Download - Cr 1061 Stein
-
8/13/2019 Cr 1061 Stein
1/4
AbstractRecently, a theoretical study has shown that theH parameter of a n-dimensional (nD) isotropic fractal (HnD) is
linked to that of its projection (H(n-1)D) by H(n-1)D = HnD + 0.5.The purpose of this contribution is to illustrate this relation on
synthetic fractal images and to apply it on 3D trabecular bone
volumes. First, we have synthesized 2D fractals using the Stein
method which leads to exact data. Their projections are
calculated giving 1D signals. It is shown that the theoretical
result for n=2 (H1D = H2D + 0.5) is well illustrated by this
example. Second, we have estimated the 3D H parameter of 21
femoral head samples and the 2D H parameter of their
projections to test if this relation can be applied. We found a
difference of 0.482 0.036 between the 2D and the 3D H
parameters. This shows that the theoretical result for n=3 (H2D= H3D + 0.5) is true for this kind of binary object. The main
advantage is that 2D bone radiographs are not life threatening,
cheap, easily obtainable and have a high resolution.
I. INTRODUCTIONFractional Brownian motion (fBm) of H parameter in ]0,1[
could be a good candidate to model 2D or 3D stochastic data
[1]. This parameter translates the roughness of the object:
the lower H is the rougher the object [2].
When an object is a 3D one, it could be a difficult issue to
obtain 3D volumes as for example in medical imaging. A 3D
scanner lead to patient irradiations and has limited
resolution. A 3D RMI has low resolution too and is an
expensive exam. At the opposite, 2D radiographs are cheap,
low irradiation and has high spatial resolution. The problem
is that it is a 2D projection of a 3D complex structure. The
assessment of volume properties from this 2D data is far
from evidence because few theoretical results link 3D
properties with 2D ones. Recently, a theoretical study has
shown that the H parameter of a nD isotropic fractal (HnD) is
linked to H(n-1)D, the self-similarity of its projection, by H(n-
1)D =HnD + 0.5 [3].
The purpose of this contribution is to illustrate this
relation on synthetic fractal images and to apply it on 3D
trabecular bone volumes.
Manuscript received October 17, 2005.
G. Lemineur, R. Harba, R. Jennane and T. Devers are with the
Laboratory of Electronics Signals Images, PolytechOrlans, Universit
dOrlans, BP 6744, 45067 Orlans Cedex 2, France (corresponding author
to provide phone: 33.(0)2.38.41.72.29; fax: 33.(0)2.38.41.72.45; e-mail:
Rachid.Harba@ univ-orleans.fr).
A. Estrade is with MAP5 laboratory, Universit Paris 5, 45 rue des
Saints Pres, 75270 Paris Cedex 6, France (e-mail: Anne.Estrade@univ-
paris5.fr).
C.L. Benhamou is with INSERM U 658, Hpital Porte Madeleine, BP
2439, 45032 Orlans Cedex 1, France (e-mail: Claude-
II. THEPROJECTIONTHEOREMWe briefly present the projection theorem. For simplicityand without loss of generality, this theorem will be shown in
the 2D case. At first, let us define a 2D fBm of H index in
]0,1[ and with variable t=(t1,t2) as in [4]:
),(dB1e
)(B 2
R1H
i
H2
t
t.
+
=
(1)
where B2= {B2() ; R2} is the 2D complex Brownianfield depending on the frequency =(1, 2), t. is the scalarproduct of t and , and || is the modulus of . Projectingthis process along parallel lines of direction , within asquare integral window of average 1, results in a Gaussianprocess {YH,(s); s R} of spectral representation:
,)(dB)(g1)(e(s)Y 1H,is
H, +
=
(2)
where B1 = {B1() ; R} is the 1D complex Brownianprocess. The spectral density gH,is given by:
( ) du)(u
(u)(g
1)u
arctgH(
22
22
H, +
+++
=
) (3)
or
( )
du.))u((1
(u)
1(g
1)uarctgH(
2
2)u
arctg(2)2/(2
2/2)2H(
2
H,
++
+++
++
+
=
HH
)
(4)
is the Fourier transform of . Therefore, the spectral
density gH,is equivalent at high frequency to:
The Projection Theorem for Multi-Dimensional Fractals
G. Lemineur, R. Harba, R. Jennane, T. Devers, A. Estrade, C.L. Benhamou
-
8/13/2019 Cr 1061 Stein
2/4
1H
C+
(5)
withC a constant term.
Let us recall that the spectral density of 1D fBm is
equivalent to 1/||H+0.5. It clearly shows that the averaged
process YH, behaves at high frequency as a 1D fBm of
index H + 0.5. It is then possible to estimate H in 2D (H2D)
from the measurement of H in 1D (H1D) since they arelinked by the following simple relation: H2D= H1D+ 0.5.
In the following section, this relation will be illustrated on
synthetic images.
III. ILLUSTRATIONONSYNTHETICIMAGESIt is not possible to directly synthesize the non stationary
isotropic 2D fBm denoted B(x) where x belongs to R2. Stein
has recently proposed a method to overcome this difficulty
[5]. First, an isotropic and stationary 2D process denoted Z
is generated using the circulant embedding method [6] with
the following covariance function:
>
-
8/13/2019 Cr 1061 Stein
3/4
IV. APPLICATIONTOTRABECULARBONEVOLUMES
The diagnosis of osteoporosis is mainly based on dual
energy X-ray absorptiometry which consists in measuring
bone mass. Characterization of the trabecular structural
properties appears to be an important adjunct to the
measurement of bone mass in determining fracture risk with
greater accuracy [7]. Using histological and stereological
analysis, it has been shown that, by combining structural
features with bone density, nearly all of the variability inmechanically measured Youngs moduli could be explained
[8][9]. A structural evaluation of the trabecular micro-
architecture and of its changes would also allow a better
understanding of the pathophysiological events that
characterize bone aging and osteoporosis. However, the
evaluation of bone structure, by non-invasive procedures,
remains a difficult issue [10]. Bone X-ray radiographs have
been suggested as a means of quantifying trabecular changes
[11][12]. This method shows potential as we know that
radiographs are feasible and are not life threatening.
In the following, we will experimentally examine whether or
not the formal link (H2D = H3D + 0.5) holds for trabecular
bone. With this aim in view, we therefore examine high
resolution micro computed tomography (-CT) images andsimulate their 2D projections. 2D and 3D self-similarity
parameters are measured and compared.
A total of 21 frozen human femoral heads were used to
derive 21 specimens of trabecular bone. Cylindrical samples
were prepared under continuous water irrigation using a
precision diamond circular saw. The samples were oriented
with respect to anatomic axes and cut to a dimension of
6 mm thickness and 8 mm diameter. All the samples were
defatted chemically (several cycles of submerging in
dichloromethane).
Images were obtained using the Skyscan 1072 high-resolution -CT. The X-ray source was set at 80kV and100A, and the magnification was fixed to get a pixel size
of 12 m. A 10241024 12-bit digital cooled CCD coupledto a scintillator was used to record the radiographic
projections. 209 projections were acquired over an angular
range of 180 (angular step of 0.9). Due to the cone beam
the radiographic images were processed with the Feldkamp
algorithm. The exposure time for one radiographic
projection was about of 5.9s so that the total scan for each
sample lasted 1 hour. All the radiographic images were
transferred to a personal computer (PIII, 933MHz) and the
image slices were reconstructed using the conebeam
reconstruction software version 2.6.
We used the central part of each image which is 4.80 mm3,
equivalent to 400x400x400 pixels. Figure 3 shows a 3D
cube and its 2D projection. The projections are calculated
along the vertical axis.
Figure 3: 3D trabecular volume and its 2D projection.
The projections of 3D bones are 2D grey level images as
seen on figure 3. Thus the variance method of Pentland can
be applied to assess the 2D H parameter. It requires that the
increments I(x) of the grey levels I(x) be computed fordifferent lags . If fractal, the following relation holds:
)),(())(( 22
xIVarxIVar DH = (8)
where x is a two dimensional time index. For a fractal, in a
Log-Log plot, the plot of previous equation is a straight line
of slope 2H2D. Such representation obtained on 2D image
presented in figure 3 is illustrated on figure 4. H parameterof the projection must be assessed at small scales, thus we
choose to estimate H over the first five pixels of each graph.
100000
1000000
10000000
100000000
1 10 100
Var(DI(lx))
Figure 4: Log-Log plot of the variance of the increments
versus the lag for the two projections of Figure 3.
To assess H3D, the box counting method is used [13]. The
original 3D structure is regularly cut in cubes, or in boxes,
-
8/13/2019 Cr 1061 Stein
4/4
of size as illustrated in figure 3. The number N() of boxes
containing at least one mineral element is:
DCN = )( . (9)
where D is the fractal or box dimension of the object and C
a constant. The plot of N() versus in a Log-Log scale is a
straight line with a slope of D as shown on figure 5. Thus,
we get:
H3D= 3 - D. (10)
1,E+00
1,E+01
1,E+02
1,E+03
1,E+04
1,E+05
1,E+06
1,E+07
1 10 100 1000
N()
Figure 5: the Log-Log plot of N() versus for the bone
volume specimen of Figure 3.
Both 2D and 3D self-similarity indexes were estimated on
the 21 bone volumes and on their projections using the
above methods. We examined experimentally if these
parameters are linked by H2D= H3D+ 0.5 as it is the case for
continuous fBm. Figure 6 shows the evolution of H2Dversus
H3Dfor the 21 bone samples as well as the straight line H2D
= H3D+ 0.5.
0,7
0,75
0,8
0,85
0,9
0,95
1
0,25 0,3 0,35 0,4 0,45 0,5
H3D
H2D
Figure 6: H2Das a function of H3Dfor the 21 bone volumes.
We notice that H2Dand H3Dare related by H2D= H3D+ 0.5.
To confirm this tendency, the mean offset, H2D- H3D, and its
standard deviation are presented in table 2.
This confirms that the above relation is true for trabecular
bone. It provides an interesting tool to quantify the 3D
architecture from its two dimensional projection. The main
advantage is that 2D bone radiographs are not life
threatening, cheap, easily obtainable and have a high spatial
resolution.
H2D- H3D
mean 0.482
Standard deviation 0.036
Table 2: mean and standard deviation for H2D- H3Dfor the
21 bone cubes.
REFERENCES
[1] B. B. Mandelbrot, J. Van Ness, Fractional Brownian Motion,Fractional Noise and Applications, SIAM, Vol. 10, pp. 422-
438, 1968.[2] A. P. Pentland, Fractal-Based Description of Natural
Scenes, IEEE Trans. on Patt. and Mach. Int., Vol. PAMI-6,
N 6, pp. 661-674, November 1984.[3] A. Bonami, A. Estrade, Anisotropic Analysis of some
Gaussian Models, The Journal of Fourier Analysis andApplications, Vol. 9, pp. 215-236, 2003.[4] S. Reed, P. Lee, T. K. Truong, Spectral Representation of
Fractional Brownian Motion in n Dimensions and its
Properties, IEEE Trans. on Inf. Theory, Vol. 41, N 5, pp.
1439-1451, 1995.[5] M. L. Stein, Fast and exact simulation of Fractional
Brownian surfaces, Journal of Computational and Graphical
Statistics, Vol. 11, N 3, pp. 587-599, 2002.[6] C. R. Dietrich and G.N. Newsam, Fast and exact simulation
of stationary gaussian processes through circulant embedding
of the covariance matrix, SIAM J. Sci. Comput., vol. 18, pp.
1088-1107, 1997.[7] S. A. Goldstein, The Mechanical Properties of Trabecular
Bone: Dependence on Anatomic location and Function,
Journ. Biomech, Vol. 20, pp. 1055-1061, 1987.[8] R. W. Goulet, S. A. Goldstein, M. J. Ciarelli, J. L. Kuhn, M.B. Brown, L. A. Feldkampt, The relation between the
structural and orthogonal compressive properties of trabecular
bone, J. Biomechanics, Vol. 27, pp. 375-389, 1994.[9] P. I. Croucher, N. J. Garrahan, J. E. Compton, Assessment of
cancellous bone structure: comparison of strut analysis,
trabecular bone pattern factor, and narrow space star
volume, J. Bone. Miner. Res, Vol 11, pp. 955-961, 1996. [10] K. G. Faulkner, C. C. Gluer, S. Majumdar, P. Lang, K.
Engelke, H. K. Genant, Noninvasive Measurements of Bone
Mass, Structure and Strength: Current Methods and
Experimental Techniques, AJR, Vol. 157, pp. 1229-1237,
1991.[11] S.D. Rockoff, J. Scandrett, R. Zacher, Quantitation of
relevant image information : automated radiographic bonetrabecular characterisation, Radiology, Vol. 101, pp. 435-
439, 1971.[12]N. D. Aggarwal, G. D. Singh, R. Aggarwall, A survey of
osteoporosis using the calcaneum as an index, Int. Orthop.,
Vol. 10, pp. 147-153, 1986.[13] B. B. Mandelbrot, The fractal geometry of nature, W. H.
Freeman, San-Fransisco, 1982.