THREE-DIMENSIONAL STRAIN FIELD IDENTIFICAION
USING SPECKLE-LESS IMAGES OF X-RAY CT
S. Kuzukami, O.Kuwazuru and N. Yoshikawa
Institute of Industrial Science
the University of Tokyo
4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505
ABSTRACT
A three-dimensional identification method for displacement and strain fields within a soft tissue is investigated to develop a
new non-invasive inspection technique from the viewpoint of evidence-based medicine. The incompressibility of soft tissue is
employed as stabilization constraint to diminish the indeterminacy of displacement field in the conventional digital image
correlation method under the speckle-less intensity distribution of medical image data. The error function of image correlation is
modified with the penalties at sample points in line with the penalty function method. The minimization of modified objective
function is archived by means of the Levenberg-Marquardt method and the conjugate gradient method. The displacement and
strain fields within a specimen are identified for a demonstration by using sets of image data of multi-slice X-ray CT. The
residuals of errors and incompressibility penalty are converged stably by virtue of adequate formulation. The identified
displacement and strain fields are reasonable considering load condition and heterogeneity of the specimen.
Introduction
The non-invasive inspection based on the medical three-dimensional imaging techniques has become essential for the
practical diagnosis [1]. In recent researches [2][3], precise and automatic shape registrations are mainly investigated in line with
image processing techniques to find out shape abnormalities in organ regarded as a reliable criterion of illness. From another
viewpoint, an abnormally high strain field in organ also seems instructive because it corresponds to mechanical fractures in
human body. If the displacement field and strain field are extracted from a shape deformation of measured by medical imaging
techniques, alternative diagnosis based on material mechanics can be established for the prediction of fracture as to screen
risky aneurysm [4]. We investigate a methodology of the three-dimensional strain identification by improving the digital image
correlation method [5][6][7].
The digital image correlation method has originally developed in a measurement technique of two-dimensional surface
displacement by comparing images of undeformed and deformed states. In typical method proposed by Peters [5], the tentative
displacement field with unknown parameters is independently defined on each small square area of the undeformed state
image. The area is called subset. The unknown parameters of tentative displacement field are modified by a pattern matching
so that a virtual deformation intensity pattern matches the actual deformation pattern on the deformed image. The virtual
deformation intensity pattern is transformed from the undeformed one by the tentative displacement field. Therefore, it is
required that the pattern on a subset is clearly distinguished notwithstanding similar patterns surrounding the subset. In the
case of two-dimensional measurement, a random speckle pattern is artificially painted over the surface of measurement target
to ensure the uniqueness of intensity pattern, as shown in Fig. 1. The red square represents one subset area. In the
three-dimensional medical images, such speckle pattern is hardly formed in the material and the intensity deviation of tissue is
usually too weak to hold the uniqueness of intensity pattern, as shown in Fig. 2.
In the full field image correlation, proposed and demonstrated by Chu [7], a continuous displacement field is solely defined all
over the identification domain and is represented by B-spline basis functions. The identification solution is determined by
pattern-matching all of the intensity data. Even under the speckle-less three-dimensional image data, the divergent solution of
rigid translation is prevented through the matches of sparsely distributed volumes of high intensity gradient, such as material
surface and boundary. Nevertheless, the displacement field in a volume of flat intensity remains indeterminate.
In this paper, we employ the incompressibility of soft tissue as a constraint of displacement to cope with the indeterminacy
under the speckle-less three-dimensional image data. It is clarified that the continuous displacement and strain field can be
directly identified over all the volume of specimen by the proposed method.
Figure 1. Speckled intensity pattern
Figure 2. Speckle-less Intensity pattern
Image Correlation with Incompressibility Constraint
We handle a data set of medical three-dimensional image obtained by multi-slice X-ray CT. It consists of pairs of a
measurement position in the material and an intensity value related to the transparency of X-ray on the position. A
measurement position is a centroid of a pixel and called intensity point. An intensity value is indicated by a 16-bit signed integer.
All of the intensity points are aligned on rectilinear grid points because cross-sectional images of CT with the same horizontal
position are taken and stacked in vertical direction. The spans of horizontal grid and vertical grid are denoted by d1 and d2,
respectively (see Fig. 3). They are determined by the settings of X-ray irradiation and CT slice span.
Figure 3. Image data of multi-slice X-ray CT
Two sets of image data are obtained for the undeformed and deformed states of a target material. The intensity point and
m
m
intensity value of m-th pair in the undeformed state image are represented by X and f(X ), respectively. Those of the p-th pair
p
p
in deformed state are represented by x and g(x ).
The unknown displacement field u(X) is defined within an arbitrary cuboid domain called identification domain in the proposed
method, where a material point of undeformed state in the domain must be ranged in the image domain of deformed state after
the deformation for intensity correlation. The u(X) is parameterized by using the basis functions of the tensor product of third
order B-spline [8] of fourth degree and the corresponding unknown coefficients. The ξ direction displacement uξ is
prescribed as,
uξ (X ) ≡
N1 N 2 N 3
∑ ∑ ∑ c ξ ijk M i , 4 ( X 1 ) M
i
j
k
j ,4
( X 2 ) M k ,4 ( X 3 )
(1)
where c ξ ijk is unknown coefficients, M i , 4 ( X l ) is l directional i-th third order B-spline of fourth degree, and N 1 , N 2 , N 3
are numbers of B-spline functions for each direction.
The material point at m-th intensity point of undeformed state X
tentative displacement as shown in Fig. 4.
m
is translated to the virtually shifted position x
m
by the
x m = X m + u( X m )
m
Xm
xm
m
a
p
(a) Intensity grid of f(X )
in Undeformed State
Figure 4.
Intensity value g(x ) at shifted
m
position x is interpolated by
using the nearest eight intensity
p
points of g(x ).
(b) Intensity grid of g(x ) in
Deformed State
Material point transfer between two sets of image data
Assuming that the intensity value does not contain any noise on the image acquisition process, the intensity value at original
m
m
position in the undeformed state f (X ) is equivalent to that at virtually shifted position in the deformed state g(x ) under exactly
identified displacement field. The tentative displacement field is determined so that the difference between two intensity values,
m
m
g(x ) - f(X ), is minimized over the identification domain. The error function of the correlation is defined by the summation of
difference squares as
2J ≡
NI
∑ {g (x m ) −
f ( X m )} 2
(2)
m
m
where N I is number of the intensity points in the identification domain. The intensity value g(x ) is not directly measured by
m
the CT. Following the conventional method [5], the g(x ) is linearly interpolated by using the nearest eight neighbor intensity
m
points of undeformed state around the possition x , which are described by the eight gray circles in Fig. Figure 4.
m
The intensity value of shifted position g(x ) never changes with respect to the modification of tentative displacement if the
intensity distribution around the shifted position is flat. In the three-dimensional X-ray CT images, large portion is occupied by
flat and speckle-less intensity distribution domain. Therefore the solution of displacement identification is hardly determinate in
such a volume because the intensity error g (xm ) − f ( X m ) is constant over the volume with respect to the modification of
unknown coefficients. The simple minimization strategy cannot yield converged solution, and what is worse, a possibility of the
incorrect solution enhances caused by noise effect in X-ray CT images.
The actual displacement field of soft tissue is almost incompressible since the tissue is largely deformed under a small stress
and it contains rich water. Therefore it seems proper manner that the incompressibility constraint is added to reduce the
indeterminacy of displacement solution. The penalty function method is employed to include the penalties of constraint
evaluated at sample points as
Ns
2 J ′ ≡ 2 J + μ ∑ {det F n − 1} 2
(3)
n
n
where F is deformation gradient tensor of unknown displacement at n-th sample point, μ is penalty coefficient, and N
number of sample points. The sample points are evenly allocated in the volume of identification domain.
s
is
Solution Search
The stationary condition of the modified objective function yields the non-linear simultaneous equations as
NI
Ns
m
n
S ≡ ∑ { g ( x m ) − f ( X m )}P m + μ ∑ {det F n − 1}Q n = 0
(4)
where S is gradient vector of the modified objective function J’ with respect to unknown coefficients, Pm of g(xm), and Qn of detFn.
The solution of the unknown coefficients cξijk to satisfy Eq. (4) is successively searched for from the tentative values of
coefficients in line with the Levenberg-Marquardt method [9]. The modification vector Δ c , whose elements correspond to the
unknown coefficients, is determined by following the linearized equations as,
NI
{∑ P m P m T +
m
Ns
∑ Q n Q nT
+ λ I}Δ c = − S
(5)
n
where λ is regularization parameter. When the regularized parameter is set to small value, the iteration time of the linearized
solution search decreases and the instability increases. The simultaneous equations (5) are solved by a modified conjugate
gradient method [10] for highly efficient parallel computation.
Strain Field Evaluation
The evaluation of displacement gradient tensor ∂ u ξ / ∂ X η is straightforward with mathematically derived first-order
derivatives of B-spline after the unknown coefficients are identified.
∂u ξ
∂X η
N1 N 2 N 3
= ∑ ∑ ∑ cξijk {δη1
i
j k
+ δη 2 M i , 4 ( X 1 )
∂M i , 4 ( X 1 )
∂X 1
∂M j , 4 ( X 2 )
∂X 2
+ δη 3 M i , 4 ( X 1 ) M
j ,4 ( X 2 )
M
j ,4 ( X 2 ) M k ,4 ( X 3 )
M k ,4 ( X 3 )
∂M k , 4 ( X 3 )
∂X 3
(6)
}
δ pq is Kronecker’s delta. In our demonstration, a large strain over 10 % within a soft tissue is observed even in a small load of
10 N. Therefore the Green-Lagrange strain tensor [11] E is evaluated from the viewpoint of finite deformation theory instead of
infinitesimal strain tensor. A strain component Eξη is given by the displacement gradient tensor ∂u ξ / ∂X η as,
2 Eξη =
∂uξ
∂X η
+
∂uη
∂X ξ
∂u γ ∂u γ
.
γ ∂X ξ ∂X η
3
+∑
(7)
Both the displacement gradient tensor and Green-Lagrange strain field are continuous because the third order B-spline
function is a piece-wise polynomial with continuity of derivatives all over the definition domain.
Experimental Demonstration
The specimen for demonstration is a spherical citrus fruit. The deformation of identification target is generated by 10 N
compression in vertical direction. Two sets of image data are obtained for both undeformed and deformed state by the load. The
absolute positions and number of intensity points are fixed for the data sets of deformed state and undeformed state.
The geometrically inclusive relationship concerning the image acquisition is described in Fig. 5. The dimension of image data
region is 40 × 40 × 20 mm, where the upper half of specimen is contained with margins. The noise effect of X-ray CT is reduced
by applying 5 × 5 median filter to intensity data on each horizontal CT slice. The horizontal span d1 of intensity point (see in Fig.
3) is smaller than the vertical span d2 in the raw image data. We decrease the density of intensity points on each horizontal CT
slice by the bi-cubic interpolation of digital image processing so as to realize spatially even distribution of intensity point ( d1 = d2
= 0.4 mm ). As a result of the preprocessing, the image data has 100 × 100 × 50 intensity points. The half view of deformed state
image after preprocessing is shown by the iso-volume visualization in Fig. 6 with a legend of intensity value to reveal the
cross-section. The cylindrical object above the specimen in Fig. 6 is the compression contactor of 10 N.
The identification domain is restricted in a central rectangular volume in the image region of undeformed state so that the
material point within the identification domain is always translated to the image region of deformed state, as shown in Fig. 5.
The identification domain contains 84 × 83 × 34 intensity points where almost all volume of specimen in the image region is
covered. The number of B-spline functions N1 × N 2 × N 3 is set as 7 × 7 × 5 . The number of sample points N s for
incompressibility constraint is 21 × 21 × 8 = 3528 . The penalty coefficient μ is fixed to 3 .2 × 10 − 7 . The optimum way to set the
value of λ is obtained as Eq. (8) through an examination of preliminary numerical experiments by using the Euclid norm of
gradient vector of the objective function ||S||.
λ = 5 × 10 −6 || S || +5 × 10 −4
(8)
All the values of cξijk are initially set to 0 so that the start displacement field of solution search becomes null. The
convergence criterion is set by ||S|| < 20.
Image Data Region
Identification Domain
Figure 5. Geometrical description of specimen and domains
Figure 6.
Intensity distribution in deformed specimen
The minimization process of the modified objective function quits at 86-th iteration of the Levenberg-Marquardt linearization.
Figure 7 shows the convergence history of three error residuals. The error function J and modified objective function J '
monotonically decrease during the iteration of minimization. The gradient vector Euclidean norm of objective function ||S||
periodically increases, though the trend is decreasing.
Iterations of Linearization
Figure 7. Convergence history
The identified displacement vector on the intensity points within the specimen is drawn in Fig. 8. The legend of figure
represents the Euclidean norm of displacement vector. Figure 9 describes the distribution of the Green-Lagrange strain tensor
E11, E22, E33 in a quarter volume of specimen. Internal dark blue regions correspond to the voids.
The contact surface on the top of soft and spherical specimen is deformed under the compressive load to fit the flat shape of
compression contactor. We observe a large deformation and compressive strain along the compression direction in the
m
displacement u(X ) of Fig. 8 and strain E33 of Fig. 9(c) on the top of specimen, respectively. Assuming the loading condition is
approximately axi-symmetric, the compression load on the specimen roughly yields tensile along the surface of specimen and
compression orthogonal to the surface around the equator of specimen because of the expansion. In the identification result,
the tensile strain is observed in both the E11 on the right side cross-section of Fig. 9(a) and the E22 on the left side cross-section
of Fig. 9(b), which correspond to the surface direction tensile. The compression strain is similarly observed in the E11 of left side
cross-section and E22 of right side cross-section, which correspond to the orthogonal surface direction compression. The
axi-symmetry of strain field seems to be weakly violated by the heterogeneity of material distribution.
[mm]
2.2
X2
X1
X3
1.1
0.0
Figure 8. Identified displacement field
0.15
0.00
-0.15
(b) E22
(a) E11
0.30
X2
X1
X3
0.00
-0.30
(b) E33
Figure 9. Identified Green-Lagrange strain field
Conclusions
A non-invasive identification method of three-dimensional strain field within the soft tissue was proposed in this study. A
formulation of the full-field digital image correlation method with incompressibility constraint was derived to cope with the
intensity flatness of medical image data. The modified objective function with penalties was constituted from the error function of
conventional image correlation and the incompressibility penalties. The displacement and strain field within a specimen was
identified to demonstrate the proposed method. Two sets of image data of a specimen were obtained by employing X-ray CT
both for the undeformed state and deformed one of 10 N compression load. The identified displacement and strain field
corresponded to their quantitative specifications predicted from the viewpoint of material mechanics.
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