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The segmentation of structures of interest in medical images can result in segmentation boundaries demonstrating two types of concavities: natural and incorrect. This study introduces a generalized method for identification and repair of incorrect

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1
Abstract
—The segmentation of structures of interest in medical images can result in segmentation boundaries demonstrating two types of concavities: natural and incorrect. This study introduces a generalized method for identification and repair of incorrect concavities. Previous methods are framed in terms of the generalized method and new techniques for concavity identification,
α
-hull
, and repair, active contours, are applied. The application of
α
-hull
s also creates a natural hierarchy on concavities that leads to more efficient computations for both the characterization and repair of concavities. The generalized method is evaluated on a database of juxtapleural nodules and compared against two other methods: convex hull and morphological closing operator. The generalized method demonstrates the best overall tradeoff between juxtapleural nodule inclusion and incorrect inclusion of non-nodule tissue from surrounding structures.
Index Terms
—Computer-aided detection (CAD), Concavity, Image Processing, Nodule, Segmentation
I.
I
NTRODUCTION
he computerized analysis of medical images often requires segmentation of structures of interest as the initial step of a larger analytical scheme. The initial segmentation step creates boundaries of the structures of interest and each of these boundaries may contain two possible types of indentations (or “concavities”). First, a concavity may result from the actual shape of the structure. These “natural concavities” correctly capture the morphology of the structure and should not be eliminated from the segmentation boundary. Second, the segmentation method may fail to include some portion of the structure of interest adjacent to the structure’s actual boundary. These are “incorrect concavities” and should be eliminated from the segmentation boundary (Fig. 1). Incorrect concavities in a segmentation boundary have two possible causes. First, image acquisition artifacts or noise may obscure the structure boundary. Common examples of this include metal artifacts in computed tomography (CT) scans and motion artifacts in thoracic magnetic resonance (MR)
Manuscript received August 11, 2008. This work was supported in part by USPHS Grant CA 102085. W. F. Sensakovic, A. Starkey, and S. G. Armato III are with the Committee on Medical Physics, The University of Chicago, Chicago, IL 60637 USA (*phone: 773-834-5107; fax: 773-702-0371; e-mail: wfsensak@uchicago.edu).
scans. Second, physical abnormalities and disease may obscure the structure boundary. For example, if the structure of interest is the lung and juxtapleural lung nodules are present, then the final segmentation boundary may incorrectly exclude these nodules. Several previous studies implemented methods to identify and repair incorrect concavities generated by specific segmentation tasks. A rolling-ball filter was implemented to repair concavities in lung segmentation boundaries in thoracic radiographs [1] and CT scans [2]. A similar method was applied to repair liver segmentation boundaries in abdominal CT scans [3]. Finally, the morphological closing operator was applied to thoracic CT scans to repair lung segmentation boundaries [4]. The purpose of this study is two-fold. First, a generalized method for the identification and repair of incorrect concavities is introduced. Existing methods are framed in terms of the generalized method, and several common techniques are applied to the task of incorrect concavity repair.
A General Method for the Identification and Repair of Concavities in Segmented Medical Images
W.F. Sensakovic*, A. Starkey, and S.G. Armato III
T
Fig. 1. Thoracic CT scan of a lung with corresponding segmentation(white outline). A juxtapleural nodule in the anterolateral portion of the lungcreates an incorrect concavity (white arrow). Several natural concavities are present along the mediastinal boundary (black arrows).
2
Second, proof of principle of the generalized method is demonstrated on several synthetic and clinical examples, and the generalized method is evaluated on a database of juxtapleural lung nodules.
II.
M
ATERIALS AND
M
ETHODS
A.
Overview of the Generalized Method
A generalized method for the identification and repair of incorrect concavities should be independent of both the structure of interest and the specific techniques applied for initial segmentation. The generalized method is composed of three steps (Fig. 2). First, the initial segmentation boundary is analyzed and incorrect-concavity candidates are identified and sorted into a hierarchical tree. Next, each incorrect-concavity candidate is characterized based on a previously defined set of features. These features may be derived from the candidate itself or calculated relative to some separate, previously defined model of the structure of interest. A threshold is applied for each feature, and those candidates that fail to satisfy these thresholds are identified as incorrect concavities. Finally, the incorrect concavities are repaired by replacing each incorrect concavity with suitable segmentation boundary points. The specific technique applied for the repair of an incorrect concavity may be based on properties of the local image, adjacent segmentation boundary points (excluding those within the concavity), or both.
B.
Step 1—Identification of Incorrect-Concavity Candidates
Given the initial segmentation of a structure of interest, the first step in the generalized method is to identify initial segmentation boundary points that form incorrect-concavity candidates. Any solid structure of interest that is segmented from a 2D medical image may be approximated by a simple polygon. The error of this approximation can be made arbitrarily small by appropriate addition of vertices to the polygon. Thus, any segmentation boundary not initially defined as a simple polygon can be represented by simple polygon vertices that approximate the initial segmentation. Let V be a simple polygon defined by n boundary points v
1
to v
n
and X the closed set of points defined by the polygon V. Definition 1: Given the point set
3
ℜ⊂
X
, the set X is globally convex if for any two points
X x x
∈
21
,
the line segment connecting x
1
and x
2
is also contained in X. Any set that is not globally convex is defined as globally concave. Definition 2: Given the point set X, the global convex hull of X, GCH(X), is defined as the intersection of all convex sets that contain X. Note that the GCH(X) is globally convex. Once the global convex hull for the set X is constructed, an incorrect-concavity candidate is identified as a connected set of boundary points of X that are not also elements of the boundary of GCH(X). Unfortunately, the global convex hull alone is inappropriate for identifying incorrect-concavity candidates. The global convex hull method identifies even the smallest concavity (e.g., a one-pixel variation due to noise) as an incorrect-concavity candidate, potentially resulting in thousands of incorrect-concavity candidates for a single segmented structure. Further, given nested concavities, only the largest concavity is identified as an incorrect-concavity candidate. A generalization of global convexity and the global convex hull may be applied to overcome these problems. Definition 3: Let
X x x
∈
21
,
and d(x
1
,x
2
) be the distance between x
1
and x
2
. The set is globally convex relative to Gaussian curvature
α
if all points
21
,
x x
such that
α
2),(
21
≤
x xd
can be connected by a path of curvature no greater than
α
that is contained in X. This definition is similar to the concept of relative convexity defined by Mezey [5]; however, it does not require that the set boundary be twice differentiable and only applies to points in X separated by a distance d or less. The Gaussian curvature of a ball of radius
α
is 1/
α
, thus a set, X that is relatively convex to
α
implies that any two points in X separated by a distance less than 2/
α
can be connected by a path in X that is the boundary of a ball with a radius of a least 1/
α
. If X is restricted to be a continuous set, then the definition of global relative convexity creates a generalized definition of global convex hull called the
α
-hull [6]. Definition 4: The global
α
-hull of X is the intersection of the complement of all balls of radius 1/
α
such that the intersection of the ball and X is empty. The global
α
-hull of X is globally convex relative to
α
. The
α
-hull fixes both deficiencies of the global convex hull for identification of incorrect-concavity candidates. If an
Apply
α
-hulls Application of Feature Thresholds Application of Interpolation MethodHierarchical Data StructureInitial SegmentationIdentification of Incorrect-Concavity CandidatesIdentification of Incorrect Concavities Incorrect Concavity Repair Refined SegmentationCharacterization
Apply
α
-hulls Application of Feature Thresholds Application of Interpolation MethodHierarchical Data StructureInitial SegmentationIdentification of Incorrect-Concavity CandidatesIdentification of Incorrect Concavities Incorrect Concavity Repair Refined SegmentationCharacterization
Fig. 2. Flowchart detailing the generalized method for identification andrepair of incorrect concavities. Note that the “Identification of IncorrectConcavities” stage eliminates natural concavities from the set of incorrect-concavity candidates.
3
initial segmentation boundary is composed of multiple small concavities that should not be identified as incorrect-concavity candidates, then careful choice of the largest
α
value can eliminate them from consideration. If the largest
α
value (
α
max
) is chosen so only these small concavities are identified as incorrect-concavity candidates at the scale of
α
max
, then subtraction of incorrect-concavity candidates identified at the
α
max
scale from the larger set of incorrect-concavity candidates identified at scales
α
i
<
α
max
will result in the elimination of small unwanted contours from consideration. If a segmentation boundary contains nested concavities, a series of
α
-hulls with successively larger values of
α
can be constructed to identify incorrect-concavity candidates at different scales as defined by
α
. This creates a natural hierarchy of incorrect-concavity candidates where the hierarchy is ordered by
α
value (Fig. 3). Note, that the value of
α
can range from 0 (corresponding to the global convex hull) to the largest value which does not identify noise as an incorrect-concavity candidate (
α
max
).
C.
Step 2—Identification of Incorrect Concavities
The
α
-hull is an effective means of identifying incorrect-concavity candidates; however, it is ill-suited in most applications for final identification of incorrect concavities because it mistakenly includes many natural concavities in the set of incorrect-concavity candidates. The number of natural concavities misidentified as incorrect-concavity candidates can be reduced by characterizing all candidates and applying appropriate thresholds to user-defined characterization features. The features for incorrect-concavity candidate characterization are chosen to differentiate incorrect concavities from natural concavities encountered in the set of incorrect-concavity candidates for a given segmentation task. Two classes of features may be applied to characterize the incorrect-concavity candidates. First, features may be chosen to characterize the shape of the incorrect-concavity candidate. Second, although not performed in this study, features may be chosen to characterize the texture of the region defined by the boundary of the incorrect-concavity candidate and the global convex hull. These two classes include a large number of possible features; therefore, this section will be limited to a discussion of several common and widely applicable two-dimensional features. The simplest features for shape characterization of incorrect-concavity candidates are “span” and “depth.” Span is the distance between the first and last boundary points of the incorrect-concavity candidate. Span allows for the elimination of natural concavities from the set of incorrect-concavity candidates based on candidate size. For example, the threshold for span may be set such that the span of the incorrect-concavity candidate formed by the mediastinum in thoracic CT lung segmentation is identified as a natural concavity while incorrect-concavity candidates formed by pulmonary vessels or juxtapleural nodules are identified as incorrect concavities (Fig. 4). Depth is the greatest perpendicular distance between the line segment that defines the span and the boundary points of the incorrect-concavity candidate. Similar to span, depth allows for the elimination of natural concavities from the set of incorrect-concavity candidates based on size. For example, the depth of an incorrect-concavity candidate formed by a pulmonary vessel likely will be large relative to the depth of an incorrect-concavity candidate formed by many diseases (e.g., pleural plaques or pulmonary nodules) (Fig. 4). Thus, appropriate selection of the depth threshold for a given segmentation task will identify incorrect-concavity candidates as either natural concavities or incorrect concavities. The “boundary length” is the number of boundary points that compose the concavity. A minimum boundary length can be imposed to eliminate candidates caused by noise. The area of the candidate region is defined by the closed boundary formed from the concavity boundary points and a line segment connecting the first and last boundary points. Similar to span and depth, the area ensures that the concavity is an appropriate size for an incorrect concavity. The simplest feature for texture characterization of incorrect-concavity candidates is mean gray level. For example, the initial segmentation of lung parenchyma may correctly exclude the bronchus. The natural concavity created by the bronchus may exhibit shape characteristics similar to incorrect concavities. If shape features alone are applied, the natural concavity created by the bronchus may be mistakenly included into the repaired segmentation boundary (Fig. 4). However, if the region formed by the natural concavity is characterized by its mean gray level in addition to shape features, then the low gray level of the region formed by the natural concavity will identify the concavity as a natural concavity and it will be removed from the set of incorrect-concavity candidates. Once the features are chosen for characterization, the thresholds for each feature must be determined. A feature threshold can be set as a fixed value or as a dynamic value relative to other features or to a previously defined structure model. Fixed values are chosen empirically by observing feature values typical of the specific segmentation task. When one feature threshold is set relative to a second feature, both features usually describe similar aspects of the candidate. For example, the threshold for depth may be set as some
α
1
α
2
α
3
< < <
α
4
α
1
α
2
α
3
< < <
α
4
α
1
α
2
α
3
< < <
α
4
Fig. 3. Synthetic image demonstrating concavities of various sizes andthe hierarchical decomposition of those concavities. The hierarchicalconcavity tree is created by the application of
α
-hulls with increasing
α
values. Note that individual concavities nested inside larger concavities areindividually resolved by larger
α
values.
4
percentage of the span. Both depth and span describe the size of the concavity, but defining the threshold for depth relative to span makes the threshold less sensitive to concavity size. Feature thresholds set relative to a previously defined structure model allow thresholds to vary depending on the position of the incorrect-concavity candidate. For example, a model for a healthy lung will demonstrate the lateral portions as almost completely convex (very shallow concavities can be formed by the ribs) with large natural concavities along the mediastinal aspect of the lung formed by the heart, trachea, and vessels. Thus, the depth feature threshold can be set to a very low value for the lateral aspect of the lung and increased near the mediastinum.
D.
Step 3—Incorrect Concavity Repair
Once incorrect concavities are identified from the set of candidates, the incorrect concavities can be repaired. The points that comprise an incorrect concavity are removed, and replacement points are calculated and incorporated into the segmentation boundary. The most basic method for replacing an incorrect concavity is to base the replacement solely on the adjacent segmentation-boundary points, which is equivalent to interpolation along the segmentation boundary. The lowest order interpolation is linear interpolation (i.e., connecting the boundary points adjacent to the concavity with a straight line). Linear interpolation is efficient, simple, and, if the span of the concavity is small (or the curvature of the segmentation boundary is low), in an accurate approximation to the actual boundary of the structure of interest. If the curvature of the segmentation boundary is high, then a linear approximation will not accurately repair the incorrect concavity and higher order interpolation can be applied (Fig. 5). Incorrect concavity repair also may utilize image information beyond the positions of adjacent segmentation-boundary points. Image gray levels may be used to guide an active contour or active surface [7] to repair the incorrect concavity. This method has the potential advantage of creating a highly accurate replacement for the incorrect concavity because both positions of adjacent segmentation-boundary points and image gray level values are used. For example, if disease is present along the boundary of a structure of interest, it may create both a protrusion in the structure-of-interest boundary and an incorrect concavity in the corresponding segmentation boundary (Fig. 6). In such a case, repair methods based only on adjacent segmentation-boundary points will cut through the protrusion and thus fail to completely include it in the final segmentation boundary. However, an appropriately defined active contour initialized between the adjacent segmentation-boundary points could correctly deform to capture the protrusion and thus correctly repair the incorrect concavity.
E.
Database Testing
Fig. 5. Comparison of interpolation methods: linear (white), piecewise hermite cubic (middle gray), and cubic spline (dark gray).
A) B)C)A)A) B)B)C)C)
Fig. 4. Examples of concavity features. A) Span for concavity formed bythe mediastinum (black line) and span for concavity formed by vessel (grayline). B) Depth for concavity formed by disease (top of image: white dottedline) and depth for concavity formed by vessel (middle of image: whitedotted line). C) Lung segmentation (grayed region with white outline)correctly excluding concavity resulting from trachea (circle). Note that depthand span are comparable to disease and thus gray-level may be used toexclude trachea created concavities.
5
The Lung Image Database Consortium (LIDC) public database of lung nodules [8] was selected to evaluate the generalized method for incorrect-concavity identification and repair. For each CT scan of the 85-patient LIDC database (one scan per patient), between one and four experienced thoracic radiologists from the LIDC provided boundaries for each nodule with longest diameter greater than 3mm and less than 30mm. Nodule boundaries were constructed in every section in which the nodule was demonstrated, as individually determined by each radiologist who chose to mark that lesion as a nodule > 3 mm. For each patient an automated method was applied to segment the lung parenchyma the surrounding soft-tissue by thresholding each section at -700 HU and performing a fill operation on the resulting lung regions. The set of all sections containing juxtapleural nodules was then extracted by finding those CT sections in which a radiologist-defined nodule boundary point existed exterior to the segmented lung region. This resulted in 47 juxtapleural nodule instances (103 juxtapleural nodule instance sections) defined across 33 patients. The generalized method was then applied to each lung segmentation boundary to identify and repair incorrect concavities. The method applied
α
-values corresponding to balls with diameters of 60mm, 30mm, 15mm, and 7mm. Depth, concavity boundary length, and area were selected as shape features to characterize incorrect-concavity candidates. The depth threshold was determined both relative to the span (i.e., depth > 50% of span) and in an absolute sense (depth > 2.5mm). The relative threshold was chosen based on a model of a nodule as a disc under the assumption that at least half the disc will be contained within the segmented lung. The absolute depth threshold was based on the radius of the smallest possible nodule (1.5mm); however, since this is roughly equal to 2 pixels at typical CT resolutions (which may be mistaken with noise in the boundary), this number was increased to a depth of 4 pixels. The concavity boundary length threshold was set to half the circumference of the smallest possible nodule (1.5 mm). The area of the concavity threshold was set to less than 25% of the area of the lung to avoid inclusion of the diaphragm. This final parameter was set because diaphragm many be misidentified as nodule, but usually is large relative to the size of the surrounding lung. These parameters were not trained on this particular database, but instead were based on a lung nodule model defined by the physical specifications of a nodule used by the LIDC. Each identified concavity was then repaired by applying both linear and cubic interpolation techniques. The results of the generalized method were compared against two existing incorrect concavity identification and repair techniques: the convex hull and morphological closing operator. As a means of comparison, morphological closing operators with several different structuring element sizes (discs with diameters of 60mm, 45mm, 30mm, 15mm and 7mm) were separately applied to the initial lung segmentation. Sensitivity-type assessment of each method was calculated by counting the number of juxtaplerual nodule instances completely included in the repaired segmentation boundary. It should be noted that if any part of the nodule protruded from the boundary, then that nodule was not categorized as included in the final segmentation. Specificity-type assessment of the method was calculated as the number of pixels in the repaired segmentation region that were not also in the initial segmentation region or in the juxtapleural nodule region (“extraneous area”). A single section was selected to demonstrate the potential application of image information beyond the positions of adjacent segmentation-boundary points in the repair of incorrect concavities (Fig. 6). The juxtaplerual nodule in this section creates an incorrect concavity in the segmented lung boundary. The nodule boundary protrudes exterior to the convex hull of the initial lung segmentation thus making it impossible to completely incorporate into the repaired segmentation boundary. Once the incorrect concavity is identified in the first two steps of the generalized method, an active contour is initialized as the linear interpolation repair of the concavity. A region of interest (ROI) encompassing the concavity and connected chest wall is automatically extracted. The ROI is filtered by setting pixels with values less than -100HU equal to -100HU. This keeps the air-tissue interface from dominating the active contour deformation. A gradient vector field active contour was then applied as the repair method, and the repaired contour was qualitatively evaluated.
Fig. 6. Application of an active contour for repair. A) Original imagewith initial segmentation boundary (dark gray) and repaired boundary (lightgray) superimposed. B) False color image demonstrating the initialsegmentation (dark gray region and line), radiologist defined juxtapleuralnodule region inside (white region) and outside (middle gray region) theinitial segmentation boundary, and the active contour repaired boundar (light gray line). Note that the nodule protrudes exterior to any possiblelinear correction (white dotted line).

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