A Hierarchical View for Level Set Method Based on Segmentation of Non-Constant Intensity Objects

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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 9, 2011

A Hierarchical view for Level Set Method based on
segmentation of Non- Constant Intensity Objects
M.Janani D.Kavitha Devi
M.Phil Scholar Assistant Professor,
P.S.G.R Krishnammal College For Women P.S.G.R Krishnammal College For Women
Coimbatore-641004. Coimbatore-641004.
. .

Abstract—Segmentation of non-constant intensity object has been The active of moving curves and surfaces, called the Level-
an important and vital issue for many applications. Segmentation Set Method. The level-set method is one computational
of non- constant intensity object is a fundamental importance in technique for tracking a propagating interface over time, which
image processing. Segmentation is difficult task in noisy images. in many problems has proven more accurate in handling
The complementary method of the Mumford shah model for
segmentation of non-constant intensity objects is been intended
topological complexities such as corners and cusps, and in
by level set method. The level set method retrieve the possible handling complexities in the evolving interface such as entropy
multiple membership of the pixels. Additive is forced through conditions and weak solutions. It is a robust scheme that is
level set method which allows the user to control the degree of relatively easily to implement. Multiple regions are captured by
non-constant intensity objects and is more secure than the soft a single contour demonstrating the topological transitions
constraint the enhanced method increase efficiency, improve the allowed by the models in level set implementation.
effectiveness of segmentation. The numerical and qualitative
analysis show that the level set algorithm provide more accurate
II. IMPROVED MUMFORD-SHAH MODEL
segmentation result with good robustness.

Keywords- level set method, non-constant intensity object, A. Mumford-shah model
terzopoulos, kass, witkins, lipschitz. The Mumford-shah model is one of the standard
segmentation models. Mumford shah functional has been
I. INTRODUCTION extensively used for image segmentation. Mumford shah
algorithm obtains simultaneous functionality of both image
Segmentation is a process of dividing an image into
smoothing and segmentation. The active contour is viewed as
meaningful, non-overlapping regions. Level set method is the
the set of discontinuities considered in the Mumford-shah
process to improve the segmentation and simultaneously
formulation. The smooth estimate of the image is continuously
solving the non-constant intensity object. Segmentation of non-
estimated based on the current position of the curve.
constant intensity object and incorporating some knowledge
about their spatial relationship is a vital task. The problem of
segmenting non-constant intensity object with possible
occlusion in a variation setting is been solved. Hard
segmentation model is that inherit the original property of the
Mumford shah formulation to segment and smooth images in a
coupled manner. Chan and Vese proposed a piece wise
constant Mumford shah model in by further Mumford shah
advances by using a level set formulation.
The Hard segmentation is to simplify and/or change the
representation of an image into something that is more
meaningful and easier to analyze. Image segmentation is
typically used to locate objects and boundaries (lines, curves,
etc.) in images.
In soft segmentation, there is the persistent control of the
intensity. In the soft segmentation, the restraint is only loosely Fig. 1 Input image
prosecuted. We call this model the soft segmentation. The soft Mumford-shah active contour model can handle image
segmentation reduces to the piecewise constant Mumford Shah containing regions with roughly two different mean. Active
segmentation model. The solution of the soft segmentation will
contours were introduced by Kass, Witkins, and Terzopoulos
approach to that of the hard segmentation. for segmenting objects in images using dynamic curves.
Munford-Shah model only can segment the image into two

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parts according to the value of the shade of gray for specific
images. The Mumford-shah model is not detecting the noisy
image. The improved technique of Mumford-shah model is
hard segmentation.
For a given image u0, the piecewise constant Mumford-shah
model seeks for a partition of Ω into Ν mutually exclusive
open segmentsΩ1,……. Ωn together with their interface C and a
set of constant c=(c1,c2,….,cn)which minimize the following Here, the over-line denotes the set closure. Although Ω1
energy functional: and Ω2 generally do not constitute a partition of Ω , we still
call the pair {Ω1 , Ω2 }a segmentation of Ω for simplicity.

The idea is to partition the image so that the intensity of u0
in each segment Ωi is well-approximated by a constant ci . The
geometry of the partition is regularized by penalizing the total
length of C. This increases the robustness to noise and avoids
spurious segments.

B. Hard mumford-shah model
Image segmentation is the process of assigning a label to
every pixel in an image such that pixels with the same label
share certain visual characteristics. Given a fixed segmentation,
it can be easily shown that the optimal constants are given by
formulas denotes the Lebesgue measure of its argument Let us Fig.3 Histogram of hard Mumford-shah model
take non-constant intensity regions in that brain MRI image.
Here, the over-line denotes the set closure. Although and
generally do not constitute a partition, we still call the pair a Given an image u0 , the hard additive model seeks for
segmentation of for simplicity. It should be clear that the segmentation {Ω1 , Ω2 } and a set of constants c=(
partition is given by (together with the boundary of these c10,c01,c11,coo) which minimize the following energy:
segments inside).Given an image, the hard additive model
seeks for a segmentation and a set of constants which minimize
the energy, subject to an additive constraint .This model
enforces a strict additive in the common region.

Subject to an additive constraint c11= c10+c01. Thus, this
model enforces a strict additive in the common region.

C. Soft mumford-shah model
The soft segmentation reduces to the piecewise constant
Mumford–Shah segmentation model. The solution of the soft
segmentation will approach to that of the hard segmentation.
Given segmentation, the optimal constants can be obtained by
the formulas. The intensity level within each region has a
certain degree of variation. A multi phase formulation with
membership functions has recently been used with a different
regularization term in for soft segmentation.
Fig.2 (a). 400 iteration of hard Mumford- shah model, (b). After noise removal
in hard Mumford-shah model

Let Ω1 and Ω2 be two open regions in Ω that represent two
computed objects. The following short hands to simplify the
notations:

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Where ᵞ ≥ 0 is a constant controlling the degree of additive. a given function. The Osher-Sethian level set formulation
In this model, c10+c01=c11 the constraint is only loosely allows the development of efficient and stable numerical
enforced. We call this model the soft additive model. schemes in which topological changes of the propagating
curve are automatically handled. The level set formulation
puts curve evolution equation into the level set formulation.
The level set method overcomes the problem of soft
segmentation model .multiphase level set image segmentation.
This method established on explicit correspondence between n
region of segmentation and a partition defined using log2n
level set functions.
Let ⱷI=ΩC R2 →R IE [1,,,,,,M],be M level set function with
M=LOG2N.
Level set reprentation

Fig.4 (a). 400 iteration of soft Mumford- shah model, (b). After noise removal in
soft Mumford-shah model
Where Ω in is a region in Ω bounded by Γ,and Ωout is
defined as the complement of Ω in, i.e. Ωout=Ωin..To avoid
unnecessary calculation and statistical errors the level set
representation is used.

Fig.5 Histogram of soft Mumford-shah model

III. LEVEL SET METHOD
The level set method is a powerful tool which can be used Fig.6 (a). 730 iteration of level set method, (b). After noise removal in level set
for numerical realization. Level set representation is an method

established technique for image segmentation .Level set
methods is to minimize a given function which aims to extract
one or several elements of interest from the background. Level
set method is referred to as a curve. In level set method, the
curves are implicitly defined as the zeros of a lipschitz
continuous function.
The level set method depended on the global information of
homogeneity region, and is more robust than curve evolution
model to detect discontinuities under noisy environment the
level set method, can successfully handle the topology
changes. Level set method has been applied to a variety of
synthetic and medical images in different modalities. The level
set method overcome the problem of soft segmentation
problem it proves to be more accurate and robust. One way to Fig.7 Histogram of level set method
represent a curve is as a level set or an equal-height contour of

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IV. EXPERIMENTAL RESULTS [3] T. Chan and L. Vese, Active contours without edges, IEEE Trans. On
image proc., 10(2):266–277, Feb. 2001.
The brain MRI image is used as an object in this paper. The [4] C.A.Cocosco, A.P.Zijdenbos, A.C.Evan, “A Fully Automatic and
brain MRI image was segmented using soft and hard Mumford- Robust Brain MRI Tissue Classification Method,” Medical image
shah model; the segmenentation image is not accurate. So we Analysis, 7, pp.513-527, 2003.
validate our model based on; (a) performance comparisons [5] A.Chakraborty, L.H.Staib, and J.S.Duncan, “Deformable Boundary
Finding in Medical Images by Integrating Gradient and Region
between the hard segmentation, soft segmentation and level set Information,” IEEE Trans. On Med. Imag., 15(6) pp. 859-870, 1996.
method; (b) non-constant intensity object segmentation. In the [6] V.Caselles, R.Kimmel, and G.Sapiro, “Geodesic Active Contours,” Int.
hard segmentation the non-constant intensity objects is not Journal of Computer Vision. 22(2), pp. 61-79, 1997.
been segmented accurately. In the soft segmentation the non- [7] Faugeras O. and Keriven R., Variation principles, surface evolution and
constant intensity objects is segmented but the output of the PDE’s level set methods, and the stereo problem. IEEE Trans image
processing., 1998, 7(3):336-344.
retrieval image is not accurate. In the level set method the non- [8] J. A. Sethian, Level set methods and fast marching methods, Cambridge:
constant intensity objects is been segmented. Cambridge University Press, 1999.
Table.1 Intensity [9] T Chan and L. Vese, “Active Contour without Edges”, IEEE
Transaction on Image Processing, 2001, 10(2), pp.266-277.
Method Original Overlapped Backgroun [10] D. Mumford. J. shah. ”Boundary detection by minimizing functional”
image image d image Proc. of IEEE CVPR 1985.
101.0906 0.8208 0.4506 [11] S. Osher. J. Sethian, “Fronts propagating with cur\aturedependent speed:
Hard
algorithms based on Hamilton-Jacobi Formulation”, Journal of
segmentation Computatzonal Physzcs. 79. 12-49. 1988.
Soft 101.0906 0.8654 0.7973 [12] Li-Tien Cheng, Paul Burchard, Barry Merriman, and Stanley Osher,
segmentation “Motion of curves constrained on surfaces using a level set approach”. J.
Level set 101.0906 0.9923 0.8112 Comput. Phys., 175:604.644, 2002.
method

Table.2 Standard deviation
Method Original Overlapped Background
image image image
Hard 101.0906 0.9291 0.5282
segmentation
Soft 101.0906 0.9499 0.7453
segmentation
Level set 101.0906 0.9822 0.8541
method

CONCLUSION
The alternative method of the Mumford shah model for
segmentation of non-constant intensity objects is been intended
by level set method. The optimized zero level set indicate their
approximate shapes and distributions clearly. Level set model
has overcome some refractory challenges in elasticity
reconstruction. The level set method is more robust than the
soft segmentation with respect to global convergence. Hard
segmentation fails to detect multiple non-constant intensity
objects. The problem of segmenting non-constant intensity
objects with possible occlusion in a variation setting is been
solved. Level set method it solves the segmentation with depth
problem that aims to recover the spatial order of non-constant
intensity objects. Segmentation of multiple objects is been
identified accurately. Finally, we demonstrate a hierarchical
implementation of our model which leads to a fast and efficient
algorithm capable of dealing with important image features.

REFERENCES
[1] J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization,
Springer, New York, 1991.
[2] W. Zhu and T. Chan, “A variational model for capturing illusory
contours
using curvature,” J. Math. Imag. Vis., vol. 27, pp. 29–40, 2007.

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