Level Set Evolution without Re‐initilization
Outline
• Parametric active contour (snake) models.• Concepts of Level set method and geometric active contours.Concepts of Level set method and geometric active contours.• A level set formulation without reinitialization.• Mumford‐Shah functional.• Piecewise constant and piecewise smooth models.
L l bi fi i d l• Local binary fitting model.
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Image Segmentation and Applicationsg g pp
• Image segmentation: extract objects of interest in images.• Image segmentation is a fundamental step in computer vision and image
analysis.analysis.• Applications of image segmentation:
1. Shape recovery, analysis, recognition…2. Measurement3 Vi li ti3. Visualization4. Medical applications: tissue measurement, diagnosis, study of anatomical
structures, computer‐integrated surgery …
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Classical Methods
Thresholding Edge detectionAn image of blood vessel
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An Advanced Method: Active Contour Model
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Parametric Active Contours (Kass et al 1987)( )
For a contour , define energy:
High energy
Low energy
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Low energy
Evolution of Active Contours
Gradient descent flow:
Advantages: • Smooth and closed contour • Sub‐pixel accuracy.
Disadvantages: • Cannot change topology.
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g p gy• Initial contour must be close to the object boundary.
Geodesic Active Contours (Caselles et al, 1997)( , )
• Minimize a weighted length of C
where
• Gradient descent flow:
• Add balloon force:
High energyLow energy
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Level Set Representation of Curvesp
l lzero level
zero level
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zero level
Level Set Method (Osher and Sethian, 1988)( , )
• Curve evolutionwhere F is the speed function N is normal vector to the curve Cwhere F is the speed function, N is normal vector to the curve C
• Level set formulation
N
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Geodesic Active Contour: Level Set Formulation
• Curve evolution of geodesic active contour:
• Level set formulation of geodesic active contours:
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Drawbacks of Geodesic Active Contour
• Unstable evolution, requires periodic reinitialization to signed distance function.• Balloon or pressure force cause boundary leakageBalloon or pressure force cause boundary leakage.• Slow evolution due to small time step.
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Variational Level Set Method without Reinitialization (Li t l 2005)et al, 2005)
Define an energy functional on level set function:
where
Level set regularization
Internal energy:• Penalize the deviation from a signed distance function
Level set regularization
External energy:D i th ti f th l l t
g
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• Drive the motion of the zero level set
External Energy for Image Segmentationgy g g
Edge indicator function for image I
I Image
Define external Energy: 0 0
Weighted length:0
Weighted area:
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Energy Functional and Gradient Flowgy
Define energy functional:
The gradient flow of the functional is the evolution equation:
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Results
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3D Segmentation of Corpus Callosumg p
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Conclusion
The proposed variational level set formulation has three main advantages over the traditional level set formulations:
•First, a significantly larger time step can be used for numerically solving h l l d ff l d h f d hthe evolution partial differential equation, and therefore speeds up the curve evolution.
•Second the level set function can be initialized with general functions that•Second, the level set function can be initialized with general functions that are more efficient to construct and easier to use in practice than the widely used signed distance function.
•Third, the level set evolution in our formulation can be easily implemented by simple finite difference scheme and is computationally more efficient.
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Region-based Methods
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Mumford‐Shah Functional
• Piece wise smooth model‐‐‐ Approximate image by piecewise smooth functions
Data fitting termSmoothing term Length term
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Active Contours without Edges (Ch & V 2001)(Chan & Vese 2001)
• Define a region‐based energy functional:
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Level Set Formulation of Chan‐Vese Model
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Results
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Piecewise Smooth Model (Vese and Chan, 2002)
Minimize the energy functional:
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Solve PDEs:
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Examplesp
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Local Binary Fitting Active Contours/Surfaces
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Local Binary Pattern in General Imagesy g
f1
f2
Assumption: image I can be locally approximated by a binary image.
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Local Binary Fittingy g
xf1
Cx
f2
C
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Level Set Formulation
The LBF energy functional on a contour C
is equivalent to the level set formulation:
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Level Set Formulation (Cont’d)( )
For extracting the entire object boundary, the local binary fitting energy is integrated over all x in the image domain:
Add two terms for regularization of the contour and the embedding level set function, and define the following energy functional:and define the following energy functional:
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Data fitting term Length term Level set regularization
Energy Minimization Using Gradient Flowgy g
The minimization of the energy functional F is achieved by solving the gradient flow:
wherewhere
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Result
Synthetic noisy image
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2D Segmentation of Real Color Imagesg g
A real image of potatoes
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2D Vessel Segmentationg
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Segmentation of White Matter in MR imagesg g
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Effect of the Level Set Regularizationg
Without level set regularization
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Final zero level contour Final level set function
Comparison with Piecewise Smooth Modelp
Comparison of computational efficiency
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Comparison with Piecewise Smooth Modelp
Our method
PS model
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3D Vessel Segmentationg
MRA Vessel Segmentation
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SummarySummary
• Variational and level set methods for image segmentation.• My recent works on variational level set methods:
1. A new level set formulation without the need for reinitialization (CVPR 05).2. A region‐based model that draws upon local image information. (CVPR 07).
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AcknowledgmentAcknowledgment
• Dr. John Gore, Vanderbilt University• Dr Zhaohua Ding Vanderbilt UniversityDr. Zhaohua Ding, Vanderbilt University• Dr. Chiu‐Yen Kao, Ohio State University• Dr. Chenyang Xu, Siemens• Dr. Kishori Konwar, Goldman Sachs
h f f• Dr. Changfeng Gui, University of Connecticut• Dr. Martin Fox, University of Connecticut
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Thank you
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