Image Segmentation Using Level Set Method With Willmore Flow
V. Ranadheer Reddy 1st
M.Tech, Digital Communication
KITS-WGL
raanadheer1992@gmail.com
G. Raghotham Reddy
Prof Rameshwar Rao
Associate Professor
Dept of ECE, KITS-WGL
grrece9@gmail.com
Professor
OU, Hyderabad.
ABSTRACT
In this paper, a new model for active contours is
proposed. This model will detect objects in a given image,
based on techniques of curve evolution, level sets. This model
can detect objects whose boundaries are not necessarily
defined by gradient. In the level set formulation, the problem
is in a mean-curvature flow like evolving the active contour,
which should stop on the desired boundary. However, the
stopping term does not depend on the gradient of the image,
as in the classical active contour models, but is instead related
to a particular segmentation of the image. By introducing an
edge-mounted Willmore flow, as well as a prior shape kernel
density estimator, to the level set segmentation framework,
the boundaries are automatically detected. Based on the
detected image partition, the area of the partition may be
evaluated. This can be applied for medical images in
particular the images of spinal vertebrae, so as to detect the
exact size of the vertebrae and estimate the size and location
of the fracture or cracks formed on bone by spinal vertebrae
segmentation. It can be used in the 3D medical imaging
softwares so as to be helpful for physicians.
challenges. Crucial steps in diagnostic imaging are Detection
and segmentation of spine from images. Although these
quantitative image analysis techniques have received
increasing interest, accurate detection and segmentation
methods are still lacking. Spine segmentation will be tough
due to the complexity of vertebrae shapes, gaps in the cortical
bone and boundaries, as well as noise, inhomogeneity, and
incomplete information as shown in image fig 1.
An important cause of lower back pains is Lumber disc
herniation. Clinical diagnosis and therapy for the lumbar disc
herniation to reduce lower back pain requires the knowledge
of the stress and strain throughout the region of lumbar. The
finite element method based on medical images is able to
analyse the biomedical characteristic of lumbar in the
compression. We are sure that accurate 2D vertebra
segmentation will help us reconstruct 3D vertebra geometric
model because 3D vertebra segmentation modelling is
fundamentally performed based on a set of axial slices. The
understanding of geometrical information about the normal
anatomy and the degenerative bony deformations of the spine
necessitates vertebra CT image segmentation for the clinical
diagnosis and the preoperative planning of spinal diseases.
Keywords
Level set, edge-mounted Willmore flow, kernel density
estimator, size, fracture, spinal vertebrae, 3D medical
imaging.
INTRODUCTION
SPINE trauma is a high morbidity and moralistic
event, causing severe psychological, social, and financial
burdens for patients, their families, and the society. Among
the incidents of spine trauma, motor vehicle accidents are
accounting 42.1% injuries reported are of spinal cord,
followed by falls (26.7%), acts of violence (15.1%) and
sporting activities (7.6%). In older adults Compression
fractures of the vertebral body are common, due to
osteoporosis. Fractures of the thoracic and lumbar spine are
more widespread than of the cervical spine, with highly
affected area in thoracolumbar region (T11 to L4). Potentially
devastating long-term consequences such as neural deficits,
permanent disability, or even death may be caused due to
Fractures and dislocations of spine. Examples of spinal
disorders due to trauma and osteoporosis in computed
tomography (CT) images. Identifying and grading severity of
spine fractures and understanding their cause will help
physicians determine the most effective pharmacological
treatments and clinical management strategies for spinal
disorders. Achievement of a firm diagnosis is one of the major
Fig. 1 Incomplete, missing or ambiguous information of the
vertebrae in CT images.
PREVIOUS WORK
In recent years, radiography has been replaced by CT imaging
as the primary modality for assessing bony structures in
human anatomy, including the spine. Thus, this section will
provide a summary of the spinal vertebrae segmentation from
CT images to highlight the need for accurate and automatic
methods for 3-D vertebrae segmentation.
Various attempts have been made on the spine
segmentation in recent years, but majority of them use 2-D
images and/or require user intervention in the process. For
example, Naegel [2] combined the watershed method and
morphological approaches to segment vertebrae. Although the
proposed method is promising in segmenting healthy bones
from high-resolution images, manual refinement is necessary
to obtain accurate segmentations and the level of refinement is
patient and resolution dependent. Ghebreab and Smeulders [3]
constructed a deformable integral spine model to segment
vertebrae. The method learns the appearance of vertebrae
boundaries a priori from a set of training images. In order to
reduce the complexity of the segmentation process through
point-based shape representation in this model generating
landmark points,. If the landmark points correspond to the
actual anatomical locations and whether they capture the
biologically meaningful variations across different subjects, it
remains unclear. The method needs step-by-step inputs from
the user and also not fully automated by which, it makes the
whole process tedious and time consuming. Ma et al. [4]
presented an automatic vertebra segmentation and
identification method on thoracic vertebra CT images. In this
a learning-based bone structure edge detection algorithm and
a hierarchical, coarse-to-fine deformable surface-based
segmentation method was used and proposed based on the
response maps from the learned edge detector. Though
satisfactory results were obtained, the segmented vertebrae
were only in 2-D and reproducibility of results in 3-D was not
known. And other limitation is the complexity and/or
inaccuracy of current segmentation methods. For example,
Lorenze and Krahnstoever [5] proposed a statistical shape
model whereby the mean shape was constructed from a set of
training samples. The initialization of the shape model for
segmentation was done manually and is highly sensitive to
dislocation. If the model is not located in the proximity of
vertebrae, segmentation may fail. More recently Klinder et al.
[6] used a mesh-based method to extract spine curves, and
then generalized Hough transform and curved planar
reformation to detect the vertebrae. The proposed approach
has a further identification step to the detected vertebrae via
rigid registration of appearance model. Although they
achieved very competitive identification rates for vertebrae,
their algorithm depends heavily on spatial registration of the
model, which is computationally very expensive. In a paper
by Mastmeyer et al. [7], a hierarchical 3-D technique was
developed to segment the vertebral bodies in order to measure
bone mineral density. The proposed framework needs
excessive user intervention to precisely locate seed points to
facilitate region growing segmentation. This process is time
consuming and impractical for unhealthy bone segmentation.
A similar approach integrating region growing segmentation
with local shape and intensity refinement for delineating
vertebrae was proposed by Kang et al. [8]. First, locally
adaptive thresholds were used to facilitate region growing
segmentations globally, followed by 3-D morphological
operations to refine the segmented surfaces. This method still
required a site specific separation of individual bones, which
remains a challenge for vertebrae segmentation. Due to the
aforementioned drawbacks of the existing spinal vertebrae
segmentation methods, we have developed a new method
capable of segmenting spinal accurately from noisy images
with missing information. The method is developed by
introducing an edge-mounted Willmore flow, as well as a
prior shape kernel density estimator, to the level set
segmentation framework. While the prior shape model
provides much needed prior knowledge when information is
missing from the image, the edge-mounted Willmore flow
helps to capture the local geometry and smoothes the evolving
level set surface.
OVERVIEW
COMPUTED TOMOGRAPHY
A computed tomography (CT) machine consists of
an X-ray source (emitter) that generates X-rays and releases
them towards the patient. The detector array at the opposite
side receives the X-rays passed and not absorbed by the
patient. The machine rotates around the patient while
releasing X-rays to get information from all directions. The
detector array (scintillator) transforms the X-rays into
proportionally strong electric currents which is represented as
image slices. By moving the table step by step a full 3D
volume can be created by combining the 2D slices together.
The advantages by using CT over a normal X-ray scan is that
CT can take images in any direction, and that the result is a
volume of data a 3D image. Another advantage is the high
contrast of the resulting images; CT can differentiate between
tissues with less than 1% density difference. But better quality
comes with a cost, increasing the quality of the images
requires an increase in the amount of radiation. So there is
always a trade of in between noise in the images and the
dosage of radiation. As mentioned before, X-rays are ionizing,
and the high amount of ionizing radiation from CT is its
biggest disadvantage. MRI is sometimes preferred over CT
for small children, since the ionizing radiation effects younger
people more. Unlike conventional X- ray imaging which is
mostly used to represent teeth and bone, CT is used more
broadly. CT is for example used to image the heart, abdomen,
acute and chronic changes in lung, detecting tumors in
different parts of body, in addition to bone fractures. Fig 2
shows a regular CT image of the head.
Fig 2: CT image of a head.
LUMBAR SPINE
The lumbar vertebrae consist of five individual
cylindrical bones that form the spine in the lower back as
shown in Fig 3. These vertebrae carry all of the upper body’s
weight while providing flexibility and movement to the trunk
region. They also protect the delicate spinal cord and nerves
within their vertebral canal. Found along the body’s midline
in the lumbar (lower back) region, the lumbar vertebrae make
up the region of the spine inferior to the thoracic vertebrae in
the thorax and superior to the sacrum and coccyx in the pelvis,
the lumbar vertebrae are stacked to form a continuous column
in order from superior (L1 or first lumbar vertebra) to inferior
(L5 or fifth lumbar vertebra). Together they create the
concave lumbar curvature in the lower back.
Connecting each vertebra to its neighboring vertebra
is an intervertebral disk made of tough fibrocartilage with a
jelly-like center. The outer layer of the intervertebral disk, the
annulus fibrosus, holds the vertebrae together and provides
strength and flexibility to the back during movement. The
jelly-like nucleus pulposus acts as a shock absorber to resist
the strain and pressure exerted on the lower back. The lumbar
vertebrae are the some of the largest and heaviest vertebrae in
the spine, second in size only to the sacrum. A cylinder of
bone known as the vertebral body makes up the majority of
the lumbar vertebrae’s mass and bears most of the body’s
weight. Posteriorly the body is connected to a thin ring of
bone known as the arch. The arch surrounds the hollow
vertebral foramen and connects the body to the bony
processes on the posterior of the vertebra. The vertebral
foramen is a large, triangular opening in the center of the
vertebra that provides space for the spinal cord, cauda equina,
and meninges as they pass through the lower back.
 Using a set of ROIs as the mask. The ROIs for each
slice are used to define the mask.
LEVEL SET
The level set method [10] has been widely used for image
segmentation [11]. For highly challenging segmentation tasks,
such as segmenting low resolution objects from medical
images, the best method have achieved good results when
coupled with before knowledge or prior shape models of the
segment is the level sets method [12]–[15]. The level set
method includes an interface in a higher dimensional function
φ (the signed distance function) as a level set φ = 0. The
equation that governs the evolution of the level set function
φ(t) is ∂φ ∂t + F|φ| = 0, where F represents the speed function.
In more recent development, the variational framework is
often considered. Under this variational framework, an energy
E(φ) is developed in relation to the speed function, and
minimizing the energy generates the Euler–Lagrange equation
and hence, providing the evolution equation through the
calculus of variation as
∂φ
∂E(φ)
=−
∂t
∂φ
(1)
In this paper, we consider the combined energies, i.e., using a
shape prior distribution estimator Es with an edge-mounted
Willmore energy Ew o
Fig 3: CT images of lumbar spine
SPINAL SEGMENTATION
We do focus on 3-D segmentation of individual spinal
vertebrae with the aim so as to help physicians to have better
visualization of the shapes and internal structures of the
vertebrae, as well as to measure quantitatively the vertebrae
size and volume of fracture for surgical procedures, e.g.,
spinal implant. The contribution of this study is in twofold,
clinically and technically: 1) providing a precise individual
vertebrae segmentation system for clinical use; 2) introducing
Willmore flow into the level set segmentation framework. To
the best of our knowledge, this is the work on 3-D
segmentation of spinal vertebrae using Willmore flow within
the level set method to auto segment the spine accurately and
may be the first best technique. It has previously successfully
applied Willmore flow to 2-D vertebrae segmentation [9]. In
the following sections, we shall describe in detail the
segmentation method used, followed by experimental results.
VERTEBRAE SEGMENTATION
MASKING
Masking involves setting the pixel values in an image to
zero, or some other "background" value. Masking can be done
in one of two ways
 Using an image as a mask. A mask image is simply an
image where some of the pixel intensity values are zero,
and others are non-zero. Wherever the pixel intensity
value is zero in the mask image, then the pixel intensity
of the resulting masked image will be set to the
background value (normally zero).
E(φ) = λEs + Ew o
(2)
where λ(0 < λ ≤ 1) is the weight parameter. Details on Es and
Ew0 will be described in the following sections. In order to
incorporate a prior dataset {φ1, φ2, . . . , φN } into the level set
segmentation framework, we adopt a shape dissimilarity
measure based on the Kernel density estimation (KDE)
discussed by Cremers et al. [13]. This nonparametric
distribution estimator overcomes the two shortcomings of
existing algorithms: 1) the assumption that the shapes are
Gaussian distributed, which is generally inappropriate when
the number of training set is small, and not practical for
modeling shapes with high complexity and structure; 2) the
signed distance functions will represent shapes, which does
not include the mean as it constitute a nonlinear space in the
region of the segmentation.
KERNEL DENSITY ESTIMATION
KDE is used for estimating the probability density function of
a random variable as it is a nonparametric approach in
statistics. The underlying theory of KDE states that data with
unknown statistical distribution converge to its actual
distribution as the number of samples approaches is infinity.
In practice, KDE provides a fundamental smoothing estimator
even with a small number of data samples. In application with
N samples of shape models, the density estimation is
formulated as a sum of Gaussian of shape dissimilarity
measures d2 (H(φ),H(φi)), i = 1, 2, . . ., N.
1 N 
P( )   e
N i 1
d  ( H ( ). H (i ))
2 2
(3)
where H(φ) is the Heaviside function, the shape dissimilarity
measure [16]–[18] is
1
( H ( )  H (i )) 2 dx
2
d 2 ( H ( ), H (i ))  
(4)
and σ2 is the mean squared nearest neighbor distance given as
2 
1 N
min d 2 ( H (i ), H ( j ))

N i , j 1
(5)
Note that the L2-norm is invariant under translation and
scaling with respect to the principal axis of the shape. Hence,
the shape dissimilarity measure d2 is also invariant under
these transformations when the prior shapes are normalized
with respect to translation and scaling accordingly [13]. The
segmentation is obtained by maximizing the conditional
probability of φ given image I
P(φ|I) =
P(I|φ)P(φ)
P(I).
2
Ew

1
2 

     M h  h  t   S  t   h  t    (9)
2

2



shape operator on φ, and S2 is the Frobenius norm of S.
In order to ensure that the smoothing effect of Willmore
energy acts around the constructed surface and does not affect
adversely the edge of vertebrae, we propose to multiply the
edge indicator function g(I) = 1/( 1+|Gσ *I |2) to the level set
evolution
2 1
Ewo

2 

  g ( I )    M h  h  t   S  t   h  t    (10)
2 2




EUCLIDEAN CURVE SHORTENING
(6)
∂φ
Since the negative logarithmic scale of the probability
distribution P(φ|I) nicely defines an energy that associates the
evolution of φ with the minimization problem, the shape
energy is formulated as
Es (φ) = −log P(φ|I).
(7)
WILLMORE FLOW
Willmore energy is a function of mean curvature, which is
a quantitative measure of how much a given surface deviates
from a round sphere. It has been applied to image in painting,
restoration of implicit surfaces [19], [20], and to studies of the
bending energy of biological cell membranes as these cell
membranes tend to position themselves to minimize Willmore
energy [21]. Willmore flow is the gradient flow of Willmore
energy. Willmore flow of a surface is the evolution of the
surface in time to follow variations of the Willmore energy.
Willmore energy was defined after the British Geometer T.
Willmore [22] and is formulated as
1
h 2 dA

M
2
= ∅||∇φ||div (
∇φ
) + ∇∅. ∇φ
||∇φ||
(11)
As before, a constant inflation term v may be added to give
the model
φt = ∅||∇φ||div (
∇φ
||∇φ||
+ v) + ∇∅. ∇φ
(12)
Hence, the variational with respect to φ becomes
Ew 
∂t
(8)
where M is a d-dimensional surface embedded in Rd+1 and h
the mean curvature on M.
In this paper, Willmore flow is integrated into the level set
segmentation framework as a geometric functional. Willmore
energy is here defined on the collection of level sets, and
Willmore flow is enabled by defining a suitable metric, the
Frobenius norm, on the space of the level sets. The Frobenius
norm of an arbitrary matrix A = (aij)mxn , which is defined as
AF = (m i=1 n j=1 |aij|2 )1/2 , coincides with the calculation for
the gradient decent. It is equivalent to the l2-norm (the
Euclidean norm) of a matrix. More importantly, it is
computationally attainable comparing to l2-norm. As
Frobenius norm is an inner-product norm, the optimization in
the variational method comes naturally. Based on the
formulation by Droske and Rumpf [23], Willmore flow or the
variational form for the Willmore energy with respect to φ is
(Once again, this inflationary constant may be taken to be
either positive (inward evolution) or negative in which case it
would have an outward or expanding effect. As in the 2-D
case, we take to be negative in the interior and positive in the
exterior of the zero level set.) Indeed, the geometric heat
equation will shrink a simple closed curve to a round point
without developing singularities, even if the initial curve is
nonconvex.
EXPERIMENTAL RESULTS
CLINICAL DATA AND GROUNDTRUTH CONSTRUCTION
The images of the lumbar spine vertebrae are taken both
of the lumbar section and spine canal so as to be segmented
with Willmore flow in MATLAB. The dataset consisting of
20 CT images of normal spinal vertebrae images of the patient
for visualization. These images are acquired from various CT
scanners such as 32- detector row Siemens definition, 64detector row Philips Brilliance, and 320-detector row Toshiba
Aquilion taken as DICOM. The in-plane resolutions for these
images range from 0.88 to 1.14 mm, with consistent slice
thickness of 2 mm. Original images have fixed sizes of
512x512, with slices varying from 45 to 98. The 3D
segmentation of the DICOM image is obtained by manual
delineations using TURTLESEG, interactive 3-D image
segmentation software [25]. By this the DICOM images are
read and taken as slice views so as to have 3D segmentation.
SEGMENTATION RESULTS
The images of the lumbar spine and spinal canal is been
segmented first in MATLAB with Willmore flow introduced
to the level set frame work. The segmented images are shown
fig 4 (a), 4(b) respectively
(a)
(a)
(b)
Fig 5: 3D segmented image of a) lumbar vertebrae (b) spine
canal using MATLAB
The segmentation of the obtained CT images DICOM of
CENOVIX is been done using the TURTLESEG software for
lumbar L4. The lumbar is been segmented using this by
manual selection of the regions in the slice views of XY, YZ,
ZX views. Thus, the 3D modelled lumbar is as shown in fig 6.
Fig 6: 3D segmented image of lumbar vertebrae using
TURTLESEG
(b)
Fig 4: Original image and segmented image of (a) lumbar
vertebrae (b) spine canal with mask using MATLAB
In previous paper [26], the segmentation is been done and
compared with previous methods. In our paper, the 3-D
segmented images of the same are obtained by applying the 3D modelling to the above segmented part to get the 3D view
of the segmented part as shown in fig 5.
The slice views of the 3D segmented images in the MATLAB
are as shown in the fig 7.
Fig 7: 3D and XY, YZ, ZX Slice views of segmented image
of lumbar vertebrae in MATLAB
CONCLUSION
We have presented a new model for the segmentation by
incorporating the Willmore flow and kernel density functions
to the levelsets framework and by this it has obtained the
accurate segmentation of the spine even with missing data.
The experimental results of this model of levelsets framework
has obtained high accurate segmentation on the CT images as
shown in the results which is clinically helpful for physicians
to give correct treatment to the patients if the fractures or
deformations are identified on the spine.
FUTURE SCOPE
This model of introducing Willmore flow to the levelset
framework, can segment the images with accurate border so,
this can be used in the 3-D segmentation, DICOM
segmentation software eg: TURTLESEG so as to select the
accurate region such that it can be grown and modeled in 3D
accurately. By which, the accurate size of segmented bones
can be known, identified so that the deformations or any
fractures present can be identified or spotted and necessary
medication can be done easily. If else by manual selection of
the region the the accuracy may be redused hence by applying
this willmore levelset framework for that selection accuracy
of obtaining correct result will be more. That means this
system can be used for detecting and estimating the exact
location and size of spine fractures and other diseases causing
deformation.
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BIOGRAPHY
V. Ranadheer Reddy, Pursuing M.Tech Digital
Communications, Department of ECE, Kakatiya Institute of
Technology and Science. Area of interest in embedded
systems, Image Processing.
G. Raghotham Reddy, Faculty of ECE Department,
Kakatiya Institute of Technology and Science.
Prof Rameshwar Rao, Prof of OU, Hyderabad.