17-11-2012, 04:03 PM
Region Based Image Segmentation using a Modified Mumford-Shah Algorithm
[attachment=40031]
Abstract.
The goal of this paper is to develop region based image segmentation
algorithms. Two new variational PDE image segmentation models are proposed.
The first model is obtained by minimizing an energy function which depends on
a modified Mumford-Shah algorithm. The second model is acquired by utilizing
prior shape information and region intensity values. The numerical experiments
of the proposed models are tested against synthetic data and simulated normal
human-brain MR images. The preliminary experimental results show the effectiveness
and robustness of presented models against to noise, artifact, and loss of
information.
Introduction
Segmentation techniques have been developed to capture the object boundary by several
different approaches; edge-based methods mainly using active contour models, regionbased
methods, or the combination of the two by using Geodesic Active Regionmodels.
The most celebrating region based image segmentation model is introduced by
Mumford and Shah in 1989 [15]. In this model, an image is decomposed into a set
of regions within the bounded open set
and these regions are separated by smooth
edges Γ. Ambrosio and Tortorelli approximated the measurement of an edge Γ length
term in the Mumford-Shah model by a quadratic integral of an edge signature function
in 1990 [1]. Chan and Vese proposed a piecewise constant Mumford-Shah model in
[4, 5] by using a level set formulation [16]. Developments of variational level set implementation
techniques based Mumford-Shah model are followed by [8, 9, 13]. The
segmentation is represented by characteristics functions using phase fields in [8, 13].
The details of phase field theory can be found in [2, 14, 18–20].
Numerical Results
In this part, the numerical results with applications to synthetic data and simulated
normal human brain MR images. The Equation (2.1) was solved by finding a steady
state solution of the evolution equations. The evolution equations are associated with
the Euler-Lagrange equations of the Equation (2.1). A finite difference scheme and the
gradient descent method is applied to discretize the evolving equations.
The Figure 1 showed the proposed model segmentation results to synthetic data.
Figure 2 showed the segmentation results by [4]. The first figure in Figure 1 and Figure
2 is the given synthetic image I with an initial contour as a solid line. In Figure 1 and
Figure 2, the second image is the segmented image and the third one is the segmented
contour as a solid line with I. Our model performs better than [4] to capture the boundary
of the region which has similar intensity. The numerical results with an application
to the simulated human brain MR image are shown from Figure 3 to Figure 6. The data
is obtained from http://www.bic.mni.mcgill.ca/brainweb. The simulated normal human
brain image with the ground truth white matter image is shown in Figure 3. In Figure
3, the first figure is the simulated brain image I, the second one is an image I with an
initial contour as a solid line, and the third one is the ground truth brain white matter.
From Figure 4 to Figure 5, the first image is the given image I with an initial contour
as a solid line, the second image is the segmented image results by our proposed model
and [4] each, and the third figure is the segmented contour in I.
Numerical Results
The Equation (3.1) was solved by finding a steady state solution of the evolution equations.
The evolution equations are associated with the Euler-Lagrange equations of the
Equation (3.1). A finite difference scheme and the gradient descent method is applied
to discretize the evolving equations. Initial value of d1 and d2 are 0.95 and 0.01 respectively.
Here λ = 1 is used in the numerical experiments. Figure 7 showed original
image, the binary image S as a prior shape information, and synthetic image I with
noise, rotation, and loss of information. The numerical results of the proposed model
using the prior shape information are shown in Figure 8. In Figure 8, the first one is
the binary image as prior shape information and the second one is the given image I.
The segmented image and segmented contour result by using the Equation (3.1) are the
third and fourth figure in Figure 8. The first figure in Figure 9 and Figure 10 is the given
image I. The second and the third figure in Figure 9 showed the numerical results of
segmented contour by [4] and the Equation (2.1) to a given Image I. Due to the noise,
rotation, and loss of information, only using the model by [4] or the Equation (2.1) was
not sufficient to get desired segmentation results. Hence the prior shape information is
necessary in the segmentation process.
Conclusions and Future Work
Two new region based variational partial differential equation (PDE) models for an image
segmentation are proposed with an application to synthetic and simulated human
brain MR images. The first model utilizes a modified piecewise constant Mumford-
Shah model. Even though this model performs better than existing model with fuzzy images, this algorithm has some limits with strong noise, rotation, and loss of information.
Therefore, the second model is obtained using a prior shape information and
region intensity value. Numerical Results show the effectiveness and robustness of the
presented model against to noise, rotation, loss of information, and artifact. In the future
work, the research will be focused on the improvements of the first model with the
robustness to the choice of the initial contour.
[attachment=40031]
Abstract.
The goal of this paper is to develop region based image segmentation
algorithms. Two new variational PDE image segmentation models are proposed.
The first model is obtained by minimizing an energy function which depends on
a modified Mumford-Shah algorithm. The second model is acquired by utilizing
prior shape information and region intensity values. The numerical experiments
of the proposed models are tested against synthetic data and simulated normal
human-brain MR images. The preliminary experimental results show the effectiveness
and robustness of presented models against to noise, artifact, and loss of
information.
Introduction
Segmentation techniques have been developed to capture the object boundary by several
different approaches; edge-based methods mainly using active contour models, regionbased
methods, or the combination of the two by using Geodesic Active Regionmodels.
The most celebrating region based image segmentation model is introduced by
Mumford and Shah in 1989 [15]. In this model, an image is decomposed into a set
of regions within the bounded open set
and these regions are separated by smooth
edges Γ. Ambrosio and Tortorelli approximated the measurement of an edge Γ length
term in the Mumford-Shah model by a quadratic integral of an edge signature function
in 1990 [1]. Chan and Vese proposed a piecewise constant Mumford-Shah model in
[4, 5] by using a level set formulation [16]. Developments of variational level set implementation
techniques based Mumford-Shah model are followed by [8, 9, 13]. The
segmentation is represented by characteristics functions using phase fields in [8, 13].
The details of phase field theory can be found in [2, 14, 18–20].
Numerical Results
In this part, the numerical results with applications to synthetic data and simulated
normal human brain MR images. The Equation (2.1) was solved by finding a steady
state solution of the evolution equations. The evolution equations are associated with
the Euler-Lagrange equations of the Equation (2.1). A finite difference scheme and the
gradient descent method is applied to discretize the evolving equations.
The Figure 1 showed the proposed model segmentation results to synthetic data.
Figure 2 showed the segmentation results by [4]. The first figure in Figure 1 and Figure
2 is the given synthetic image I with an initial contour as a solid line. In Figure 1 and
Figure 2, the second image is the segmented image and the third one is the segmented
contour as a solid line with I. Our model performs better than [4] to capture the boundary
of the region which has similar intensity. The numerical results with an application
to the simulated human brain MR image are shown from Figure 3 to Figure 6. The data
is obtained from http://www.bic.mni.mcgill.ca/brainweb. The simulated normal human
brain image with the ground truth white matter image is shown in Figure 3. In Figure
3, the first figure is the simulated brain image I, the second one is an image I with an
initial contour as a solid line, and the third one is the ground truth brain white matter.
From Figure 4 to Figure 5, the first image is the given image I with an initial contour
as a solid line, the second image is the segmented image results by our proposed model
and [4] each, and the third figure is the segmented contour in I.
Numerical Results
The Equation (3.1) was solved by finding a steady state solution of the evolution equations.
The evolution equations are associated with the Euler-Lagrange equations of the
Equation (3.1). A finite difference scheme and the gradient descent method is applied
to discretize the evolving equations. Initial value of d1 and d2 are 0.95 and 0.01 respectively.
Here λ = 1 is used in the numerical experiments. Figure 7 showed original
image, the binary image S as a prior shape information, and synthetic image I with
noise, rotation, and loss of information. The numerical results of the proposed model
using the prior shape information are shown in Figure 8. In Figure 8, the first one is
the binary image as prior shape information and the second one is the given image I.
The segmented image and segmented contour result by using the Equation (3.1) are the
third and fourth figure in Figure 8. The first figure in Figure 9 and Figure 10 is the given
image I. The second and the third figure in Figure 9 showed the numerical results of
segmented contour by [4] and the Equation (2.1) to a given Image I. Due to the noise,
rotation, and loss of information, only using the model by [4] or the Equation (2.1) was
not sufficient to get desired segmentation results. Hence the prior shape information is
necessary in the segmentation process.
Conclusions and Future Work
Two new region based variational partial differential equation (PDE) models for an image
segmentation are proposed with an application to synthetic and simulated human
brain MR images. The first model utilizes a modified piecewise constant Mumford-
Shah model. Even though this model performs better than existing model with fuzzy images, this algorithm has some limits with strong noise, rotation, and loss of information.
Therefore, the second model is obtained using a prior shape information and
region intensity value. Numerical Results show the effectiveness and robustness of the
presented model against to noise, rotation, loss of information, and artifact. In the future
work, the research will be focused on the improvements of the first model with the
robustness to the choice of the initial contour.