0.85
0.86
0.87
0.88
0.89
0.86
0.87
0.88
0.89
kr −→
kr −→
(a) d 1 = 2, d 2 = 1.9, and d 3 = 1.8
(b) d 1 = 2, d 2 = 1.9, and φ = 0.8
Fig. 14. Bifurcations of three-phase periodic points observed in asymmetric three-coupled
system
branching because of the asymmetric system. There is a stable three-phase periodic point in
each parameter region shaded by a pattern. The shape and size of the whole shaded parameter
region where there are three-phase periodic points are similar to those in Fig. 13.
As seen in Fig. 14(b), we also computed the bifurcations of three-phase periodic points
observed at d 1 = 2, d 2 = 1.9, and φ = 0.8 on the ( kr, d 3)-plane. As we can see from the figure,
there are several stable three-phase periodic points even if the value of d 3 is set as small as 1.5.
This suggests that our dynamic image-segmentation system can work for an image with three
regions having different gray levels.
4. Application to Dynamic Image Segmentation
We demonstrated successful results for dynamic image segmentation carried out by our
system with appropriate parameter values according to the results analyzed from the two- and
Discrete-Time Dynamic Image-Segmentation System
419
three-coupled systems. Our basic concept was that we assigned system parameters to certain
values such those in-phase oscillatory responses, which are unsuitable for dynamic image
segmentation. They do not appear but a multiphase periodic point with as many phases as
image regions does occur.
4.1 Image with two image regions
Let us consider a dynamic image segmentation problem for the 8-bit gray-level image with
256 × 256 pixels shown in Fig. 15(a). This is a slice from the X-ray CT images of the human
head from the Visible Human Dataset (Ackerman, 1991). Using a thresholding method, we
transformed the gray-level CT image into a binary image in preprocessing with ∀i, pi =
{ 0, 255 }, where pi denotes the i th pixel value. Here, the black and white correspond to 0
and 255. The process image contains the two white image regions shown in Fig. 15(b). The
upper region corresponds to teeth and the mandible bone, and the lower regions indicate the
cervical spine.
We need a neuronal network system to segment the binary image consisting of 256 × 256
neurons and a global inhibitor. The DC-input value to the i th neuron, di, was set to 2.0 for
neurons corresponding to pixels in the two white image regions based on di = 2 pi/255.
Therefore, we can design system-parameter values according to the analyzed results for the
symmetric two-coupled system in Figs. 5 and 8.
Based on the information in the bifurcation diagrams, we set the two unfixed parameters to
kr = 0.885 and φ = 0.8, which correspond to a parameter point in the left neighborhood of
NS 11 in Fig. 5, so that no in-phase oscillatory responses appear from any initial values but a
fixed point or an out-of-phase 36-periodic point does occur. Note that, any of the out-of-phase
periodic points in Fig. 8(b) are available for dynamic image segmentation, and the period
of the periodic point used in dynamic image segmentation corresponds to the period each
segmented image appeared in output images that were exhibited in a time series.
The binarized image was input to our dynamic image segmentation system with 256 × 256
neurons and a global inhibitor. According to an out-of-phase 36-periodic point, our system
output images in the time series shown in Fig. 15(c), i.e., images were dynamically segmented
successfully. Note that the output images sequentially appeared from the top-left to the
bottom-right, and they also began to appear in each line from the left; moreover, output
images corresponding to state variables in the transient state were removed. We confirmed
from the series of output images that the period where each image region appeared was 36.
4.2 Image with three image regions
We considered the 8-bit gray-level image with 128 × 128 pixels shown in Fig. 16(a). It has three
image regions: a ring shape, a rectangle, and a triangle. To simplify the problem, the color in
each image region was made into a monotone in which the pixel values were 255, 242, and 230
so that these values corresponded to d 1 = 2, d 2 = 1.9, and d 3 = 1.8 according to di = 2 pi/255.
To dynamically segment the target image, we needed a neuronal network system consisting
of 128 × 128 neurons and a global inhibitor. The DC-input value to the i th neuron, di, was
set to 2.0 for neurons corresponding to pixels in the ring shape, 1.9 for those in the rectangle,
and 1.8 for those in the triangle. The neuronal network system with 128 × 128 neurons could
be regarded as an asymmetric three-coupled system. Therefore, according to the analyzed
results in Fig. 14, e.g., we set the unfixed parameter values to kr = 0.875 and φ = 0.8 such that
a three-phase 25-periodic point occurred in the asymmetric three-coupled system.
420
Discrete Time Systems
(a) CT image
(b)
Binarization
(c) Output images in time series
Fig. 15. Results of dynamic image segmentation based on out-of-phase oscillatory response
In trials for randomly given initial values, we achieved a successful result where three image
regions appeared separately, as shown in Fig. 16(b). In addition, because the output images
were generated according to a three-phase 25-periodic point observed in the asymmetric
three-coupled system, we confirmed that the period where each image region appeared was
25. Note that we removed output images corresponding to state variables in the transient
state. Our neuronal network system could work for a simple gray-level image with three
image regions.
4.3 Image with many image regions
To segment an image with an arbitrary number of image regions using our dynamic
image-segmentation system in one process, it is necessary for a multiphase periodic oscillatory
response with as many phases as image regions to appear. As far as our investigations
were concerned, however, it was difficult to generate a multiphase periodic point with many
phases. Therefore, we proposed an algorithm that successively and partially segments an
image.
Here, according to the previously mentioned results obtained from analysis, we considered
a successive algorithm that partially segmented many image regions using two- and
three-phase oscillatory responses. We let the gray-level image with five image regions in
Fig. 17(a) be the target that should be segmented. To simplify the segmentation problem, we
Discrete-Time Dynamic Image-Segmentation System
421
(a) Target image
(b) Output images in time series
Fig. 16. Results of dynamic image segmentation based on three-phase oscillatory response
assumed that all pixels in an identical image region would have the same gray levels. Based
on the analyzed results for two- and three-coupled systems, we set the system parameters to
certain values such that no in-phase oscillatory responses occurred but a fixed point or a two-
or three-phase oscillatory responses appeared.
We could obtain the three segmented images in Figs. 17(b)–(d) in the first step from the input
image in Fig. 17(a) if a three-phase oscillatory response appeared. Note that the segmented
images were extracted from output images in a time series by removing duplicate images and
all black images.
Each segmented image in Figs. 17(b)–(d) became an input image for our system in the next
step. We obtained the two images in Figs. 17(e) and 17(f) in this step from the input image
in Fig. 17(b) using a two-phase oscillatory response; as well as this process, we also obtained
the two images in Figs. 17(g) and 17(h) from the input image in Fig. 17(c) according to the
two-phase response; whereas we obtained no output images from the input image in Fig. 17(d)
because the system to segment the image corresponded to a single neuronal system, and a
fixed point always appeared under the system-parameter values we assigned. Therefore, the
segmentation of the image in Fig. 17(d) was terminated in this step.
The four images with only one image region in Figs. 17(e)–(h) are input images in the third
step. As previously mentioned, we obtained no output images for an input image with only
one image region. Therefore, our successive algorithm was terminated at this point in time.
Thus, we could segment an image with an arbitrary number of image regions based on the
successive algorithm.
422
Discrete Time Systems
(a)
Segmentation by three-phase oscillatory response
(b)
(c)
(d)
Segmentation by two-phase oscillatory response
(e)
(f)
(g)
(h)
Appearance of fixed point corresponding to non-oscillatory response
no image
no image
no image
no image
no image
Fig. 17. Schematic diagram of successive algorithm using our dynamic image-segmentation
system
5. Concluding remarks
We introduced a discrete-time neuron model that could generate similar oscillatory responses
formed by periodic points to oscillations observed in a continuous-time relaxation oscillator
model. The scheme of dynamic image segmentation was illustrated using our neuronal
network system that consisted of our neurons arranged in a 2D grid and a global inhibitor.
Note that we suggested that a neuronal network system where neurons are arranged in a 3D
grid can be applied to segmenting a 3D image.
Images were dynamically segmented according to the responses of our system, and therefore,
knowing about the bifurcations of the responses allowed us to directly set system-parameter
values such that appropriate responses for dynamic image segmentation would appear. We
derived reduced models that simplified our analysis of bifurcations observed in our neuronal
network system and we found parameter regions where there was a non-oscillatory response
or a periodic oscillatory response in the reduced models. According to the analyzed results,
we set system parameters to appropriate values, and the designed system could work for two
sample images with two or three image regions. Moreover, to segment an image with many
image regions, we proposed a successive algorithm using our dynamic image-segmentation
system.
We encountered three main problems that should be solved to enable the practical use of our
dynamic image-segmentation system:
1. Development of a method that can form appropriate couplings between neurons for a
textured image and a gray-level image containing gradation.
Discrete-Time Dynamic Image-Segmentation System
423
2. Development of a method that can give initial values to neurons and a global inhibitor so
that an appropriate response will always appear.
3. Development of a method or system that can provide fast processing using our system to
segment a large-scale image and a 3D image within practical time limits.
To solve the first problem, we proposed a dynamic image-segmentation system with a
method of posterization (Zhao et al., 2003) used as preprocessing (Fujimoto et al., 2009a;
2010). However, their method of posterization involves high computational cost and a
large memory, we are considering a neuronal network system with plastic couplings as
weight adaptation (Chen et al., 2000). We proposed a solution to the second problem with
a method that avoids the appearance of non-oscillatory responses (Fujimoto et al., 2011a).
However, toward an ultimate solution, we are investigating parameter regions such that no
inappropriate responses appear through bifurcation analysis. An implementation to execute
our dynamic image-segmentation system on a graphics processing unit is in progress as a
means of rapid processing.
6. Acknowledgments
This study was partially supported by the Ministry of Education, Culture, Sports, Science and
Technology, Japan, Grant-in-Aid for Young Scientists (B), Nos. 20700209 and 22700234.
7. References
Ackerman, M. J. (1991). The visible human project, J. Biocommun. 18(2): 14.
Aihara, K. (1990). Chaotic neural networks, in H. Kawakami (ed.), Bifurcation Phenomena
in Nonlinear Systems and Theory of Dynamical System, World Scientific, Singapore,
pp. 143–161.
Aihara, K., Takabe, T. & Toyoda, M. (1990).
Chaotic neural networks, Phys. Lett. A
144(6,7): 333–340.
Chen, K., Wang, D. L. & Liu, X. (2000). Weight adaptation and oscillatory correlation for image
segmentation, IEEE Trans. Neural Netw. 11(5): 1106–1123.
Doi, K. (2007). Computer-aided diagnosis in medical imaging: Historical review, current
status and future potential, Comput. Med. Imaging Graph. 31(4,5): 198–211.
FitzHugh, R. (1961).
Impulses and physiological states in theoretical models of nerve
membrane, Biophys. J. 1(6): 445–466.
Fujimoto, K., Musashi, M. & Yoshinaga, T. (2008). Discrete-time dynamic image segmentation