Active Contour Models: Snakes

Discussion by Bob Sumner

Why snakes?

The shape of many objects is not easly represented by rigid primitives. For example:

Natural objects, such as bananas, have similar recognizable shapes. But no two bananas are exactly the same.
In medical imaging, objects are similar but not exact. An exact representation of a vein's shape, for example, cannot be given.
Some objects, such as lips, change over time.

What is a snake?

Energy minimising spline
Deforms to fit local minima
Local, not global, so initial location must be provided

Online examples

Other online information

Irfan's lecture notes

A snake is defined as an energy function. To find the best fit between a snake and an object's shape, we minimize the energy.

Where the snake is parametrically defined as

: Internal spline energy caused by stretching and bending.
: Measure of the attraction of image features such as contours.
: Measure of external constraings either from higher level shape information or user applied energy.

The internal energy provides a smoothness constraint. This can be further defined as:

is a measure of the elasticity of the snake.

is a measure of the stiffness of the snake.

The first order term makes the snake act like a membrane; the constant controls the tension along the spine (stretching a balloon or elastic band). The second order term makes the snake act like a thin plate; the constant controls the rigidity of the spine (bending a thin plate or wire). If then the function is discontinuous in its tangent, i.e. it may develop a corner at that point. If then this also allows a break in the contour, a positional discontinuity.

The image energy is derived from the image data. Considering a two dimensional image, the snake may be attracted to lines, edges or terminations.

where is an appropriate weighting function.

Commonly, the line functional is defined simply by the image function,

so that if is large positive the spline is attracted to light lines (or areas) and if large negative then it is attracted to dark lines (or areas). The use of the terminology ``line'' is probably misleading.

The edge functional is defined by

Hence, the spline is attracted to large image gradients. i.e. parts of the image with strong edges.

Finally, the termination functional allows terminations (i.e. free ends of lines) or corners to attract the snake.

The constraint energy is determined by external constraints. This energy my come in the form of a spring attached by the user. Or, the constraint energy may come from higher knowledge about the images in question.

[Note that much of the content of this page was borrowed from http://www.icbl.hw.ac.uk/marble/vision/medium/snakes/snakes.htm]