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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 9, SEPTEMBER 2003
Limits on the Accuracy of 3-D Thickness
Measurement in Magnetic Resonance
Images—Effects of Voxel Anisotropy
Yoshinobu Sato*, Member, IEEE, Hisashi Tanaka, Takashi Nishii, Katsuyuki Nakanishi, Nobuhiko Sugano,
Tetsuya Kubota, Hironobu Nakamura, Hideki Yoshikawa, Takahiro Ochi, and Shinichi Tamura, Member, IEEE
Abstract—Measuring the thickness of sheet-like thin anatomical
structures, such as articular cartilage and brain cortex, in threedimensional (3-D) magnetic resonance (MR) images is an important diagnostic procedure. This paper investigates the fundamental
limits on the accuracy of thickness determination in MR images.
We defined thickness here as the distance between the two sides
of boundaries measured at the subvoxel resolution, which are the
zero-crossings of the second directional derivatives combined with
Gaussian blurring along the normal directions of the sheet surface. Based on MR imaging and computer postprocessing parameters, characteristics for the accuracy of thickness determination
were derived by a theoretical simulation. We especially focused on
the effects of voxel anisotropy in MR imaging with variable orientation of sheet-like structure. Improved and stable accuracy features were observed when the standard deviation of Gaussian blurring combined with thickness determination processes was around
2 2 times as large as the pixel size. The relation between voxel
anisotropy in MR imaging and the range of sheet normal orientation within which acceptable accuracy is attainable was also clarified, based on the dependences of voxel anisotropy and the sheet
normal orientation obtained by numerical simulations. Finally, in
vitro experiments were conducted using an acrylic plate phantom
and a resected femoral head to validate the results of theoretical
simulation. The simulated thickness was demonstrated to be wellcorrelated with the actual in vitro thickness.
Index Terms—Articular cartilage, brain cortex, point spread
function, quantitative image analysis, spatial resolution, thickness
determination, three-dimensional imaging.
Manuscript received July 4, 2002; revised March 21, 2003. This work was
supported in part by the Japan Society for the Promotion of Science (JSPS)
Research for the Future Program JSPS-RFTF99I00903 and JSPS Grant-in-Aid
for Scientific Research (B)(2)15300059. The Associate Editor responsible for
coordinating the review of this paper and recommending its publication was J.
Duncan. Asterisk indicates corresponding author.
*Y. Sato is with the Division of Interdisciplinary Image Analysis, Osaka University Graduate School of Medicine, 2-2-D11, Yamada-oka, Suita, Osaka, 5650871, Japan (e-mail: [email protected]).
H. Tanaka, K. Nakanishi, and H. Nakamura are with the Department of Radiology, Osaka University Graduate School of Medicine, Suita, Osaka, 565-0871,
Japan.
T. Nishii, N. Sugano, and H. Yoshikawa are with the Department of
Orthopaedic Surgery, Osaka University Graduate School of Medicine, Suita,
Osaka, 565-0871, Japan.
T. Kubota is with Tsuyama National College of Technology, 624-1, Numa,
Tsuyama, Okayama 708-8509, Japan.
T. Ochi is with the Division of Applied Medical Engineering, Osaka University Graduate School of Medicine, Suita, Osaka, 565-0871, Japan.
S. Tamura is with the Division of Interdisciplinary Image Analysis, Osaka
University Graduate School of Medicine, Suita, Osaka, 565-0871, Japan.
Digital Object Identifier 10.1109/TMI.2003.816955
I. INTRODUCTION
T
HICKNESS measurement of sheet-like or plate-like thin
anatomical structures in magnetic resonance (MR) images
is an important procedure in clinical practice. For example, diagnosis of joint diseases requires the evaluation for the distribution of articular cartilage thicknesses [1]–[8], and diagnosis
of specific neuropsychiatric disorders needs the assessment of
cortical thicknesses in the brain [9]–[11]. While several methods
for thickness quantification have been proposed [5], [6], [8], [9],
inherent limits on accuracy arising from finite resolution have
not been adequately evaluated. Previously, Sato et al. examined
the limits on accuracy resulting from partial volume averaging
with emphasis on the effect of anisotropic voxels by software
simulation [8]. However, the modeling of MR image acquisition was insufficient and validation with actual MR images was
not carried out in that study.
The aim of the present study is, therefore, to provide a theoretical procedure for ascertaining the inherent limits on the
accuracy of thickness determination in MR images. A preliminary report was communicated earlier [12]. In the present paper,
thickness is defined as the distance between the two sides of
boundaries of sheet structures measured at the subvoxel resolution, which are the zero-crossings of the second directional
derivatives along the normal directions of the sheet surface. We
focus on the effects of the parameters relevant to computer postprocessing as well as those of MR imaging parameters on thickness accuracy. For the effects of computer postprocessing parameters, the dependences of the standard deviation (SD) in
Gaussian blurring combined with thickness determination processes are examined, and its role in obtaining improved and
stable accuracy characteristics is elucidated. To determine the
effects of MR imaging parameters, we especially address the
question as to how the accuracy depends on the orientation of
sheet structures when a voxel shape is anisotropic. Sheet structures of interest are often distributed within specific orientation
ranges. If they are situated on an approximated cylindrical surface, the voxel shape can be highly anisotropic to balance the
signal-to-noise ratio (SNR) and thickness accuracy. As a result,
the resolution in the plane orthogonal to the axis of the cylinder
is much higher than that along the axis. However, if their orientations are randomly distributed or unknown, isotropic (cubic)
voxel shape might be the best. We establish a method for evaluating the dependences of sheet orientation, voxel anisotropy,
and Gaussian SD on the accuracy of thickness measurement
0278-0062/03$17.00 © 2003 IEEE
SATO et al.: LIMITS ON THE ACCURACY OF 3-D THICKNESS MEASUREMENT IN MAGNETIC RESONANCE IMAGES
by numerical simulations based on mathematical modeling of
sheet structures, MR imaging, and postprocessing. The simulation method is validated by in vitro experiments with the actual
measurements. The characteristics of accuracy determination
obtained by the simulations are useful in delivering objective
criteria for designing optimal imaging and postprocessing protocols, provided that the orientation distribution of sheet structures and the fixed volume of a voxel directly related to SNR are
known.
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(a)
II. THEORY
A. Modeling a Sheet Structure
A three-dimensional (3-D) sheet structure orthogonal to the
axis is modeled as
(1)
where
, and
(b)
(2)
, and
in which represents the thickness of the sheet. ,
are the MR signal intensities of the sheet and both sides
of backgrounds, respectively [Fig. 1(a)]. Let ( , ) be a pair of
latitude and longitude which represents the normal orientation
of the sheet given by
(3)
The 3-D sheet structure with orientation
Fig. 1. Modeling 3-D sheet structures. (a) Bar profile of MR values along
sheet normal direction with thickness . L , L , and L denote sheet object,
left-side, and right-side background levels, respectively. (b) Three-dimensional
r .
sheet structures with thickness and normal orientation ~
of the sheet structure with orientation
given by
and thickness
is
(9)
where
denotes the convolution operation.
is written as
(4)
, in which
denotes a 3 3 matrix reprewhere
senting rotation which causes the normal orientation of the sheet
, i.e., the axis, to correspond to
[Fig. 1(b)].
B. Modeling MR Image Acquisition
The one-dimensional (1-D) point spread function (PSF) of
MR images [13] is given by
(5)
is the number of samples in the frequency domain,
where
represents the sampling interval in the spatial domain.
and
Equation (5) is well-approximated [14] by
(6)
C. Thickness Determination Procedure
In this paper, we restrict the scope of our investigation to the
sheet model described in Section II-A (Fig. 1), that is, a sheet
. We destructure with constant thickness and orientation
fine 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. 1 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 [16]. 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 blur, for example, is given by
ring for the MR image
(10)
where
where
(7)
(11)
The 3-D PSF is given by
(8)
,
, and
are sampling intervals along the axis,
where
axis, and 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 [15]. Thus, the MR image
is the isotropic 3-D Gaussian function
in which
is repwith SD . The second directional derivative along
resented as
(12)
, in which
denotes a 3 3 matrix reprewhere
senting rotation which causes the normal orientation of the sheet
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, i.e., the axis, to correspond to
larly, the first directional derivative along
[Fig. 1(b)]. Simiis represented as
(13)
Practically, the second and the first directional derivatives can
be calculated in a computationally efficient manner using the
Hessian matrix and gradient vector, respectively. The second
directional derivative along the normal direction of the sheet
structure is given by
(14)
where
is the Hessian matrix given by
(15)
Similarly, the first directional derivative along the sheet normal
is given by
(a)
(16)
where
is the gradient vector given by
(17)
Thickness of sheet structures can be determined by analyzing
and
along straight
1-D profiles of
line given by
(18)
is a parameter representing the position on the
where
and
straight line. By substituting (18) for in
,
(b)
(19)
and
(20)
are derived, respectively. Fig. 2(a) shows a schematic diagram
for the 1-D profile processing. Both sides of the boundaries for
sheet structures can be defined as the points with the maximum
among those satisfying the conand minimum values of
. Let
have its maximum and
dition given by
and
, respectively. The meaminimum values at
sured thickness, is defined as the distance between the two
detected boundary points, which is given by
(21)
The procedures for thickness determination from discrete real
3-D MR images are described later in Section III-B3.
D. Frequency Domain Analysis of MR Imaging and Thickness
Determination
In order to elucidate the effects of MR imaging and postprocessing parameters, observations in the frequency domain
and
are helpful. All the processes to obtain
from the original sheet structure
are
modeled as linear filtering processes excepting the magnitude
operator applied in (9).
(c)
Fig. 2. 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.
1) Modeling a Sheet Structure: The Fourier transform of
, is given
3-D sheet structure orthogonal to the axis,
by
(22)
denotes the unit
where represents the Fourier transform,
. Note that
impulse, and
when
and
in
.
The Fourier transform of 3-D sheet structure whose normal is
,
, is given by
(23)
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where
, in which
denotes a 3 3 matrix repaxis correspond to
resenting rotation which enables the
[Fig. 3(a)].
has
In the 3-D space of the frequency domain,
energy only in the 1-D subspace represented as a straight line
given by
(24)
where is a parameter representing the position on the straight
line. By substituting (24) for in (23), the following is derived:
(25)
represents energy distribution along (24). Analysis
where
, is sufficient to exof the degradation of 1-D distribution,
amine the effects of MR imaging and postprocessing parameters
is the
in the subsequent processes. It should be noted that
.
1-D sinc function when
2) Modeling MR Image Acquisition: The Fourier transform
of MR PSF is given by
(a)
(26)
where
[Fig. 3(b)], and
otherwise.
(27)
By substituting (24) for in (26) to obtain 1-D frequency
, the following is derived
component affecting
(b)
(28)
Thus, the Fourier transform of MR image of the sheet structure,
is given by
(29)
represents the inverse Fourier transform. If
is a nonnegative function,
is
and all the processes can
given by
be described as linear filtering processes. Deformation of the
original signal due to truncation is clearly understandable in
the frequency domain [Fig. 3(c)].
3) Gaussian Derivatives of MR Imaged Sheet Structure: The Fourier transform of the second derivative of
Gaussian of is given by
where
(30)
and that of the second directional derivative along
sented as
is repre-
(31)
, in which
denotes a 3 3 matrix
where
axis correspond to
representing rotation which enables the
. The 1–D frequency component of
affecting
is given by
(32)
(c)
Fig. 3. Frequency domain analysis of sheet structure modeling, MR imaging,
and thickness determination. (a) Modeling a sheet structure. In the frequency
domain, a sheet structure is basically modeled as the sinc function whose width
is inversely proportional to the thickness in the spatial domain. (b) Modeling MR
imaging. It is assumed here that
. The voxel size determines
the frequency bandwidth of each axis, which is also inversely proportional to
the size in the spatial domain. (c) Modeling MR image acquisition of a sheet
structure. In the frequency domain, imaged sheet structure is essentially the
band-limited sinc function.
1 =1 =1
Similarly, the 1-D component of the first directional derivative
, is obtained.
of Gaussian,
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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 9, SEPTEMBER 2003
Finally, the Fourier transforms of
and given by
and
are derived
(33)
Hence, it can be assumed that the resolution along the axis is
lower than that in the - plane and that pixels in the - plane
, and a measure of voxel anisotropy
are square, i.e.,
. In the simulations, we assumed that
can be defined as
(35)
and
(34)
respectively.
The 1-D profiles along the sheet normal direction of the
Gaussian derivatives of MR imaged sheet structures [(19) and
(20)] are obtained by inverse Fourier transform of (33) and (34),
and then thickness is determined according to the procedure
shown in Fig. 2(a). While simulating the MR imaging and
Gaussian derivative computation described in Section II-C
essentially requires 3-D convolution in the spatial domain,
only 1-D computation is necessary in the frequency domain,
which drastically reduces computational cost. In the following
sections, we examine the effects of various parameters, which
are involved in the sheet model, MR imaging resolution, and
thickness determination processes, on measurement accuracy.
Efficient computational methods of simulating MR imaging
and postprocessing thickness determination processes are
essential, and thus simulating the processes by 1-D signal
processing in the frequency domain is regarded as the key to
comprehensive analysis.
III. MATERIALS AND METHODS
A. Numerical Simulation
In order to examine the effects of various parameters on
the accuracy of thickness determination, numerical simulation
based on the theory described in Section II was performed.
The parameters used in the simulation were classified into
,
,
, and
for
the following three categories: ,
,
, and
for determining MR
defining sheet structures,
imaging resolution, and Gaussian SD, , used in computer
postprocessing for thickness determination.
was obWe assumed that the estimated sheet thickness
tained under the condition that the sheet normal orientation
was known. The numerical simulation was performed in the frequency domain exactly in the same manner as described in Sec,
, ,
tion II-D. Based on sheet structure parameters ,
, MR imaging parameters
,
, and
, and postand
and
were obtained by
processing parameter ,
1-D computation in the frequency domain according to (33) and
and
were obtained by
(34), respectively. And then,
and
, respectively.
inverse Fourier transform of
and
, thickness was estimated using (21).
Using
Finally, estimated thickness was compared with the actual
thickness to reveal the limits on accuracy. It should be noted
that only 1-D computation was necessary for 3-D thickness determination in our numerical simulation.
In the simulation, the effect of anisotropic resolution of MR
be the pixel
volume data was the focus. Let
size within the slices. Resolution of MR volume data is typically
anisotropic because they usually have lower resolution along the
third direction (orthogonal to the slice plane) than within slices.
without loss of generality, and thus,
(36)
We performed the above described numerical simulation with
,
, ,
,
, and .
different combinations of ,
B. In Vitro Experiments
To validate the numerical simulation, the postprocessing
method for thickness determination was used to measure real
MR images of two different objects, an acrylic plate phantom
and a resected femoral head.
1) Plate Phantom: A phantom of sheet-like objects with
known thickness was used. It consisted of four acrylic plates
) with thickness
, 1.5, 2.0, and 3.0
of 80 80 (
(mm), placed parallel to each other with an interval of 30 mm
[Fig. 4(a)]. The phantom was submerged in a water bath so that
the background (water) showed higher intensity as contrasted to
low intensity objects (acrylic plates). Three–dimensional MR
images (TR/TE/flip angle/matrix/FOV/slice thickness: 12.8
ms/5.6 ms/5/256 256/160 mm/1.5 mm) of the phantom were
obtained using a fast spoiled gradient-echo sequence (FSPGR).
and
The voxel size was
. Thus
(37)
Thirteen data sets of 3-D MR images were acquired with different normal positions of the phantom plates, eight with variand
) and fixed
able (
(
), and five with variable (
and
) and fixed (
). In the obtained MR images, we oband
. Fig. 4(b) and (c) shows
served
examples of the MR images. We compared actually measured
thickness from the real MR data with the computational thickness calculated by the numerical simulations.
2) Femoral Head Specimen: The resected femoral head
used in the study was approximately spherical in shape, with
articular cartilage distributed on its surface [Fig. 5(a)]. The
cartilage was the subject of the experiments, in which we
assumed it was distributed on a spherical surface. The articular
cartilage was imaged as a bright sheet structure distributed
on the spherical head surface. The thickness of the cartilage
was then measured along the normal direction of the spherical
surface approximating the femoral head. The cartilage did not
completely satisfy the restrictions on the sheet model assumed
in Section II, that is, constant thickness and orientation (Fig. 1).
However, it did not contain large variations in its thickness and
orientation and, thus, approximately satisfied the restrictions.
3-D MR images of sagittal sections were obtained using
3-D-spoiled gradient-echo sequences (SPGR) [7] under the
following two conditions.
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(a)
(a)
(b)
(b)
(c)
Fig. 5. Resected femoral head and its MR images. In subfigures (b) and (c),
the horizontal and vertical axes of the images correspond, respectively, to the x
axis and z axis in the left frame and the x axis and y axis in the right frame.
(a) Physical appearance. Side (left) and top (right) views are shown. (b) MR
images with isotropic imaging. The voxel size was
:
. (c)
MR images with routine imaging. The voxel size was
:
and
:
. When the normal orientations were close to perpendicular to
the x-y plane, i.e., , imaged cartilage appeared to be more blurred and
thicker (shown by arrows).
1 = 1 5 mm
(c)
Fig. 4. Acrylic plate phantom and its MR images. (a) Physical appearance. (b)
and (c) MR images. The horizontal and vertical axes of the images correspond
to the x axis and z axis, respectively. The voxel size was
:
and
:
. As can be easily observed by naked eye, the acrylic plate
with appears to be imaged slightly thicker in and than and .
1 = 1 5 mm
= 1 mm
=0
=0
1 = 0 625 mm
= 45
=0
• Isotropic imaging—MR imaging with isotropic (cubic)
[Fig. 5(b)]. Thus, voxel
voxels:
.
anisotropy is equal to
• Routine imaging—MR imaging with anisotropic
(noncubic) voxels performed in clinical routine:
,
[Fig. 5(c)]. Thus,
.
voxel anisotropy is equal to
' 90
1 = 1 = 0 7 mm
1 = 0 625 mm
To obtain an acceptable SNR, the imaging time with isotropic
voxels was three times as long as that employed in routine
imaging with anisotropic voxels. In each MR image, the matrix
size was 256 256. In the obtained MR images, we observed
,
, and
. In this case,
approximately
true thickness was unknown. In Section IV, we show that
measured thickness with isotropic voxels can be regarded as
a good approximation of true thickness under the condition
and
. We evaluated
that
thickness measured from the MR data with routine imaging by
using the thickness from the MR data with isotropic imaging as
a reference approximating true thickness .
3) Procedures for Thickness Determination From Real MR
Images: The theory described in Section II is for continuous
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images. However, real MR images are discrete samples of continuous ones. Therefore, a thickness determination method from
discrete samples is described below.
Sinc interpolation: Sinc interpolation is known as the
ideal interpolation to recover original continuous signal from
its discrete samples, when the original signal is band-limited
with respect to the sampling frequency. In order to reduce the
effects of discretization, 3-D MR images were interpolated
using sinc interpolation [17], [18] so that a) the signal sampling
was isotropic in all three directions and b) the image matrix size
was doubled [Fig. 2(b)], i.e., sinc interpolation was performed
, where is the unit voxel interval for
to satisfy
(
, and
interpolated discrete volume data set
, , and are integer). It was difficult to use value smaller
due to large computational cost and memory
than
requirement. It should be noted, however, that the volume data
were inherently more blurred in the direction
of
than in the - plane even if the signal sampling was isotropic
in all three directions, that is, the , , and directions.
Detection of sheet structures: In the real MR images, detection of sheet structures was needed before thickness quantification. To facilitate the detection, sheet enhancement filtering
with multiscale integration was performed according to the following equation:
(38)
(in which
, 2, 3), and
is the
where
denotes
normal direction of the sheet at position .
at combined with
the second directional derivatives along
Gaussian blurring of scale (SD) . In (38), was multiplied for
the normalization of each scale [19], [20]. (See Appendix for
.)
the procedure of computing
was
In the experiments using the acrylic plate phantom,
fixed to the known sheet normal vector irrespective of position . In the experiments using resected femoral head,
where is the manually specified center position
of the sphere approximating the femoral head.
were thresholded, with the
The filtered images
threshold values being determined through operator interaction.
Using connectivity analysis, the approximated segmented 3-D
, were extracted.
regions of the sheet structures,
Thickness determination: The extracted 3-D sheet regions
were thinned to a width of one voxel by nonmaximum suppression along the sheet normal directions of the sheet
. A sheet thickness was assigned to
enhanced images
each voxel of the thinned sheet regions.
For each voxel of the thinned regions, the 1-D profile of
was reconstructed
the second directional derivative
along the sheet normal direction, where
from
denotes the length parameter along , and is the SD of
Gaussian blurring introduced in Section II-C. (See Appendix
.) Similarly, binary
for detailed procedure for obtaining
were reconstructed along the same
profiles
directional line for the binary images of the segmented sheet
. The reconstruction of 1-D profiles
regions
and
was performed at the subvoxel resolution
by using a trilinear interpolation for the second directional
and a nearest-neighbor interpolation for
derivative
the segmented sheet regions
, respectively. The
two sides of sheet structure edge were localized in two steps
[Fig. 2(c)]: finding the initial point for the subsequent search
, and searching for the zero-crossing of
using
. The initial point of one side of edge, , was given by
if it
the maximum value of that satisfied
existed. Otherwise, the edge localization process terminated.
The initial point of the other side of edge, , was given by the
.
minimum value of that satisfied
Based on the initial point of the search, if
, search
operated toward the direction of increasing along the profile
for the zero-crossing position . Otherwise search did toward
, search
the direction of decreasing. Similarly, if
operated toward the direction of decreasing along the profile
for the zero-crossing position . Otherwise search did toward
the direction of increasing. The thickness, , was given by
.
IV. RESULTS
A. Numerical Simulation
The unit of dimension for the following simulation results
, i.e.,
was assumed as described in the prewas
, ) were norvious section. Thus, other parameters ( , ,
, and voxel anisotropy was represented as
malized by
. In the simulation, we mainly used
and
for the bar profile if not specified.
These parameter values were determined so that the bar profile
was symmetric and the magnitude operator in (9) did not affect
the results. (The effects of the magnitude operator and asymmetrical bar profile are described in paragraph Section IV-A4
in this section.) Table I summarizes the parameter values used
in the numerical simulations described below.
1) Effects of Gaussian Standard Deviation in Postprocessing: Fig. 6 shows the effects of the SD, , in Gaussian
blurring. In Fig. 6(a), the relations between true thickness and
,
,
measured thickness are shown for three values (
is equal to one, i.e., in the case of isotropic voxel.
1) when
, which is the
The relation is regarded as ideal when
diagonal in the plots of Fig. 6(a). For each value, the relations
were plotted using two values of sheet normal orientation (0 ,
45 ) while was fixed to 0 . Strictly speaking, voxel shape
is equal to 1 because
is not perfectly isotropic even when
the shape is not spherical. Thus, slight dependence on was
observed.
more clearly,
In order to observe the deviation from
. Fig. 6(b) shows the plots
we defined the error as
, considerable ringing
of error instead of . With
, error magnitude
was
was observed for error . With
). With
,
significantly large for small (around
was
however, ringing became small and error magnitude
.
gave a good comsufficiently small around
promise optimizing the trade-off between reducing the ringing
and improving the accuracy for small . Actually, error magniis guaranteed to satisfy
for
with
tude
, while
for
with
and,
for
with
. Based on this result, we used
in the following experiments if not specified.
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TABLE I
PARAMETER VALUES USED IN NUMERICAL SIMULATIONS
(a)
(b)
Fig. 6. Effects of Gaussian SD, in postprocessing for thickness determination with isotropic voxel. The unit is
thickness and measured thickness T . (b) Relations between true thickness and error T .
0
2) Effects of Voxel Anisotropy in MR Imaging: Fig. 7(a)
and voxel
shows the effects of sheet normal orientation
anisotropy
on measured thickness . The relations between
measured thickness and sheet normal orientation for six
values of true thickness (1, 2, 3, 4, 5, 6) were plotted when
(1, 2, 4) were
three different values of voxel anisotropy
for
used. The relations were regarded as ideal when
any , which is the horizontal in the plots of Fig. 7(a). When
, the relations were highly close to the ideal for
.
When
and
, significant deviations from the
and
, respectively.
ideal were observed for
Fig. 7(b) shows the plots of the maximum at which error
is guaranteed to satisfy
,
,
magnitude
for
with varied voxel anisotropy . These
and
plots clarify the range of where the deviation from the ideal is
sufficiently small. There was no significant difference between
and different values of (for
).
the plots for
1
.
= 0 . (a) Relations between true
3) Using Anisotropic Gaussian Blurring Based on Voxel
Anisotropy: We have assumed that Gaussian blurring combined with derivative computation is isotropic as shown in
(11). Another choice is to use anisotropic Gaussian blurring
corresponding to voxel anisotropy, which is given by
(39)
and
are determined so as to satisfy
and, thus,
because we assumed
. Fig. 8(a) shows plots of measured thickness obtained using
and
.
anisotropic Gaussian blurring when
The plots using anisotropic Gaussian blurring were closer to the
and any , while those using isotropic one were
ideal for
and
.
closer for
4) Effects of Bar Profile Shapes: In the above sections, the
circumstances in which the magnitude operator in (9) did not
affect the measurement and bar profile was symmetrical were
where
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(a)
1
1 =2
1
j j 01 j j 02
2
=0
1 =1
1 =2
j j 04 =2
(b)
Fig. 7. Effects of voxel anisotropy
in MR imaging on measured thickness T . The unit is
.
= and . (a) Relations between measured
thickness T and sheet normal orientation with different values. The relations are shown for three
values:
(left),
(middle), and
(right). (b) Plots of maximum at which error magnitude E is guaranteed to satisfy E < : , E < : , and E < : for while voxel anisotropy
is varied (where E
T
).
j j
= 0
1 =4
1
Fig. 8. Effects of anisotropic Gaussian blurring based on voxel anisotropy
and bar profile shapes. (a) Relations between true thickness and measured
thickness T with the use of anisotropic Gaussian blurring based on voxel
= and =
. (b) Relations between
anisotropy. true thickness and measured thickness T for four different types of bar
profile shapes.
=2
=3
2
= (2
2)1
discussed. In order to examine the effects of the magnitude operator and asymmetrical bar profile, we further considered the
and
.
following bar profiles. Type 0:
and
. Type 2:
Type 1:
and
. Type 3:
,
, and
. In Type 0, the profile was the same as the one used
in the above presented results. The magnitude operator affected
measured thickness in Types 1 and 2. The background level was
in Type 1, while the bar level was
(contrast was inverted) in Type 2. The bar profile was asymmetrical,
, in Case 3. Fig. 8(b) shows the relations between
i.e.,
measured thickness and sheet normal orientation for the
four different types of bar profiles when voxel anisotropy was
and true thickness was
. The differences in measured thickness among the four types of bar profile are revealed
in Fig. 8(b). While the effects of asymmetrical profiles were sufficiently small (in Type 3), those of the magnitude operator were
considerable (in Types 1 and 2). Thickness was underestimated
by around 0.1 in Type 1 and overestimated by around 0.1 in Type
compared with Type 0.
2 for
B. Validating the Numerical Simulation by In Vitro
Experiments
1) Plate Phantom: Fig. 9 shows the averages and the SDs
of the actually measured (in vitro) thickness from the MR data
of the phantom imaged with different and and the plots of
the simulated thickness representing the dependences on sheet
Fig. 9. Comparison of simulated thickness and in vitro thickness determined
from MR images of acrylic plate phantom.
:
,
:
,
and =
. For in vitro thickness, its average and SD values
are indicated by symbols and error bars. (a) Dependences on sheet normal
orientation . (b) Dependences on sheet normal orientation . Note that the
dependence on is theoretically equivalent to the dependence on when the
anisotropy is
=
.
= (2
2)1
1 = 0 625 mm 1 = 1 5mm
1 1 =1
normal orientation and . Fig. 9(a) and (b) shows the plots of
, respectively.
the dependences of and with
Good agreement between the simulated and the in vitro thicknesses was observed in both cases although the in vitro thicknesses was slightly greater than the simulated thickness. The
biases, i.e., the difference between the simulated thickness and
the average of in vitro thickness, were predominantly around 0.1
of
), and the SDs of
mm or less (except for
the in vitro thickness were mostly within 0.1 mm (except for
of
and
of
). It should
SATO et al.: LIMITS ON THE ACCURACY OF 3-D THICKNESS MEASUREMENT IN MAGNETIC RESONANCE IMAGES
Fig. 10. Comparison of simulated thickness and in vitro thickness determined
from MR images of resected femoral head. The dependence on sheet
:
,
:
, and
normal orientation is shown.
=
. For in vitro thickness, its average and SD values are
indicated by symbols and error bars, respectively.
= (2
2)1
1 = 0 625 mm 1 = 1 5 mm
be noted that the dependence on is theoretically equivalent to
.
the dependence on when the anisotropy is
2) Femoral Head Specimen: Fig. 10 shows the averages and
SDs of in vitro thickness at different obtained from MR data
,
, and
with routine imaging (
) and the plots of the simulated thickness representing the dependence on sheet normal orientation . The averages and SDs were computed for the in vitro thickness values
measured at each regardless of (because the effect of was
shown to be sufficiently small in the numerical simulation). To
. Let be
obtain in vitro thickness, we used
the reference thickness obtained from MR data with isotropic
and
). The difimaging (
,
ference between the reference and true thicknesses,
for
and
was shown to be within
through the numerical simulation and, thus, can be a good approximation for true thickness . We regarded the true thickness
corresponding to the in vitro thickness as if the reference
thickness at the same position as where was measured sat. As shown in Fig. 10, the in vitro
isfied
thickness was well-correlated with the simulated thickness. For
of
and
example, underestimation around
and overestimation in
of
were observed in the in vitro thickness as well as in the simulated
thickness. The biases between in vitro and simulated thicknesses
were mostly within 0.1 mm and the SDs of in vitro thickness
were around 0.2 mm or less.
V. DISCUSSION
We have formulated a theoretical simulation method to evaluate the accuracy of thickness determination of sheet structures
in 3-D MR images. Sheet structures, MR imaging processes,
and postprocessing for thickness determination were modeled
and simulated. One important aspect of our simulation was that
these processes, which should be essentially performed in 3-D,
were reduced to 1-D signal analysis in the frequency domain
without loss of information. Thus, the computation of the simulation was simplified and its cost was drastically reduced.
The simulation results clarified quantitative dependences of
the accuracy of measured thickness on true thickness , sheet
normal orientation , and bar profile shapes in sheet structures,
1085
voxel anisotropy
in MR imaging, and Gaussian SD
in thickness determination. For example, in order to obtain
a reasonable accuracy for thickness as small as possible (e.g.,
satisfies
for
), as
error
when voxel
shown in Fig. 7(b), the range of should be
,
when
, and
anisotropy
when
using
. The
validity of the simulation results was confirmed by the in vitro
experiments.
The main findings of this work are as follows. 1) It is effective
to integrate appropriate Gaussian blurring into postprocessing
for thickness determination. In the simulation,
was shown to be preferable for obtaining improved and stable
accuracy characteristics, although without effect on noise reduc, the minimum thickness to be meation. 2) Given pixel size
in
sured with acceptable accuracy is approximately
. Based on this finding, optimal
the case of
pixel size can be deduced to measure accurately the required
minimum thickness. 3) The relation between voxel anisotropy
and the maximum value of sheet orientation (the
angle between the sheet normal and the - plane) to be measured with acceptable accuracy is shown in Fig. 7(b). Based on
this result, required voxel anisotropy or slice thickness can be
reasoned to measure accurately the thickness of sheet structures
with desired maximum angle between the sheet normal and the
- plane. 4) While there is no significant effect of asymmetrical profile of MR values along sheet normal on thickness accuracy, measured thickness is overestimated (underestimated)
by around 10% in the case that the MR value of the background
(sheet object) in the profile is equal or close to zero, due to the
operator to obtain the magnitude from complex MR values.
The effect of sheet normal orientation on measured thickness was nonmonotonic, i.e., depending on , measured thickness varied nonmonotonically with the increase of . For ex, overestimation and unample, in Fig. 7(a), when
and at
,
derestimation were observed around
, and underestimation was observed
respectively, with
with
. In the frequency domain,
around
sheet structures are basically represented as the sinc function
while the voxel shape determines the bandwidth to be transmitted. The sinc function has alternate negative and positive
side lobes in addition to the main lobe. Depending on the bandwidth, underestimation (overestimation) occurs when negative
(positive) side lobes are dominant in the transmitted frequency
components. Integrating Gaussian blurring into the band-limited sinc function of sheet structures reduces the unwanted effect
of the side lobes while it increases unwanted systematic overestimation in small thickness due to blurring. In the simulation,
we found that a good compromise to balance this trade-off was
. One important requirement in the application of Gaussian convolution is that the sampling interval of the
convolution kernels needs to satisfy the Nyquest criterion. Let
be the sampling interval.
needs to meet
in
order to approximately satisfy the Nyquest criterion [19]. When
and
, the Nyquest criterion is not sat. Thus,
isfied because
it is important that the image data are interpolated without additional blurring, i.e., using sinc interpolation so that the signal
1086
sampling is isotropic in all three directions and the interval
satisfies
. When
,
, which approximately satisfies the Nyquist criterion.
Previous works have addressed similar issues, which include
vessel diameter measurement from MR angiography and
thickness measurement from CT. These studies also evaluate
the accuracy limits by using numerical simulations based on the
point spread function of imaging modalities. Our work differs
in the following two aspects. First, we use zero-crossings of the
second directional Gaussian derivatives as the boundary definition to determined the thickness of thin sheet structures while
others evaluate the diameter or thickness using the full-width
at half-maximum (FWHM) [14], [21], [22]. FWHM is a
useful measure for evaluating imaging systems themselves. In
sophisticated image analysis systems, however, zero-crossings
of the second derivatives (equivalent of the maxima of the first
derivatives) combined with Gaussian blurring are needed for
accurate and robust boundary localization. Our study provides
comprehensive evaluations that includes computer postprocessings for thickness determination as well as imaging processes,
and demonstrates the importance of parameter optimization for
both imaging and postprocessing. Although zero-crossings of
the second derivatives are used in [21] for the evaluation of
thickness accuracy from CT images, the effects of combining
Gaussian blurring are not evaluated, which is shown to have
considerable effects on stabilizing the accuracy as well as on
noise reduction in our study. Second, we intensively investigate
the dependence of voxel anisotropy of imaging systems and the
orientation of thin sheet structures. In the previous works [14],
[21], [22], the tilt of vessels or thin structures relative to the plane is not evaluated. The relations among voxel anisotropy,
orientation of sheet structures, and the thickness accuracy are
revealed in our work.
Our findings are of potential use in providing an objective
criterion for optimal determination of the voxel shape and size
in MR imaging, i.e., slice thickness and field of view (FOV).
For example, when the distribution of orientation and thickness
ranges of the concerned anatomical sheet structures are available, the slice thickness and FOV to attain acceptable accuracy
with the least imaging time can be determined based on our results. Furthermore, the relations between the measured thickness and the sheet orientation [as shown in Fig. 7(a)] can be
used as a calibration table to adjust the error in the measured
thickness. However, the effects of noise are not addressed in the
present work. It should be noted that the true thickness estimated
based on the calibration table is easily affected by noise when
the intervals of the plots shown in Fig. 7(a) are narrow.
The results obtained in the present study are valid under the
restrictions that the sheet structure has constant thickness and
orientation as well as is not influenced by peripheral structures
(Fig. 1). Sheet structures in in vivo MR images often violate
the restrictions. For example, the brain cortex is highly curved,
which results in large variations in its sheet orientation. Further,
pieces of cortex are nearby each other and the femoral cartilage of the knee is nearby the tibial cartilage. In these cases,
the adjacent structures influence each other. The results in this
paper are not validated for the above situations. Another impor-
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 22, NO. 9, SEPTEMBER 2003
tant characteristics of cartilage assessment from MR images is
the so-called “magic angle phenomenon” [23], [24], whereby
its signal intensity is influenced by the angle between the static
magnetic field and the local fiber orientation of the cartilage. It
appears brighter at approximately 54 . Our results do not take
this effect into account. In spite of the above issues, our results
would give an insight into the inherent limits for the accuracy
of thickness quantification in in vivo MR images.
To determine the thickness from in vivo MR images, segmentation, that is, boundary detection, of the sheet structure is
needed before thickness determination. There are different approaches in the boundary detection. Our results are valid for
the boundary detection based on zero-crossings of the second
derivatives, or equivalently maxima of the first derivatives, combined with Gaussian blurring [4], [8], [25]. When the segmentation is based on other approaches such as thresholding of original images, the results presented in this paper are not validated.
However, the theoretical simulation procedures conducted in
this paper can be modified so as to incorporate a different segmentation method and derive its specific accuracy characteristics. Another important aspect of this study is that the usefulness
of numerical simulation in evaluating the accuracy characteristics was validated. In addition, different principles exist in the
thickness definition for both sides of the detected boundaries. As
long as the simple sheet model with constant thickness and orientation assumed in this paper (Fig. 1) is considered, variation in
thickness definition does not affect measured thickness in general. For example, the distance along the sheet normal direction
and the shortest distance between two boundaries give the same
value for this simple model. However, variation in thickness
definition affects measured thickness for sheet structures with
large changes in thickness and orientation such as highly curved
sheets. Thus, the scope of our results is basically restricted to the
simple sheet model for which variation in thickness definition
does not considerably affect measured thickness.
VI. CONCLUSION
The fundamental limits on the accuracy of thickness determination in MR images were investigated. A simulation method
was established to derive the characteristics of thickness determination accuracy, considering both imaging and postprocessing parameters, especially voxel anisotropy and Gaussian
blurring combined with the second derivatives. The effect of
sheet structure orientation on accuracy with anisotropic (noncubic) voxels was clarified via the simulation. It was also found
that postprocessing Gaussian blurring played an important role
in realizing accurate and well-performed thickness determination. The simulations were validated by comparison with the
actual results obtained from in vitro experiments.
Future work will include the formulation of a method for
automatically segmenting a sheet structure, estimating its orientation, determining its thickness, and adjusting the thickness
based on the calibration table derived from simulation results.
When thickness quantification is applied to in vivo MR images,
the segmentation of sheet structures of interest becomes as important as the quantification itself. We are planning to develop
SATO et al.: LIMITS ON THE ACCURACY OF 3-D THICKNESS MEASUREMENT IN MAGNETIC RESONANCE IMAGES
a unified framework for both segmentation and quantification
based on multi-scale second and first derivative analysis [25].
APPENDIX
THICKNESS DETERMINATION OF SHEET STRUCTURES
DISCRETE 3-D MR IMAGES
IN
A computationally feasible procedure for discrete MR data is
described below. This procedure reproduces thickness determination for continuous images as much as possible.
be 3-D MR images, in which signal sampling is
Let
isotropic in all three directions and its interval is . Here,
, and , , and are integers. The partial second derivative combined with Gaussian blurring for the MR image
with sampling interval is given by
(40)
where represents discrete convolution. The Hessian matrix of
is given by
(41)
Thus, the directional second derivative along the normal direction of the sheet structure is given by
(42)
is satisfied,
As long as the condition
can be almost completely recovered from
using
sinc interpolation. Due to the limitation of computational cost
at any
and memory space, however, it is difficult to obtain
continuous position . Practically, trilinear interpolation can be
at any position
used to obtain an approximated value of
be the approxin a computationally efficient manner. Let
and it is given by
imated value of
(43)
where
is the trilinear interpolation function given by
(44)
in which
is a triangle function defined as
otherwise.
(45)
is obtained using sinc interpolation and
Similarly,
discrete convolution followed by trilinear interpolation.
Thickness of sheet structures can be determined by analyzing
and
along straight
1-D profiles of
line given by (18). By substituting (18) for in
and
(46)
and
(47)
1087
are derived, respectively. Both sides of the boundaries for sheet
structures can be defined as the points with the maximum and
minimum values of
among those satisfying the condition
. The distance between the two detected
given by
boundary points is defined as the measured thickness, . We
confirmed by experiments that the combination of sinc interpolation to reduce the sampling interval to half and discrete convolution followed by trilinear interpolation reasonably approximates the ideal continuous reconstruction for the purpose of this
study.
ACKNOWLEDGMENT
The authors would like to thank L. Wang for making the
acrylic plate phantom.
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