Modified Multiphase Level Set Image Segmentation Search for Energy Formulation with Bias Correction

International Journal of Science and Research (IJSR), India Online ISSN: 2319-7064
Volume 2 Issue 9, September 2013
www.ijsr.net
Modified Multiphase Level Set Image Segmentation
Search for Energy Formulation with
Bias Correction
N. UmaDevi1,
S. Revathi2
1
Research Supervisor, Head, Department of Computer Science and Information Technology, Sri Jayendra Saraswathy Maha Vidyalaya
College of Arts and Science, Coimbatore-5, India
2
Research Scholar, Sri Jayendra Saraswathy Maha Vidyalaya College of Arts and Science, Coimbatore-5, India
Abstract: Level set methods are a popular way to solve the image segmentation problem. The solution contour is found by solving an
optimization problem where a cost functional is minimized. The purpose of this paper is the level set approach to simultaneous tissue
segmentation and bias correction of Magnetic Resonance Imaging (MRI) images. A modified level set approach to joint segmentation
and bias correction of images with intensity in homogeneity. A sliding window is used to transform the gradient intensity domain to
another domain, where the distribution overlap between different tissues is significantly suppressed. Tissue segmentation and bias
correction are simultaneously achieved via a multiphase level set evolution process. The proposed methods are very robust to
initialization, and are directly compatible with any type of level set implementation. Experiments on images of various modalities
demonstrated the superior performance over state-of-the-art methods.
Keywords: Bias correction, Gradient, Levelset, Contour.
1. Introduction
One popular approach for solving the image segmentation
problem is to use the calculus of variations. The objective of
the segmentation problem is defined with energy functional,
and minimization of this functional represents the resulting
segmentation. The solution of the segmentation problem is
the contour which extremizes this functional. The standard
way of solving this optimization problem is by gradient
descent search in the solution space, which typically suffers
from many unwanted local optima and poor convergence.
Classically, these problems have been circumvented by
modifying the energy functional. The functional depends on
properties of the image such as gradients, curvatures and
intensities, as well as regularization terms, e.g. smoothing
constraints. Early variation methods, such as Snakes and
Geodesic Active Contours [1]–[4], often have boundary
based terms such as an edge map. A parameterized curve,
the active contour, is evolved according to the minimization
of the cost functional until it converges to an equilibrium
state representing the resulting segmentation. Later variation
methods often include terms which are more region-based,
allowing segmentation of objects without distinctive edges.
These methods are often based on the Mumford-Shah
segmentation model [5] where the image is partitioned into
piecewise smooth regions with short boundaries. Chan and
Vese used this model together with implicit contours
represented by level sets [6]–[9]. This made the optimization
problem easier and it naturally handles segmentation
topology changes.
2. Related Work
Level set methods are a popular way to solve the image
segmentation problem in computer image analysis. A
contour is implicitly represented by the zero level of a
signed distance function, and evolved according to a motion
equation in order to minimize a cost function. This function
defines the objective of the segmentation problem and also
includes regularization constraints. Gradient descent search
is the defacto method used to solve this optimization
problem. It deforms an initial contour in the steepest
(gradient) descent of the energy. The equations of motion for
the contour, and the corresponding energy gradients, are
derived using the Euler-Lagrange equation [10] and the
condition that the first variation of the energy functional
should vanish at a (local) optimum. Then, the contour is
evolved to convergence using these equations. The use of a
gradient descent search commonly leads to problems with
convergence to local optima and slow convergence in
general. The problems are accentuated with noisy data or
with a non-stationary imaging process, which may lead to
varying contrasts for example. The problems may also be
induced by inaccurate initial conditions for certain
applications. Traditionally, the energy functional have been
modified to avoid these problems by, for example, adding
regularizing terms to handle noise, rather than to analyze the
performance of the applied optimization method. This is
however discussed in [11], [12], where the metric defining
the notion of steepest descent (gradient) has been studied.
By changing the metric in the solution space, local optima
due to noise are avoided in the search path.
There are many and much more advanced, alternatives to
gradient descent. For non-convex functional, there are global
optimization techniques such as subdivision schemes with
much better performance [13]. Their high complexity,
however, makes them applicable for small problems only.
Stochastic methods such as the Monte-Carlo (MC) family
are another alternative. Simulated annealing implemented
with the Metropolis algorithm and local updates have been a
popular choice [14], [15]. Unfortunately its convergence to a
global solution is logarithmically slow, limiting its
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usefulness but spurring the development of more advanced
MC methods, such as basin hopping [16].
In spite of these more advanced methods, optimization using
gradient descent search is still very common and in active
use. This is partly due to its simple implementation, but also
to its direct applicability to general non-convex functional.
The focus of this paper is to show that the performance of
these standard searches can easily be improved upon by very
simple modifications with, (i) Momentum Method and (ii)
Rprop method. These methods are commonly used in the
machine learning community. In a series of 2D/3D
experiments using real and synthetic data with ground truth,
the modifications are shown to reduce the sensitivity for
local optima and to increase the convergence rate.
2.1. Momentum Method
Turning to gradient descent with Momentum, we will adopt
some terminology from the machine learning community
and choose a search vector according to:
�� = −η(1 − �)∇f� + ωs� − 1 (1)
Where η is the learning rate and ω ∈ [0, 1] is the momentum.
Note that ω = 0 gives standard gradient descent sk = −η∇fk,
while ω = 1 gives “infinite momentum” sk = sk−1. The
intuition behind this strategy is that the current solution xk
has a momentum, which prohibits sudden changes in the
velocity. These will effectively filter out high frequency
changes in the cost function and allow for larger steps in
favorable directions. Using appropriate parameters, the rate
of convergence is increased while local optima may be
overstepped.
2.2. Rprop Method
In standard implementations of steepest descent search, αk =
α is a constant not adapting to the shape of the cost-surface.
Therefore if we set it too small, the number of iterations
needed to converge to a local optimum may be prohibitive.
On the other hand, a large value of α may lead to oscillations
causing the search to fail. The optimal α does not only
depend on the problem at hand, but vary along the cost-
surface. In shallow regions of the surface a large α may be
needed to obtain an acceptable convergence rate, but the
same value may lead to disastrous oscillations in
neighboring regions with larger gradients or in the presence
of noise. In regions with very different behaviors along
different dimensions it may be hard to find an α that gives
acceptable convergence performance.
The Resilient Propagation (Rprop) algorithm [17] was
developed to overcome these inherent disadvantages of
standard gradient descent using adaptive step-sizes k called
update-values. There is one update-value per dimension in x,
i.e. dim(k) = dim(xk). However, the defining feature of
Rprop is that the size of the gradient is never used. Only the
signs of the partial derivatives are considered in the update
rule. Another advantage of Rprop, very important in
practical use, is the robustness of its parameters; Rprop will
work out- of-the-box in many applications using only the
standard values of its parameters [18].
Rprop provides a modification which uses individual (one
per parameter) adaptive step sizes and considers only the
signs of the gradient components. This modification makes
Rprop less sensitive to local optima and avoids the harmful
influence of the magnitude of the gradient on the step size.
Individual adaptive step sizes also allow for cost functional
with very different behaviors along different dimensions
because there is no longer a single step size that has to be
appropriate for all dimensions. The Rprop method succeeds
in reaching the global minimum neighborhood within ten
iterations. It passes the area with small gradients quickly
since it only uses the signs of the gradient components, not
the magnitude.
3. Our Contribution
This paper proposes a modified variation level set approach
to bias correction and segmentation for images corrupted
with intensity in homogeneities. A unique feature of this
method is that the computed bias field is intrinsically
ensured to be smooth by the data term in the variation
formulation, without any additional effort to maintain the
smoothness of the bias field, and it can approximate bias
fields of more general profiles, such as those in 3T or 7T
MRI images.
Magnetic resonance imaging (MRI),is a medical imaging
technique used in radiology to visualize internal structures of
the body in detail. MRI makes use of the property of nuclear
magnetic resonance (NMR) to image nuclei of atoms inside
the body. To produce an image, an MRI machine uses a
powerful magnet to generate a magnetic field. MRI provides
good contrast between the different soft tissues of the body,
which makes it especially useful in imaging the brain,
muscles, the heart, and cancers compared with other medical
imaging techniques such as computed tomography (CT) or
X-Rays. Unlike CT scans or traditional X-Rays, MRI does
not use ionizing radiation.
MRI scans require a magnetic field with two properties,
uniform field density and strength. The magnetic field
cannot vary more than 1/10,000 of 1% and field strength
ranges (depending on the scanner) from 0.2 to 3 Teslas in
strength in scanners currently used clinically, with research
scanners investigating higher field strengths such as 7
Teslas. The Tesla (T) Images unit of magnetic flux density
in the International System of Units, equal to the magnitude
of the magnetic field vector necessary to produce a force of
one Newton on a charge of one coulomb moving
perpendicular to the direction of the magnetic field vector
with a velocity of one meter per second. The lower field
strengths (3T) can be achieved with permanent magnets,
which are often used in "open" MRI scanners for
claustrophobic patients. Higher field strengths (7T) can be
achieved only with superconducting magnets.
3.1. Geometric Model of Images with Intensity in
Homogeneity
Intensity in-homogeneities in MRI images can substantially
reduce the accuracy of segmentation and registration.
Despite advances of correcting spatial intensity in-
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homogeneities of modern scanner software, the advent of
multichannel phased array coils and 7T+ scanners have
increased (again) the importance of this problem for post-
scan processing since images may exhibit substantial
intensity in-homogeneities across space .(image on left side).
In this approach, low-order polynomials are used to model
low-frequency variations across 3T image space; these
polynomials are fitted to a subset of voxels that have been
labeled as belonging to white matter. After estimating the
low-frequency intensity fluctuations (producing a "bias
field", image on lower right side), they are removed from the
data producing voxels with much more homogeneous
intensities (image on upper right side) that improve
visualization and also are better starting points for
subsequent segmentation tools.
The image intensity can be approximated by Gaussian
distribution with mean and variance. However, using only a
single Gaussian model is not accurate enough to describe the
statistical characteristics of the image intensity. In order to
accurately model the intensity distribution, the intensity
distribution in each object domain should be ascribed to a
Gaussian model.
3.2. Bias Correction
A bias field is a low frequency smooth undesirable signal
that corrupts MRI images because of the in-homogeneities in
the magnetic fields of the MRI machine. Bias field blurs
images and thus reduces the high frequency contents of the
image such as edges and contours and changes the intensity
values of image pixels so that the same tissue has different
gray level distribution across the image. A low level
variation will not have great impact on clinical diagnosis.
However it degrades the performance of image processing
algorithms such as segmentation and classification or any
algorithm that is based on the assumption of spatial
invariance of the processed image. A preprocessing step is
needed to correct for the effect of bias field before doing
segmentation or classification.
Methods of bias correction can be categorized into two
classes: prospective methods and retrospective methods.
Prospective methods aim to avoid intensity in-homogeneities
in the image acquisition process. These methods, while
capable of correcting intensity in-homogeneity induced by
the imaging device, are not able to remove subject-induced
intensity in-homogeneity. In contrast, retrospective methods
only rely on the information in the acquired images.
Therefore, they can also remove intensity in-homogeneities
regardless of their sources.
The bias field in a brain MRI image can be assumed to be
formed of two components, i.e. one observed image and an
unknown field, as follows:
I = bJ + n (2)
Where I: observed image, J: true image to be restored, b: an
unknown bias field and n: additive Gaussian noise with
zero-mean. The aim is to obtain the value of bias field
component from the observed image and correct the
intensity in-homogeneity. In general, the bias field b is
assumed to be slowly varying in the entire image domain.
Ideally, the intensity J in each tissue should take a specific
value, reflecting the physical property being measured. This
property, in conjunction with the spatially coherent nature of
each tissue type, implies that the true signal J is
approximately a piecewise constant map.
While bias field correction is needed for good segmentation,
many approaches have exploited the idea that a good
segmentation helps to estimate the bias field. Dawant et al. it
manually selected some points inside white matter and
estimated the bias field as the least-squares spline fit to the
intensities of these points [19]. They also presented a
slightly different version where the reference points are
obtained by an intermediate classification operation, using
the estimated bias field for final classification.
3.3. Energy Formulation
The intensity in homogeneity caused by the bias field
measure intensities are not separable by using traditional,
intensity based classification methods. A new method for
joint segmentation and bias correction. An observation that
the intensities in a relatively small region are separable,
which can be varied by the below assumptions of (i) The
bias field is slowly varying in the entire image domain. (ii)
The true image intensities approximately a constant within
each class of tissue, being a partition of bias field. Directly
minimizing the energy with the partition variable is not
convenient. The multiple levels set functions to represent a
partition. The energy minimization can thus be performed by
solving a level set evolution. The gradient descent search
flow for an energy functional consisting of texture
(proportional to length) and bulk energies (proportional to
region), is developed.
The local intensity clustering property explained above
exemplifies that the intensities in the neighborhood can be
classified into N clusters, with centers mi ≈ b (x) ci, i= 1, 2,
3, 4…N. Hence, standard membership function to classify
these local intensities can be applied. In particular, for the
intensities Image (X) in the neighborhood NGy, the
membership function includes boundary condition and
threshold function is an iterative process to minimize the
pixel intensity which can be presented as follows,
EFy = � �|Image(X) − mi|�
u�(x)dx
�
��Ω�⋂��
(3)
Where EF: Energy formulation; mi: cluster center of the i-th
cluster, ui: membership function of the region Ωi i.e. to be
determined, ui(x) =1, for x € Ωi and ui(x) = 0, for x € Ωi.
The kernel function K is chosen as a truncated Gaussian
function defined by
K(u) = �
�
�
e�ǀ�ǀ
�
/��
,
0, other wise
for ǀuǀ ≤ ρ (4)
Where a is a normalization constant such that ∫K (µ) = 1, σ is
the standard deviation (or the scale parameter) of the
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Gaussian function, and ρ is the radius of the neighborhood
NGy.
The radius of the neighborhood NGy should be selected
appropriately according to the degree of the intensity in-
homogeneity. For more localized intensity in-homogeneity,
the bias field b varies faster, and therefore the approximation
is valid only in a smaller neighborhood. A smaller should
be used as the radius of the neighborhood NGy, and for the
truncated Gaussian function, the scale parameter
(variance) should also be smaller.
The objective is integrated over the entire domain and
incorporated into a variation level set formulation. The
energy minimization is performed via a level set evolution
process. By minimizing this energy, this method is able to
simultaneously segment the image and estimate the bias
field, and the estimated bias field can be used for bias
correction. Moreover, it is robust to initialization, and
therefore allows automatic applications.
3.4. Multiple Level Set Evolutions
The signiﬁcant number of problems in science reduces to the
study of the evolution of curves, which are usually the
boundaries between different media. During the evolution
process, the Multiple Level Set Functions (MLSFs) need to
be periodically regularized in order to keep numeric
stability. The constant kernel values in each iteration, which
can be seen as a special case of the method. The MLSF
keeps the property of Euclidean distance function, so it can
make the evolution stable. The size of the kernel should not
be too large. The contours are constants with different signs
inside and outside. With this initialization, the convergence
speed is much faster than that by distance function, and the
tissue segmentation results are also much more accurate.
In the multiple-phase case: the image domain Ω is
segmented into two disjoint regions Ω1 and Ω2. The level set
function is used to represent the two regions Ω1 and Ω2.
The regions Ω1 andΩ2 can be represented with their
membership functions defined by M1( ) = H( ) and by M2(
) = 1-H( ), respectively, where H is the Heaviside
function. Thus, for the case of N=2, the energy can be
expressed as the following level set formulation:
ε = �(∑ � K(y − x)ǀI(x) − b(y)c�ǀ�
M�(∅(x))dx)dy�
�� (5)
By exchanging the order of integrations,
ε = �(∑ (� K(y − x)ǀI(x) − b(y)c�ǀ�
dy)M�(∅(x))dx�
�� (6)
The constants c1,...,cN are represented with a vector c=
(c1,...,cN). The level set function; the vector c and the bias
field b are the variables of the energy , which can be
written as ( , c, and b). The energy can be rewritten as:
�(∅, �, �) = � ∑ ��(�)��∅(�)��
�� (7)
By minimizing this energy, the result of image segmentation
given by the level set function and the estimation of the
bias field b will be obtained. The energy minimization is
achieved by an iterative process: In each iteration, minimize
the energy � ( , c, b) with respect to each of its variables ,
c and b, given the other two updated in previous iteration.
The solution to the level set energy minimization with
respect to each variable is given the image region.
4. Implementation
The level set algorithm has also been implemented in
Matlab: Version 2010a. The implementation steps are:
Step 1: Pre-processing: In this step, the pre-processing stage
checks whether the input image contains number of
channels. Ifthere are 3 channels then the image is converted
into grayscale as indexed image. For regularizing the level
set function after the each iteration the kernel value is
mentioned along with the channel. Assign the Point Spread
Function (PSF) with corresponding pixel intensity variance.
Step 2: Contour Plot: The contour plot is defined as
separating the background and foreground region in the MRI
image.
Step 3: Regularization: In Regularization process, (i) The
typical image is passed from the degradation function (in
practice this degradation function) then an additive noise is
also added to get an artificial corrupted image. Using these
two images an optimal filter mask is found. (ii) To find m x
n coefficients of the filter mask, iteratively. For
implementing fitness function must be defined as a criterion
for maximization. Absolute difference between the degraded
image and the image estimated by filter mask is the natural
fitness function.
Step 4: Bias Correction: In bias correction process, the
image membership values are convoluted according to the
image dimensions. The convolution process is establishes
the unfairness pixels present in the instance.
The proposed method is demonstrated in the two phase case
(i.e., N=2) as well as multiphase case. The parameter is set
to 4 for the experiments. The multiphase method is applied
to 10 medical images with intensity in-homogeneities. As a
pre-processing step, a convolution with a Gaussian kernel is
applied to smoothen the image. The scale parameters of the
Gaussian kernel are chosen as 2. The initial contours are
plotted on the original images. The corresponding results of
segmentation, bias field estimation are shown in ensuing
pages. These results demonstrate desirable performance in
segmentation.
5. Results
To evaluate the algorithm on both synthetic data and real
datasets in 3T and 7T (Tesla) with the metrics: PSNR,
coefficient of variation, segmentation time calculation. The
narrow band parameter N controls the size of the region
close to the zero level set where calculations are performed.
This is a computational optimization which is valuable
especially in 3T. There are three classes of tissues: white
matter (WM), gray matter (GM), and background. The
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histogram of the bias corrected image by MLS method has
three well-separated peaks, which correspond to the
background, GM, and WM from left to right, respectively,
while the peak of the GM is not distinct since its volume is
relatively small. It should be noted that it is very flexible to
initialize the LSFs in MLS method. The initial contours can
be set inside, outside or across the object boundary. The 7T
MRI image, whose intensity in-homogeneity is so severe
that even the experts are difficult to tell the structures, by
applying MLS method; the image quality can be
significantly improved, Many regions (e.g. the regions inside
the red circles and ellipses) that are difficult to distinguish
can be much better viewed after the bias correction.
Figure 1: 3T MRI image of brain with MLS Method
Figure 2: 7T MRI image of brain with MLS Method
MLS method has been tested on 7T MRI images. At 7T,
significant gains in image resolution can be obtained due to
the increase in signal-to-noise ratio. However, susceptibility-
induced gradients scale with main field, while the imaging
gradients are currently limited to essentially the same
strengths as used at lower field strengths (i.e., 3T). This
appears as a localized and stronger bias, which is
challenging to traditional methods for bias correction.
The test image is matched with database to identify high
frequency regions. The PSNR fraction measure of quality of
reconstruction of lossy compression codecs (e.g., for image
compression). The signal in this case is the original data, and
the noise is the error introduced by compression.
Table: 1 MRI images with Comparison of PSNR Values in
dB
MRI Images Momentum Rprop MLS method
3T Brain Image 15.32 18.27 46.26
7T Brain Image 20.56 24.53 48.57
Figure 3: PSNR Calculations
Above Table1shows comparison of MRI Image with PSNR
Values an approximation to human perception of
reconstruction quality. Although a higher PSNR generally
indicates that the reconstruction is of higher quality.
The coefficient of variation (CV) is a normalized measure of
dispersion of a probability distribution or frequency
distribution. The coefficient of variation represents the ratio
of the standard deviation to the mean, and it is a useful
statistic for comparing the degree of variation from one data
series to another, even if the means are drastically different
from each other. Coefficient of Variance is defined as a
quotient between standard deviation and mean value of
selected tissue class. A good algorithm for bias correction
and segmentation should give low CV values for the bias
corrected intensities within each segmented region.
Our MLS method is compared with the existing Leem put
etal. and Wells et al [20]-[22] methods. Coefficient of
variance (CV) metricis used to evaluate the performance of
the algorithms for bias correction and segmentation. Two
simulated images obtained from Brain Web in the
linkhttp://www.bic.mni.mcgill.ca/brainweb/,one corrupted
with bias field without noise and the other corrupted with
both bias and noise. The CV values for the two images are
listed in Table 2 and Table 3. It can be seen that the CV
values of MLS method is lower than those of Leemput’s and
Wells’s methods, which indicates that the bias corrected
images obtained in this method is more homogeneous than
those of the other two methods.
Table 2: Co-efficient of Variation of MRI Image with bias
and noise
Tissue Wells Leemput MLS method
Image with Bias and
Noise
White matter 6.97% 7.92% 6.45%
Grey matter 13.68% 14.68% 12.03%
Figure 4: Co-efficient of variation of MRI image with bias
and noise
10 10
1 1
2
0 0
20 log ( ) 10 log ( )
1
[ ( , ) ( , )]
I
m n
i j
PSNR MAX MSE
MSE I i j K i j
mn
− −
= =
= ⋅ − ⋅
= −∑ ∑
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Table 3: Co-efficient of Variation of MRI Image with bias
MRI Images Momentum Rprop MLS method
3TBrianImage 16.24 sec 14.45 sec 12.23 sec
7TBrain Image 18.35 sec 16.54 sec 14.26 sec
Figure 5: Co-efficient of variation of MRI image with bias
Above Table 2 shows the comparison of the coefficient of
variation of MRI Image with bias and noise for two tissue
methods and Table 3 with bias for two tissue methods.
Execution speed is another performance metric that is used
to analyze the performance of the algorithms. It is defined as
the time taken by the algorithm to produce the segmented
image. The results obtained are shown in Table 4.
Table 4: Segmentation Time (In seconds)
Image with
Bias
Tissue Wells Leemput MLS method
White matter 6.33% 6.81% 5.45%
Grey matter 16.11% 16.97% 15.55%
Figure 6: Segmentation Time calculation
From the Table 4, it is evident that the MLS Method
produces quick result when compared to the traditional
Momentum and Rprop Methods.
6. Conclusion
In this paper we have worked to the multiphase level set
with bias correction and energy formulation. We presented a
unified framework of bias correction and segmentation. A
unique advantage of this method is that the smoothness of
the computed bias field is intrinsically ensured by the data
term in the distinction formulation. This method is able to
capture bias of quite general profiles, and can be used for
images of various modalities. Moreover, it is robust to
initialization, thereby allowing automatic applications.
Comparisons with two well-known bias correction methods
demonstrate the advantages of the proposed method.
7. Future Work
In future we have planned to find the solutions by changing
the unfairness object correction method of solving the
optimization problem rather than modifying the instance
evolution and energy functional using hardware basis.
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Author Profile
N. Umadevi is working as head in the department of
Computer Science and Technology cum NSS
Programme officer, Sri Jayendra Saraswathy Maha
Vidyalaya College of Arts and Science of industrial
experience. Her area of interest is image processing and Data
Mining. Her publications include 5 international Journals. She has
presented paper in 5 international conferences and 3 national
conferences
S. Revathi received her B. Sc and M. Sc degrees in
computer Science from Sri Jayendra Saraswathy Maha
Vidyalaya College of Arts and Science in 2010 and
2012, respectively. At present she is doing her M. Phil
in Computer Science. Her area of interest is digital
Image Processing under the guidance of Mrs. N. Umadevi working
as head in Department of Computer Science and information
Technology, Sri Jayendra Saraswathy Maha Vidyalaya College of
Arts and Science, Singanallur, Coimbatore.
Paper ID: 12013129 333

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