Geodesic Active Contours

1
Geodesic Active Contours
2
Introduction

• Proposed by
• Vincent Caselles
• Universitat Pompeu Fabra (Spain)
• http//www.iua.upf.es/vcaselles/
• Ron Kimmel
• Technion University (Israel)
• http//www.cs.technion.ac.il/ron/
• Guillermo Sapiro
• University of Minnesota
• http//www.ece.umn.edu/users/guille/
• Geodesic Active Contours.
• International Journal of Computer Vision, Vol.
  22, No. 1,
• pp 61-79, 1997

?
?
3
Energy based active contours

• Classical active contour models
• deforming initial contour C0 towards the boundary
  of the object to be detected
• First two terms control the smoothness of
  contours (internal energy), Third term is
  responsible for attracting contour towards edge
  (external energy)
• Solving this equation involves in finding C that
  minimize E

4
Energy based active contours

• Let ß 0,
• Regularization effect on the geodesic curve comes
  from curvature based curve flows
• Experiments show this curvature driven curve
  motions to be very effective
• To derive the relation between this model and
  geometric curve evolution ones.

5
Energy based active contours

• Let g(r) be steadily decreasing function,
• g(r) ? 0 as r ? 8
• Drawbacks
• The approach is not topology independent.
• Too many parameters
• When considering more than one object

6
Geodesic Curve Flow

• Lets ENJOY!

7
Geodesic

• Definition
• A geodesic is a locally length-minimizing curve

by mathworld, http//mathworld.wolfram.com
8
Geodesic

• Property
• The shortest distance between two points is the
  length of a geodesic between the points
• Examples
• In the plane, the geodesics are straight lines
• On the sphere, the geodesics are great circles
  (like the equator)

9
Riemannian Geometry

• Metric gij
• The length of a segment of a curve parameterized
  by t,
• from a to b, is defined as
• Ex) A 2D Euclidean metric gij
• Curvature k
• The amount by which an geometric object deviates
  from being flat, more deviates bigger k.
• Ex) For a plane curve C(t) (x(t), y(t)), the
  curvature k is
• where the dot denote
  differentiation respect to t

10
Geodesic Curve Flow

• Let us define
• Then

11
Maupertuis Principle
If
Curves C(q) is in Euclidean space which are
extremal corresponding to the Hamiltonian H ,
and have a fixed energy level E0(law of
conservation of energy)
then
Curves C(q) are geodesics, with non-natural
parameter, with respect to the new metric
12
Geodesic Curve Flow

• The Maupertuis Principle explains,

Minimizing E(C) with H E0,
Minimizing (C (C1, C2)), finding geodesic curve
measured by gij

Let E0 Eint-Eext 0,
13
Geodesic Curve Flow

• We have transformed

The Problem of minimizing energy
Min
The Problem of geodesic computation in a
Riemannian space, according to a new metric
14
Geodesic Curve Flow

• In order to minimize
• we should follow the curve evolution equation
• (a way of minimizing via the steepest-descent
  method)

This differential equation is derived by the
Euler-Lagrange of
NOTE
15
Geodesic Curve Flow

• Curve evolution equation
• Shows how each point in the active contour C
  should move in order to decrease the length.

16
The Level-set Geodesic Flow

• To find the geodesic curve
• We compute steepest-descent flow
• But, the geodesic flow is represented using the
  level-set approach

17
Level Set Representation

• C is level set function u.
• Level Set
• A set points of u C
• Especially, if u0, C is zero level set
• Parameter free, Topology free
• Accurate, Stable
• ?
• Moving to normal direction
• Topological change of C(t)
• can be handled

18
Level Set Representation

• Geodesic problem is equivalent to searching for
  the steady state solution(?u/?t0) of the
  following evolution equation (u(0,C)u0(C) )
• This equation is the Euler-Lagrange of
• with C represented by a level-set of u, and the
  curvature is computed on the level-sets of u

Geodesic computation
19
250years ago
20
Maupetuis Principle
The stream engine
An energy minimization problem is equivalent to
finding a geodesic curve in a Riemannian space
21
150years ago
22

• The Riemannian geometry

The Civil war
The solution is given by a geodesic curve in a
Riemannian space.
23
33years ago
24
Lena Image
The Playboy Nov 1972
The Lennas issue is Playboy's best selling
issue ever
25
17years ago
26
Active contour models
The 88 Seoul Olympic
Solving this equation involves in finding Curve
that minimize Energy
27
10years ago
28
Geodesic Active Contours
MS Windows 95
Solving this equation involves in finding Curve
that minimize Energy
29
The existing model

• Caselles et al.(93) proposed the following model
  for boundary detection
• means that each one of
  the level-sets C of u is evolving according to

Geometric smoothing
Constant velosity similar to the balloon force
30
Curvature Flow

• Curvature flow
• The flow decreases the total curvature
• geometric smoothing property
• smoothing
• shortening

31
Constant Velocity

• The balloon force
• An area minimizing force
• The contour will propagate inwards by
  minimization of the interior
• Its crucial in order to capture non-convex
  shapes small gaps

32
The stopping function

• Back to energy based active contours
• g(I) The stopping function
• The main goal is actually to stop the evolving
  curve when it arrives to objects boundaries.

Internal energy
External energy
33
Return to Geodesic framework

• is naturally incorporated in
  geodesic framework
• This term attracts the curve to the boundaries of
  the objects.
• ?g points toward the middle of the boundaries
• We can remove the constant velocity term with the
  aid of ?g ?u.

34
Boundary Detection

• It will lead the propagating curve into the
  boundary and force it to stay there.
• Its possible to detect boundaries with high
  differences in their gradient values.

35
Constant velocity

• We can add constant velocity term to Geodesic
  framework
• To increase the speed of convergence
• To help to avoid certain local minima

36
(Almost) Done

• means that the level-sets move according to
• This is the level-sets representation of the
  modified solution of the following geodesic
  problem

37
Geodesic active contour
This page is from sapiros presentation
38
Geodesic Model

• The solution of the geodesic active contours
  model satisfy the existence and uniqueness.
• Based on the theory of viscosity solutions

39
Experimental Results

• The characteristics of this image
• Separated by only a few pixels
• Shadows
• But, stronger attraction force provided by the
  term
• towards the real boundaries.
• Inward motion to detect two objects

40
Experimental Results

• The characteristics of this image
• Separated by only a few pixels
• Shadows
• Outward motion to detect two objects

41
Experimental Results

• Tumor detection
• Inward geodesic flow
• The difficulties are caused by the triangular
  shaped portion at the top left part.

42
Experimental Results

• The geometric model without new gradient term
• Very sensitive to variation of the gradient along
  the object boundaries and the noise in the image
• Curve did not stop at the correct position
• So, it needs more complicated stopping conditions
• Use A-priori knowledge
• In geodesic model on the other hand, the stopping
  is obtained automatically and new gradient term
  solves many hard problems.

43
Experimental Results

• Tracking in color movies

44
Experimental Results

• 3D extension of the geodesic flow
• The computation of minimal surfaces

45
Concluding Remarks

• A geodesic formulation for active contours
• The connections between classical energy based
  contours and geometric curve evolution
• Introduce a new term to the curve evolution
  models
• Improve the detection of boundaries with large
  differences in their gradient
• Free the model from the need to estimate crucial
  parameters
• Active contour approach with topology independent

46
(No Transcript)