding to specific values of {σθ,σφ}.

In a second experiment we analyzed the robustness to im-

age noise by synthetically introducing random noise of

σn∈ [0−10], and setting {σθ,σφ} ∈ [0.15−0.3] and po∈

[5 − 20]%. Finally, we evaluated the effect of occlusions in

an interval po∈ [0 − 50]%, with {σθ,σφ} ∈ [0.15 − 0.3]

and σn∈ [1 − 3].

For each set of parameters we performed 50 trials. The

graphs on Fig. 5 depict the mean 2D reprojection error, ex-

pressed in pixels, and the mean 3D reconstruction error, ex-

pressed in voxels. Observe that our algorithm consistently

outperforms [18] in all experiments. This difference is spe-

cially remarkable when dealing with occlusions, for which

we obtain reconstruction errors below 5 voxels even when

a 50% of the model is occluded. Observe in the top right

graph of Fig. 5 that these amounts of error correspond to

very good approximations. It is fair to mention, though,

that [18] is a general algorithm easily adaptable to different

domains, fromarticulatedstructurestodeformablesurfaces,

while our algorithm is specifically designed to handle tree-

like and articulated structures. In Fig. 5 we also plot the

results that would be recovered using a rigid registration

and, as expected, the errors are significantly larger. Note

that the error values for this case are scaled by a factor 1/2

for displaying purposes.

6.2. Real Data

We also evaluated our approach on real CT data and X-

ray data collected during ordinary pre-operative diagnosis

and percutaneous intervention. We collected CT data of 7

patients, using a Philips Brilliance iCT, at the 75% of the

heart cycle, with slice thickness 0.67 or 0.8mm, and pixel

resolution between 0.38×0.38 and 0.45×0.45 mm. We col-

lected a total of 17 X-ray sequences, 10 of Left Coronary

Artery (LCA) and 7 of Right Coronary Artery (RCA), us-

ing a single plane Philips INTEGRIS Allura Flat Detec-

tor. Image and camera calibration was performed using the

catheter width and the geometrical information on the C-

Arm position. For each sequence, one image in which the

contrast liquid was sufficiently visible was selected.

For each of pair CT scan/X-ray image, we then extracted

3D and 2D features as described in Sect. 4. In all experi-

ments, we represented the segmented CT volume as a tree

with 75-nodes, and extracted 500 feature points from each

X-ray image. Starting with the initial tree of the CT scan,

we then iteratively fit the model and established 3D-to-2D

matches. In all experiments we achieved convergence in

less than 5 iterations, taking about 8 seconds per iteration.

Note that this represents in fact a significantly faster algo-

rithm compared to competing methods. For instance, [8]

reports computation times of about 7 minutes per image.

The 2D registration and 3D reconstruction results of a

few sample experiments are depicted in Fig. 6. Observe that

even when the segmentation contains false positive regions,

or does not detect some branches of the coronary tree our

approach is able to provide an accurate solution.

Sincethe3Dgroundtruthdoesnotexistforthedeformed

artery tree, we quantitatively evaluated the performance of

our algorithm based on the 2D reprojection error with re-

spect to ground truth centerlines manually annotated by an

expert physician. This error is shown in the bar plot of

Fig. 6-right, which summarizes the results for all the 17

experiments. Observe that our non-rigid approach clearly

outperforms a method that rigidly registers the original CT

Page 8

scan. In fact, considering an average calibrated pixel res-

olution of 0.22 mm, the median error of our method is of

about 1.9 mm. This compares very well with the 1 mm er-

ror reported in [11], especially considering that they restrict

their evaluation to X-ray images acquired at systole and di-

astole time instants, where the coronary tree deformation is

minimal compared to other cardiac cycle instants we con-

template.

7. Conclusion

We have presented a novel approach to estimate the 3D

structure of the coronary tree from single X-ray images. In

order to handle the large amounts of deformation, noise

and occlusions present in this kind of images we have in-

troduced a generative model based on a recursive parame-

terization that progressively increases its complexity. We

have integrated this model within a Kalman filter frame-

work which, making use of very weak priors on the struc-

ture, iteratively guides the matching process while recover-

ing coarse-to-fine levels of deformation.

The formulation we propose is fairly general, and al-

lows integrating additional features. As part of future work,

we consider exploiting motion coherence for tracking heart

beat sequences in real time. Moreover, we believe that the

inextensibility constraints between neighboring nodes may

be relaxed, thus allowing to handle stretchable structures.

Acknowledgments

This work has been partially funded by the project La

Marat´ o de TV3 082131; by the Spanish Ministry of Science

and Innovation under projects DPI2008-06022, DPI2010-

17112, Consolider Ingenio 2010 CSD2007-00018; and by

the EU project GARNICS FP7-247947. The work of C.

Gatta is supported by a Beatriu de Pinos Fellowship.

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Monocular

Coronary artery segmentation