Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
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Seminar report : Active Contour Models
(Snakes)
By : Li Huang
(ID : 01552457)
Erdong Cheng (ID : 00272027)
1
Introduction
Active contours or snakes are computer-generated curves that move within images to
find object boundaries. They are often used in computer vision and image analysis to
detect and locate objects, and to describe their shape. For example, a snake might be
used edge detection, corner detection, motion tracking, and stereo matching; one
might be used to find the outline of an organ in a medical image; or one might be used
to automatically identify characters on a postal letter.
1.1
What is a Snake?
The concept of snake was first introduced in1988 [9] and has later been developed by
different researchers. A snake is an energy-minimizing spline guided by external
constraint forces and influenced by image forces that pull it toward feature such as
lines and edges. Snake are active contour models: they lock onto nearby edges,
localizing them accurately. Scale-space continuation can be used to enlarge the
capture region surrounding a feature. Snakes provide a unified account of a number of
visual problems, including detection of edges, lines, and subjective contours; motion
tracking; and stereo matching.
1.2
Applications
 Actor driven facial animation.
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Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
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
Traffic monitoring – time to collision.

Medical Diagnosis.
1.3 Procedure
1.3.1. Initialization
Place the snake near the contour of interest
• Discrete initialisation => Control points (e.g. spline)
• Enough control points
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Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
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1.3.2 Evolution
Snake shape and location affected by forces iteratively
• Internal forces
• Image forces
• External constraint forces
Internal contour forces
Make the snake tend to be more continuous and smooth
Image forces
Draws the contour towards the closest image edges.
External forces
Force the snake in some a priori known direction or shape.
2
Basic Snake Behaviour
The basic snake model is a controlled continuity spline under the influence of image
forces and external constraint forces. The internal spline forces serve to impose a
piecewise smoothness constraint. The image forces push the snake toward salient
image feature like lines, edges, and subjective contours. The external constraint forces
are responsible for putting the snake near the desired local minimum. These forces
can, for example, come from a user interface, automatic attentional mechanisms, or
high-level interpretations.
Representing the position of a snake parametrically by v(s) = (x(s), y(s)), To obtain
the best fit between the snake and the object, we minimise the energy. Specifically, a
snake is defined as
E snake 

1

0
1

0
Esnake (v ( s )) ds
1
1
0
0
Eint (v ( s )) ds   Eimage (v ( s )) ds   E forces (v( s )) ds
Where Eint represent the internal energy of the spline due to bending, Eimage gives rise
to the image forces, and gives rise to the external constraint forces.
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Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
----------------------------------------------------------------------------------------------------------------------------
External and image forces which determine the snake
2.1
Internal Energy
The internal energy provides a smoothness constraint. This can be further defined as
dv 2
d 2v 2
Eint   ( s ) |
|   ( s) | 2 |
ds
d s
The spline energy is compose of a first-order term controlled by α(s) and a secondorder term controlled by β(s). The first-order term makes the snake act like a
membrane and the second-order term makes it act like a thin plate. Adjusting the
weights α(s) and β(s) controls the relative importance of the membrane and thin-plate
terms. If β(s) = 0 then the function is discontinuous in its tangent, i.e. it may develop a
corner at that point. If α(s) = β(s) = 0 then this also allows a break in the contour, a
positional discontinuity.
|
|
dv 2
| called Continuity Force (minimize the distance between points)
ds
d 2v 2
| called Curvature Force (enforce smoothness and avoid oscillation of the snake)
d 2s
2.2
Image Forces
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Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
----------------------------------------------------------------------------------------------------------------------------
In order to make snakes useful for early vision we need energy functionals that attract
them to salient features in images. In this section, we present three different energy
functionals which attract a snake to lines, edges, and terminations. The total image
energy can be expressed as a weighted combination of the three energy functionals.
Eimage   lineEline   edge Eedge   term Eterm
By adjusting the weights, a wide range of snake behavior can be created.
2.2.1 Line Functional
The simplest useful image functional is the image intensity itself. If we set
Eline  I ( x, y)
then depending on the sign of  line , the snake will be attracted either to light lines
or dark lines. Subject to its other constraints, the snake will try to align itself with the
lightest or darkest nearby contour.
2.2.2 Edge Functional
Finding edges in an image can also be done with a very simple energy functional. If
we set Eedge   | I ( x, y) |2 . then the snake is attracted to contour with large image
gradients. An example of the use of this functional is shown in figure. In the upper
left, a user has placed two snakes on the edges of the pear and potato. He has then
pulled part of the snake off the pear with a spring. The remaining pictures show what
happens when he lets go. The snake snaps back rapidly to the boundary of the pear.
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Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
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2.2.3 Termination Functional
The termination functional allows terminations (i.e. free ends of lines) or corners to
attract the snake. (Consider the perception of illusory contours.). In order to find
terminations of line segments and contours, we use the curvature of level lines in a
slightly smoothed image. Let C ( x, y )  G ( x, y ) * I ( x, y ) 2 be a slightly smoothed
version of the image. Let   tan 1 (C y / Cx ) be the gradient angle and let n  (cos  , sin  )
and n  ( sin  , cos ) be unit vectors along and perpendicular to the gradient
direction . Then the curvature of the level contour in C( x, y) can be written
2
2
  2C / n2 C yy C x  2C xy C x C y  C xx C y
Eterm 


3/ 2
n
C / n
(C x2  C y2)
By combining Eedge and Eterm we can create a snake that is attracted to edges or
terminations. Figure shows an example of such a snake exposed to a standard
subjective contour.
Right: Standard subjective contour illusion.
Left: Edge/termination snake in equilibrium on the subjective contour
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Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
----------------------------------------------------------------------------------------------------------------------------
Above left: Dynamic subjective contour illusion. (Sequence is left to right, top to
bottom. Above Right: Snake attracted to edges and terminations. As moving
horizontal line slides to the right, the snake bends until it falls off the lines. Bringing
the line close enough makes the snake reattach.
2.3 External Forces
The external energy function attracts the deformable contour to interesting features,
such as object boundaries, in an image. Any energy expression that accomplishes this
attraction can be considered for use.
Image gradient and intensity are obvious (and easy) characteristics to look at (another
could be object size or shape). Therefore, the following external energy function is
investigated:
where
regions and
the constants,
is an expression that attracts the contour to high or low intensity
is an energy term that moves the contour towards edges. Again,
and , are provided to adjust the relative weights of the terms.
Page 7 of 10
Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
----------------------------------------------------------------------------------------------------------------------------
Image Intensity Energy
Each element in the intensity energy matrix,
is assigned the intensity value
of the corresponding image point in the neighborhood of :
Then, if
versa.
is positive, the contour is attracted to regions of low intensity and vice-
Image Gradient Energy
The image gradient energy function attracts the deformable contour to edges in the
image. An energy expression proportional to the gradient magnitude will attract the
contour to any edge:
When active contours are used to find object boundaries, an energy expression that
discriminates between edges of adjacent objects is desirable. The key to such an
expression is that the gradients at the edges of the objects have different directions.
Further, the direction of the gradient at the object's edge should be similar to the
direction of the unit normal of the contour. This situation is illustrated in Figure .
Figure : An example of the movement of a deformable contour due to gradient
energy. Because the gradient direction at the edge of the object of interest is similar to
the outward unit normal direction of the contour, the active contour algorithm moves
the snake point from
to
even though the gradient magnitudes at
both points are similar.
Page 8 of 10
Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
----------------------------------------------------------------------------------------------------------------------------
The value for each element in the directional gradient energy matrix,
, can
therefore be defined by a dot product between the unit normal of the deformable
contour and the image gradient:
where
3
is the unit normal of the contour at point
.
advantages and disadvantages of snakes
Snakes have several advantages over classical feature extraction techniques:






They can be controlled interactively by using appropriately placed springs and
volcanoes.
They are easy to manipulate because the external image forces behave in an
intuitive manner.
They are autonomous and self-adapting in their search for a minimal energy
state.
They can be made sensitive to image scale by incorporating Gaussian
smoothing in the image energy function.
They are relatively insensitive to noise and other ambiguities in the images
because the integral operator is an inherent noise filter.
They can be used to track dynamic objects in temporal as well as the spatial
dimensions.
Snakes are not without their drawbacks, chief among which are the following:



3
They can often get stuck in local minima states; this may be overcome by
using simulated annealing techniques at the expense of longer computation
times.
They often overlook minute features in the process of minimizing the energy
over the entire path of their contours.
Their accuracy is governed by the convergence criteria used in the energy
minimization technique; higher accuracies require tighter convergence criteria
and hence, longer computation times.
Conclusion

Low-level” feature detection processes are effective up to a point but cannot
be expected to retrieve entire geometric structures.

Snakes are becoming increasingly effective.
– Edges, lines, and contours can all be found by essentially the same
mechanism.
– Motion tracking and stereo matching can be done in the same
framework.
Page 9 of 10
Seminar report: Active Contour Models
By : Li Huang (ID:01552457)&Erdong Cheng(ID:00272027)
159.731 Machine Vision
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References:
http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/
MARBLE/medium/snakes/snakes.htm
http://julien.jomier.com/courses/comp258/lecturesnakes/snakes-kass.pdf
http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/R
AMANI1/node31.html
http://www.cc.gatech.edu/classes/cs7322_97_spring/participants
/irfan/lectures/lecture.06/snakes.html
http://www.uib.no/med/avd/miapr/arvid/vedad_diploma.pdf
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