R-snakesLyubomir Zagorchev, Ardeshir Goshtasby,

Martin Satter

Speaker: HongxingShi

2007.11.01

Image and Vision Computing 25 (2007) 945–959

Author

• Ardeshir Goshtasby· Postion

Professor of Department of Computer Science and

Engineering, Wright State University

· Education

B.E. Electronics Engineering, University of Tokyo

M.S. Computer Science, University of Kentucky

Ph.D. Computer Science, Michigan State University

· Journal Special Issues Edited

Pattern Recognition on Image Registration

Computer Vision and Image Understanding

Information Fusion on Image Fusion

Introduction and background

• What is snake?· An energy minimizing countours

· Continuously deform to minimize its energy

· Slither while deforming, like snake

Energy

• Contours

• Internal force · Constrain the smoothness of contours

• External force · Push the contours toward image features

( ) ( ( ), ( )), here [0,1]Q u x u y u u

int ( ( ))E Q u

ext ( ( ))E Q u

Situations of curves’ definition

• DiscreteDefined by a sequence of points

• ContinuousDefined by a parametric curve, such as B-Splines ,

NURBS,RaG curves, and so on…

Energy definition

• Total energy

• Internal energy

• External energy

Compute

• From the calculas of variations, obtain:

• After discrezing….

Result

• The equations can be written

where A is a pentadiagonal matrix.

B-snakes

• Definition

• Energy function

B-snakes

• Minimize the total energy

• Obtain

Rational Gaussian curves

• Definition

• Control points

• Blending functions

Rational Gaussian curves

• Where is the weight, and

• are standard deviation

• are nodes

• The blending function can be varied.

RaG curves

• are (a)0.05 (b)0.1 (c)0.15

Closed RaG curves

• Replaced the Gaussian function with

• are (a) 0.045 (b) 0.06 (c) 0.085

RaG snakes

• As the situation of discrete snakes:

• Let

RaG snakes

• Discrete, and obtain

• The extern energy( ( )) ( ( ), ( ))ext j x j y jE Q u f u f u

Synthetical images

Parameters in synthetic image

CT images

Parameters in CT images

Parameters analyse

• Standard deviation and the number of nodes n – bigger , smoother shapes– larger n ,smoother shapes

• Experientially, we let

1/ 2n

Standard deviations: (c)-(h)0.015, 0.025, 0.035, 0.045, 0.055, 0.065

Standard deviations: (c)-(h)0.015, 0.025, 0.035, 0.045, 0.055, 0.065

Standard deviations: (c)-(h)0.015, 0.025, 0.035, 0.045, 0.055, 0.065

Standard deviations: (c)-(h)0.015, 0.025, 0.035, 0.045, 0.055, 0.065

Coarse-to-fine segmention

• How to select the n, number of nodes?– First, with a small n and a large , find a co

arse boundary

– Then, increase n until the finest resolution image is segmented

Conclusions

• Advantages over B-snakes– The stiffness can be varied to recover shapes co

ntaining smooth as well as detailed parts– The stiffness can be continuously varied to track a

boundary from coarse to fine

• Disadvantage– Complexity

References

• [1] L Zagorchev, A Goshtasby, M Satter, R-snakes, Image and Vision Computing, 2007

• [2] M. Kass, A. Witkin and D. Terzopoulos, Snakes: Active contour models, International journal of computer vision. 321-331, 1988

• [3] A. Goshtasby, Geometric modelling using rational Gaussian curves and surfaces, Computer Aided Design 27 (5) (1995) 363–375

Q&A