IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 6, JUNE 2004
661
Surface Normal Overlap: A Computer-Aided
Detection Algorithm With Application to Colonic
Polyps and Lung Nodules in Helical CT
David S. Paik*, Christopher F. Beaulieu, Geoffrey D. Rubin, Burak Acar, R. Brooke Jeffrey, Jr., Judy Yee,
Joyoni Dey, and Sandy Napel
Abstract—We developed a novel computer-aided detection
(CAD) algorithm called the surface normal overlap method that
we applied to colonic polyp detection and lung nodule detection
in helical computed tomography (CT) images. We demonstrate
some of the theoretical aspects of this algorithm using a statistical
shape model. The algorithm was then optimized on simulated CT
data and evaluated using a per-lesion cross-validation on 8 CT
colonography datasets and on 8 chest CT datasets. It is able to
achieve 100% sensitivity for colonic polyps 10 mm and larger at
7.0 false positives (FPs)/dataset and 90% sensitivity for solid lung
nodules 6 mm and larger at 5.6 FP/dataset.
Index Terms—Colonic polyp, computed tomography colonography (CTC), computer-aided detection (CAD), cross-validation,
lung nodule, statistical shape model.
I. INTRODUCTION
I
N the United States, lung cancer and colon cancer are the
first and second leading cancer killers, respectively. Early
detection of colonic polyps and lung nodules, the precursors
to these diseases, has been shown to improve survival [1]–[4].
Clinically significant colonic polyps and lung nodules are resolvable given the spatial resolution of helical computed tomography (CT). However, the accuracy and efficiency of viewing
hundreds of source axial images per exam are limited by human
factors, such as attention span and eye fatigue.
In response to this challenge, a variety of computer-aided
diagnosis (CAD) methods have been developed to improve
both the accuracy and the efficiency of detecting lesions in
this and other difficult three–dimensional (3-D) diagnostic
problems. Among them, many different approaches to CAD for
CT lung nodule detection and for CT colonic polyp detection
have been developed, several of which are described next.
Manuscript received September 22, 2003; revised February 5, 2004. This
work was supported in part by the National Institutes of Health (NIH) under
Grant R01-CA72023 and Grant P41-RR09784. The Associate Editor responsible for coordinating the review of this paper and recommending its publication
was G. Wang. Asterisk indicates corresponding author.
*D. S. Paik is with the Department of Radiology, Stanford University, Stanford, CA, USA 94305-5450 USA (e-mail: paik@smi.stanford.edu).
C. F. Beaulieu, G. D. Rubin, R. B. Jeffrey, Jr., J. Dey, and S. Napel are with the
Department of Radiology, Stanford University, Stanford, CA, USA 94305-5450
USA.
B. Acar is with the Department of Electrical and Electronic Engineering,
Bogazici University, 34342 Bebek, Istanbul, Turkey.
J. Yee is with the Department of Radiology, University of California at San
Francisco, San Francisco, CA 94143 USA.
Digital Object Identifier 10.1109/TMI.2004.826362
For detecting lung nodules, Giger et al. [5] developed a
two–dimensional (2-D) multilevel thresholding detection algorithm that creates a tree structure of image components. Rules
were applied to shape features in order to identify nodules. 94%
per-nodule sensitivity was achieved with 1.25 false positives
(FPs) per patient. Armato et al. [6], [7] applied multilevel
thresholding and a rolling ball algorithm toward detecting
lung nodules. Shape and attenuation features were classified
using linear discriminant analysis and the algorithm achieved
70% per-nodule sensitivity with 1.5 FPs per axial section.
Brown et al. [8] have presented an algorithm for both detection
and surveillance of lung nodules in CT. Region-growing
and morphological operators were used to create candidate
locations. Attenuation, location, volume, and shape features
were matched to model objects in a semantic net with fuzzy
membership that serves as a generic a priori anatomic model.
In the initial detection task, 86% per-nodule sensitivity was
achieved with 11 FPs per patient. Lee et al. [9] used both
genetic algorithm-based and semicircular template matching to
identify initial candidates and attenuation, shape, and gradient
feature rules to reduce FPs. They achieved 72% per-nodule
sensitivity with 31 FPs per patient. Erberich et al. [10] applied
the Hough transform (HT) for both 2-D circles and 3-D spheres
using a rule-based classifier and achieved 30%–40% per-nodule
sensitivity with a “large amount of false positive nodules.”
Several approaches to colonic polyp CAD in CT colonography have also been proposed. Vining et al. [11] developed
a method that measures abnormal wall thicknesses using
heuristics. They report 73% per-polyp sensitivity with a range
of 9–90 FPs per patient. Other approaches have analyzed the
morphology of the mucosal surface. Summers et al. [12], [13]
have developed a method that uses size, attenuation, and curvatures calculated with convolution-based partial derivatives to
find polyps. They achieved 64% per-lesion sensitivity with 3.5
FPs per patient. Yoshida et al. [14]–[16] use shape index and
curvedness (computed with partial derivatives), directional gradient concentration, and quadratic discriminant analysis. Using
both prone and supine datasets, they achieve 100% per-patient
sensitivity with 2.0 FPs per patient (per-polyp sensitivity not
stated). Kiss et al. [17] combined surface normal and sphere
fitting methods to achieve 100% per-polyp sensitivity with 8.2
FPs per patient. In addition, secondary CAD algorithms that
are designed to reduce the FP rate of primary CAD algorithms
have been proposed. Göktürk et al. [18] applied support vector
machines to shape and attenuation features to reduce FPs and
0278-0062/04$20.00 © 2004 IEEE
662
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 6, JUNE 2004
reported a 50% increase in specificity at a constant sensitivity
level. Acar et al. [19] have applied edge displacement fields
to reduce FPs and reported a 23% increase in specificity at a
constant sensitivity level. Both of these FP reduction methods
were evaluated using initial versions [20] of the work presented
in this paper.
These previously described CAD algorithms for both lung
nodules and colonic polyps have achieved varying levels of accuracy although they all leave room for improvement. Additionally, many of them represent analogous approaches, using similar feature vectors and similar classifiers. The purpose of this
work was to create a new and effective approach to CAD by developing new features and by optimizing them toward two clinical applications.
We present in this paper 1) a novel multi-purpose CAD algorithm that we call the surface normal overlap (SNO) method, 2)
a theoretical analysis of this algorithm using a statistical model
of anatomic shape, 3) an optimization and analysis method for
this algorithm using simulated CT data and a per-lesion crossvalidation, and 4) preliminary evaluations of the detection performance of the CAD algorithm in lung nodule and colonic
polyp detection, using the free-response receiver operating characteristic (FROC) paradigm. We propose to use this algorithm
as the first step in a larger overall detection scheme and, thus, we
strive for high sensitivity at a reasonable FP rate, thus allowing
secondary FP reduction algorithms, such as some of those described above, and/or radiologist visualization to improve specificity.
II. CAD ALGORITHM
The following sections describe the processing steps of the
SNO method, of which the end result is a list of the coordinates
of the center of each suspicious region, sorted in decreasing
prospect of being a lesion.
A. Pre-Processing and Segmentation
Because both colonic polyps and lung nodules are generally
not much denser than water, high density structures (e.g., bone)
are removed by clamping voxel intensities to be no greater than
that of water. Next, the CT volume data are made isotropic by
tri-linear interpolation to 0.6 mm 0.6 mm 0.6 mm voxels
. This is done in order to reduce any bias
to produce
between lesions at different orientations and also to reduce any
bias between datasets with different voxel sizes.
Next, segmentation is performed automatically to identify
either the colon lumen or the lung parenchyma. A binary
image, , is created by thresholding all air intensity voxels
including air outside the body. This
is followed by a negative masking of all air intensity voxels
morphologically connected to any of the edges of the data
volume (air outside the body), thus leaving only voxels with
air density within the body. In the case of CT colonography,
the inferior portions of the lungs are usually captured and are
removed using a negative mask of a 3-D region-filling seeded
with air intensity regions with a width or depth of greater
than 60 mm in the most superior axial slice. Finally, small air
in the colon datasets,
in the lung
pockets (
datasets) are also negatively masked from .
and is used to
Next, a binary image, , is derived from
limit the remaining computations to voxels near the air-tissue
interfaces in the colon or lung. This 1) reduces computational
requirements and 2) eliminates FPs arising within soft tissue
structures outside the region of interest. begins as the surface
voxels of
and is then morphologically dilated by 5 mm to
produce a thickened region that contains the air-tissue interfaces
of interest (see Fig. 1, rows 1–2).
B. Gradient Orientation
The gradient orientation step computes the image gradient
vector,
, at high-contrast edges in order to determine
the 3-D orientation of the image surface normals. We have modified the Canny edge detector [21] to limit calculations to only
those voxels contained in . Each partial derivative is computed using two one-dimensional (1-D) Gaussian convolution
kernels and one 1-D derivative-of-Gaussian convolution kernel,
,
, and
(standard deviawhich are parameterized by
,
, and
discrete samples respectively.
tions) with
Our implementation additionally takes advantage of the
, by only computing these
greatly reduced search space,
separable 1-D convolutions where strictly necessary. The
minimum locus of voxels necessary to correctly calculate the
1-D convolutions is calculated by morphologically dilating .
to denote the floor function (greatest integer less
Using
than or equal to ), the convolutions are performed as follows.
The separable convolutions in the direction are calculated for
dilated
voxels in the direction and
each voxel in
voxels in the -direction. The separable
then dilated
convolutions in the -direction are then calculated for each
voxel in
dilated
voxels in the -direction. The
separable convolutions in the -direction are then calculated
for each voxel in . Nonmaximum suppression and hysteresis
thresholding (thresholds of 100 HU and 200 HU) follow the
separable convolutions The resulting surface normal vectors,
.
which point inward into the tissue, are denoted as
C. Surface Normal Overlap
The surface normal overlap step is critical for detecting lesions. Each voxel in
accumulates a score proportional to the
number of surface normals that pass through or near it. Generally speaking, both colonic polyps and lung nodules tend to
have some convex regions on their surfaces and thus, the inward
, near these features
pointing surface normal vectors,
tend to intersect or nearly intersect within the tissue. Pulmonary
vessels in the lungs and haustral folds in the colon also have
convex surfaces, but since they have a dominant curvature along
a single direction (as opposed to high curvature in two directions as is common on the surfaces of polyps and nodules), the
score for vessels and folds is generally less than that for nodules
, counts the number
and polyps. A 3-D array, denoted
of surface normals that pass through or near to each voxel in
(see Fig. 1, row 3). Each voxel in
corresponds to
. In order to limit the contributions from
a voxel in
normal vectors from very distant structures, the length of the
, the scale
projected surface normal vectors was defined as
PAIK et al.: SURFACE NORMAL OVERLAP
663
Fig. 1. Intermediate steps of the CAD algorithm. Left column: CT colon data with a polyp. Right column: CT lung data with a nodule. Top row: cross sectional
slice through an example lesion. Middle row: limited search space (S ) from segmentation shown with semitransparent overlay. Bottom row: cross sectional slice
through summed overlapping Gaussian profile cylinders shown in grayscale with white denoting highest CAD score.
of the largest spatial features of interest. Prior to any evaluation,
was set to 10 mm.
Providing robustness to variations from perfectly spherical
objects is critical to the success of this algorithm in real patient
data. Our algorithm provides robustness both in the radial direction (objects with nonconstant distance from surface points
to center) and in the transverse direction (objects with nonuniform magnitude of curvature). Robustness in the radial direction
is provided by the fact that normal vectors can intersect at dif-
), thus allowing many
ferent distances from the surface (up to
nonspherical but roughly globular objects to have a significant
response.
Robustness in the transverse direction is provided by allowing
skewed surface normal vectors (those that do not intersect but
. This is accomnearly intersect) to be additive in
plished by projecting cylinders of a finite width in the direction
of the surface normal rather than by projecting line segments.
Because surface normals that come closer to intersecting are
664
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 6, JUNE 2004
assumed to be more likely generated by the same convex surface patch, the projected cylinders are given a transverse profile that gradually decreases in intensity at greater radial distances, thereby providing robustness in the transverse direction
and again, allowing many nonspherical but roughly globular objects to have a significant response. The profile was chosen to
.
be Gaussian with a scale of
For computational efficiency, the entire surface normal
overlap step is implemented by first scan converting (i.e.,
discretizing into voxels) a line segment for each surface normal
and summing it into
. Then, the
in
Gaussian profile of the cylinders is achieved by a sequence of
. The
linearly separable 1-D convolutions to produce
convolution is given by
Fig. 2. Cross section through the stochastic shape model. Dotted circle is the
nominal sphere/cylinder of radius R. Solid contour is the shape after deviation
from the nominal model. The deviated surface patch shown as a small oval and
.
the deviated surface normal direction as direction
PQ
(1)
include
The discrete kernels are chosen so that , , and
sampled at 0.6 0.6 0.6 mm to cover 95% of the
Gaussian curve. The computational burden imposed by the convolution calculation is minimized using morphological operators similar to the gradient orientation calculation, as in Section II-B.
D. Candidate Lesion Selection
The local maxima of
are selected as candidate
lesion locations. However, complex anatomic structures with
multiple convex surface patches may generate multiple local
was defined to be the smallest scale of the features
maxima.
that might generate distinct local maxima and was set to 10
mm prior to any evaluation. Local maxima are considered in
descending order, and if a local maximum occurs within
of an already accepted local maximum, the lesser value is
assumed to be part of the same structure and is rejected. After
this spatial filtering, the remaining local maxima are sorted in
decreasing order and recorded as the potential lesion locations.
is given by
The score for a potential lesion at location
, and we refer to this as a “CAD hit.
III. THEORETICAL ANALYSIS
In this section, we present several theoretical analyzes of
SNO. To facilitate them, we have created a statistical anatomic
shape model that balances the complex variability of human
anatomy with sufficient simplicity to allow for analytic insight.
We then use this theoretical model to compare the behavior of
the SNO method to the 3-D Hough transform for spheres in
distinguishing lung nodules from vessels and colonic polyps
from haustral folds.
A. Stochastic Anatomic Shape Model
This shape model begins with a simple parametric shape and
then adds stochastically-governed variation in order to produce
realistic anatomic shape. The nominal model for lung nodules
and colonic polyps are spheres and hemispheres, respectively,
while the nominal model for vessels and haustral folds are cylinders and half-cylinders, respectively. In order to account for
anatomic variability, infinitesimal surface patches on the surface are then allowed to simultaneously vary from their nominal
position at radius in an implicit manner that preserves continuity between patches. In this analysis, radial position deviation is represented with random variable , and surface normal
direction variation is represented with random variable (see
Fig. 2).
We model each surface patch as deviating from its nominal
position in the radial direction with a Gaussian distribution on
, with a mean of 1 and a standard deviation of
. We then
model each surface normal vector as deviating from its nominal direction with two independent and identically distributed
and , with zero mean and standard
Gaussian variables,
deviation of . These two displacements are in the plane perpendicular to the radial direction, a unit distance away from the
surface normal. For convenience, we represent directional devi, which has a Rayleigh
ation by its magnitude
distribution, and by its angle, , which has a uniform distribution on the interval [0, ). The probability density functions
and , respectively, and
for and are parameterized by
are given by
and
Thus, the variability of a shape of radius is represented by
the random variables , , and .
As each surface patch varies from its nominal position and direction, the solid angle subtended by the patch stays constant but
the area of the patch changes due to 1) the magnification factor
at different radial distances and 2) a cosine inverse proportionality as it is tilted away from its nominal direction. We let be
the area of the nominal surface patch and be the area of each
surface patch after variation. For spheres, the surface patches
.
are indexed by , and the relationship is
For cylinders, the surface patches are indexed by around the
axis of the cylinder and by down the length of the axis, and
the relationship is
.
PAIK et al.: SURFACE NORMAL OVERLAP
665
Fig. 3. Examples of theoretical model parameter estimation. The surface normals that belong to the structure are shown as lines on the solid surface and the
nominal sphere/cylinder model is shown as partially translucent with perspective. (a) polyp, (b) haustral fold, (d) lung nodule, (e) pulmonary vessel. Histograms
of and normalized to a unit Gaussian and unit Rayleigh are shown for each shape class and compared to the parametric models in (c) and (f).
m
u
B. Model Parameter Estimation
In order to make quantitative comparisons using this theoretical model, the parameters controlling the degree of variation from the nominal shapes,
and , were estimated directly from the patient datasets. This process involved: 1) performing edge detection on the datasets; 2) identifying the surface normal vectors that belong to the nodule, polyp, vessel or
fold; 3) finding the nominal sphere or cylinder that fit those surface normal vectors; 4) computing the value of and for each
and from those sample
surface normal; and 5) estimating
populations.
All polyps 5 mm and larger and all nodules 3 mm and larger
were used for parameter estimation. From each of the 8 colon
datasets and each of the 8 lung datasets, eight folds or vessels
were selected prospectively and manually and then, selected for
parameter estimation. Thus, our analyzes included 18 polyps
and 64 selected folds in the colon, and 84 nodules and 64 selected vessels in the lung. Section V-C contains full details about
these datasets.
Edge detection was performed as described in Section II-B.
Isolation of the surface normals belonging to the structure of interest was performed as follows. The center of the structure of
interest, , was chosen manually, and all surface normal vectors
whose bases were further than two radii away were eliminated.
Next, if a line segment between and a surface normal intersected an air intensity voxel (
), the surface normal
was eliminated. Then, surface normals pointing more than 90
away from were eliminated. Finally, the largest contiguous
region of surface normals was kept as belonging to the structure
of interest. This algorithm was quite successful in isolating the
structures of interest; Fig. 3 provides some examples.
The nominal shape was derived using a least squares fit of a
sphere or cylinder to the bases of the surface normals (i.e., directional information was not used), which leads to an estimate of
. Using this nominal shape, both and were computed diwas estimated using
rectly for each surface normal. Finally,
was estimated
the maximum likelihood estimate. However,
differently because a few surface normals were at nearly 90
away from either the center of the sphere or the axis of the
cylinder, leading to nearly infinite values of and, thus giving
inaccurate estimates due to MLE sensitivity to outliers. This
happened particularly around the “skirt” of polyps and haustral
folds where it is difficult to make binary decisions about what
belongs to the polyp or fold and what does not. Instead, we estiby using the method-of-moments but substituting the
mated
more robust median estimator for the mean. By setting the em, equal to the point at which the CDF is
pirical median,
0.5, we get
which leads to
The results of the parameter estimation are shown in Fig. 4.
C. Algorithm Models
In order to understand the theoretical performance of the SNO
algorithm and to compare it to the Hough transform for spheres,
we applied this anatomic shape model for polyps, folds, nodules,
and vessels with differing degrees of variation from the nominal
model. Specifically, we compared the expectation of the CAD
666
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 6, JUNE 2004
(a)
(a)
(b)
(b)
(c)
(c)
Fig. 4. Boxplots showing the minimum, maximum, interquartile range, and
median for the theoretical shape model parameters: (a) R, (b) , and (c) s .
scores over the random variables , , and . See Appendix for
the CAD score formulas and their derivations.
D. Theoretical Comparison of SNO and HT
In order to compare the theoretical performance of the SNO
and HT algorithms, we varied the shapes from perfect spheres
and cylinders to more realistic anatomic shapes. The range
of realistic shape variability was estimated as described in
Fig. 5. Results from the theoretical model. In (a) and (b), and s are
simultaneously varied from 0 to twice their median values. (a) Colon CAD
scores as a function of and s . (b) Lung CAD scores as a function of and s . (c) CAD scores as a function of lesion radius, R, at the median values
of and s .
Section III-B. Fig. 5(a)–(b) presents resulting CAD scores as
a function of deviation from ideal shape for polyps and folds,
and for nodules and blood vessels. These plots demonstrate the
robustness of SNO to deviation from ideal shapes whereas HT
fails to discriminate between shapes with realistic amounts of
shape variability. Fig. 5(c) presents the scores of both polyps and
PAIK et al.: SURFACE NORMAL OVERLAP
Fig. 6.
lung.
667
(a)
(b)
(c)
(d)
(a) Boxplot showing SNO scores using the theorical model in the colon, (b) HT scores in the colon, (c) SNO scores in the lung, and (d) HT scores in the
nodules as a function of lesion size, revealing that larger lesions
lead to a smoothly increasing response with SNO. However, HT
produces a response that varies tremendously for nearly identical lesion sizes.
and were then used to produce
The estimated values of
a CAD score for each shape. These scores are shown in Fig. 6.
Wilcoxon rank sum tests were performed to test the difference
between TP and FP scores. For SNO, there were significant dif) and between
ferences between polyp and fold (
nodule and vessel (
). However, for HT, there were
)
not significant differences between polyp and fold (
).
nor between nodule and vessel (
IV. GRADIENT ORIENTATION OPTIMIZATION
Using simulated CT phantoms, we optimized the gradient orientation kernel scale parameters,
,
, and , in order to
yield the most accurate gradient orientations. This step was critical because errors in estimating the gradient direction can di. The selection of
minish surface normal overlap in
is particularly important because too
values for , , and
small a value will lead to errors from noise and very localized
perturbations in the surface whereas too large a value will lead
to insensitivity to smaller lesions. The optimization of the pais described later in Section V.
rameter
A. Phantom Model and Error Metric
A series of hemispherical phantom objects were “scanned”
using software that simulates CT scanning including forward
projections, partial volume effects, correlated CT noise, helical
interpolation, and filtered backprojection reconstruction [22].
The simulations were performed with a 3-mm slice thickness,
), 0.7 0.7 mm
pitch of 2 (
pixels in plane, and 1-mm reconstruction interval. For each
phantom, a water-equivalent density sphere was embedded
halfway into a water-equivalent density, randomly oriented, flat
wall to simulate a prototypical colonic polyp or prototypical
lung nodule on the chest wall. The diameters of the spheres,
, ranged from 5 to 15 mm at 1-mm increments, chosen to
demonstrate the effects of changes in size from the prototypical
10-mm lesion. For each sphere size, there were 10 phantoms,
each with a different wall orientation, randomized subvoxel
offset, and randomized CT detector noise, leading to a total of
110 phantoms.
The error metric used to evaluate the accuracy of gradient orientations, , was defined to be the mean perpendicular distance
, to the true center of
from the surface normal vector,
the sphere. In order to include only detected gradient orientations from the hemisphere and not those from the flat wall, only
of the sphere center
gradients located within 1.05
(see Fig. 7).
entered into the calculation of
B. Gradient Orientation Kernel Scale
and executed the CAD algorithm with
We let
simultaneously varying
all three
from 0.05 to 4.0 mm in 0.05-mm increments. The errors,
from the 10 phantoms at a given
and
, were then
averaged. The results of this optimization are plotted in Fig. 8,
led to the least error across all
showing that
lesion sizes.
668
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 6, JUNE 2004
Fig. 7. Left: simulated CT phantom of a prototypical lesion consisting of a
sphere embedded halfway into a flat wall. Right: an oblique CT cross-section
through a phantom lesion showing how e is calculated as the mean distance
(black line segments) from normal vectors (arrows) to the true sphere center
(white dot).
Fig. 9. The accuracy of the gradient orientation step on tri-linearly interpolated
data is relatively independent of kernel scale anisotropy as varies from by a factor of 0.5–2.0. d
ranges from 5 to 15 mm and was held
constant at 1 mm while was varied.
Fig. 8. The accuracy of the gradient orientations is dependent on the kernel
, and the size of the lesion (d
ranging from 5 to 15 mm).
scale, = 1 mm. The
The error metric is minimized across all lesion sizes at rippling effect is an artifact due to discontinuous jumps in the number of samples
in the convolution kernel, n , n , and n , at different values of .
C. Gradient Orientation Kernel Anisotropy
We also investigated the effects of anisotropic resolution in
helical CT on gradient orientations. In general, helical CT has
lower effective resolution through-plane than in-plane, regardless of reconstruction interval. Thus, one might expect setting
would compensate for this effect. To test this, we
(based on the results of
fixed
the first optimization) and let
vary from 0.05–4.0 mm in
0.05-mm increments. The errors, , from the 10 phantoms at
and , were then averaged, as in Section III-B.
a given
Fig. 9 plots the results of this optimization, showing that the
accuracy of the gradient orientations on tri-linearly interpolated
anisotropic data is almost independent of the anisotropy of the
kernel scale between ratios of 0.5 and 2.0. As a result, all subsequent experiments were carried out with
.
V. CAD PERFORMANCE EVALUATION
This section describes two experiments that were performed
in order to evaluate the performance of the CAD algorithm in
detecting real colonic polyps and in detecting solid lung nodules. In Section III, the goal of the gradient orientation optimization was to produce as accurate gradient orientations as possible
error metric. However, the goal of optimizing the
using the
, is to maximally differentiate between
cylinder scale,
lesions and FPs. A wider cylinder scale will increase the robustness to deviations from perfectly spherical shapes but will also
decrease the differentiability between lesions and FP structures.
Since this effect is dependent on the variability of true lesion
and FP shapes, the CT simulations were insufficient. Instead, a
cross-validation was performed to evaluate lesion detection per.
formance with prospectively chosen values of
A. Cross-Validation
When cross-validation is used to evaluate a classifier, the
dataset is split into
sets (sometimes referred to as “folds”).
sets and then evalThe classifier is trained de novo on
uated on the remaining
independent set(s); this is repeated
sets and
sets and the
for all possible divisions into
results are averaged in a reasonable manner [23]. This type of
evaluation gives an unbiased estimate of performance and has
a lower standard error than traditional holdout methods [24].
. However, in a detection
In this evaluation, we chose
problem such as this, splitting the dataset (CAD hits) into sets
at the granularity of lesions is problematic because the result
based on the training sets)
of training (e.g., selecting
changes the dataset (e.g., CAD hit locations) and thus, changes
the sets themselves.
In order to retain the independence between training and test
sets, the sets were selected on the basis of distinct anatomic features rather than on the basis of CAD hit locations. The locus
of all CAD hit locations was computed for each possible value
. Two CAD hits were considered to be the same
of
anatomic feature if they were within 10 mm of each other. The
sets (7 in the colon dataset, 46 in the lung dataset) were made
to be disjoint by having one true positive (TP) lesion and equal
numbers of randomly selected FP anatomic features. CAD hits
PAIK et al.: SURFACE NORMAL OVERLAP
were distributed among the sets according to the anatomic feature to which they belonged. Thus, to within the 10-mm constraint, no two sets contained CAD hits on the same anatomic
feature.
For an error metric in training, we selected the value of
that maximized
, which we define as the normalized partial area under the FROC curve from 0–20 FPs/dataset
indicates perfect
and from 90%–100% sensitivity.
indicates that less
detection performance whereas
than 90% of lesions are detected by the time 20 FPs/dataset is
is that it is an area
reached. The geometric interpretation of
that represents how close the curve comes to perfect performance (i.e., 100% sensitivity at 0 FPs/dataset). No probabilistic
are drawn in this paper, but it should
conclusions based on
has not been
be noted that the probabilistic interpretation of
fully explored and its interpretation in a probabilistic context
is unclear.
B. Automated CAD Scoring
Because the results of a CAD evaluation can be especially
numerous, we implemented a method for automatically scoring
each CAD hit as either a TP or FP, thus eliminating subjectivity
and clerical errors. Because there is some spatial variance in
what is declared to be the center of a lesion in the gold standard (Section V-C2), the scoring algorithm must allow for some
small amount of spatial mis-registration between gold standard
lesion locations and TP CAD hits. Thus, we defined any CAD
hit as a TP if it was within half the lesion’s measured diameter
from the lesion’s measured center. Lesion diameters were measured manually from the CT images during the setting of the
gold standard using a multi-planar digital caliper tool.
To determine the overall performance, all of the CAD hits
within the test sets were determined to be a TP or a FP and then
pooled and sorted in descending order of score. In the event that
multiple CAD hits are scored as TP for a given lesion, only the
highest scoring hit was considered a TP; lower scoring hits were
ignored. TP CAD hits on lesions below the size range of interest
were not considered FPs nor did they increase the sensitivity.
At a given score threshold, sensitivity was calculated as the percentage of lesions within the size range of interest that had been
identified by a TP CAD hit above that score threshold. The FP
rate was calculated as the total number of FP CAD hits divided
by the number of datasets.
C. Detection Evaluation
1) Data Collection:
Colon: From a database of 116 CT colonography exams
performed at either Stanford University or at the San Francisco
VA hospital, 8 exams were selected for this study in order to
include a reasonably large number of colonic polyps and to
balance the number of patients with and without large polyps.
Exams with excessive image artifact or retained water were excluded. Case selection was done without regard to polyp conspicuity or shape. These 8 patients were given rectal air contrast and scanned in the supine position with single- or multidetector helical CT (GE HiSpeed/CTi or LightSpeed, General
Electric Medical Systems, Milwaukee, WI) with effective section width of 2.5–3.75 mm and 50% overlapping reconstruction.
669
Fig. 10. Per-lesion cross validation training. The range of values of that was selected is shown in the shaded areas. For reference, A across all
the datasets (i.e., no cross-validation) is shown by the solid lines. Note that the
(shaded area) near
method is able to prospectively choose values of the true optimum (maxima of solid lines) with low variance.
Immediately following CT scanning, each patient also underwent fiber-optic colonoscopy (FOC). These results were correlated to the CT images with a total of 7 “clinically significant”
polyps (
) found in 4 of 8 patients and a total of 11
small polyps (5-9 mm) found in 3 of 8 patients. A wide range
polyp shapes were present in the datasets
2) Gold Standard:
Colon: A study coordinator with extensive experience in
CTC and blinded to CAD results carefully reviewed the CTC
data and recorded the location and diameter of polyps found
by FOC into the gold standard database. Only one significant
polyp (measured as 15 mm by FOC) was unable to be located
in the CT images, most likely due to retained water. A total of
10 small polyps (1 was 8 mm and 9 were 5–6 mm measured by
FOC) were unable to be located in the CT images.
Lung: The gold standard in the chest was established by
consensus of two radiologists interpreting the axial CT data.
Both the location and diameters of nodules were recorded into
the gold standard database.
The CAD algorithm was then executed on the 8 colon and
8 lung datasets using the optimized gradient orientation from
Section III and cross-validated as described above.
3) Results:
Colon: In the colon datasets, the value of
based on the cross-validation training sets was in the range
2.2–2.6 mm with a mean of 2.5 mm. Fig. 10 shows the range
of these values of
compared to the performance on
all of the datasets (
computed on all datasets over all values
of
, not just on training sets). Note that the latter is
shown in this figure for reference only and was never used in
training or evaluation. The mean performance across the test
sets in detecting “clinically significant” colonic polyps was as
follows.
in diameter were detected
at 4.6 FPs/dataset. 90% were detected at 6.0 FPs/dataset. 95%
were detected at 6.5 FPs/dataset. 100% were detected at 7.0
FPs/dataset.
Fig. 11 shows a FROC plot of these results.
A manual analysis of the 50 highest scoring FPs in each colon
dataset (400 total) revealed that 86% were due to haustral folds,
670
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 6, JUNE 2004
Fig. 11. Per-lesion cross validation evaluation. FROC results for both colonic
polyp and ling nodule detection.
5% were due to the colon wall between adjacent loops, 4% were
due to a failure in segmentation in one dataset that captured the
air trapped in the blanket beneath the patient. Finally, each of
the following classes contributed 1% or less: stool, insufflation
catheter, small bowel, and the ileocecal valve.
based
Lung: In the lung datasets, the value of
on the cross-validation training sets was in the range 0.6–0.8
mm with a mean of 0.6 mm (see Fig. 10). The mean performance across the test sets in detecting “clinically significant”
solid lung nodules was as follows.
in
diameter were detected at 1.3 FPs/dataset. 90% were detected
at 5.6 FPs/dataset. 95% were detected at 63 FPs/dataset. 100%
were detected at 165 FPs/dataset (see Fig. 11).
A manual analysis of the 50 highest scoring FPs in each lung
dataset (400 total) revealed that 69% were due to pulmonary
vessels, 13% were due to bronchi, 6% were due to vessels or
bronchi in the mediastinum, 6% were calcified nodules, 2%
were due to bulges on the pleural surface, 2% were small indeterminate opacities, and each of the following classes contributed 1% or less: mass, metal artifact, and a single 2.9-mm
noncalcified nodule.
VI. DISCUSSION
A. CAD Algorithm
The surface normal overlap algorithm was originally inspired
by the Hough transform for spheres [25] but differs in some
which counts the
important ways. First, the array
number of overlapping or nearly overlapping surface normals is
similar to the Hough transform accumulator array in that it sums
“votes” for objects that could produce those normals, but it does
not require all of the votes to correspond to a single parameterized sphere, as does the Hough transform. Second, the Hough
transform, in its various forms, is highly specific for one type of
shape (e.g., spheres). Even the variant known as the generalized
Hough transform, which avoids parametric representations, requires a specific model. This specificity is desirable when the
shape to be detected can be precisely defined in advance, but
the specificity is problematic when it cannot. In contrast, the
SNO algorithm does not use an explicit model of a single type
of shape, but instead uses an implicit model to represent an entire gamut of shapes much larger than the set of spheres detected
by the Hough transform for spheres. This property is extremely
important when the objects to be detected can have significant
variability in shape such as with lung nodules or colonic polyps.
Rather than specifying the exact shape to be detected, the SNO
algorithm defines a fuzzy constraint on surface normal orientation in order to define the varied set of shapes to be detected,
both by allowing angular mismatches (transverse robustness)
and by allowing edges at different radial distances to sum (radial robustness). For comparison, Erberich et al. have applied
the 3-D Hough transform for spheres toward lung nodule detection in CT but reported only 30%–40% sensitivity at a high FP
rate [10].
The voxel intensity clamping pre-processing step is used to
eliminate edges due to bone but will also make calcified lung
nodules have a similar response as noncalcified nodules. While
the presence of calcification in lung nodules may help distinguish between benign and malignant nodules, the goal of this
algorithm is detection, not classification. For classification purposes, the original voxel intensities can easily be restored following the detection of suspicious regions.
While we have presented and evaluated the SNO CAD
method as being preceded by a specific segmentation scheme,
we emphasize that the only purposes of the segmentation step
are 1) to reduce computation by targeting only anatomical
regions of interest (ROIs) and 2) to eliminate hits from regions
disjoint from the anatomical ROI. Unlike some other CAD
approaches requiring extremely accurate segmentation (e.g.,
inclusion of juxtapleural nodules), the goal of our segmentation
step is to provide a volumetric region that contains all possible
image edges that could be due to the presence of lesions and
not to fully delineate their edges. Thus, our relatively simplistic
segmentation algorithm is sufficient and is not limiting factor
in the overall detection performance. However, it is possible
that gross errors in segmentation could adversely affect performance. In colon CAD, for example if the most superior axial
slice contains a large enough portion of the transverse colon,
it could be assumed to be lung and, therefore, erroneously
eliminated. Thus far, we have not had any such gross failures
in segmentation; however, we note that, ultimately, any robust
segmentation method could be substituted for ours. One
limitation of this study is that we did not formally optimize the
pre-processing and segmentation as we did many of the other
parameters of the algorithm. This stage was used to create a
thick boundary region containing easily detected high contrast
edges and it was very robust to a wide variety of parameter
settings. Our experience was that these parameters had little
effect or no on the overall performance.
We also emphasize that, at this stage, this algorithm is not
intended to be used independent of visual interpretation by a
radiologist. At the present stage of development (and perhaps,
well into the foreseeable future), this type of algorithm should
be seen as an aid for improving radiologist performance. In this
regard, although our algorithm generated more than one FP (on
average) per data set in order to achieve high sensitivity, it does
not indicate that the majority of patients will have FP detections
once reviewed in conjunction with a radiologist. If many of the
PAIK et al.: SURFACE NORMAL OVERLAP
FP hits are recognized as such and are discarded by the radiologist, overall performance may be acceptable. However, this
remains to be shown by future evaluations.
The difference in performance between detecting colonic
polyps and detecting lung nodules is of interest. Although neither FROC curve completely dominates the other, the “average”
lung nodule (i.e., near 50% sensitivity) was easier to detect (i.e.,
at fewer FPs/dataset) than the “average” colonic polyp, despite
the relatively larger size of polyps. The difference is partially
accounted for by the difference in lesion morphology—the
gross shape of the nodules (not at the fine level of spiculations)
in these datasets tended to be more globular than that of polyps,
which tended to have more complex surfaces due to the gradual
rise and fall of the mucosal surface around a polyp. This may
be attributable to the relatively isotropic growth pattern of lung
nodules in lung parenchyma as compared to the anisotropic
growth pattern of colonic polyps, which emerge and protrude
from the colon wall. Another factor in the different detection
performance is that nodules usually have detectable edges on
their entire surface compared to polyps, which have detectable
edges only on their outer half and thus, have half the number of
overlapping surface normals. The difference in performance is
also accounted for by the completely dissimilar sources of FPs
(e.g., background anatomy), which are very different in both
appearance and quantity between the lung and colon.
The hardest to find nodules (i.e., near 100% sensitivity) were,
however, harder to find than the hardest to find polyps. This
was due to several exceptional nodules whose appearance was
different than most other nodules. These four nodules accounted
for the range of sensitivity from 91%–100% and were the only
nodules that were detected at greater than 8 FPs/dataset. These
included three small, elliptical nodules on the chest wall (6 3
mm, 7 4 mm, and 6 4 mm) and one very irregular nodule at
the apex of the lung (20 16 mm); see Fig. 12.
B. Theoretical Analysis
While both SNO and HT perform similarly on perfect spheres
and cylinders, the results shown in Fig. 5(a)–(b) demonstrate
that HT rapidly loses its ability to distinguish between sphere
and cylinder as the shape variability approaches realistic levels.
On the other hand, SNO retains its shape discrimination under
much greater levels of shape variability. Additionally, Fig. 5(c)
demonstrates that slight differences in size can lead to very different HT scores due to the interaction of and .
The theoretical CAD scores (see Fig. 6) suggest that SNO
is better able to distinguish between the presented shapes than
does HT. While HT is valued for its specificity to the parametric
model (e.g., spheres), it is the ability to detect shapes that vary
from the nominal shape model that makes SNO particularly suitable for discriminating anatomic shapes.
In the theoretical shape model, note that statistical dependence between neighboring surface patches is not assumed since
this would be very unrealistic. However, not assuming independence limits the analysis to the score at the center of the shape
instead of the maximum score over the whole shape (expectation of maximum is not maximum of the expectations, see Appendix ). Although HT scores may be higher off center, this
671
Fig. 12. The four lung nodules that were hardest to detect accounting for the
sensitivity from 91%–100% and accounting for all detected nodules at greater
than 8 FPs/dataset. Nodule sizes are (a) 6 3 mm, (b) 7 4 mm, (c) 6 6 mm,
and (d) 20 16 mm.
2
2
2
2
would require nearly spherical subportions of the surface in
order to yield higher scores off center.
The theoretical model does assume independence between
and . The correlation coefficients between and for polyps,
folds, nodules, and vessels were 0.22, 0.04, 0.06,
, respectively. Although very low correlation does not rule out dependence, it helps to justify this first order approximation.
In our formulation of the SNO method, we have chosen to
project Gaussian-profiled cylinders for each surface normal.
One limitation of this work is that this choice of projected
shape may not be optimal with respect to the theoretical model.
This is an area of future work on this algorithm that we plan
to investigate further.
C. CAD Algorithm Optimization
The design of the simulated phantoms was an important factor
in the optimization of the gradient orientations. The simulated
hemisphere on a flat wall model was designed to find the optimal balance between the decreased noise from greater blurring and the increased sensitivity to small objects from lesser
blurring. The use of a hemisphere on a flat wall is an obvious
first order model for colonic polyps. We had originally tried
optimizing gradient orientations for lung nodules on spherical
phantoms. However, it was necessary to include other background anatomic structures (e.g., flat wall) in order to realistically model the effect of a large gradient orientation convolu). With a large kernel, nearby but distion kernel (large
tinct anatomic structures would contribute to the convolution
and cause error in the gradient orientation. This was balanced
against the effect of a small kernel, which had a decreased noise
).
reduction benefit due to less blurring (small
672
The hemisphere on a flat wall also serves as a model for lung
nodules in contact with the chest wall, which are not the most
common type of lung nodule but are anecdotally more difficult
to detect than contact-free nodules by this CAD algorithm. We
chose to optimize the gradient orientation step for this type of
lung nodule because we wanted the algorithm to perform as well
as possible on these difficult to detect lesions.
The results of the gradient orientation kernel anisotropy optimization were initially unexpected by the authors. The Canny
edge detector is designed so that the blurring is performed by the
Gaussian and derivative of Gaussian kernels. Because most CT
images are inherently blurred more in the longitudinal direction
(i.e., -direction), we originally hypothesized that
would compensate and produce more accurate gradient orientations. However, the experiment showed that this effect was
nearly nonexistent on tri-linearly interpolated data. We have not
tested the effect of using an anisotropic kernel on higher order
interpolated data, which may yield different results.
There were several other algorithm parameters that were not
formally optimized. For instance, the hysteresis thresholds used
in edge detection were not optimized. However, in both the
colon and lung, image contrast is excellent and edge detection
was observed to be very robust. Also note that these thresholds do not affect the direction of detected gradients, which are
,
only dependent on the convolution. Another example is
the length of the projected cylinder. We have anecdotally observed both polyp-to-fold and nodule-to-vessel distances to be
. Direct visualization of
typically 15–20 mm, greater than
has demonstrated that cylinder overlap from neighboring structures is generally not a large problem.
D. CAD Algorithm Evaluation
The results of this preliminary evaluation of lung nodule
detection were based on a dataset with a large proportion of
the nodules are due to one patient with metastatic disease
). Although we cannot distinguish the
(
performance of our algorithm on primary bronchogenic carcinoma from the performance on metastases, we believe that the
detection of both primary and metastatic nodules is important.
For those patients with pulmonary metastases secondary to
colorectal cancer, many gynecological cancers, head and neck
cancers, renal cell cancer, malignant melanoma, and sarcomas,
pulmonary resection is an important primary therapy with a
5-year survival rate of 21%-68% [26], [27]. Also, we note that
we did not evaluate the efficacy of our algorithm for detecting
ground glass opacities in the lung. Further studies are needed
to evaluate the performance characteristics on various types of
lesions.
A limitation of the evaluation of colonic polyp detection is
that it uses only supine data. Generally, both prone and supine
images are used for CT colonography, but we evaluated the
algorithm using only supine images because the problem of
matching CAD results between prone and supine images is still
unsolved. Additionally, treating prone and supine images of the
same patient as independent would violate the assumption of
independence that allows the cross-validation estimate of performance to be unbiased. Another point regarding the polyp
evaluation is that not all FOC determined polyps were found
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 6, JUNE 2004
by the gold standard setter in the supine CT images (
), probably due to retained water and/or other factors.
Therefore, the CAD algorithm’s failure to identify these polyps
is not a failure of the algorithm (since they were not visible in
the images) but rather a failure of the CT colonography patient
preparation and/or data collection. Thus, they were not counted
against the algorithm in this evaluation.
While cross-validation mitigates the problem of over-fitting
to the data, it does not remove biases that may be present in
the entire dataset. While effort was to avoid bias in the case
selection, performance with this algorithm on other datasets
may vary with factors such as patient population, image quality,
scanner parameters, etc. Although a greater number of cases
were available, we did not utilize the entire database for this
evaluation because the cross-validation technique required
executing the algorithm on each case over a large number of
, which was computationally prohibitive.
values of
For binary classification problems, the area under the ROC
curve, , has been widely used as a performance metric. Anal, has been deogously, the area under the AFROC curve,
scribed as a performance metric for multiple detection probcalculated by a binormal FROC
lems. We experimented with
curve fitting procedure [28] but found the curve fitting to be unreliable on this data. The FP image model assumes that the operator generally makes less than one FP per image. Data points
much beyond one FP per image become nearly indistinguishable due to the Poisson assumption and thus, the fitted curves are
most unreliable in this region even though it may be of greatest
interest for evaluating a CAD algorithm that will subsequently
be reviewed by a human reader. We chose to use partial area
under the FROC curve as a cost function for training because it
was not prone to curve fitting errors under parametric assumptions. This is analogous to the ROC partial area index [29]. In
particular, we found that using a partial area index was important so that the training optimized for the hardest to find polyps
and nodules rather than the average polyps and nodules.
E. Comparison to Other CAD Algorithms
The SNO method differs from many of the previously proposed lung nodule CAD algorithms [5]–[8] in that rather than
using a variety of basic shape descriptors such as perimeter,
area, volume, sphericity, compactness, elongation, etc., we
focus on a single shape measure that is tuned for the specific
application(s). Other approaches use some type of idealized
shape model [9], [10] but achieve poorer performance, perhaps
due to the lack of flexibility in such an explicit model rather
than an implicit model that describes an entire gamut of shapes.
The approach of McNitt-Gray et al. [30] exemplifies another
important aspect of computer-aided diagnosis, the classification between benign and malignant. While our work does not
specifically address this problem, we envision our work as part
of a larger overall CAD scheme that will at some point also
include classification.
The approach of Vining et al. [11] to polyp detection is notable because it attempts to detect polyps based on wall thickness rather than mucosal surface morphology. However, thus
far, no other groups have reported success with this type of approach. The approaches of both Summers et al. [12], [13] and
PAIK et al.: SURFACE NORMAL OVERLAP
of Yoshida et al. [14]–[16] share in common the use of partial derivatives to compute principle curvatures. However, the
differences in how they are combined and classified vary and
may partially account for the differences in performance. Also,
Yoshida et al.add gradient concentration (GC) and directional
gradient concentration (DGC) in order to improve performance.
These two measures also compute the confluence of gradient
vectors toward a common point, although they are quite different than the SNO method in actual formulation. However,
direct comparison of performance of GC and DGC to this work
is difficult because many other features are combined and also,
per-polyp sensitivity is not reported. Quantitative comparisons
are precluded by differences in study designs and by the relatively small number of datasets used in both this work and other
published works.
While most CAD algorithms are described for a single clinical application, the CAD algorithm described in this paper was
found to be promising at more than one task. It performed favorably compared to many of the aforementioned CAD schemes
although differences in patient populations, CT technology, and
analysis methods preclude strict quantitative comparisons.
673
HT has a value of , the accumulator bin size, that is applied to
polyps and folds alike and another value of that is applied to
nodules and vessels alike.
A. SNO: Nodules and Polyps
The SNO score can be computed in terms of the weight,
,
of the surface normals in a given surface patch, the area of
of surface normals per
each surface patch, and the density
unit area. The expected SNO score of a polyp or nodule is given
by
The weight,
, of each surface normal due to convolution, has contributions from the entire length in the direcin Fig. 2). Using (1) and the relationship
tion (along
, the expected SNO score for a given sphere
radius is as shown in the equation at the bottom of the page.
After factoring, the integral become unity and we get
VII. CONCLUSION
We have 1) developed a novel CAD algorithm, the surface
normal overlap method, for both colonic polyp detection and
solid lung nodule detection, 2) demonstrated the theoretical
traits of this algorithm using a statistical shape model, 3) optimized its performance using a CT simulations and a per-lesion
cross-validation method, and 4) provided a preliminary evaluation of its performance in both detection tasks,. The approach
we have presented is generalized in that it is able distinguish
between focal lesions such as polyps and solid nodules and
background anatomy such as blood vessels and haustral folds.
While the CAD algorithm demonstrated in this paper has
shown promise for both lung nodule and colonic polyp detection, we ultimately envision it as the first stage of a larger CAD
scheme where a set of suspicious locations is passed on to a
second stage, possibly comprised of more computationally intensive classifier(s) that would aim to decrease the FP rate.
Regardless of dependence, the expectation of a sum of
random variables is the sum of the expectations of each random
variable, and we obtain
Reducing further, substituting for , and using the relationship
, we get
Because we model polyps as hemispheres, we use
to get
APPENDIX
In the following sections, the formulas for the expected CAD
score (SNO or HT) of the various types of anatomic objects
(polyps, folds, nodules, and vessels) are derived using the theoretical model. Note that each of the four anatomic object classes
, and , which control the size and
has its own value of ,
degree of shape variability. The SNO method has a value of
that is applied to polyps and folds alike and another
value of
that is applied to nodules and vessels alike.
B. SNO: Vessels and Folds
For the sake of this analysis, a local coordinate system is
chosen with the CAD hit at the origin, the cylinder axis in the
-direction. The index variable varies along and the index
variable varies as a function of angle around the axis.
674
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 23, NO. 6, JUNE 2004
A surface patch that whose position varies along the -direction has a normal vector that is known to pass through
Using (1) and
, the distance from the line
to the CAD
hit at the origin, we get the equation at the bottom of the page.
As before, the integral becomes unity after factoring and the
expectation of a sum is switched for a sum of expectations
This becomes
The expectation of a sum is switched for a sum of expectations
Reducing further and using the relationship
we get
We define
such that
and we get
,
Because we model polyps as hemispheres, we use
to get
D. HT: Vessels and Folds
Similar to before, we start with
We add subscripts to emphasize the dependence of
, and , and we get
on
,
Because we model a fold as a half-cylinder, we use
to get
C. HT: Nodules and Polyps
if
otherwise
The surface normal passes through and and the point
is the point along the surface normal which corresponds to the
quantized value
We get
We model the Hough transform for spheres using a function
that is 1 when a surface normal vector corresponds to a
given accumulator bin of width , and 0 otherwise. This function examines , the distance in
to the true center given
, the quantized value of in the accumulator
if
otherwise
The expectation of a sum is switched for a sum of expectations
We define
such that
In the continuous case,
is only a function of
and we get
and, thus
PAIK et al.: SURFACE NORMAL OVERLAP
675
Because we model a fold as a half-cylinder, we use
to get
E. Noise Limit
In order to calculate the response due to noise, we examine
the response due to a single edge element. We assume a surface
and calculate each algorithm’s response. For
patch of 1
somewhere and, thus
SNO, the patch will have
For HT, a single patch will have
and, thus
In order to make scores comparable with in a constant, we
and
present results by normalizing all SNO scores by
all HT scores by
.
ACKNOWLEDGMENT
The authors would like to thank Dr. D. Naidich, Dr. P. S.
Desmond, Dr. A. Pineda, and the members of the 3-D Medical
Imaging Laboratory in the Department of Radiology at Stanford
University for helpful discussions.
REFERENCES
[1] J. D. Potter, M. L. Slattery, R. M. Bostick, and S. M. Gapstur, “Colon
cancer: A review of the epidemiology,” Epidemiologic Rev., vol. 15, pp.
499–545, 1993.
[2] S. J. Winawer, A. G. Zauber, M. N. Ho, M. J. O’Brien, L. S. Gottlieb, S.
S. Sternberg, J. D. Waye, M. Schapiro, J. H. Bond, and J. F. Panish, “Prevention of colorectal cancer by colonoscopic polypectomy. The national
polyp study workgroup,” New Eng. J. Med., vol. 329, pp. 1977–1981,
1993.
[3] G. M. Strauss and L. Dominioni, “Perception, paradox, paradigm:
Alice in the wonderland of lung cancer prevention and early detection,”
Cancer, vol. 89, pp. 2422–2431, 2000.
[4] T. L. Petty, “Screening strategies for early detection of lung cancer: The
time is now,” JAMA, vol. 284, pp. 1977–1980, 2000.
[5] M. L. Giger, K. T. Bae, and H. MacMahon, “Computerized detection of
pulmonary nodules in computed tomography images,” Investigat. Radiol., vol. 29, pp. 459–465, 1994.
[6] S. G. Armato 3rd, M. L. Giger, C. J. Moran, J. T. Blackburn, K. Doi, and
H. MacMahon, “Computerized detection of pulmonary nodules on CT
scans,” Radiographics, vol. 19, pp. 1303–1311, 1999.
[7] S. G. Armato 3rd, M. L. Giger, and H. MacMahon, “Automated detection of lung nodules in CT scans: Preliminary results,” Med. Phys., vol.
28, pp. 1552–1561, 2001.
[8] M. S. Brown, M. F. McNitt-Gray, J. G. Goldin, R. D. Suh, J. W. Sayre,
and D. R. Aberle, “Patient-specific models for lung nodule detection
and surveillance in CT images,” IEEE Trans. Med. Imag., vol. 20, pp.
1242–1250, Dec. 2001.
[9] Y. Lee, T. Hara, H. Fujita, S. Itoh, and T. Ishigaki, “Automated detection of pulmonary nodules in helical CT images based on an improved
template-matching technique,” IEEE Trans. Med. Imag., vol. 20, pp.
595–604, July 2001.
[10] S. G. Erberich, K. Song, H. Arakawa, H. K. Huang, W. Richard, K. S.
Hoo, and B. W. Loo, “Knowledge-based lung nodule detection from helical CT [abstract],” Radiology, vol. 205P, p. 617, 1997.
[11] D. J. Vining, Y. Ge, D. K. Ahn, and D. R. Stelts, “Virtual colonoscopy
with computer-assisted polyp detection,” in Computer-Aided Diagnosis
in Medical Imaging, K. Doi, H. MacMahon, M. L. Giger, and K. R.
Hoffman, Eds. Amsterdam, The Netherlands: Elsevier Science B.V.,
1999, pp. 445–452.
[12] R. M. Summers, C. F. Beaulieu, L. M. Pusanik, J. D. Malley, R. B.
Jeffrey Jr., D. I. Glazer, and S. Napel, “Automated polyp detector for
CT colonography: Feasibility study,” Radiology, vol. 216, pp. 284–290,
2000.
[13] R. M. Summers, C. D. Johnson, L. M. Pusanik, J. D. Malley, A. M.
Youssef, and J. E. Reed, “Automated polyp detection at CT colonography: Feasibility assessment in a human population,” Radiology, vol.
219, pp. 51–59, 2001.
[14] H. Yoshida and J. Nappi, “Three-dimensional computer-aided diagnosis
scheme for detection of colonic polyps,” IEEE Trans. Med. Imag., vol.
20, pp. 1261–1274, Dec. 2001.
[15] H. Yoshida, Y. Masutani, P. MacEneaney, D. T. Rubin, and A. H.
Dachman, “Computerized detection of colonic polyps at CT colonography on the basis of volumetric features: Pilot study,” Radiology, vol.
222, pp. 327–336, 2002.
[16] J. Nappi and H. Yoshida, “Automated detection of polyps with CT
colonography: Evaluation of volumetric features for reduction of
false-positive findings,” Academic Radiol., vol. 9, pp. 386–397, 2002.
[17] G. Kiss, J. V. Cleynenbreugel, M. Thomeer, P. Suetens, and G. Marchal,
“Computer-aided diagnosis in virtual colonography via combination of
surface normal and sphere fitting methods,” European Radiol., vol. 12,
pp. 77–81, 2002.
[18] S. B. Gokturk, C. Tomasi, B. Acar, C. F. Beaulieu, D. S. Paik, R. B.
Jeffrey Jr., J. Yee, and S. Napel, “A statistical 3-D pattern processing
method for computer-aided detection of polyps in CT colonography,”
IEEE Trans. Med. Imag., vol. 20, pp. 1251–1260, Dec. 2001.
[19] B. Acar, C. F. Beaulieu, S. B. Gokturk, C. Tomasi, D. S. Paik, R. B.
Jeffrey Jr., J. Yee, and S. Napel, “Edge displacement field-based classification for improved detection of polyps in CT colonography,” IEEE
Trans. Med. Imag., vol. 21, pp. 1461–1467, Dec. 2002, to be published.
[20] D. S. Paik, C. F. Beaulieu, R. B. Jeffrey Jr., G. D. Rubin, and S. A. Napel,
“Detection of polyps in CT colonography: A comparison of a computer
aided detection algorithm to 3D visualization methods [abstract],” Radiology, vol. 213P, p. 197, 1999.
[21] J. Canny, “A computational approach to edge detection,” IEEE Trans.
Pattern Anal. Machine Intell., vol. PAMI-8, pp. 679–698, 1986.
[22] C. R. Crawford, private communication, 1998.
[23] R. Kohavi, “A study of cross-validation and bootstrap for accuracy estimation and model selection,” in Proc. 14th Int. Joint Conf. Artificial
Intelligence, 1995, pp. 1137–1145.
[24] C. E. Metz, “Evaluation of CAD methods,” in Computer-Aided Diagnosis in Medical Imaging, K. Doi, H. MacMahon, M. L. Giger, and K. R.
Hoffman, Eds. Amsterdam, The Netherlands: Elsevier Science B.V.,
1999, pp. 543–554.
[25] P. V. C. Hough, “Methods and Means for Recognizing Complex Patterns,” U.S. Patent 3 069 654, 1962.
[26] “Long-term results of lung metastasectomy: Prognostic analyzes based
on 5206 cases. The international registry of lung metastases,” J. Thoracic Cardiovasc. Surg., vol. 113, pp. 37–49, 1997.
[27] V. W. Rusch, “Pulmonary metastasectomy. Current indications,” Chest,
vol. 107, pp. 322–331, 1995.
[28] D. P. Chakraborty, “Maximum likelihood analysis of free-response receiver operating characteristic (FROC) data,” Med. Phys., vol. 16, pp.
561–568, 1989.
[29] Y. Jiang, C. E. Metz, and R. M. Nishikawa, “A receiver operating characteristic partial area index for highly sensitive diagnostic tests,” Radiology, vol. 201, pp. 745–750, 1996.
[30] M. F. McNitt-Gray, E. M. Hart, N. Wyckoff, J. W. Sayre, J. G. Goldin,
and D. R. Aberle, “A pattern classification approach to characterizing
solitary pulmonary nodules imaged on high resolution CT: Preliminary
results,” Med. Phys., vol. 26, pp. 880–888, 1999.