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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-

Laurent.Benhamou@chr-orleans.fr).

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

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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:

>

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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,

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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.

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