Level set based Image
Segmentation
Hang Xiao
Jan12, 2013
Outline
• Part1 : Level set method
• Part2 : Image segmentation
Part1 : Level set method
In mathematics, a level set of a real-valued function
f of n variables is a set of the form
STANLEY OSHER
1988
Front propagating problems
Ocean waves
Image contour
Burning flames
Interface Propagation
F = F(L, G, I)
Inside
Outside
Outside
F(L,G,I) : speed function
L : local properties, e.g. curvature and normal direction
G: Global properties, e.g. heat
I: Independent properties, independ of the shape of the front, e.g.
underlying fluid velocity
Arrival time function
Assume F > 0, hence the front moves “outward”
T(x,y) : the arrival time,
characterize the position of the
expanding front
Distance = speed * time
1D :
2D:
: the initial location of the
interface
Time-depend level set function
The stationary level set function
F is constant to t
Time-depend level set function
F is inconstant to t
Level set function properties
Front formula:
F
By the chain rule,
Inside
Outside
Hamilton-Jacobi equation
Outside
Level set method work flow
Initialize level set function Φt=0
Compute speed F
t++
Compute level set function in
next state Φt+1
Reinitialize level set function
Compute final zero level set
Arrival time
e.g.
Example – simple curve evolution
Simple curve
Complex curves
Level set
Curvature
Speed
Property: Any simple curve will become a ball
in the force of curvature
Example – simple curve evolution
Example – 3D surface
3D curvature
Gaussian curvature
Mean curvature
Mean curvature
Gaussian curvature
Other examples
The particle level set method was used to
represent the interface separating water
from air as water is being poured into a
glass
Level set method is used to simulate a ball
catching on fire
Part2 : Image segmentation
Image segmentation models
• Explicit method
• Implicit method
Inside
Outside
Interface front
Evolution equation
Energy minimization
Outside
The Chan-Vese Segmentation Model
Assume the image is piecewise constant
Energy : Sum of difference
Add more regularizing terms
The Chan-Vese Segmentation Model
Energy function
Energy Minimization
Substituting into the Euler-Lagrange equation,
and applying Green’s identity and Green’s theorem
Effect of Parameters
• μ controls the importance of the length of C in the
minimization.
– if μ ≫ 1, then a few large closed curves will retain in the steady state,
compared to many small ones. This may be useful in ignoring noise, or
grouping objects of similar characteristics
• ν controls the the importance of area inside C.
– If ν ≫ 1, like the geometric active contour model, it forces C to move
strictly inward. Also, the speed at which C evolves inward increases.
• The relative balance between λ1 and λ2 determines which
side, inside or outside, has higher importance in minimizing
the regional variance.
– This is useful in segmenting blurred images: for example, if one wishes
to completely enclose the blurred object, λ2 > λ1 will ensure this.
Chan-Vese Examples
Parameters: μ=1, ν=0, λ1 = λ2 =1, iterations = 41
Chan-Vese Examples
Change in topology
Parameters: μ=1, ν=0, λ1 = 1, λ2 =2, iterations = 45
Chan-Vese Examples
Blurred and Noised Image
Parameters: μ=1, ν=0, λ1 = 1, λ2 =2, iterations = 45
Chan-Vese Examples
Chromatic Resemblance
Parameters: μ=10^8, ν=0, λ1 = λ2 = 1, iterations = 40
Chan-Vese Examples
Restriction on Curve Evolution Direction
Parameters: μ=1, ν=0, λ1 = λ2 = 1, iterations = 40
Chan-Vese Examples
Restriction on Curve Evolution Direction
Parameters: μ=1, ν=10^6, λ1 = λ2 = 1, iterations = 40
Detection of lines and curves not necessarily closed
Example of image for which the averages “inside”
and “outside” are the same
Detection of a simulated
minefield, with contour
without gradient.
Time steps (dt) evalution
E1: false negative error
E2: false positive error
Conclusion: The image can be segmented
well in one iteration for timesteps greater
than about 0.001.
Overlap error after one iteration for small or large dt
Iteration number evalution
Overlap error vs. number of iterations for
a small timestep
Overlap error vs. number of iterations for
a large timestep
Conclusion:
1). When the timestep is small the segmentation converges.
2). However, if we take the timestep too large, the results diverge after a few
iterations even though the segmentation is initially good.
Top: The noisy image
Bottom : the thresholding segmentation result
max(E1, E2) = 0.5625, min(E1, E2) = 0.1222
Evaluation on noisy image:
1) Use the true segmentation from the noiseless image
(ground truth) for forming the overlap error
2) Perform five iterations of the algorithm for each fixed
value of c1, c2
3) Vary one parameter at a time and plot the overlap
errors against that parameter.
Overlap errors vs. length penalty μ.
The best segmentation is obtained for μ
∼ 94.
Overlap error vs. λ1.
Changing λ1 from 1 does not improve the segmentation, since our foreground
and background regions both have uniform intensity before noise is added. The
results for varying λ2 are similar.
Chan-Vese VS. Mumford-Shah
Chan-Vese
function
2001
Mumford-Shah
function 1992
MS model  CV model
The Chan-Vese Segmentation Model
The Chan-Vese Segmentation Model
Energy function
Energy Minimization
Substituting into the Euler-Lagrange equation,
and applying Green’s identity and Green’s theorem
How to add area constant constraint?
Generalizing to Vector-Valued Images
Texture Segmentation
1. Generate sparse texture features by
nonlinear diffusion filtering
Brox, Weickert ’04, ’06
Texture Segmentation
2. Vector-Image based level set methods
Brox, Weickert ’04, ’06
Texture Segmentation
efficient coarse-to-fine scheme
Brox, Weickert ’04, ’06
Coupling Multiple Active Surfaces
(Dufour 2004)
Needs N level set functions for N objects segmentation
Multi-Phase Level Set methods
Need log2N Level set functions for segmenting N objects
Multi-Phase Level Set methods
Results on a synthetic image, with a triple junction, using the 4-phase piecewise
constant model with 2 level set functions. We also show the zero level sets of φ1
and φ2
Multi-Phase Level Set methods
Brox, Weickert ’04, ’06
One Level Set without Coupling Based
Multiple object segmentation
Pros. And Cons. For level set method
• Advantage
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Capture Range
Effect of Local Noise
No Need of Elasticity Coefficients
Suitability for Medical Image Segmentation
Normal Computation
Integration of Regional Statistics
Flexible Topology
Extension to high dimension
Incorporation of Regularizing Terms e.g. volume
Handling Corners
Resolution Changes
Jasjit S. et. al. 2002
Pros. And Cons. For level set method
• Disadvantage
– Initial Placement of the Contour
– Hard to stop
– Expensive running time
Jasjit S. et. al. 2002
Thanks!!!