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the perpendicular on the axis. The localization of edges consists of two steps; finding the initial point for the subsequent search using b(s), and then searching for the zero-crossing of f (s). The initial point, p0, is given by s of the first encountered point satisfying b(s) = 0, starting the search from s = 0, that is, the axis point, to the direction r"a. Given the initial point of the search, if f ( p0) < 0, search outbound from the axis point along the profile for the zero-crossing position p; otherwise, search inbound. After the zero-crossing position q in the opposite direction −r"a is similarly determined, the width (diameter) is given by | p q|.

10.4.2.1 Sheet Case

At every voxel within the candidate regions, the directional second derivative is taken orthogonal to both e"1 and e"2, that is, along e"3, in its corresponding moving frame. This spatially variable directional derivative is written as

f

(x ; σ )

r(x)$

 

2 f (x ; σ )r(x),

(10.40)

sheet

"

e

= " "

"

e " "

 

where r"(x") is the unit vector whose direction is parallel to the medial surface normal of the moving frame. Using a method analogous to that employed in the line case, the profiles f (s) and b(s) of fsheet (x" ; σe) and bsheet (x") are reconstructed for the directions r", respectively. These profiles are then used to determine the edge locations p and q in the two directions r" and −r", respectively, and finally the width (thickness) is obtained as | p q|.

10.4.3 Simulational Evaluation of Medial Axis Detection

We evaluated the medial axis detection performance using synthesized 3-D images of lines with pill-box cross-sections. A simulated partial volume effect was incorporated when synthesizing the images. We focused on the effects of the filter scale σ f used in medial axis detection on the detection various widths of line structures.

Synthesized 3-D images of a line with a circular axis were generated. The

diameter of the line, D, was varied between 2.0 and 8 2( 11.3) voxels. The radius of the circular axis was proportional to D (we used 4 × D). Gaussian noise with 25% standard deviation in the intensity height of pill-box crosssections was added to the images. After line and sheet enhancement filtering with

Hessian-Based Multiscale Enhancement and Segmentation

563

(a)

(b)

(c)

Figure 10.12: Medial axis detection from synthesized 3-D images. All units are voxels. (a) Volume-rendered images of original synthesized 3D images with Gaussian noise. Upper: D = 2.0. Lower: D = 11.3. (b) Examples of successful axis detection where the filter scale σ f is appropriate for the line diameter D. The detected axis points are shown as bright points. Upper: D = 2.0, σ f = 1.4. Lower: D = 11.3, σ f = 4.0. (c) Examples of undesirable axis detection. Upper:

D= 2.0, σ f = 4.0. When diameter D is smaller than that appropriate for the filter scale σ f , many true axis points are overlooked. Lower: D = 11.3, σ f = 1.4. When

Dis larger than that appropriate for σ f , many false axis points are detected.

integration of scales appropriate for the line diameter [7, 11], the candidate regions were extracted by thresholding and extracting large connective com-

ponents. The medial axis points were detected within these regions using the

procedures described in Section 10.4.1 with two values for the filer scale σ f , 2 ( 1.4) and 4.0 voxels. The same candidate regions were used for both values of σ f .

Figure 10.12 shows the volume rendering of the synthesized 3-D images and typical axis detection results. The detection was successful using appropriate combinations of line diameter D and filter scale σ f (Fig. 10.12(b)). Many axis points are overlooked when the filter scale is larger than appropriate, while a number of false detections are made when the filter scale is smaller than appropriate (Fig. 10.12(c)).

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Ratio

1

 

s = 1.4

 

 

 

 

 

 

Axis Detection

0.75

 

sff = 4.0

 

 

0.5

 

 

 

 

 

0.25

 

 

 

 

 

False

0

 

 

 

 

 

2

2.82

4

5.66

8

11.31

 

Line_Diameter (voxels)

 

 

 

(a)

 

 

(voxels)

2

 

sf

=1.4

 

 

 

 

 

 

1.5

 

sf

=4.0

 

 

 

 

 

 

 

 

Error

1

 

 

 

 

 

 

 

 

 

 

 

Position

0.5

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

2

2.82

4

5.66

8

11.31

Line_Diameter (voxels)

(c)

Ratio

2

 

sf

= 1.4

 

 

 

 

 

 

Detection

1.5

 

sf

= 4.0

 

 

1

 

 

 

 

 

 

 

 

 

 

 

True Axis

0.5

 

 

 

 

 

0

2.82

4

5.66

8

11.31

 

2

Line_Diameter (voxels)

 

 

 

(b)

 

 

(degree)

10

 

sf

=1.4

 

 

8

 

 

 

 

s =4.0

 

 

 

 

f

 

 

 

6

 

 

 

 

 

Error

 

 

 

 

 

4

 

 

 

 

 

Angle

2

 

 

 

 

 

0

 

 

 

 

 

 

 

 

 

 

 

 

2

2.82

4

5.66

8

11.31

Line_Diameter (voxels)

(d)

Figure 10.13: Performance evaluation of medial axis detection. See text for the definitions of true and false detections. (a) False positive detection ratio, which is the ratio of the number of false detections to all the detections. The ratio was zero for all the line diameters at σ f = 4.0 voxels. (b) True positive detection ratio, which is the ratio of the number of true detections to all the analytically determined points. (c) Average position error of axis points regarded as true detections. The distance between detected points and analytically determined points was used as the error. (d) Average angle error of the directions of axis points regarded as true detections.

Figure 10.13 shows the performance evaluation results. Detected axis points were evaluated by comparing them with analytically determined axis points. We regarded a detected point as a true detection if the distance between its position and one of the analytically determined points was within two voxels; otherwise, detected points were regarded as false. The false and true positive detection ratios are shown in Figs. 10.13(a) and 10.13(b); the plots verify the observations in Fig. 10.12. The positions and directions of the detected axis points regarded as true detections were compared with analytically determined ones (Figs. 10.13(c) and 10.13(d)). These graphs clarify the effect of σ f on accurate and reliable axis detection.

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565

Figure 10.14: Diameter estimation of bronchial airways from CT images.

(a) Original CT images. The bronchial airway regions, which are darker than the surrounding structures, are shown by arrows. (b) Detection of medial axes at three different scales. Left: σ f = 1.4 voxels. Middle: σ f = 2.0 voxels. Right:

σ f = 2.8 voxels. (c) Diameter estimation at the three different scales. (A color version of this figure will appear on the CD that accompanies the volume.)

10.4.4 Examples

10.4.4.1 Bronchi Diameter Quantification from 3-D CT Data

The line width quantification method was applied to chest CT images taken by a helical CT scanner to determine the diameters of bronchi. The original voxel dimensions were 0.29 × 0.29 × 1.0 (mm3). In order to make the voxel isotropic, sinc interpolation was applied along the z-direction. The volume size used in the experiment was 90 × 70 × 80 (voxels) after interpolation.

Figure 10.14(a) shows the original CT images. After the initial region extrac-

tion by thresholding the line filtered images, the medial axis was detected using

 

 

 

σ f =

2, 2, and 2 2 voxels. Figure 10.14(b) shows the results of axis detection

at the three different scales. Note that the axis points of thin structures were detected only at the smaller two scales while those of large structures (the right segment) were stably extracted at the larger two scales. Figure 10.14(c) shows the results of diameter estimation using σe = 1.2 voxels based on the medial axes at these three scales.

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Figure 10.15: Thickness estimation of cartilages from MR images. (a) Original MR images. The acetabular (pelvic side) cartilages are shown by arrowheads and the femoral head cartilages by arrows. (b) Thickness distribution of acetabular cartilages. The bone regions are volume-rendered in white. (c) Thickness distribution of femoral cartilages. (A color version of this figure will appear on the CD that accompanies the volume.)

10.4.4.2Hip Joint Cartilage Thickness Quantification from 3-D MR Data

The sheet width quantification method was applied to MR images of a hip joint [37, 38] to determine the thickness of hip joint cartilage. The original voxel dimensions were 0.62 × 0.62 × 1.5 (mm3). Sinc interpolation was applied along the z-direction to make the voxel isotropic, and then further applied along all the three directions to make the resolution double. The resultant sampling pitch was 0.31 (mm) in all the three directions. The volume size used in the experiment was 256 × 256 × 100 (voxels) after interpolation.

As shown in Fig. 10.15(a), cartilages are thin structures; thickness distributions are considered to be particularly important in the diagnosis of joint diseases. The initial cartilage regions were extracted from the enhanced images by the sheet filter. The medial surfaces were extracted using σ f = 1.4 voxels. Figures 10.15(b) and 10.15(c) show the results of thickness distribution estimated using σe = 1.2 voxels. We also obtained the thickness distributions using

σe = 1.0 voxel for comparison purposes. The average thickness estimated using

σe = 1.2 voxels was Tf = 4.24 voxels and Ta = 3.50 voxels for the femoral and

Hessian-Based Multiscale Enhancement and Segmentation

567

acetabular cartilages, respectively, compared with

Tf = 4.16 voxels and

Ta = 3.39 voxels using σe = 1.0 voxel.

In the related work [10], hip joint cartilages were assumed to be distributed on a sphere approximating the femoral head. The user needs to specify the center of the sphere, and the cartilage thickness is then estimated along radial directions from the specified center. The method applied here does not use the sphere assumption, and thus can potentially be applied to badly deformed hip joints as well as to articular cartilages of other joints.

10.5Analysis for Sheet Width Quantification Accuracy

In this section, we present a systematic approach to the accuracy validation of width quantification. Especially, we investigate inherent limits on the accuracy of sheet width measurement described in the previous section arising from finite resolution. We focus on MR imaged structure, and especially address the question as to how the accuracy depends on the orientation of sheet structures when a voxel shape is anisotropic in MR imaging. In the following, a theoretical procedure for ascertaining the inherent limits on the accuracy of sheet width (thickness) measurement in MR images is presented.

10.5.1Mathematical Modeling of MR Imaging and Width Measurement Processes

10.5.1.1 Modeling a Sheet Structure

A 3-D sheet structure orthogonal to the x-axis is modeled as

 

s0(x" ; τ ) = Bar(x ; τ ),

 

 

(10.41)

where x

(x, y, z)$, and

 

 

 

 

 

 

 

 

 

" =

 

 

 

 

 

 

 

 

 

 

 

 

=

 

L

x < 21 τ

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Bar(x ; τ )

 

L0

 

21 τ

x

 

21 τ,

(10.42)

 

 

 

L+

x > 2 τ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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(a)

(b)

Figure 10.16: Modeling 3-D sheet structures. (a) Bar profile of MR values along sheet normal direction with thickness τ . L0, L, and L+ denote sheet object, left-side, and right-side background levels, respectively. (b) 3-D sheet structures with thickness τ and normal orientation r"θ ,φ . ( c 2004 IEEE)

in which τ represents the thickness (width) of the sheet. L0, L, and L+ are the MR signal intensities of the sheet and both sides of backgrounds, respectively (Fig. 10.16(a)). Let (θ , φ) be a pair of latitude and longitude which represents the normal orientation of the sheet given by

r

(cos θ cos φ, cos θ sin φ, sin θ )$.

(10.43)

"θ ,φ =

 

 

 

 

 

The 3-D sheet structure with orientation r"θ,φ is written as

 

 

s(x ; τ, r

)

s (x ; τ ),

(10.44)

 

"

"θ ,φ

 

= 0 "

 

where x" = Rθ ,φ x", in which Rθ,φ denotes a 3 × 3 matrix representing rotation which enables the normal orientation of the sheet s0(x" ; τ ), i.e. the x-axis, correspond to r"θ ,φ (Fig. 10.16(b)).

10.5.1.2 Modeling MR Image Acquisition

The 1-D point spread function (PSF) of MR images [39] is given by

 

1 sin(π

x

)

 

 

 

 

 

 

m(x ; x) =

 

 

 

 

x

,

(10.45)

Nx sin(π

 

x

)

 

 

 

 

 

 

 

 

Nx x

 

 

where Nx is the number of samples in the frequency domain, and x represents the sampling interval in the spatial domain. Eq. (10.45) is well approximated

Hessian-Based Multiscale Enhancement and Segmentation

569

[40] by

 

x ; x .

 

m(x ; x) = Sinc

(10.46)

 

 

1

 

 

where

 

 

 

 

 

Sinc(x ; w) =

sin(π wx)

(10.47)

 

 

.

 

π wx

The 3-D PSF is given by

 

 

 

 

 

m (x" ; x, y, z) = m (x ; x) · m (y ; y) · m (z ; z),

(10.48)

where x, y, and z are sampling intervals along the x-axis, y-axis, and z-axis,

respectively.

In actual MR imaging, the magnitude operator is applied to the complex number obtained at each voxel by FFT reconstruction, whose effects are not negligible [41]. Thus, the MR image of the sheet structure with orientation r"θ ,φ and thickness τ is given by

f (x") = |s(x" ; τ, r"θ ,φ ) m(x" ; x, y, z)|,

(10.49)

where denotes the convolution operation.

10.5.1.3 Thickness Determination Procedure

In this chapter, we restrict the scope of our investigation to the sheet model described in section 10.5.1 (Fig. 10.16), that is, a sheet structure with constant thickness τ and orientation r"θ ,φ . We define the thickness measured from the MR imaged sheet structure as the distance between both sides of image edges along the sheet normal vector. As long as the sheet model shown in Fig. 10.16 is considered, other definitions of measured thickness, for example, the shortest distance between both sides of the image edges, generally give the same thickness value. We define the image edges as the zero-crossings of the second directional derivatives along the sheet normal vector, which is equivalent to the Canny edge detector [42]. Gaussian blurring is typically combined with the second directional derivatives to adjust scale as well as reduce noise.

The partial second derivative combined with Gaussian blurring for the MR image f (x"), for example, is given by

fxx(x" ; σ ) = gxx(x" ; σ ) f (x"),

(10.50)

gxx(x" ; σ ) =

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Sato

where

2

∂ x 2 Gauss(x" ; σ ), (10.51) in which Gauss(x" ; σ ) is the isotropic 3-D Gaussian function with SD σ . The second directional derivative along r"θ ,φ is represented as

f

(x ; σ, r

)

=

g

xx

(x

; σ )

 

f (x),

(10.52)

 

"

"θ ,φ

 

 

"

 

"

 

where x" = Rθ ,φ x", in which Rθ,φ denotes a 3 × 3 matrix representing rotation which enables the normal orientation of the sheet s0(x" ; τ ), i.e. the x-axis, correspond to r"θ ,φ (Fig. 10.16(b)). Similarly, the first directional derivative along r"θ,φ is represented as

f

(x ; σ, r

)

=

g

x

(x

; σ )

 

f (x),

(10.53)

 

"

"θ ,φ

 

 

"

 

"

 

Practically, the first and second directional derivatives can be calculated in a computationally efficient manner using the Hessian matrix 2 f (x" ; σ ) and gradient vector f (x" ; σ ), respectively. The first and second directional derivatives along the normal direction r"θ ,φ of the sheet structure are given by

 

f

(x ; σ, r

 

 

)

r$

 

f (x ; σ ),

 

(10.54)

 

 

"

"θ ,φ

 

= "θ ,φ

 

"

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

f

(x ; σ, r

)

 

 

r$

2 f (x ; σ )r

,

(10.55)

 

 

"

"θ ,φ

 

= "θ ,φ

 

"

"θ ,φ

 

 

respectively.

Thickness of sheet structures can be determined by analyzing 1-D profiles of f (x" ; σ, r"θ ,φ ) and f (x" ; σ, r"θ ,φ ) along straight line given by

x" = s · r"θ ,φ ,

(10.56)

where s is a parameter representing the position on the straight line. By substituting Eq. (10.56) for x" in f (x" ; σ, r"θ ,φ ) and f (x" ; σ, r"θ ,φ ),

f (s)

=

f (s

r

 

; σ, r

 

),

(10.57)

 

 

 

· "θ ,φ

"θ ,φ

 

 

and

 

 

 

 

 

 

 

 

 

f

(s)

=

f (s

r

; σ, r

),

(10.58)

 

 

 

· "θ ,φ

 

"θ ,φ

 

 

 

are derived respectively. Figure 10.17(a) shows a schematic diagram for the 1-D profile processing. Both sides of the boundaries for sheet structures can be

Hessian-Based Multiscale Enhancement and Segmentation

571

(a)

(b)

(c)

Figure 10.17: Thickness determination procedure using zero-crossings of the second directional derivatives along sheet normal direction. (a) Basic concept of thickness determination procedure. (b) Interpolation of discrete MR data.

(c) Zero-crossing search procedure. ( c 2004 IEEE)