IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 12, NO. 5, SEPTEMBER/OCTOBER 2006
Lines of Curvature for Polyp Detection in Virtual Colonoscopy
Lingxiao Zhao, Charl P. Botha, Member, IEEE, Javier O. Bescos, Roel Truyen, Frans M. Vos, and Frits H. Post
Abstract— Computer-aided diagnosis (CAD) is a helpful addition to laborious visual inspection for preselection of suspected colonic
polyps in virtual colonoscopy. Most of the previous work on automatic polyp detection makes use of indicators based on the scalar
curvature of the colon wall and can result in many false-positive detections. Our work tries to reduce the number of false-positive
detections in the preselection of polyp candidates.
Polyp surface shape can be characterized and visualized using lines of curvature. In this paper, we describe techniques for generating
and rendering lines of curvature on surfaces and we show that these lines can be used as part of a polyp detection approach. We have
adapted existing approaches on explicit triangular surface meshes, and developed a new algorithm on implicit surfaces embedded in
3D volume data. The visualization of shaded colonic surfaces can be enhanced by rendering the derived lines of curvature on these
surfaces.
Features strongly correlated with true-positive detections were calculated on lines of curvature and used for the polyp candidate
selection. We studied the performance of these features on 5 data sets that included 331 pre-detected candidates, of which 50 sites
were true polyps. The winding angle had a significant discriminating power for true-positive detections, which was demonstrated by
a Wilcoxon rank sum test with p < 0.001. The median winding angle and inter-quartile range (IQR) for true polyps were 7.817 and
6.770−9.288 compared to 2.954 and 1.995−3.749 for false-positive detections.
Index Terms—Medical visualization, virtual colonoscopy, polyp detection, line of curvature, implicit surface.
F
1 INTRODUCTION
Colonic polyps are an important precursor of colon cancer, which is
among the leading causes of cancer deaths in the western world [24].
A polyp is a benign growth of the colon lining. It typically presents as
a sphere protruding from the colon wall. Early detection and removal
of polyps significantly decrease the incidence of colon cancer. For
this purpose, virtual colonoscopy has been developed as a procedure
to inspect the interior wall of the human colon by using CT or MRI-
scans.
Virtual colonoscopy is a minimally-invasive technique, which
causes much less discomfort to the patient than traditional optical
colonoscopy [3, 22]. The CT or MRI-scans are processed by iso-
surface extraction or by direct volume rendering (DVR) to allow for
visual inspection by a radiologist. However, a thorough, visual exam-
ination of the complete colon wall is rather time consuming, which
makes the method unattractive for large-scale screening. Therefore,
computer-aided techniques have been proposed to pre-detect and high-
light colonic polyps in order to reduce the examination time and cost,
especially in mass screening of low-incidence populations [26].
A considerable amount of work has been done in the field of auto-
matic polyp detection; many schemes make use of curvature. Curva-
ture is an important quantity from differential geometry [6], which is
widely used in computer vision and visualization applications to char-
acterize the shape of 3D surfaces. It can be represented by a scalar
• Lingxiao Zhao is with Data Visualization Group, Delft University of
Technology, E-mail: zlx@graphics.tudelft.nl.
• Charl P. Botha is with Data Visualization Group, Delft University of
Technology, E-mail: c.p.botha@tudelft.nl.
• Javier O. Bescos is with Philips Medical Systems Nederland BV, Best,
E-mail: javier.olivan.bescos@philips.com.
• Roel Truyen is with Philips Medical Systems Nederland BV, Best, E-mail:
roel.truyen@philips.com.
• Frans M. Vos is with Quantitative Imaging Group, Delft University of
Technology and Dept. of Radiology, Academic Medical Centre,
Amsterdam, E-mail: f.m.vos@tudelft.nl.
• Frits H. Post is with Data Visualization Group, Delft University of
Technology, E-mail: frits.post@ewi.tudelft.nl.
Manuscript received 31 March 2006; accepted 1 August 2006; posted online 6
November 2006.
For information on obtaining reprints of this article, please send e-mail to:
tvcg@computer.org.
(e.g. mean or Gaussian curvature), and by two vectors, indicating the
directions of principal curvatures at a given point.
Indicators based on scalar curvature values have been frequently
used in previous work for computer-aided diagnosis (CAD) of colonic
polyps. Yoshida et al. [33] made use of 3D geometric features called
the volumetric shape index and curvedness to develop a CAD scheme
for polyp detection. Na¨ppi and Yoshida [18] proposed to use feature-
guided analysis for achieving high sensitivity and a low false positive
rate in their CAD scheme. Huang et al. [8] developed a two-stage
curvature estimation method on triangular surface meshes and per-
formed a filtering based on the sphericity index to identify potential
polyps. Van Wijk et al. [31] introduced the technique of normalized
convolution to measure curvature features in 3D volume data for au-
tomatic polyp detection. Accurate and noise-insensitive curvature cal-
culation is essential for any such scheme. Representations based on
point-sampling will in general be more sensitive to noise, whereas ag-
gregation within a region of interest will enhance the robustness at the
cost of some sensitivity. Using scalar curvature by itself for polyp
detection can result in a large number of false-positive detections.
The potential of principal curvature direction fields has not yet been
explored in current polyp detection techniques. On a surface, the two
principal curvature directions define two orthogonal vector fields, and
these can be visualized by lines of curvature, which are lines every-
where tangent to one of these vector fields. We will call these curves
streamlines of curvature. They have also been used for surface shape
analysis in engineering design [4]. However, to the best of our knowl-
edge, no attempt has been made to apply streamlines of curvature for
colonic polyp characterization in medical visualization. We hypothe-
size that the patterns in streamlines of curvature are a good indicator
of specific features to detect polyps, both visually and automatically.
In our work, the proposed CAD process proceeds in three steps:
pre-detection of polyp candidates, candidate selection, and finally en-
hanced visualization. Polyp candidates are pre-detected using an ex-
isting polyp detection scheme. These polyp candidates include many
false-positive detections. The main contribution of our work is to pro-
pose a new additional polyp candidate selection approach based on
a use of streamlines of curvature, which helps to reduce the number
of false-positive detections. We will present methods to generate pat-
terns of streamlines of curvature on the colon wall. We have improved
existing algorithms for explicit triangle mesh surfaces, and developed
a new method for implicit surfaces embedded in a 3D volume data
representation. The basis of our work is a robust technique for the
computation of principal curvature directions so that the streamlines
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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 12, NO. 5, SEPTEMBER/OCTOBER 2006
are as smooth as the surface itself. Subsequently, we ensure that a
streamline accurately traces the true principal curvature directions of
the surface. Note that constraining the streamlines to the surface will
require extra operations compared to unconstrained streamline gener-
ation in a 3D vector field. For a good view of the local surface shape, a
set of streamlines with controlled spacing must be generated, to reveal
the important shape features, and to cover all important details. There-
fore, the seeding and spacing of the streamlines is governed by local
surface curvature. We extract features strongly correlated with true-
positive detections from the generated streamlines. These new features
provide a new basis for an automatic polyp detection algorithm. The
streamlines of curvature support the perception of surface morphology
in addition to the traditional shading. The shape is summarized using
the streamlines that have specific patterns at polyp positions.
The rest of the paper is organized as follows. A brief survey of rel-
evant research is given in section 2. In section 3, a relation between
principal curvature directions and polyp surface characteristics is es-
tablished. In section 4, our approach to generate surface-constrained
streamlines of curvature and the seeding strategy is described. We start
with an improvement of the existing algorithm on triangle meshes and
then we describe a new scheme based on implicit iso-surfaces. The
performance and applicability of these two methods are compared. In
section 5, new features are defined on streamlines of curvature and
used to select true-positive polyp detections. An experimental study
of our polyp selection scheme and enhanced visualization of the colon
wall are given in section 6. At the end of the paper, we draw some
conclusions and suggest items for future work.
2 RELATED RESEARCH
Our work is related to many other research fields. The main purpose is
to generate and render streamlines of curvature on the colonic surface
and use them for improving automatic pre-detection of polyps.
Drawing from computer vision and visualization, surface curvature
estimation is the basis of our curvature streamline generation. Trian-
gle meshes are problematic for surface curvature estimation. We often
do not have an analytical function describing the continuous surface
approximated by the mesh. Concepts in differential geometry are no
longer applicable in this discrete case. Approaches have been devel-
oped to approximate original surface curvature [27]. Mesh quality
significantly influences the estimation results. Current triangle mesh
extraction techniques often introduce strong artifacts. Most estimation
methods suffer from mesh irregularities and data noise. Computing
curvature directly from 3D volume data is based on derivatives of 3D
images. These derivatives are usually estimated using kernel methods
[30]. Parameters of the kernel must be properly set to achieve accurate
and reliable results [5].
For triangle meshes, some methods [12] fit an analytic surface to a
neighborhood of a mesh vertex and calculate curvature from the fitted
function. Other methods estimate curvature for a surface region us-
ing total curvature defined in [19]. Lin and Perry [13] used the angle
deficit to measure the Gaussian curvature. Meyer et al. [15] defined
discrete differential geometry operators on triangle meshes. Taubin’s
method [28] is unique among other methods. It calculated curvatures
at mesh vertices using the curvature tensor. For implicit iso-surfaces in
3D volume data, Thirion and Gourdon’s method [29] was based on the
implicit surface representations defined in differential geometry. The
Hessian matrix of 3D volume data was introduced as another tool in
[16].
Lines are often used to express shape features. Reflection lines on
surfaces give designers crucial feedback on the quality of their product
[21]. Interrante et al. [9] made use of line textures to enhance the
perception of surface shape. In [1], streamlines in 2D optical flow
were used to examine interesting features for polyp detection.
Streamlines of curvature can be traced on the surface and should be
well distributed for efficient shape characterization. Nagy et al. [17]
presented an interactive technique based on streamlines of curvature
in 3D volume. Girshick et al. [7] introduced a simple scheme to trace
streamlines of curvature on triangle meshes. In their work, streamlines
were evenly spaced by using a special seeding method [11]. Mebarki
et al. [14] proposed a fast farthest point seeding algorithm that can
trace longer streamlines in 2D vector fields. Other work made use
of local surface features to control the streamline density. Verma et
al. [32] discussed characteristics of a good seeding strategy and pre-
sented a method using local flow topology. Alliez et al. [2] applied
streamlines of curvature for surface remeshing. They traced stream-
lines in the parameter space of a surface after the original surface was
flattened. Curvatures were employed to optimize the streamline den-
sity in order to preserve surface information after remeshing. However
their work did not apply to implicit iso-surfaces.
3 POLYP CHARACTERIZATION USING LINES OF CURVATURE
Fig. 1. Distinctive patterns on polyp surfaces. Left: Circular pattern
in maximum curvature directions. Right: Focusing pattern in minimum
curvature directions.
The idea of using streamlines of curvature for polyp candidate se-
lection is inspired by visualizing the two vector fields of surface princi-
pal curvature directions. Our work starts with curvature estimation on
triangular surface meshes, both for scalar curvature and for principal
curvature directions.
A robust curvature estimation approach, named the Normal Vector
Voting (NVV) method [20], is applied on triangular colonic surfaces
extracted from real CT scans using Marching Cubes. We will discuss
the NVV method later in Section 4.1. Visual inspection of principal
curvature directions shows patterns on polyp surfaces that may dis-
criminate polyps from healthy tissue (Figure 1). In the case that mesh
surface normals point towards the interior of the colon, the maximum
curvature directions around the polyp present a circular pattern while
the minimum curvature directions present a focusing pattern. This re-
lation can be explained with a generalization of the polyp surface.
Fig. 2. A polyp surface has two parts: ideally, the cap is spherical and
convex, the neck is a closed-ring and anticlastic surface.
A polyp surface consists of two parts, the top part and the bottom
part (Figure 2). We call the top part the polyp cap and the bottom part
the polyp neck. The polyp cap is the most protruding area on a polyp
surface. It is ideally represented as a spherical or ellipsoidal surface.
The polyp neck is the transition area from the background to the polyp
cap. It is typically a closed-ring and anticlastic surface, i.e. all points
on the polyp neck are hyperbolic. Considering the local geometric
properties, surface principal curvature directions on the polyp cap do
not always show characteristic patterns. Sometimes, a polyp can not
be easily identified only according to its cap. Most existing polyp
detection methods use indicators based on scalar curvatures, e.g. the
volumetric shape index [33] and the sphericity index [8]. Such indi-
cators only work on the polyp cap. Therefore, they sometimes can
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ZHAO et al.: LINES OF CURVATURE FOR POLYP DETECTION IN VIRTUAL COLONOSCOPY
introduce many false-positive detections. The polyp neck is a distinc-
tive part of the polyp. Every polyp has a neck as long as it protrudes
from the colon wall. Around such an anticlastic area, principal cur-
vature directions present particular patterns of polyps, i.e. circular or
focusing patterns.
Streamlines of curvature can be used to visualize principal curva-
ture directions on the colonic surface. Therefore, specific patterns in
streamlines of curvature can help to discriminate polyps from back-
ground. The circular pattern in maximum curvature directions around
the polyp neck suggests potential. Since the polyp neck is typically
a closed-ring area, approximately closed streamlines are expected to
locally represent the polyp narrow part or neck. For this reason, we
propose to supplement existing polyp detection methods with an anal-
ysis on the polyp neck.
4 GENERATING AND RENDERING LINES OF CURVATURE ON
COLONIC SURFACES
Streamlines of curvature support polyp candidate selection and en-
hance polyp visualization when they are superimposed on the colon
wall. This section describes techniques to generate these curves on
explicit triangle meshes as well as on implicit iso-surfaces.
4.1 Lines of Curvature on Explicit Triangle Mesh Surfaces
The first step to generate streamlines of curvature on triangle meshes
is curvature estimation. Smooth and regular principal curvature direc-
tions are desired to correspond with surface shape. The NVV method
developed by Page et al. [20] is a robust curvature estimation approach
on general triangle meshes even with low quality. It uses the geodesic
neighborhood as an improvement of other types of neighborhoods, e.g.
the k-ring (k = 1,2,3, . . .) neighborhood. It is defined to be a surface
region bounded by a user-specified geodesic distance from the central
mesh vertex. A k-geodesic neighborhood not only enlarges the com-
putation area but also supports balanced directional sampling around
a mesh vertex. This is very useful to reduce the effects of mesh irreg-
ularity and noise.
The NVV method provides robust curvature estimation on trian-
gle meshes, for both scalar curvature values and principal curvature
directions. We extended and improved the streamline tracing tech-
nique described in [7] on triangular colonic surfaces, by integrating
the NVV method and a simplified computation. This algorithm traces
the streamline on the mesh by projecting each integration onto the
local mesh triangle. When the streamline reaches a mesh edge, the in-
tersection point with that edge is added and the local mesh triangle is
updated. We found calculating intersection points of mesh-constrained
streamlines and edges to be complicated by the limited floating point
precision. The inaccuracy caused by this accumulates as an error dur-
ing streamline integration. To alleviate this problem and simplify the
computation of intersection points, we transform the problem to 2D.
We define a local 2D coordinate system for each triangle plane. A
triangle vertex is set to be the origin. One axis coincides with a trian-
gle edge. Intersections with edges can be easily computed in this 2D
coordinate system and then transformed to 3D.
We characterize the local surface shape using collections of stream-
lines. Therefore it is important that streamlines of curvature are dis-
tributed regularly over the triangle mesh using a seeding strategy. The
evenly distributed streamlines in [7] may lose details of surface shape.
An adaptive and efficient way was presented in [2]. Although the ap-
plication is different, curvature controlled seeding contributes much to
polyp surface characterization. Streamlines of curvature on a colonic
surface mesh are shown in Figure 3. In this case, the appearance and
accuracy of streamlines are significantly affected by the mesh quality
and estimated principal curvature directions.
4.2 Lines of Curvature on Implicit Iso-surfaces
Implicit surfaces are widely used in Virtual Colonoscopy. Compared
to an explicit triangle mesh, an implicit surface provides a smoother
and clearer visualization of the colon wall.
An implicit surface is embedded in a 3D volume as an iso-surface.
It is usually visualized using volume ray casting techniques. In this
Fig. 3. Streamlines of maximum curvature are generated and rendered
on the triangular colonic surface.
section, we describe a new approach to generate streamlines of cur-
vature on implicit iso-surfaces. We first discuss iso-surface curvature
estimation. Then we describe our curvature streamline tracing on im-
plicit iso-surfaces. We have adapted the strategy of curvature con-
trolled seeding and spacing for this case.
4.2.1 Curvature Estimation on Iso-surfaces
We use the method proposed by Van Vliet and Verbeek [30] to estimate
principal curvatures on an iso-surface in 3D volume data. There are
three steps in this algorithm.
First, we calculate the gradient vector g and the Hessian matrix H
at a position P on the iso-surface in 3D volume based on the following
definitions:
g =
( fx, fy, fz
)
H =
( fxx fxy fxz
fxy fyy fyz
fxz fyz fzz
)
Entries in these definitions are partial derivatives of the 3D image:
fx = ∂ f∂x and fxx = ∂
2 f
∂x2
These derivatives can be estimated in a 3D volume by convolution with
the derivative of a Gaussian function with a specific kernel width σ .
Second, the Hessian matrix H is rotated to align one axis with the
local gradient direction g. The rotated Hessian matrix can be written
as:
Hr =




fgg · · · · · ·
.
.
. fuu fuv
.
.
. fuv fvv




=
( fgg · · ·
.
.
. Ht
)
where fgg is the second order derivative along the gradient direction,
u and v are the other two axes in the local coordinate system and Ht is
a 2D Hessian matrix in the plane uv.
Finally we perform eigen analysis on the 2D Ht. Local principal
curvature values at P can be computed based on the local gradient
magnitude and two eigenvalues λ1 and λ2:
k1 = −λ1‖g‖ and k2 =
−λ2
‖g‖
The two eigenvectors correspond to principal directions in the tangent
plane uv at P. They must be transformed to the original 3D coordinate
system.
Since most 3D CT images involve noise, we must choose a proper
Gaussian convolution kernel width σ . In our implementation, σ is
set to be 2.0 millimeters and kernel size is 11 voxels with regard to
covering noise and computational efficiency.
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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 12, NO. 5, SEPTEMBER/OCTOBER 2006
4.2.2 Tracing Lines of Curvature on Implicit Iso-surfaces
Tracing a streamline of curvature on an implicit iso-surface is differ-
ent from a triangle mesh. The position of the surface is not explicitly
known. The work in [7, 17] also included approaches tracing stream-
lines of curvature in 3D volume. But such streamlines are not con-
strained to iso-surfaces. In our approach, we restrict streamline points
to the iso-surface in an implicit way when tracing through the 3D vol-
ume.
I
i1
i
i2 d
g D
Fig. 4. Left: Streamline tracing on iso-surfaces. Right: Projection of
initially integrated point on iso-surfaces.
A streamline is traced using stepwise integration from a seed point
in forward and reverse directions. For each integration step, first we
calculate principal curvature directions at the current streamline point
on the iso-surface using the method in Section 4.2.1. Currently we use
linear interpolation to estimate the gradient, the Hessian and the data
value at an arbitrary position in the 3D volume. Higher order inter-
polation, e.g. cubic interpolation, may give a better result. Then the
streamline is propagated one step further following the local principal
curvature direction. This is a first order integration and higher order
schemes, e.g. second and fourth order Runge-Kutta methods, are other
improvements. Higher order interpolation will work better only with
higher order integration. Since the principal curvature directions are
tangents to the iso-surface at the current point, the next point initially
integrated in first order generally is not on the iso-surface (See Figure
4). Therefore, we project it back onto the iso-surface. First, the gra-
dient vector g and data value i are calculated at the initial point P. If
i is smaller than the isovalue I of the implicit iso-surface, the projec-
tion direction ~D is g. Otherwise, ~D is −g. Then we iteratively move P
along ~D for a step length d and compute i at P after each step. i1 and i2
are defined to be two consecutive values of i. If one of them is smaller
than I and the other is larger, we stop this procedure and compute P′
as the projection of P on the implicit iso-surface using:
P′ = P− ( i1−Ii1−i2 ·d) ·~D
Streamline tracing and projection on implicit iso-surfaces are ex-
plained in Figure 4. A reasonable value for the projection step length
d is 120 ×VoxelLongestDiagonal, where VoxelLongestDiagonal is the
longest diagonal of a volume voxel. If we use a very small step length
in first order integration or use the fourth order Runge-Kutta method
to trace a streamline in 3D volume, this projection technique may not
be necessary.
L
Fig. 5. Use curvatures to adapt streamline integration step length on
iso-surfaces.
On a highly curved surface area, a long streamline integration step
may miss important surface information. This can be improved by us-
ing the surface curvature to adapt the step length. We developed the
approach shown in Figure 5. The osculating circle is used to approx-
imate the relative local iso-surface normal curve. When tracing in a
principal curvature direction, the integration step length L is depen-
dent on the corresponding curvature value k:
L =
√
εL
(
2
|k| + εL
)
where εL is specified by the user to bound the distance how far the
point is allowed to move away from the iso-surface. Our implementa-
tion sets εL = 110 ×VoxelLongestDiagonal. We also specify a longest
step length Lmax = 0.3mm when tracing on a flat surface area.
4.2.3 Curvature Controlled Seeding on Implicit Iso-surfaces
We modified the curvature-adapted seeding strategy in [2] to distribute
streamlines on implicit iso-surfaces. The basic idea of using curvatures
to control streamline density on surfaces is to use minimum curvature
value to determine the spacing distance between streamlines following
maximum curvature direction, and vice versa. We follow a procedure
similar to the evenly spaced seeding [11]. We first start a streamline
from a chosen point on the surface. For each integration step of trac-
ing, a pair of new seeds placed perpendicularly to the integration seg-
ment at the local curvature controlled spacing distance (Figure 6) is put
into a seed queue. Then we iteratively pick another seed in the seed
queue and start another new streamline. Except for the first stream-
line, every time a new streamline is generated, we must remove invalid
seeds from the queue before new valid seeds are added. This seeding
procedure stops when there is no valid seed. A seed is valid if the
distance between it and any streamline point is not less than its local
curvature controlled spacing distance. A streamline stops when it is
too close to other streamlines or reaches a certain maximum number
of integrations.
T
g
V2dsV1 S
Fig. 6. A pair of seeds is initially placed perpendicularly to each stream-
line segment. Then they are projected onto the iso-surface.
We developed a method to place the seed on the implicit iso-surface.
For each streamline integration (Figure 6), we initially place a pair
of seeds perpendicular to the current streamline segment on the local
tangent plane. At every streamline point S, we construct a normal
plane containing the streamline integration direction T and the local
gradient vector g. We then compute two vectors V1 and V2, orthogonal
to and pointing outwards from the normal plane:
V1 = T ×G and V2 = G×T
The next problem is to decide the spacing distance ds to place the ini-
tial seeds along V1 and V2. In the case shown in Figure 7, the distance
from the initial seed point to the iso-surface is bounded by ε1. Then the
spacing distance ds will be specified based on the controlling curvature
k:
ds =
√
ε1
(
2
|k| + ε1
)
These two initial seeds generally are not exactly on the iso-surface.
Therefore they are projected using the projection technique described
in Section 4.2.2.
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ZHAO et al.: LINES OF CURVATURE FOR POLYP DETECTION IN VIRTUAL COLONOSCOPY
12
Fig. 7. Use curvatures to control streamline spacing distance and seed
validating distance.
Invalid seeds in the potential seeds queue are removed after a new
streamline is traced. To validate a seed, we specify another error
boundary ε2. In Figure 7, an osculating circle approximates the in-
tersection arc of the iso-surface between the local streamline point S
and a seed on the iso-surface. The optimal spacing distance between
these two points is the arc length. The Euclidean distance is used as
an approximation to simplify the computation. Its path is offset from
the circle arc by a small distance. This distance is bounded by ε2.
Thus the local curvature controlled spacing distance is dependent on
the controlling curvature k:
dv = 2×
√
ε2
(
2
|k| − ε2
)
dv can also be used to validate when to stop the tracing of a streamline.
As ε2 can be computed based on ε1, we only need to specify one error
boundary ε1:
ε2 = 1|k|
(
1−
√
2+|k|×ε1
2×(1+|k|×ε1)
)
In our experiments, ε1 = 110 ×VoxelLongestDiagonal yielded the best
result.
Fig. 8. Streamlines of curvature on the implicit iso-surface embedded
in an artificial 3D volume. Left: A synthetic colon model with many
polyp-like bumps. Central: streamlines of maximum curvature. Right:
streamlines of minimum curvature.
When tracing in one principal curvature direction, the controlling
curvature k corresponds to the other principal curvature direction. Fig-
ure 8 shows the results of our algorithm on an artificial 3D volume
data. On both triangle meshes and implicit iso-surfaces, a streamline
of curvature is rendered as a polyline. In order to make a streamline
more visible, a thin tube is rendered around it.
4.3 Comparison
We compared our two techniques described to generate streamlines
of curvature on triangle meshes and implicit iso-surfaces. We only fo-
cused on the streamline tracing and spacing. Surface curvature estima-
tion was performed as part of the preprocessing stage and we assumed
comparable and robust curvature estimations in both cases. Linear in-
terpolation and first order integration were used. We tested on both
synthetic data sets and real data sets. Our workstation had a 2.60GHz
Pentium 4 CPU with 1GB of memory. The graphics card was NVIDIA
GeForce4 MX440 with 64MB.
We first recorded computation time in each implementation. Nine
synthetic data sets and five real data sets were used. Streamlines were
generated for each detection patch with a radius of 16 millimeters.
Table 1. Comparison of Computation Time: Streamlines are generated
per detection patch.
triangle meshes Iso-surfaces
Average Number of Streamlines 43.3 41.2
Average Number of Points 1781 1072
Average Computation Time (sec.) 6.60 3.78
Spacing distance between streamlines was less than 2 millimeters. Ta-
ble 1 gives a first result for both techniques. It indicates that our tech-
nique applied to implicit iso-surfaces offers a faster computation for
streamline tracing and spacing on a certain part of a surface. The main
reason is that mesh-constrained streamlines have more integrations in-
cluding many intersections with mesh edges.
Fig. 9. Comparison of curvature streamline appearance on the trian-
gle mesh and the iso-surface of a medical 3D volume data: Left two
show streamlines of minimum curvature, right two show streamlines of
maximum curvature.
Another comparison criterion was the appearance of the stream-
lines. Since we want to use streamlines of curvature as a tool to char-
acterize specific features of polyps, the shape of the streamline is most
important. In Figure 9, streamlines of curvature are generated on a tri-
angle mesh and an iso-surface in a real medical data. This 3D volume
data is 512× 512× 255. Judged by visual inspection, our technique
for implicit iso-surfaces gives better results. Streamlines in this case
appear to be smoother, longer and more characteristic.
Both triangle meshes and implicit iso-surfaces offer relative advan-
tages in different visualization applications. These two different tech-
niques should be employed on corresponding surface representations.
If both of them are applicable, the technique for implicit surfaces is
preferred. It favors our improved CAD process in virtual colonoscopy.
5 FEATURE CALCULATION ON LINES OF CURVATURE
Streamlines of curvature with specific patterns, i.e. (almost) closed
streamlines, are inspected around the polyp neck. The polyp neck
can be identified by detecting such characteristic curves on the colonic
surface. To do this, we generate streamlines of curvature for each pre-
detected polyp candidate area and select streamlines on each polyp
neck. Then we calculate features on the selected streamlines to detect
specific patterns.
5.1 Selecting Lines of Curvature on the Polyp Neck
We generate collections of streamlines on the colonic surface. Only
those streamlines presenting distinctive patterns characteristic of the
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polyp neck are used for the diagnosis, others may even confuse the
result.
We present a method to automatically select important streamlines.
In our case, we are most interested in streamlines on the polyp neck
area. We combined the use of curvature magnitudes with streamline
selection. Considering the fact that the polyp neck area is an anticlas-
tic surface (Section 3), the two principal curvature values everywhere
on such a surface have opposite signs. In other words a streamline
generated around the polyp neck should have many hyperbolic points.
We define a quantity, named hyperbolic percentage HP, as a feature to
select streamlines on the polyp neck:
HP = nhN ×100%
where nh is the number of hyperbolic points and N is the total num-
ber of points on a streamline. A hyperbolic point on the surface has
principal curvature values with opposite signs.
The HP is computed for each streamline generated. Then we select
the first N streamlines with the largest nh. Each selected streamline
must have a HP ≥ 50%. Our experience shows that N = 3 or 5 offers
simplified computation and guarantees that streamlines on the polyp
neck are always selected.
5.2 Calculating Streamline Features for Polyp Characteri-
zation
We want to find (almost) closed streamlines of maximum curvature in
the region of the polyp neck. Therefore, our problem seems to be the
topic of detecting (almost) closed streamlines on 3D surfaces.
The detection of specific patterns of streamlines, e.g. closed stream-
lines and swirling streamlines, is a topic in flow visualization tech-
niques. Our polyp candidate selection scheme is similar to the de-
tection of flow vortices using streamlines. In 2D flow fields, Portela
[23] introduced the winding angle of streamlines to detect vortices.
Sadarjoen et al. [25] proposed to use the sum of signed angles along
a streamline as a simplification. In 3D cases, winding angles are no
longer meaningful. Portela [23] suggested reducing the problem from
3D to 2D by projecting local vectors onto the swirl plane. Jiang et al.
[10] presented an algorithm to detect swirling features based on the
geometry of streamlines.
Swirling streamline patterns used to detect vortices in fluid flow
are different from the curvature line patterns to detect polyps. We
want to find (almost) closed streamlines, but we can use similar tech-
niques such as the winding-angle technique [25]. In 2D flow fields,
the winding angle is defined as the cumulative change of direction of
the streamline segments. It measures the rotation of a streamline. A
swirling streamline must have a winding angle of at least 2pi . This
feature is used to detect swirling patterns of vortices in 2D flow fields.
However the winding angle is inherently limited to 2D.
Fig. 10. Adapt the winding angle method: the changing angle of stream-
line direction is projected onto the local tangent plane.
We generalized the winding angle concept to space curves. Since
streamlines of curvature are traced on 3D colonic surfaces, surface nor-
mals could be obtained at streamline points. For each streamline point,
the changing angle of streamline direction is projected onto the local
tangent plane (Figure 10). This projected angle is signed according
to the right-hand rule. The sum of such signed angles along a closed
streamline may be less than 2pi . Therefore, a streamline is considered
to be closed if it has two points within a certain very small distance
from each other, however the arc length of the streamline in between
should be above a certain threshold. The winding angle is computed
for each selected streamline, and then the largest winding angle is used
as an important feature per pre-detected polyp candidate.
Only polyps larger than 5mm in diameter are significant for clinical
diagnosis. There are also small bumps on the colon wall, which are
not real polyps. Our streamline generation algorithm also generates
(almost) closed streamlines around them. To separate them from true
polyp detections, we also measure the size of the area enclosed by
the (almost) closed streamline in terms of the mean radius. The mean
radius of a closed streamline is defined as the average distance from
the mean center of the streamline to its points. The mean radius of
the selected streamline that has the largest winding angle is used as an
additional feature per candidate area in our polyp candidate selection.
6 RESULTS
In this section we document the results of a study that we performed on
5 patient data sets to demonstrate the utility of our streamline selection
and streamline-based feature calculation for polyp detection strategies.
We also show renderings which illustrate how our surface-constrained
curvature streamlines enhance the visual perception of colonic surface
shape in virtual colonoscopy.
6.1 Polyp Candidate Selection Study
We are planning to integrate our streamline selection and streamline-
based feature calculation in a complete polyp detection protocol. The
first stage of the protocol is a polyp pre-detection phase [33] that yields
all true-positive detections (as confirmed by medical diagnosis), and
also a large number of false-positive detections. Suspect locations are
first detected on the colonic surface based on the volumetric shape
index. During the second stage, the number of false-positive detections
will be reduced using various classification techniques.
We assessed the value of our streamline selection and the wind-
ing angle calculation to discriminate between true- and false-positive
polyp detections in a large number of candidate areas, with the goal
of integrating these techniques in the second stage of polyp detection
and classification. CT scans were performed on five patients with a
Philips Mx8000 multislice scanner. The average voxel size of the 3D
volume image is 0.77mm×0.77mm×1.60mm. The preprocessing al-
gorithm [33] detected 331 polyp candidate areas in total. An expert
opinion of an experienced radiologist was used to classify these candi-
date areas into true- and false-positive detections, where true-positive
detection indicates a definite polyp and false-positive detection a non-
polyp. Of the 331 candidate areas, 50 sites (15.1%) were classified as
true-positive detections. True-positive detections were found in all 5
patients.
Clustering Polyp Candidates
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7 8
Mean Radius (mm)
W
in
di
ng
A
ng
le

(ra
dia
n)
False Positives
True Positives
Fig. 11. Features of Streamlines: Mean Radius and Alternative Winding
Angles. True- and false-positive detections are visually clustered.
We applied our streamline selection and winding angle calculation
method on all 331 candidate areas. The winding angle was signifi-
cantly higher for true-positive detections than for false-positive detec-
tions (Wilcoxon rank sum test, p < 0.001). The median winding an-
gle and inter-quartile range (IQR) for the true-positive detections was
7.817 and 6.770–9.288 compared to 2.954 and 1.995–3.749 for the
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ZHAO et al.: LINES OF CURVATURE FOR POLYP DETECTION IN VIRTUAL COLONOSCOPY
false-positive detections. Figure 11 shows clustering of the true- and
false-positive detections. These results indicate that streamline selec-
tion and winding angle measurements in candidate areas could make a
valuable contribution to an approach for discriminating between true-
and false-positive polyp candidate areas as detected by the preprocess-
ing algorithm.
6.2 Enhanced Visualization
The visualization of the shaded colon wall can also benefit from
streamlines of curvature (images on the left of Figure 12). This can
greatly enhance the perception of colonic surface features, e.g. polyps.
We also show the results of our automatic streamline selection method
in the images on the right of Figure 12. As can be seen in these im-
ages, one or more of the selected streamlines are situated on the polyp
neck.
7 CONCLUSIONS AND FUTURE WORK
Our work explored the potential of surface principal curvature di-
rections to characterize specific surface shape features, e.g. colonic
polyps in virtual colonoscopy. Streamlines of curvature were used to
visualize such vector fields. The most important elements are robust
methods for curvature calculation, surface-constrained streamline in-
tegration, and the use of adaptive seeding and spacing of the stream-
lines. We developed a new scheme for curvature controlled spacing of
streamlines on implicit iso-surfaces. Both the implicit and explicit
surface implementations showed good results for surface curvature
streamlines. In particular, our approach is useful for implicit surface
characterization based on 3D volume data.
(Almost) closed streamlines can be generated on the polyp neck
area. Therefore we expect that such characteristic patterns are a good
indicator of colonic polyps in virtual colonoscopy. Streamlines are
selected based on the local surface geometry and new features are cal-
culated on selected streamlines. A statistical analysis indicated that
such streamline features are highly correlated with true-positive de-
tections of a preprocessing polyp detection method. The polyp neck
area is considered as an addition to the polyp cap, which is often the
main focus of current polyp detection techniques. Our newly defined
features on curvature streamlines are expected to reduce the number of
false-positive polyp candidates in a pre-detection. We proposed a new
polyp candidate selection scheme that can be easily combined with
other techniques to improve current computer-aided detection. An en-
hanced visualization of the colon wall by curvature line patterns can
also improve the perception of colonic surface shape for radiologists.
There are a number of promising avenues for future research to-
wards a robust polyp detection technique. For the techniques de-
scribed, the use of higher order integration and interpolation methods
can possibly lead to more accurate results. The streamline-enhanced
visualization must be compared with the standard visualization in a
clinical environment. Our polyp candidate selection scheme will also
be tested with a large number of clinical CT scans to obtain a general-
ized specification of its performance.
ACKNOWLEDGEMENTS
This work is supported in part by Philips Medical Systems Nederland
BV, Best, the Netherlands. The authors thank Dr. Frans A. Gerritsen
and Ir. C. van Wijk for kind advices and helpful discussions, as well
as providing the authors with CT scans of patients.
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