- Email: [email protected]
Reconstruction of food microstructure via statistical correlation functions. The use of lineal-path distribution functions
Accepted Manuscript Reconstruction of food microstructure via statistical correlation functions. The use of lineal-path distribution functions A. Derossi, C. Severini, T. De Pilli PII: DOI: Reference:
S0260-8774(14)00227-1 http://dx.doi.org/10.1016/j.jfoodeng.2014.05.020 JFOE 7810
To appear in:
Journal of Food Engineering
Received Date: Revised Date: Accepted Date:
3 March 2014 21 April 2014 25 May 2014
Please cite this article as: Derossi, A., Severini, C., De Pilli, T., Reconstruction of food microstructure via statistical correlation functions. The use of lineal-path distribution functions, Journal of Food Engineering (2014), doi: http:// dx.doi.org/10.1016/j.jfoodeng.2014.05.020
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1 2
Reconstruction of food microstructure via statistical correlation functions. The use of lineal-path distribution functions.
3 4 5 6
Authors: Derossi, A., Severini, C. De Pilli, T.
7 8 9 10
Department of Science of Agriculture, Food and Environment, University of Foggia, Italy
11 12 13 14 15 16 17
18
19
20
21
22
23
24
Corresponding authors: [email protected], Department of Science of Agricultural, Food and Environment, University of Foggia, Via Napoli 25, Italy. Phone +39 0881 589245.
1
Abstract
2
The reconstruction of microstructure of biological and synthetic materials is of great importance for
3
theoretical and practical application. The obtaining of this purpose through limited statistical
4
information is a challenge at which some pioneering researchers are focusing their efforts. For the
5
first time, a reconstruction method was used to rebuild bread microstructure by using the
6
information of lineal-path distribution function (LPF). The method was a powerful tool to
7
reconstruct bread microstructure; a perfect match between lineal-path function of reference and
8
reconstructed images were obtained. The reconstructed images progressively improve, compared to
9
the reference, increasing the number of LPFs used during reconstruction. The use of LPFs in
10
different direction did not allowed to obtain a perfect match between the bread and the
11
reconstructed images because, LPF did not contain sufficient microstructure information however,
12
the essential features of bread were captured proving the possibility to reconstruct food
13
microstructure.
14
15
16
Keywords: morphological descriptors; microstructure; random media; reconstruction; distribution
17
function; bread
18
19
20
21
22
1
1. INTRODUCTION
2
Random heterogeneous materials are ubiquitous in nature and in synthetic systems. Examples are
3
soil, porous media, biological tissues, sandstone, fiber, rock formations, tree patterns in forests,
4
cosmological structure, etc. (Lu and Torquato, 1992a; Jiao et al., 2009). The precise understanding
5
of the microstructure of these systems is of crucial importance for practical application due to the
6
strict relation between microstructure information and the bulk properties of materials such as
7
conductivity, permeability, mechanical and electromagnetic properties, heat and mass transfer (Lee
8
and Torquato, 1989; Lu and Torquato, 1992a; Torquato and Lu, 1993; Russ, 2005; Baniassadi et al.;
9
2012; Li et al., 2012). Most of heterogeneous materials may be considered as two-phase random
10
systems composed from two different phases or from one phase in different states. In food science a
11
lot of product may be considered as two-phase systems in which a void phase (pores) and a solid
12
matrix phase (cell membranes, proteins, crystals, globules, etc.) exists. A typical examples are bread
13
characterized from pores and solid matrix (crumb), and sausages, composed by fat globules
14
(discontinuous phase) and protein elements (continuous phase), or emulsions composed by water
15
and oil phases. The importance of microstructure on the quality of food is not a new idea (Aguilera
16
2005; Parada and Aguilera, 2007; Datta 2007). Aguilera (2005) was the first scientist who
17
highlighted as all macroscopic properties of food are governed from elements in the range of 10-
18
100 Mm. In an interesting paper, he clearly analyzed as food microstructure greatly affects
19
nutritional values, sensorial properties, transport phenomena as well as microbial stability. Later,
20
Datta (2007) who studying the food as porous media, showed as the intrinsic permeability, k, which
21
is a parameter strictly related with pore connectivity, significantly affects the mass transfer during
22
processing. Furthermore, Parada and Aguilera (2007) showed the importance of microstructure on
23
the bioavailability of several nutrients. In light of above considerations, one of the most important
24
challenge is to ascertain what is the essential information to obtain the quantitative characterization
25
of food microstructure, its exact theoretical and experimental quantification and its relation with the
1
macroscopic properties. With this aim some authors, on the basis of a rigorous theory, developed a
2
series of statistical correlation functions able to extract microstructure information from two-phase
3
random systems (Torquato et al., 1988; Lu and Torquato, 1992b; Torquato and Lu, 1993; Torquato,
4
2002; Quintavalla, 2006). One of these basic functions is the n-point correlation function
5
Sn(x1,x2,…….xn), which is the probability to find n points at position x1, x2,…….xn all in one of the
6
two phases of the system (Torquato et al. 2000). However, since infinite n-point functions are not
7
attainable in practice, low-order version, such as a two-point correlation S2(x1, x2) and three-point
8
function S3(x1, x2, x3), are commonly used (Smith and Torquato, 1988; Quitavalla, 2006; Jao et al.
9
2007). The lineal-path distribution function, Li(z) is another interesting statistical descriptor, which
10
refers the probability that a segment of length z completely falls in the phase i (Lu and torquato,
11
1992a, Lu and Torquato 1992b; Torquato et al., 2002; Singh et al., 2008). This function contains
12
important connectedness information along a straight line and also it gives some information in
13
stereology. Further correlation functions such as chord-length distribution function, p(z), pore size
14
distribution function, P(z), two-point cluster function, C2(x1,x2), have been developed and validated
15
for several digitized model systems (identical hard disks, identical overlapping disks, periodic rods,
16
Debye overlapping disks, etc.) and for some materials such as sandstone, magnetic gels, Boron
17
modified Ti-alloys (Rintoul et al., 1996; Singh et al.,2008; Chan and Covindaraju, 2004) while very
18
few papers showed the use of these function to obtain microstructure information of food (Derossi
19
et al., 2012; Derossi et al., 2013a; Derossi et al., 2013b). The reconstruction of microstructure of
20
random heterogeneous systems from limited microstructure information is an interesting inverse
21
problem. If achieved, this result should enable to generate an accurate bi- or three-dimensional
22
structure of any material from which should be possible to estimate their macroscopic properties.
23
Moreover, the study of reconstruction will give light on the type of microstructure information
24
contained into the statistical correlation functions. Although different approaches to reconstruct
25
random media have been proposed (Joshi, 1974; Quiblier, 1984; Adler et al., 1990; Berk, 1991) we
26
focus our attention on the simulated annealing method proposed from Rintoul and Torquato (1997).
1
The method is based on the finding of a state of minimum ‘energy’ by interchanging the phase of
2
two the pixels in a digitized random model system (simulated), having the same porosity fraction of
3
the reference material (Yeong and Torquato, 1998). The ‘energy’ is defined as the sum of square of
4
the difference between the distribution functions of simulated and reference systems. Although this
5
method has been used for several digitized model systems showing a good ability in the
6
reconstruction of two-phase random systems (Rintoul and Torquato, 1996; Yeong and Torquato,
7
1997; Li et al., 2012; Baniassadi et al., 2012) this problem is still understudied and never it was
8
used for the reconstruction of food microstructure. On the basis of these considerations, the aim of
9
this paper was to study the reconstruction of bread microstructure by using the information
10
contained in the lineal-path distribution function extracted, from 2D images, in different direction.
11 12
2. MATERIAL AND METHODS
13 14
2.1 Theoretical background
15
A random medium is a domain of space V(W) ߳ R3 (the realization W is taken from some probability
16
space 7) where the volume V is characterized from two-phases: phase 1 in the region ϝ1 with a
17
volume fraction Ѱ1, and phase 2 in the region ϝ2 with a volume fraction Ѱ2. For a given realization
18
W, the characteristic function I(x) of phase 1 may be reported as:
19
(ܫ௫) = ൜
1, 0,
݂݅ ݁ݏ݄ܽ ݄݁ݐ ݊݅ ݏ݈݈݂ܽ ݔ1 (߳ ᎇ ) ݁ݏ݅ݓݎ݄݁ݐ
Eq.1
20
Where ᎇ is the region occupied by phase i (Equal to 1 or 2) (Lu and Torquato, 1992). Also, a
21
useful system to study heterogeneous material is a two-phase fully penetrable spheres which is
22
obtained adding polydispersed spheres of radius R until a specific porosity fraction is reached. Lu
23
and Torquato (1992a), based on a rigorous theory, showed that for these systems the lineal-path
24
distribution function is defined as
మ(శభ)
ଵା[ഏೃ(శమ)]
(ܮ௭) = ߶ଵ
1
Eq. 2
2
Where R is the average radius of the disks, z is the length of the segment (pixels or mm) and m is
3
the parameter of Schultz size distribution which is equal to:
4
݂(ܴ) =
ଵ ௰(ାଵ)
ቂ
ାଵ ାଵ ቃ ۃோۄ
ܴ ݁ ݔቂ
ି (ାଵ)ோ ቃ ۃோۄ
Eq. 3
5
Where ߁ is the gamma function.
6
2.2 Raw material and images acquisition
7
Bread was chosen as model because their structure may be considered as a two-phase systems in
8
which the void (pores) is the phase 1 and the solid matrix (crumb) is the phase 2. Also, since the
9
position of pores produced during yeast fermentation is extremely affected from several random
10
variables such as air incorporation, dough preparation, bread structure may be considered a random
11
system. Particularly, we used “Altamura” bread (Oropan s.p.a., Italy), purchased locally because of
12
the large diffusion of bread in the south Italy. The loaves were cut to obtain slices with a thickness
13
of 1 cm and the images were acquired by using a flat scanner mod. Hp 3600 (Hp, Scanjet) covering
14
the samples with a black box to obtain a good contrast between the background and the samples,
15
and to guarantee constant lightness conditions. The images were acquired by positioning the top of
16
the slice sample parallel with the light of scanner (x axis) with the aim to avoid the effect of
17
sample’s position on lineal-path distribution function. A resolution of 600 dpi = 0.004233
18
mm/pixels was used and the images were saved in TIFF format. Binary image were obtained by
19
applying the Otsu’s method (Sezgin et al. 2003; Jao et al., 2010), choosing the threshold through the
20
functions “rgb2gray” and “graytresh” available in the image analysis Toolbox of Matlab R2012b
21
(Mathworks, USA).
22
23
2.3 Computation of lineal-path distribution function
1
The binary images of bread may be represented by a common two-dimensional arrays:
2
ܫଵ,ଵ ܫ = ܫଶ,ଵ ܫଷ,ଵ
ܫଵ,ଶ ܫଶ,ଶ ܫଷ,ଶ
ܫଵ,ଷ ܫଶ,ଷ ܫ,
Eq. 4
3
where Ii,j (i = j = 1, 2, 3, 4.….., n) only assumes value 0 (black pixels) or 1 (white pixels); black
4
pixels refer to the pores of bread (phase 1), while the white pixels refer to the crumb (phase 2).
5
Lineal-path distribution function (LPF) was extracted in four different directions: 1) LPF0 along
6
horizontal direction (x axis, 0°); 2) LPF90 along vertical direction (y axis, 90°); 3) LPF45 along
7
diagonal direction (45°); 4) LPF135 along diagonal direction (135°). The extraction of LPF in
8
horizontal and vertical direction was obtained by using the algorithm developed and validated by
9
Derossi et al. (2012a), Derossi et al. (2013a) and Derossi et al. (2013b). In addition, a modified
10
version of the these algorithms was developed to extract LPF function along diagonal directions
11
(45° and 135°).
12
2.4 Reconstruction procedure
13
For simplicity, we consider the reconstruction procedure of a general two-phase random system
14
carried out by using the microstructure information provides from a general statistical correlation
15
function, f(r). Also, we defined the correlation function of the phase j (equal to 1 or 2) of our
16
“reference” systems as f0(r) and the lineal-path function of the reconstructed model system as fs(r)
17
which will evolve toward f0(r). Once the fs(r) is calculated we can define a new variable E as
18
follows: ଶ
19
= ܧσ ߚ ൫݂௦ (ݎ ) െ ݂ (ݎ )൯ Eq. 5
20
Here E may be considered as the energy in the simulated annealing method, B is an arbitrary weight
21
of the function f(r) and ri is the distance between two points of the system. To enable the digitized
22
model system to evolve toward the reference system we have had the aim to minimize the value of
1
E which is a property that decreases when the differences between the two correlation functions
2
reduce. More precisely, once the first value of E0 is calculated, an interchange of the states of two
3
pixels of different phase is performed (in other words, we arbitrarily change a white pixels of phase
4
1 with a black pixel of phase 2) allowing to preserve the volume fraction of both phase during the
5
reconstruction procedures. Then a new value of energy, E1, is calculated and from these the
6
difference of energy between two different states $E = E1 – E0 is calculated; if $E is negative the
7
interchange of the two pixels is accepted, otherwise another interchanges is performed. This
8
procedure is carried out until the value of energy becomes less than a small tolerance or when an
9
equilibrium condition in which a large number of consecutive unsuccessful interchanges, occurs (~
10
20.000). In our case, since lineal-path function was calculated in four directions, equation 5
11
becomes: (,௧,௬,௨)
= ܧσ σ σ௧ σ௬ σ௨ ߚ(,௧,௬,௨) ቂܨܲܮ௦
12
(,௧,௬,௨)
(ݎ ) െ ܨܲܮ
(ݎ )ቃ
ଶ
Eq. 6
13
where q is the number of LPFs extracted in different directions, Bj,t,y,u are the weights of LPFs in
14
each direction and ri is the length of a segment of 1, 2, 3 …….i pixels. Since no available software
15
able to perform the reconstruction procedure on the basis of above method, an algorithm able to
16
carried out the different steps of the annealing procedure above described was developed in Matlab
17
ver R2012b (Mathworks). We put equal weight on the importance of lineal-path distribution
18
function calculated in different direction such that B(j,t,y,u) = 1.
19
2.5 Accuracy and efficiency of the reconstruction
20
The accuracy of the reconstruction procedure was evaluated not only by visually comparing the
21
bread and reconstructed images, but also by the lineal-path distribution functions; this was
22
necessary because as reported from several authors (Yeong and Torquato, 1998; Jao et al., 2009)
23
even though two systems may show the same distribution function their microstructure could not be
24
match well.
1
3. RESULTS AND DISCUSSION
2
To show the ability of the reconstruction procedure, figure 1 reports the evolution of the energy
3
when the information contained into LPF0 and LPF90 were combined to rebuild a reference
4
(digitized) system of 32*32 pixels. Particularly, the system is characterized from nine equidistance
5
squares of 8*8 pixels (phase 1) with a porosity fraction, F of ~ 56%. Figure 1a shows the changes
6
of energy during the first 10,000 attempts while the complete evolution of energy (after 5*105 steps)
7
is shown in figure 1b. The energy abruptly decreased during the first 3,000 steps showing a value of
8
0.0419 which progressively reduced until 3.66*10-4 after 3*105 attempts. As expected, as energy
9
decreases as the random image evolves toward the reference one (Figs. 1d-h). When the energy was
10
0.0419 the reconstructed image (fig.1e) dramatically deviates from the reference, while after 104
11
attempts it begins to evolve toward the reference image reaching an energy of 0.0033 after 5*104
12
steps (fig. 1g). Then, a slow reduction of energy occurs until 3.66*10-4 observed after 5*105
13
attempts where the system may be considered at equilibrium. In this state, though the images cannot
14
be considered perfectly the same, the reconstruction contains the overall features of the reference
15
image; in fact 6 black squares are clearly observed while the other 3 contains only some defects. To
16
show the accuracy of the reconstruction, the comparison of lineal path distribution functions of
17
phase 1 along the horizontal direction is depicted in figure 2. It is clear that when the energy was of
18
3.66*10-4, an excellent agreement between the L(x) of reference and reconstructed image was
19
obtained. These preliminary results states that the algorithm developed for the reconstruction of
20
images works precisely. In the next section of the paper the reconstruction of bread structure
21
through the use of different statistical descriptors is shown, focusing the effort on the void phase
22
(pores). Particularly, a region of interest (ROI) of a bread slice image with a dimension of 100*100
23
pixels was arbitrarily chosen. This was necessary because the time consumed during the
24
reconstruction procedure, exponentially increases as a function of the number of statistical
25
descriptors used as well as with the dimension of the image. In light of these considerations, and
1
with the aim to study the ability in reconstructing of bread microstructure only by limited statistical
2
information of lineal-path function, we believe that 2D image of 100*100 pixels may be considered
3
sufficiently representative of bread crumb. Figure 3 shows the results of reconstruction obtained
4
individually using the aforementioned statistical descriptors. After 150,000 attempts, energy of
5
3.34*10-7 was obtained when the lineal path distribution function along the horizontal axis (LPF0)
6
was used as microstructure descriptor (Table 1). This value may be considered highly acceptable for
7
our purposes. It is higher than the values used from Li et al. (2012) who, studying the reconstruction
8
of digitized image, reached an energy value of 0.001 while it was similar to E value of 10-7 used
9
from Jao et al. (2009). Furthermore, this value was practically constant in the last 20,000 attempts
10
proving that the system reached it equilibrium state (Yeong and Torquato, 1998) after ~130,000
11
attempts (data not shown). In this condition a perfect match between the lineal-path functions of
12
binary and reconstructed image was obtained (data not shown). However, in the next section of the
13
paper we will consider as acceptable an energy value of ~ 10-5. Though the energy obtained using
14
LPF0 was very low, the figure 3b and 3d deeply deviate because the microstructure information
15
contained into LPF0 is not sufficient to precisely describe the bread structure. This is in accordance
16
with Jao et al., (2009) who reported as an infinite set of n-point correlation functions should be
17
required to completely characterize random texture. The authors studying different random systems
18
such as galaxy cluster and sphere packaging, showed as combining the information of two or more
19
statistical correlation functions the quality of reconstruction significantly improved. Similar results
20
were found by using as descriptors the lineal-path function along the vertical axis (LPF90) but in this
21
case after 150,000 attempts an energy value of 0.0230 was obtained which is higher than the value
22
obtained using LPF0. A number of 300,000 steps were necessary to reach the state of equilibrium
23
with an E value of 9.46*10-6 (table 1), at which the image of figure 3e was obtained. This greater
24
number of attempts necessary to reach the equilibrium was a consequence of the spatial structure of
25
the void phase, which is clearly characterized by pores with a maximum length in vertical direction.
26
However, although the LPF90 of binary and reconstructed images did match perfectly (figure 3h),
1
their visual aspect dramatically deviated. Again, the only information contained into the LPF90 it
2
was insufficient to obtain a precise reconstruction of the bread structure. Similar results were
3
obtained by using as microstructure descriptors LPF45 and LPF135 by which energy values of
4
1.46*10-6 and 1.26*10-5 were respectively obtained at equilibrium (table 1) (about 20,000 of
5
unsuccessfully attempts). Since the results proved as the information contained in the LPF along
6
only one direction, were not sufficient to obtain a good reconstruction, the qualitative and
7
quantitative agreement between real and reconstructed bread microstructure was assessed
8
combining the information of two statistical descriptors. For each combination of two LPF’s, the
9
energy values was reached at equilibrium were in the range of 10-5 – 10-6, stating a very good match
10
between the lineal-path functions of the bread structure and reconstructed system (Table 1). The
11
reconstructed images obtained through the use of two lineal-path distribution functions in different
12
directions are shown in figure 4. In general, all combination of LPF functions improved the
13
reconstructed images which were more similar to the bread sample. Using both the couples of
14
descriptors LPF0 - LPF45 and LPF0 - LPF135 images not similar but complementary were obtained as
15
well as when LPF90 - LPF45 and LPF90 - LPF135 were used during reconstruction. However in these
16
cases, the reconstructed images still deviate from the bread structure showing some “harshness”
17
along the perimeter of the pores in comparison with the binary image of bread sample. Combining
18
the information contained in the lineal-path functions extracted in diagonal directions (LPF45 and
19
LFP135, figure 4g) an improvement in describing the overall bread microstructure, was obtained. It
20
is reasonable to suppose that, through the microstructure information contained into LPF at 45° and
21
135°, it was possible to reconstruct pores with a more “roundness” along their perimeters, allowing
22
to obtain shapes more similar to the real pores of bread sample. However, this image shows smaller
23
crumb elements (solid matrix) when compared with bread microstructure, highlighting as some
24
deviation still are present. When two or more statistical correlation functions are used for the
25
reconstruction, the effect of each one on the total energy values should be considered and analyzed,
26
with the aim to gain information on their importance in the characterization of the microstructure.
1
With this purpose, figure 5 shows the evolution of the energy values associated to each LPF0 and
2
LPF90 when their information was combined during reconstruction procedure. We point out that the
3
values shown in the figure are the single energies of each LPF before that equation 6 was applied
4
(i.e. before that algorithm decides to accept or reject an interchange of a white with a black pixel).
5
From these, therefore, it is possible to appreciate their change during reconstruction procedure. The
6
values associated to LPF in vertical direction (LPF90) were significantly higher than those in
7
horizontal direction (LPF0), showing as our sample contains more microstructure information along
8
the y plane (fig 5a). This is also confirmed from the greater variability of the E values of LPF90 in
9
comparison with LPF0. On average, arbitrarily interchanging a black with a white pixel, the energy
10
related to LPF90 showed significant fluctuation while, in the case of LPF0, only slight changes were
11
observed. To make this clear, a small fraction of figure 5a is shown (figure 5b). This is because the
12
microstructure of our sample is characterized from a great number of segments which mostly extend
13
in vertical direction. Of course, under these conditions and taking into account the eq.6, the LPF90
14
played a key role in the decision of the acceptance or rejection of each single attempts of
15
reconstruction (i.e. the successful or unsuccessful reconstruction attempt). These results state the
16
higher weight of microstructure information contained into LPF90 rather than LPF0 for the
17
reconstruction of our sample. Furthermore, combining the information of three or of all the LPFs
18
function analyzed, energy values in the range of 1.76*10-5 and 6.50*10-5 were obtained stating the
19
high correlation between lineal-path distribution function of bread and of reconstructed
20
microstructure (table 1). As example, figure 6 shows the LPF0 and LPF90 of both bread structure
21
and reconstructed system, when the reconstruction procedures was performed combining all LPFs.
22
It is clear that when the system reached its equilibrium state (energy of 2.6*10-5) a perfect match
23
was obtained both in horizontal and vertical direction; the same results were obtained for LPF in
24
angular directions (data not shown). With the aim to evaluate the ability in the reconstructing
25
images, figure 7 shows the results obtained when some combination of three LPF or all the four
26
LPF were used. Combining the information of three LPFs, some improvement of the reconstructed
1
images were observed. In comparison whit the images obtained combining two LPFs (figure 4), a
2
less number of pores each of which with bigger dimension were depicted, allowing to obtain an
3
overall enhancement of the images which moved toward the real bread microstructure. When all
4
LPFs were used during reconstruction, four main pores were obtained in comparison with the five
5
main pores obtained for the bread structure. Yet, studying the changes of the single energy values of
6
each LPF, it was shown that the most important contribute on the total value of E was given from
7
the LPF90 (data not shown). This result states as the most part of microstructure of our sample
8
resides in vertical direction. According to this result, it is possible to note (in the left side of the
9
image) the presence of a pore which extends across the vertical axis similarly with the right side of
10
the reference image; furthermore, the dimension of the solid matrix (crumb) phase appears to be
11
similar to the bread structure in several sections of the image. Of course, some deviation still are
12
present as the absence of a clear roughly “round” pores in the centre of the image but this is due to
13
from the nature of reconstruction procedures and from the type of correlation functions used. Since
14
LPF contains microstructure information along a straight line, during reconstruction the algorithm
15
tends to align the pixels in order match the lineal-path function of the reference with the
16
reconstructed image. However, the results show as the salient features of the microstructure of
17
bread sample were captured though the combined used of LPF function of solid matrix phase and/or
18
the use of more statistical correlation functions such as two-point correlation function or two-point
19
cluster function would useful to improve the reconstruction by the addiction of different
20
microstructure information.
21
4. CONCLUSIONS
22
For the first time the reconstruction of food microstructure by using some statistical correlation
23
function was studied. The reconstruction procedure was proved to be a useful method by which a
24
random image is pushed toward the bread structure iteratively interchanging the phase of two
25
pixels. In all cases, the rebuild allowed to obtain energy values of the systems always of 10-5 stating
1
the perfect match between the LPFs of the reference and reconstructed images. However, visually
2
speaking, using lineal-path distribution function individually in horizontal, vertical and angular (45°
3
and 135°) the reconstructed images deeply deviated from the bread samples, while combining the
4
information contained in two LPFs, the obtained images began to move toward bread
5
microstructure. It was shown that studying the changes of energy values associated at each LPFs
6
function it is possible to highlight the different weight of each one of these on the reconstruction
7
procedure. In our case, the LPF in vertical direction had the most importance in the decision of the
8
accepted or rejected interchange of the phase of two pixels. Adding more statistical information, the
9
reconstruction procedure improved, approaching the microstructure of bread samples and
10
reconstructed image. When all LPF were combined for the reconstruction, although the
11
reconstructed images still showed some deviation from the reference one, the salient features of the
12
bread microstructure were captured, proving as the reconstruction of bread microstructure is
13
reasonable to obtain. Of course, in the next future different types of statistical correlation function
14
should be used to obtain a better reconstruction of food microstructure.
15
16
5. REFERENCES
17
Adler, P.M., Jacquin, C.G., & Quiblier, A. (1990). International Journal Multiphase Flow, 16, 691.
18
Aguilera J.M. (2005). Why food microstructure?. Journal of Food Engineering, 67, 3-11.
19
Baniassadi, M., Mortazzavi, B., Amani Hamedani, H., Garmestani, H., Ahzi, S., Fathi-Torbaghan,
20
M., Ruch, D., & Khaleel, M. (2012). Three dimensional reconstruction and homogenization of
21
heterogeneous materials using statistical correlation functions and FEM. Computational Materials
22
Science, 51, 372-379.
23
Berk, N.F. (1991). Scattering properties of the leveled-wave model of random morphologies.
24
Physical Review A, 44, 5069.
1
Chan, T.P., & Govindaraju, R.S. (2004). Estimating soil water retention curve from particle-size
2
distribution data based on polydisperse sphere systems. Vadose Zone Journal, 3, 1443-1454.
3
Datta, A.K. (2007). Porous media approaches to studying simultaneous heat and mass transfer in
4
food processes. I: Problem formulations. Journal of Food Engineering, 80, 80-95.
5 6
Derossi, A., De Pilli, T., & Severini, C. (2012a). Statistical description of fat and meat phases of
7
sausages by the use of lineal-path distribution function. Food Biophysics, 7, 258-263.
8
Derossi, A. De Pilli, T., & Severini, C. (2013a). Statistical description of food microstructure.
9
Extraction of some correlation functions from 2D images. Food Biophysics, 8, 311-320.
10
Derossi, A., De Pilli, T., & Severini, C (2013b) Use of lineal-path distribution function as sta
11
statistical descriptor of the crumb structure of bread. Food Biophysics, DOI 10.1107/s11483-013-
12
9289-0.
13
Jao, Y, Stillinger, F.H., & Torquato, S. (2007). Modeling heterogeneous materials via two-point
14
correlation function: Basic principles. Physical Review E, 76, 031110-1 – 031110-15.
15
Jiao, Y. Stillinger, F.H., & Torquato, S. (2010). Geometrical ambiguity of pair statistics: Point
16
configuration. Physical Review E, 81-92.
17
Jiao, Y., Stilinger, F.H., & Torquato, S. (2009). A superior descriptors of random textures and its
18
predictive capacity. PNAS, 106 (42), 17634-17639.
19
Joshi, M. (1974). PhD thesis, University of Kansas, Lawrence.
20
Lee, S.B., & Torquato, S. (1989). Measure of clustering in continuum percolation: Computer-
21
simulation of the two-point cluster function. Journal of Chemical Physics, 91 (2), 1173-1178.
1
Li, D.S., Tschopp, M.A., Khaleel, M., & Sun, X. (2012). Comparison of reconstructed spatial
2
microstructure images using different statistical descriptors. Computational Materials Science, 51,
3
437-444.
4
Lu, B., & Torquato, S. (1992a). Lineal-path function for random heterogeneous materials, II. Effect
5
of polydispersivity. Physical Review A, 45 (10), 7292-7301.
6
Lu, B., & Torquato, S. (1992b). Lineal-path function for random heterogeneous materials. Physical
7
Review A, 45 (2), 922-929.
8
Parada, J., & Aguilera, J.M. (2007). Food microstructure affects the bioavailability of several
9
nutrients. Journal of Food Science, 72 (2), 21-32.
10
Quiblier, J.A. (1984). A new three-dimensional modeling techniques for studying porous media.
11
Journal of Colloid Interface Science, 98, 84.
12
Quintavalla, J. (2006). Measures of clustering in systems of overlapping particles. Mechanics of
13
Materials, 38, 849-858.
14
Rintoul, M.D., & Toruato, S. (1997). Reconstruction of the structure of dispersion. Journal of
15
Colloid and Interface Science, 186, 467- 476.
16
Rintoul, M.D., Torquato, S., Yeong, C., Keane, D.T., Erramilli, S., Jun, Y.N., Dabbs, D.M., &
17
Aksay, I.A. (1996). Structure and transport properties of a porous magnetic gel via x-ray
18
microtomography. Physical Review E, 54 (3), 2663-2669.
19
Russ, J.C. (2005). Image analysis of food microstructure. Boca Raton, LA, CRC press.
20
Sezgin, M., & Sankur, B. (2003). Survey over image thresholding techniques and quantitative
21
performance evaluation. Journal Electron Imaging, 13 (1), 146–165, (2003)
1
Singh, H., Gokhale, A.M., & Lieberman, S.I., & Tamirisakandala, S. (2008). Image based
2
computations of lineal path probability distributions for microstructure representation. Material
3
Science and Engineering A, 474, 104-111.
4
Smith, P., & Torquato, S. (1988). Computer simulation results for the two-point probability
5
function of composite media. Journal of Computational Physics, 76, 176-191.
6
Torquato, S. (2002). Statistical Descrption of microstructures. Annual Review of Material Research,
7
32, 77-111.
8
Torquato, S., & Lu, B. (1993). Chord-length distribution function for two-phase random media.
9
Physical Review E, 47 (4), 2950-2953.
10
Torquato, S., Beasley, J.D., & Chiew, Y.C. (1988). Two-point cluster function for continuum
11
percolation. Journal of Chemical Physic, 88 (10), 6540 – 6547.
12
Yeong, C.L.Y., & Torquato, S. (1998). Reconstructing random media. Physical Review E, 57 (1),
13
495-506.
14 15
This paper has not been published previously and it is not under consideration for publication
16
elsewhere. Moreover the paper is approved by all authors and tacitly or explicitly by the responsible
17
authorities where the work was carried out and that, if accepted, it will not be published elsewhere
18
including electronically in the same form, in English or in any other language, without the written
19
consent of the copyright-holder. The authors declare that they have no actual or potential conflict of
20
interest including any financial, personal or other relationships with other people or organization
21
within three years of beginning the submitted work that could inappropriately influence, or be
22
perceived to influence, their work.
23
1
Figure captions
2 3 4 5 6
Fig 1 - Changes of energy (E) as a function of attempts. a) Evolution of E during the first 104; b) Evolution of E during 5*105 steps. c) reference images; d) random image; e) reconstructed image after 3,000 attempts; f) reconstructed image after 104 attempts; g) reconstructed image after 5*104 attempts; h) reconstructed image after 5*105 attempts. Reference systems is a digitized image of 32*32 pixels with a porosity fraction,
F1 ~ 56%.
7 8 9 10
Fig 2 – Lineal-path distribution function of phase 1 for the reference and reconstructed image when an E of 3.66*10-4 was reached (after 5*105 attempts). (Filled squares) - reference image; (dotted line) – reconstructed image.
11 12 13 14 15 16
Fig 3 – Reference and reconstructed images of bread microstructure obtained by using different descriptors. a) original image; b) binary image; c) random image; d) LPF0; e) LPF90; f) LPF45; g)LPF135; h) Lineal path distribution of functions along vertical axis of original and reconstructed images obtained by using the descriptor LPF90. (dotted line), L(z) of reconstructed image; (void triangles), L(z) binary image of bread structure.
17 18 19 20
Fig 4 – Reference and reconstructed images of bread microstructure by using the combination of two statistical descriptors. a) Reference image; Reconstructed images: b) LPF0+LPF90; c) LPF0+LPF45; d) LPF90+LPF45; e) LPF0+LPF135; f) LPF90+LPF135; g) LPF45+LPF135.
21 22 23
Fig 5 – Energy values as function of the number of attempts during reconstruction of the microstructure of bread samples.
24 25 26 27
Fig 6 - Lineal path function extracted in horizontal (a) and vertical direction (b) of reference and reconstructed images obtained combining all LFPs during reconstruction procedures. (dotted line), bread sample; (open symbol), reconstructed image.
28 29 30 31 32 33
Fig 7 - Reference and reconstructed images obtained by combining different statistical descriptors. a) Reference image; reconstructed image: b) LPF0+LPF90+LPF45; c) LPF0+LPF45+LPF135; d) LPF0+LPF90+LPF45+LPF135.
1 2
Table 1 – Energy values at equilibrium obtained reconstructing the microstructure of bread sample by using different statistical descriptors. Descriptors LPF0 LPF45 LPF90 LPF135
Energy value 3.34*10-7 1.46*10-6 9.47*10-6 1.26*10-5 Combination of two LPF 2.56*10-5 3.08*10-6 1.25*10-5 6.85*10-6 1.62*10-5 3.58*10-5
LPF0+LPF90 LPF0+LPF45 LPF90+LPF45 LPF0+LPF135 LF90+LPF135 LPF45+LPF135 Combination of three LPF
3.76*10-5 6.50*10-5 1.76*10-5 2.50*10-5
LPF0+LPF90+LPF45 LPF0+LPF90+LPF135 LPF0+LPF45+LPF135 LPF90+LPF45+LPF135 Combination of four LPF LPF0+LPF90+LPF45+LPF135
2.60*10-5
3 4 5 6 7 8 9 10 11 12 13 14
Highlights Statistical correlation functions were used to reconstruct bread microstructure by applying an annealing method; Lineal-path distribution function (LPF) in different directions were used to reconstruct 2D images of bread; The reconstruction method is a powerful tool for reconstruction and its use could be extended for several foods; Although LPF of real and reconstructed images perfectly match, visually speaking some deviation were detected.
Fig 1
a)
b)
c)
d)
e)
f)
g)
h)
Fig 2
Fig 3
a)
c)
h)
d)
b)
e)
f)
g)
Fig 4
a)
b)
e)
c)
f)
d)
g)
Fig 5
a)
b)
Fig 6
a)
b)
Fig 7
a)
c)
b)
d)
S0260-8774(14)00227-1 http://dx.doi.org/10.1016/j.jfoodeng.2014.05.020 JFOE 7810
To appear in:
Journal of Food Engineering
Received Date: Revised Date: Accepted Date:
3 March 2014 21 April 2014 25 May 2014
Please cite this article as: Derossi, A., Severini, C., De Pilli, T., Reconstruction of food microstructure via statistical correlation functions. The use of lineal-path distribution functions, Journal of Food Engineering (2014), doi: http:// dx.doi.org/10.1016/j.jfoodeng.2014.05.020
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1 2
Reconstruction of food microstructure via statistical correlation functions. The use of lineal-path distribution functions.
3 4 5 6
Authors: Derossi, A., Severini, C. De Pilli, T.
7 8 9 10
Department of Science of Agriculture, Food and Environment, University of Foggia, Italy
11 12 13 14 15 16 17
18
19
20
21
22
23
24
Corresponding authors: [email protected], Department of Science of Agricultural, Food and Environment, University of Foggia, Via Napoli 25, Italy. Phone +39 0881 589245.
1
Abstract
2
The reconstruction of microstructure of biological and synthetic materials is of great importance for
3
theoretical and practical application. The obtaining of this purpose through limited statistical
4
information is a challenge at which some pioneering researchers are focusing their efforts. For the
5
first time, a reconstruction method was used to rebuild bread microstructure by using the
6
information of lineal-path distribution function (LPF). The method was a powerful tool to
7
reconstruct bread microstructure; a perfect match between lineal-path function of reference and
8
reconstructed images were obtained. The reconstructed images progressively improve, compared to
9
the reference, increasing the number of LPFs used during reconstruction. The use of LPFs in
10
different direction did not allowed to obtain a perfect match between the bread and the
11
reconstructed images because, LPF did not contain sufficient microstructure information however,
12
the essential features of bread were captured proving the possibility to reconstruct food
13
microstructure.
14
15
16
Keywords: morphological descriptors; microstructure; random media; reconstruction; distribution
17
function; bread
18
19
20
21
22
1
1. INTRODUCTION
2
Random heterogeneous materials are ubiquitous in nature and in synthetic systems. Examples are
3
soil, porous media, biological tissues, sandstone, fiber, rock formations, tree patterns in forests,
4
cosmological structure, etc. (Lu and Torquato, 1992a; Jiao et al., 2009). The precise understanding
5
of the microstructure of these systems is of crucial importance for practical application due to the
6
strict relation between microstructure information and the bulk properties of materials such as
7
conductivity, permeability, mechanical and electromagnetic properties, heat and mass transfer (Lee
8
and Torquato, 1989; Lu and Torquato, 1992a; Torquato and Lu, 1993; Russ, 2005; Baniassadi et al.;
9
2012; Li et al., 2012). Most of heterogeneous materials may be considered as two-phase random
10
systems composed from two different phases or from one phase in different states. In food science a
11
lot of product may be considered as two-phase systems in which a void phase (pores) and a solid
12
matrix phase (cell membranes, proteins, crystals, globules, etc.) exists. A typical examples are bread
13
characterized from pores and solid matrix (crumb), and sausages, composed by fat globules
14
(discontinuous phase) and protein elements (continuous phase), or emulsions composed by water
15
and oil phases. The importance of microstructure on the quality of food is not a new idea (Aguilera
16
2005; Parada and Aguilera, 2007; Datta 2007). Aguilera (2005) was the first scientist who
17
highlighted as all macroscopic properties of food are governed from elements in the range of 10-
18
100 Mm. In an interesting paper, he clearly analyzed as food microstructure greatly affects
19
nutritional values, sensorial properties, transport phenomena as well as microbial stability. Later,
20
Datta (2007) who studying the food as porous media, showed as the intrinsic permeability, k, which
21
is a parameter strictly related with pore connectivity, significantly affects the mass transfer during
22
processing. Furthermore, Parada and Aguilera (2007) showed the importance of microstructure on
23
the bioavailability of several nutrients. In light of above considerations, one of the most important
24
challenge is to ascertain what is the essential information to obtain the quantitative characterization
25
of food microstructure, its exact theoretical and experimental quantification and its relation with the
1
macroscopic properties. With this aim some authors, on the basis of a rigorous theory, developed a
2
series of statistical correlation functions able to extract microstructure information from two-phase
3
random systems (Torquato et al., 1988; Lu and Torquato, 1992b; Torquato and Lu, 1993; Torquato,
4
2002; Quintavalla, 2006). One of these basic functions is the n-point correlation function
5
Sn(x1,x2,…….xn), which is the probability to find n points at position x1, x2,…….xn all in one of the
6
two phases of the system (Torquato et al. 2000). However, since infinite n-point functions are not
7
attainable in practice, low-order version, such as a two-point correlation S2(x1, x2) and three-point
8
function S3(x1, x2, x3), are commonly used (Smith and Torquato, 1988; Quitavalla, 2006; Jao et al.
9
2007). The lineal-path distribution function, Li(z) is another interesting statistical descriptor, which
10
refers the probability that a segment of length z completely falls in the phase i (Lu and torquato,
11
1992a, Lu and Torquato 1992b; Torquato et al., 2002; Singh et al., 2008). This function contains
12
important connectedness information along a straight line and also it gives some information in
13
stereology. Further correlation functions such as chord-length distribution function, p(z), pore size
14
distribution function, P(z), two-point cluster function, C2(x1,x2), have been developed and validated
15
for several digitized model systems (identical hard disks, identical overlapping disks, periodic rods,
16
Debye overlapping disks, etc.) and for some materials such as sandstone, magnetic gels, Boron
17
modified Ti-alloys (Rintoul et al., 1996; Singh et al.,2008; Chan and Covindaraju, 2004) while very
18
few papers showed the use of these function to obtain microstructure information of food (Derossi
19
et al., 2012; Derossi et al., 2013a; Derossi et al., 2013b). The reconstruction of microstructure of
20
random heterogeneous systems from limited microstructure information is an interesting inverse
21
problem. If achieved, this result should enable to generate an accurate bi- or three-dimensional
22
structure of any material from which should be possible to estimate their macroscopic properties.
23
Moreover, the study of reconstruction will give light on the type of microstructure information
24
contained into the statistical correlation functions. Although different approaches to reconstruct
25
random media have been proposed (Joshi, 1974; Quiblier, 1984; Adler et al., 1990; Berk, 1991) we
26
focus our attention on the simulated annealing method proposed from Rintoul and Torquato (1997).
1
The method is based on the finding of a state of minimum ‘energy’ by interchanging the phase of
2
two the pixels in a digitized random model system (simulated), having the same porosity fraction of
3
the reference material (Yeong and Torquato, 1998). The ‘energy’ is defined as the sum of square of
4
the difference between the distribution functions of simulated and reference systems. Although this
5
method has been used for several digitized model systems showing a good ability in the
6
reconstruction of two-phase random systems (Rintoul and Torquato, 1996; Yeong and Torquato,
7
1997; Li et al., 2012; Baniassadi et al., 2012) this problem is still understudied and never it was
8
used for the reconstruction of food microstructure. On the basis of these considerations, the aim of
9
this paper was to study the reconstruction of bread microstructure by using the information
10
contained in the lineal-path distribution function extracted, from 2D images, in different direction.
11 12
2. MATERIAL AND METHODS
13 14
2.1 Theoretical background
15
A random medium is a domain of space V(W) ߳ R3 (the realization W is taken from some probability
16
space 7) where the volume V is characterized from two-phases: phase 1 in the region ϝ1 with a
17
volume fraction Ѱ1, and phase 2 in the region ϝ2 with a volume fraction Ѱ2. For a given realization
18
W, the characteristic function I(x) of phase 1 may be reported as:
19
(ܫ௫) = ൜
1, 0,
݂݅ ݁ݏ݄ܽ ݄݁ݐ ݊݅ ݏ݈݈݂ܽ ݔ1 (߳ ᎇ ) ݁ݏ݅ݓݎ݄݁ݐ
Eq.1
20
Where ᎇ is the region occupied by phase i (Equal to 1 or 2) (Lu and Torquato, 1992). Also, a
21
useful system to study heterogeneous material is a two-phase fully penetrable spheres which is
22
obtained adding polydispersed spheres of radius R until a specific porosity fraction is reached. Lu
23
and Torquato (1992a), based on a rigorous theory, showed that for these systems the lineal-path
24
distribution function is defined as
మ(శభ)
ଵା[ഏೃ(శమ)]
(ܮ௭) = ߶ଵ
1
Eq. 2
2
Where R is the average radius of the disks, z is the length of the segment (pixels or mm) and m is
3
the parameter of Schultz size distribution which is equal to:
4
݂(ܴ) =
ଵ ௰(ାଵ)
ቂ
ାଵ ାଵ ቃ ۃோۄ
ܴ ݁ ݔቂ
ି (ାଵ)ோ ቃ ۃோۄ
Eq. 3
5
Where ߁ is the gamma function.
6
2.2 Raw material and images acquisition
7
Bread was chosen as model because their structure may be considered as a two-phase systems in
8
which the void (pores) is the phase 1 and the solid matrix (crumb) is the phase 2. Also, since the
9
position of pores produced during yeast fermentation is extremely affected from several random
10
variables such as air incorporation, dough preparation, bread structure may be considered a random
11
system. Particularly, we used “Altamura” bread (Oropan s.p.a., Italy), purchased locally because of
12
the large diffusion of bread in the south Italy. The loaves were cut to obtain slices with a thickness
13
of 1 cm and the images were acquired by using a flat scanner mod. Hp 3600 (Hp, Scanjet) covering
14
the samples with a black box to obtain a good contrast between the background and the samples,
15
and to guarantee constant lightness conditions. The images were acquired by positioning the top of
16
the slice sample parallel with the light of scanner (x axis) with the aim to avoid the effect of
17
sample’s position on lineal-path distribution function. A resolution of 600 dpi = 0.004233
18
mm/pixels was used and the images were saved in TIFF format. Binary image were obtained by
19
applying the Otsu’s method (Sezgin et al. 2003; Jao et al., 2010), choosing the threshold through the
20
functions “rgb2gray” and “graytresh” available in the image analysis Toolbox of Matlab R2012b
21
(Mathworks, USA).
22
23
2.3 Computation of lineal-path distribution function
1
The binary images of bread may be represented by a common two-dimensional arrays:
2
ܫଵ,ଵ ܫ = ܫଶ,ଵ ܫଷ,ଵ
ܫଵ,ଶ ܫଶ,ଶ ܫଷ,ଶ
ܫଵ,ଷ ܫଶ,ଷ ܫ,
Eq. 4
3
where Ii,j (i = j = 1, 2, 3, 4.….., n) only assumes value 0 (black pixels) or 1 (white pixels); black
4
pixels refer to the pores of bread (phase 1), while the white pixels refer to the crumb (phase 2).
5
Lineal-path distribution function (LPF) was extracted in four different directions: 1) LPF0 along
6
horizontal direction (x axis, 0°); 2) LPF90 along vertical direction (y axis, 90°); 3) LPF45 along
7
diagonal direction (45°); 4) LPF135 along diagonal direction (135°). The extraction of LPF in
8
horizontal and vertical direction was obtained by using the algorithm developed and validated by
9
Derossi et al. (2012a), Derossi et al. (2013a) and Derossi et al. (2013b). In addition, a modified
10
version of the these algorithms was developed to extract LPF function along diagonal directions
11
(45° and 135°).
12
2.4 Reconstruction procedure
13
For simplicity, we consider the reconstruction procedure of a general two-phase random system
14
carried out by using the microstructure information provides from a general statistical correlation
15
function, f(r). Also, we defined the correlation function of the phase j (equal to 1 or 2) of our
16
“reference” systems as f0(r) and the lineal-path function of the reconstructed model system as fs(r)
17
which will evolve toward f0(r). Once the fs(r) is calculated we can define a new variable E as
18
follows: ଶ
19
= ܧσ ߚ ൫݂௦ (ݎ ) െ ݂ (ݎ )൯ Eq. 5
20
Here E may be considered as the energy in the simulated annealing method, B is an arbitrary weight
21
of the function f(r) and ri is the distance between two points of the system. To enable the digitized
22
model system to evolve toward the reference system we have had the aim to minimize the value of
1
E which is a property that decreases when the differences between the two correlation functions
2
reduce. More precisely, once the first value of E0 is calculated, an interchange of the states of two
3
pixels of different phase is performed (in other words, we arbitrarily change a white pixels of phase
4
1 with a black pixel of phase 2) allowing to preserve the volume fraction of both phase during the
5
reconstruction procedures. Then a new value of energy, E1, is calculated and from these the
6
difference of energy between two different states $E = E1 – E0 is calculated; if $E is negative the
7
interchange of the two pixels is accepted, otherwise another interchanges is performed. This
8
procedure is carried out until the value of energy becomes less than a small tolerance or when an
9
equilibrium condition in which a large number of consecutive unsuccessful interchanges, occurs (~
10
20.000). In our case, since lineal-path function was calculated in four directions, equation 5
11
becomes: (,௧,௬,௨)
= ܧσ σ σ௧ σ௬ σ௨ ߚ(,௧,௬,௨) ቂܨܲܮ௦
12
(,௧,௬,௨)
(ݎ ) െ ܨܲܮ
(ݎ )ቃ
ଶ
Eq. 6
13
where q is the number of LPFs extracted in different directions, Bj,t,y,u are the weights of LPFs in
14
each direction and ri is the length of a segment of 1, 2, 3 …….i pixels. Since no available software
15
able to perform the reconstruction procedure on the basis of above method, an algorithm able to
16
carried out the different steps of the annealing procedure above described was developed in Matlab
17
ver R2012b (Mathworks). We put equal weight on the importance of lineal-path distribution
18
function calculated in different direction such that B(j,t,y,u) = 1.
19
2.5 Accuracy and efficiency of the reconstruction
20
The accuracy of the reconstruction procedure was evaluated not only by visually comparing the
21
bread and reconstructed images, but also by the lineal-path distribution functions; this was
22
necessary because as reported from several authors (Yeong and Torquato, 1998; Jao et al., 2009)
23
even though two systems may show the same distribution function their microstructure could not be
24
match well.
1
3. RESULTS AND DISCUSSION
2
To show the ability of the reconstruction procedure, figure 1 reports the evolution of the energy
3
when the information contained into LPF0 and LPF90 were combined to rebuild a reference
4
(digitized) system of 32*32 pixels. Particularly, the system is characterized from nine equidistance
5
squares of 8*8 pixels (phase 1) with a porosity fraction, F of ~ 56%. Figure 1a shows the changes
6
of energy during the first 10,000 attempts while the complete evolution of energy (after 5*105 steps)
7
is shown in figure 1b. The energy abruptly decreased during the first 3,000 steps showing a value of
8
0.0419 which progressively reduced until 3.66*10-4 after 3*105 attempts. As expected, as energy
9
decreases as the random image evolves toward the reference one (Figs. 1d-h). When the energy was
10
0.0419 the reconstructed image (fig.1e) dramatically deviates from the reference, while after 104
11
attempts it begins to evolve toward the reference image reaching an energy of 0.0033 after 5*104
12
steps (fig. 1g). Then, a slow reduction of energy occurs until 3.66*10-4 observed after 5*105
13
attempts where the system may be considered at equilibrium. In this state, though the images cannot
14
be considered perfectly the same, the reconstruction contains the overall features of the reference
15
image; in fact 6 black squares are clearly observed while the other 3 contains only some defects. To
16
show the accuracy of the reconstruction, the comparison of lineal path distribution functions of
17
phase 1 along the horizontal direction is depicted in figure 2. It is clear that when the energy was of
18
3.66*10-4, an excellent agreement between the L(x) of reference and reconstructed image was
19
obtained. These preliminary results states that the algorithm developed for the reconstruction of
20
images works precisely. In the next section of the paper the reconstruction of bread structure
21
through the use of different statistical descriptors is shown, focusing the effort on the void phase
22
(pores). Particularly, a region of interest (ROI) of a bread slice image with a dimension of 100*100
23
pixels was arbitrarily chosen. This was necessary because the time consumed during the
24
reconstruction procedure, exponentially increases as a function of the number of statistical
25
descriptors used as well as with the dimension of the image. In light of these considerations, and
1
with the aim to study the ability in reconstructing of bread microstructure only by limited statistical
2
information of lineal-path function, we believe that 2D image of 100*100 pixels may be considered
3
sufficiently representative of bread crumb. Figure 3 shows the results of reconstruction obtained
4
individually using the aforementioned statistical descriptors. After 150,000 attempts, energy of
5
3.34*10-7 was obtained when the lineal path distribution function along the horizontal axis (LPF0)
6
was used as microstructure descriptor (Table 1). This value may be considered highly acceptable for
7
our purposes. It is higher than the values used from Li et al. (2012) who, studying the reconstruction
8
of digitized image, reached an energy value of 0.001 while it was similar to E value of 10-7 used
9
from Jao et al. (2009). Furthermore, this value was practically constant in the last 20,000 attempts
10
proving that the system reached it equilibrium state (Yeong and Torquato, 1998) after ~130,000
11
attempts (data not shown). In this condition a perfect match between the lineal-path functions of
12
binary and reconstructed image was obtained (data not shown). However, in the next section of the
13
paper we will consider as acceptable an energy value of ~ 10-5. Though the energy obtained using
14
LPF0 was very low, the figure 3b and 3d deeply deviate because the microstructure information
15
contained into LPF0 is not sufficient to precisely describe the bread structure. This is in accordance
16
with Jao et al., (2009) who reported as an infinite set of n-point correlation functions should be
17
required to completely characterize random texture. The authors studying different random systems
18
such as galaxy cluster and sphere packaging, showed as combining the information of two or more
19
statistical correlation functions the quality of reconstruction significantly improved. Similar results
20
were found by using as descriptors the lineal-path function along the vertical axis (LPF90) but in this
21
case after 150,000 attempts an energy value of 0.0230 was obtained which is higher than the value
22
obtained using LPF0. A number of 300,000 steps were necessary to reach the state of equilibrium
23
with an E value of 9.46*10-6 (table 1), at which the image of figure 3e was obtained. This greater
24
number of attempts necessary to reach the equilibrium was a consequence of the spatial structure of
25
the void phase, which is clearly characterized by pores with a maximum length in vertical direction.
26
However, although the LPF90 of binary and reconstructed images did match perfectly (figure 3h),
1
their visual aspect dramatically deviated. Again, the only information contained into the LPF90 it
2
was insufficient to obtain a precise reconstruction of the bread structure. Similar results were
3
obtained by using as microstructure descriptors LPF45 and LPF135 by which energy values of
4
1.46*10-6 and 1.26*10-5 were respectively obtained at equilibrium (table 1) (about 20,000 of
5
unsuccessfully attempts). Since the results proved as the information contained in the LPF along
6
only one direction, were not sufficient to obtain a good reconstruction, the qualitative and
7
quantitative agreement between real and reconstructed bread microstructure was assessed
8
combining the information of two statistical descriptors. For each combination of two LPF’s, the
9
energy values was reached at equilibrium were in the range of 10-5 – 10-6, stating a very good match
10
between the lineal-path functions of the bread structure and reconstructed system (Table 1). The
11
reconstructed images obtained through the use of two lineal-path distribution functions in different
12
directions are shown in figure 4. In general, all combination of LPF functions improved the
13
reconstructed images which were more similar to the bread sample. Using both the couples of
14
descriptors LPF0 - LPF45 and LPF0 - LPF135 images not similar but complementary were obtained as
15
well as when LPF90 - LPF45 and LPF90 - LPF135 were used during reconstruction. However in these
16
cases, the reconstructed images still deviate from the bread structure showing some “harshness”
17
along the perimeter of the pores in comparison with the binary image of bread sample. Combining
18
the information contained in the lineal-path functions extracted in diagonal directions (LPF45 and
19
LFP135, figure 4g) an improvement in describing the overall bread microstructure, was obtained. It
20
is reasonable to suppose that, through the microstructure information contained into LPF at 45° and
21
135°, it was possible to reconstruct pores with a more “roundness” along their perimeters, allowing
22
to obtain shapes more similar to the real pores of bread sample. However, this image shows smaller
23
crumb elements (solid matrix) when compared with bread microstructure, highlighting as some
24
deviation still are present. When two or more statistical correlation functions are used for the
25
reconstruction, the effect of each one on the total energy values should be considered and analyzed,
26
with the aim to gain information on their importance in the characterization of the microstructure.
1
With this purpose, figure 5 shows the evolution of the energy values associated to each LPF0 and
2
LPF90 when their information was combined during reconstruction procedure. We point out that the
3
values shown in the figure are the single energies of each LPF before that equation 6 was applied
4
(i.e. before that algorithm decides to accept or reject an interchange of a white with a black pixel).
5
From these, therefore, it is possible to appreciate their change during reconstruction procedure. The
6
values associated to LPF in vertical direction (LPF90) were significantly higher than those in
7
horizontal direction (LPF0), showing as our sample contains more microstructure information along
8
the y plane (fig 5a). This is also confirmed from the greater variability of the E values of LPF90 in
9
comparison with LPF0. On average, arbitrarily interchanging a black with a white pixel, the energy
10
related to LPF90 showed significant fluctuation while, in the case of LPF0, only slight changes were
11
observed. To make this clear, a small fraction of figure 5a is shown (figure 5b). This is because the
12
microstructure of our sample is characterized from a great number of segments which mostly extend
13
in vertical direction. Of course, under these conditions and taking into account the eq.6, the LPF90
14
played a key role in the decision of the acceptance or rejection of each single attempts of
15
reconstruction (i.e. the successful or unsuccessful reconstruction attempt). These results state the
16
higher weight of microstructure information contained into LPF90 rather than LPF0 for the
17
reconstruction of our sample. Furthermore, combining the information of three or of all the LPFs
18
function analyzed, energy values in the range of 1.76*10-5 and 6.50*10-5 were obtained stating the
19
high correlation between lineal-path distribution function of bread and of reconstructed
20
microstructure (table 1). As example, figure 6 shows the LPF0 and LPF90 of both bread structure
21
and reconstructed system, when the reconstruction procedures was performed combining all LPFs.
22
It is clear that when the system reached its equilibrium state (energy of 2.6*10-5) a perfect match
23
was obtained both in horizontal and vertical direction; the same results were obtained for LPF in
24
angular directions (data not shown). With the aim to evaluate the ability in the reconstructing
25
images, figure 7 shows the results obtained when some combination of three LPF or all the four
26
LPF were used. Combining the information of three LPFs, some improvement of the reconstructed
1
images were observed. In comparison whit the images obtained combining two LPFs (figure 4), a
2
less number of pores each of which with bigger dimension were depicted, allowing to obtain an
3
overall enhancement of the images which moved toward the real bread microstructure. When all
4
LPFs were used during reconstruction, four main pores were obtained in comparison with the five
5
main pores obtained for the bread structure. Yet, studying the changes of the single energy values of
6
each LPF, it was shown that the most important contribute on the total value of E was given from
7
the LPF90 (data not shown). This result states as the most part of microstructure of our sample
8
resides in vertical direction. According to this result, it is possible to note (in the left side of the
9
image) the presence of a pore which extends across the vertical axis similarly with the right side of
10
the reference image; furthermore, the dimension of the solid matrix (crumb) phase appears to be
11
similar to the bread structure in several sections of the image. Of course, some deviation still are
12
present as the absence of a clear roughly “round” pores in the centre of the image but this is due to
13
from the nature of reconstruction procedures and from the type of correlation functions used. Since
14
LPF contains microstructure information along a straight line, during reconstruction the algorithm
15
tends to align the pixels in order match the lineal-path function of the reference with the
16
reconstructed image. However, the results show as the salient features of the microstructure of
17
bread sample were captured though the combined used of LPF function of solid matrix phase and/or
18
the use of more statistical correlation functions such as two-point correlation function or two-point
19
cluster function would useful to improve the reconstruction by the addiction of different
20
microstructure information.
21
4. CONCLUSIONS
22
For the first time the reconstruction of food microstructure by using some statistical correlation
23
function was studied. The reconstruction procedure was proved to be a useful method by which a
24
random image is pushed toward the bread structure iteratively interchanging the phase of two
25
pixels. In all cases, the rebuild allowed to obtain energy values of the systems always of 10-5 stating
1
the perfect match between the LPFs of the reference and reconstructed images. However, visually
2
speaking, using lineal-path distribution function individually in horizontal, vertical and angular (45°
3
and 135°) the reconstructed images deeply deviated from the bread samples, while combining the
4
information contained in two LPFs, the obtained images began to move toward bread
5
microstructure. It was shown that studying the changes of energy values associated at each LPFs
6
function it is possible to highlight the different weight of each one of these on the reconstruction
7
procedure. In our case, the LPF in vertical direction had the most importance in the decision of the
8
accepted or rejected interchange of the phase of two pixels. Adding more statistical information, the
9
reconstruction procedure improved, approaching the microstructure of bread samples and
10
reconstructed image. When all LPF were combined for the reconstruction, although the
11
reconstructed images still showed some deviation from the reference one, the salient features of the
12
bread microstructure were captured, proving as the reconstruction of bread microstructure is
13
reasonable to obtain. Of course, in the next future different types of statistical correlation function
14
should be used to obtain a better reconstruction of food microstructure.
15
16
5. REFERENCES
17
Adler, P.M., Jacquin, C.G., & Quiblier, A. (1990). International Journal Multiphase Flow, 16, 691.
18
Aguilera J.M. (2005). Why food microstructure?. Journal of Food Engineering, 67, 3-11.
19
Baniassadi, M., Mortazzavi, B., Amani Hamedani, H., Garmestani, H., Ahzi, S., Fathi-Torbaghan,
20
M., Ruch, D., & Khaleel, M. (2012). Three dimensional reconstruction and homogenization of
21
heterogeneous materials using statistical correlation functions and FEM. Computational Materials
22
Science, 51, 372-379.
23
Berk, N.F. (1991). Scattering properties of the leveled-wave model of random morphologies.
24
Physical Review A, 44, 5069.
1
Chan, T.P., & Govindaraju, R.S. (2004). Estimating soil water retention curve from particle-size
2
distribution data based on polydisperse sphere systems. Vadose Zone Journal, 3, 1443-1454.
3
Datta, A.K. (2007). Porous media approaches to studying simultaneous heat and mass transfer in
4
food processes. I: Problem formulations. Journal of Food Engineering, 80, 80-95.
5 6
Derossi, A., De Pilli, T., & Severini, C. (2012a). Statistical description of fat and meat phases of
7
sausages by the use of lineal-path distribution function. Food Biophysics, 7, 258-263.
8
Derossi, A. De Pilli, T., & Severini, C. (2013a). Statistical description of food microstructure.
9
Extraction of some correlation functions from 2D images. Food Biophysics, 8, 311-320.
10
Derossi, A., De Pilli, T., & Severini, C (2013b) Use of lineal-path distribution function as sta
11
statistical descriptor of the crumb structure of bread. Food Biophysics, DOI 10.1107/s11483-013-
12
9289-0.
13
Jao, Y, Stillinger, F.H., & Torquato, S. (2007). Modeling heterogeneous materials via two-point
14
correlation function: Basic principles. Physical Review E, 76, 031110-1 – 031110-15.
15
Jiao, Y. Stillinger, F.H., & Torquato, S. (2010). Geometrical ambiguity of pair statistics: Point
16
configuration. Physical Review E, 81-92.
17
Jiao, Y., Stilinger, F.H., & Torquato, S. (2009). A superior descriptors of random textures and its
18
predictive capacity. PNAS, 106 (42), 17634-17639.
19
Joshi, M. (1974). PhD thesis, University of Kansas, Lawrence.
20
Lee, S.B., & Torquato, S. (1989). Measure of clustering in continuum percolation: Computer-
21
simulation of the two-point cluster function. Journal of Chemical Physics, 91 (2), 1173-1178.
1
Li, D.S., Tschopp, M.A., Khaleel, M., & Sun, X. (2012). Comparison of reconstructed spatial
2
microstructure images using different statistical descriptors. Computational Materials Science, 51,
3
437-444.
4
Lu, B., & Torquato, S. (1992a). Lineal-path function for random heterogeneous materials, II. Effect
5
of polydispersivity. Physical Review A, 45 (10), 7292-7301.
6
Lu, B., & Torquato, S. (1992b). Lineal-path function for random heterogeneous materials. Physical
7
Review A, 45 (2), 922-929.
8
Parada, J., & Aguilera, J.M. (2007). Food microstructure affects the bioavailability of several
9
nutrients. Journal of Food Science, 72 (2), 21-32.
10
Quiblier, J.A. (1984). A new three-dimensional modeling techniques for studying porous media.
11
Journal of Colloid Interface Science, 98, 84.
12
Quintavalla, J. (2006). Measures of clustering in systems of overlapping particles. Mechanics of
13
Materials, 38, 849-858.
14
Rintoul, M.D., & Toruato, S. (1997). Reconstruction of the structure of dispersion. Journal of
15
Colloid and Interface Science, 186, 467- 476.
16
Rintoul, M.D., Torquato, S., Yeong, C., Keane, D.T., Erramilli, S., Jun, Y.N., Dabbs, D.M., &
17
Aksay, I.A. (1996). Structure and transport properties of a porous magnetic gel via x-ray
18
microtomography. Physical Review E, 54 (3), 2663-2669.
19
Russ, J.C. (2005). Image analysis of food microstructure. Boca Raton, LA, CRC press.
20
Sezgin, M., & Sankur, B. (2003). Survey over image thresholding techniques and quantitative
21
performance evaluation. Journal Electron Imaging, 13 (1), 146–165, (2003)
1
Singh, H., Gokhale, A.M., & Lieberman, S.I., & Tamirisakandala, S. (2008). Image based
2
computations of lineal path probability distributions for microstructure representation. Material
3
Science and Engineering A, 474, 104-111.
4
Smith, P., & Torquato, S. (1988). Computer simulation results for the two-point probability
5
function of composite media. Journal of Computational Physics, 76, 176-191.
6
Torquato, S. (2002). Statistical Descrption of microstructures. Annual Review of Material Research,
7
32, 77-111.
8
Torquato, S., & Lu, B. (1993). Chord-length distribution function for two-phase random media.
9
Physical Review E, 47 (4), 2950-2953.
10
Torquato, S., Beasley, J.D., & Chiew, Y.C. (1988). Two-point cluster function for continuum
11
percolation. Journal of Chemical Physic, 88 (10), 6540 – 6547.
12
Yeong, C.L.Y., & Torquato, S. (1998). Reconstructing random media. Physical Review E, 57 (1),
13
495-506.
14 15
This paper has not been published previously and it is not under consideration for publication
16
elsewhere. Moreover the paper is approved by all authors and tacitly or explicitly by the responsible
17
authorities where the work was carried out and that, if accepted, it will not be published elsewhere
18
including electronically in the same form, in English or in any other language, without the written
19
consent of the copyright-holder. The authors declare that they have no actual or potential conflict of
20
interest including any financial, personal or other relationships with other people or organization
21
within three years of beginning the submitted work that could inappropriately influence, or be
22
perceived to influence, their work.
23
1
Figure captions
2 3 4 5 6
Fig 1 - Changes of energy (E) as a function of attempts. a) Evolution of E during the first 104; b) Evolution of E during 5*105 steps. c) reference images; d) random image; e) reconstructed image after 3,000 attempts; f) reconstructed image after 104 attempts; g) reconstructed image after 5*104 attempts; h) reconstructed image after 5*105 attempts. Reference systems is a digitized image of 32*32 pixels with a porosity fraction,
F1 ~ 56%.
7 8 9 10
Fig 2 – Lineal-path distribution function of phase 1 for the reference and reconstructed image when an E of 3.66*10-4 was reached (after 5*105 attempts). (Filled squares) - reference image; (dotted line) – reconstructed image.
11 12 13 14 15 16
Fig 3 – Reference and reconstructed images of bread microstructure obtained by using different descriptors. a) original image; b) binary image; c) random image; d) LPF0; e) LPF90; f) LPF45; g)LPF135; h) Lineal path distribution of functions along vertical axis of original and reconstructed images obtained by using the descriptor LPF90. (dotted line), L(z) of reconstructed image; (void triangles), L(z) binary image of bread structure.
17 18 19 20
Fig 4 – Reference and reconstructed images of bread microstructure by using the combination of two statistical descriptors. a) Reference image; Reconstructed images: b) LPF0+LPF90; c) LPF0+LPF45; d) LPF90+LPF45; e) LPF0+LPF135; f) LPF90+LPF135; g) LPF45+LPF135.
21 22 23
Fig 5 – Energy values as function of the number of attempts during reconstruction of the microstructure of bread samples.
24 25 26 27
Fig 6 - Lineal path function extracted in horizontal (a) and vertical direction (b) of reference and reconstructed images obtained combining all LFPs during reconstruction procedures. (dotted line), bread sample; (open symbol), reconstructed image.
28 29 30 31 32 33
Fig 7 - Reference and reconstructed images obtained by combining different statistical descriptors. a) Reference image; reconstructed image: b) LPF0+LPF90+LPF45; c) LPF0+LPF45+LPF135; d) LPF0+LPF90+LPF45+LPF135.
1 2
Table 1 – Energy values at equilibrium obtained reconstructing the microstructure of bread sample by using different statistical descriptors. Descriptors LPF0 LPF45 LPF90 LPF135
Energy value 3.34*10-7 1.46*10-6 9.47*10-6 1.26*10-5 Combination of two LPF 2.56*10-5 3.08*10-6 1.25*10-5 6.85*10-6 1.62*10-5 3.58*10-5
LPF0+LPF90 LPF0+LPF45 LPF90+LPF45 LPF0+LPF135 LF90+LPF135 LPF45+LPF135 Combination of three LPF
3.76*10-5 6.50*10-5 1.76*10-5 2.50*10-5
LPF0+LPF90+LPF45 LPF0+LPF90+LPF135 LPF0+LPF45+LPF135 LPF90+LPF45+LPF135 Combination of four LPF LPF0+LPF90+LPF45+LPF135
2.60*10-5
3 4 5 6 7 8 9 10 11 12 13 14
Highlights Statistical correlation functions were used to reconstruct bread microstructure by applying an annealing method; Lineal-path distribution function (LPF) in different directions were used to reconstruct 2D images of bread; The reconstruction method is a powerful tool for reconstruction and its use could be extended for several foods; Although LPF of real and reconstructed images perfectly match, visually speaking some deviation were detected.
Fig 1
a)
b)
c)
d)
e)
f)
g)
h)
Fig 2
Fig 3
a)
c)
h)
d)
b)
e)
f)
g)
Fig 4
a)
b)
e)
c)
f)
d)
g)
Fig 5
a)
b)
Fig 6
a)
b)
Fig 7
a)
c)
b)
d)