Short Presentation of [1]. [1] C. Panagiotakis and A. Argyros, Parameter-free Modelling of 2D Shapes with Ellipses, Pattern Recognition, 2015. For more details, please visit https://sites.google.com/site/costaspanagiotakis/research/EFA

1. Parameter-free Modelling of 2D Shapes with Ellipses
Parameter-free Modelling of 2D
Shapes with Ellipses
Costas Panagiotakis1,2
and Antonis Argyros2,3
1
Dept. of Business Administration, TEI of Crete, Greece
2
Institute of Computer Science, FORTH, Crete, Greece
3
Computer Science Department, University of Crete, Greece
WebPage: https://sites.google.com/site/costaspanagiotakis/research/EFA

2. Parameter-free Modelling of 2D Shapes with Ellipses
• Develop a method that approximates a
given 2D shape with an automatically
determined number of ellipses under the
Equal Area constraint.
• Equal Area constraint: The total area
covered by the ellipses has to be equal to
the area of the original shape.
• We want to achieve:
• Automatic selection of the number of
ellipses
• Automatic estimation of the parameters
of the ellipses
• Good balance between model
complexity and shape coverage under
the Equal Area constraint.
Goal

3. Parameter-free Modelling of 2D Shapes with Ellipses
• The first parameter-free method that automatically estimates an
unknown number of ellipses that best fit a given 2D shape under the
Equal Area constraint.
• Novel definition of shape complexity that exploits the shape skeleton.
• Good balance between model complexity and shape coverage.
• Experiments on more than 4,000 2D shapes show the effectiveness of
the proposed methods.
• The proposed solutions agree with human intuition.
Research highlights

4. Parameter-free Modelling of 2D Shapes with Ellipses
• Input: A binary image I that represents a
2D shape of area A.
• Output: A set E of k ellipses Ei so that
the sum of the areas of all ellipses is A.
• Goal: Compute the number k and the
parameters of ellipses Ei, so that the
trade-off between shape coverage and
model complexity is optimised.
• Shape coverage α(E) is the percentage of
the 2D shape points that are under some
of the ellipses in E.
Background
Fig 2. The proposed solution k = 6
with 96.6% shape coverage.
Fig 1. The given binary image

5. Parameter-free Modelling of 2D Shapes with Ellipses
Shape complexity and model selection
• A new shape complexity measure C based on the Medial Axis Transform
(MAT) of the shape.
• Model Selection: The Akaike Information Criterion (AIC) is used to define
a novel, entropy-based shape complexity measure that balances the
model complexity and the model approximation error:

6. Parameter-free Modelling of 2D Shapes with Ellipses
Methods
• In order to minimise the AIC, two variants are proposed and
evaluated:
• (a) AEFA (Augmentative Ellipse Fitting Algorithm): Gradually
increases the number of considered ellipses starting from a single
one.
• (b) DEFA (Decremental Ellipse Fitting Algorithm): decreases the
number of ellipses starting from a large, automatically defined set.

7. Parameter-free Modelling of 2D Shapes with Ellipses
AEFA
(a)-(e): The solutions proposed by AEFA using one to five ellipses. (f) the six circles in SCC
that initialise GMM-EM for k = 6. (g) The solution of AEFA for k = 6. (h) the solution in case
that circles were selected only based on their size, only. (i) the association of pixels to the
final solution of AEFA for k = 6 ellipses. (j) the AIC and BIC criteria for different values of k.
Captions show the estimated values of shape coverage.

8. Parameter-free Modelling of 2D Shapes with Ellipses
DEFA
(a)-(f): The intermediate solutions proposed by DEFA using 11, 8, 7, 6, 5 and 4 ellipses.
Captions show the estimated values of shape coverage . (g) the skeleton of the 2D shape. (h)
the association of pixels to k = 8 ellipses which is the final solution estimated by DEFA. (i) the
AIC and BIC criteria for different values of k.

9. Parameter-free Modelling of 2D Shapes with Ellipses
• MPEG7 (standard): 1400 images
• LEMS (standard): 1462 images
• SISHA (SImple SHApe): 32
images to evaluate scale, shear
and noise effects.
• SISHA-SCALE
• SISHA-SHEAR
• SISHA-NOISE
Datasets
•1st
shape of SISHA-SCALE
•1st
shape of SISHA-SHEAR
Shapes of the SISHA dataset

10. Parameter-free Modelling of 2D Shapes with Ellipses
Quantitative results
• Pr(m/AIC): the percentage of images of the datasets where the method m clearly outperforms the two others
under the AIC.
• Pr(m/α): the percentage of images of the datasets where the method m clearly outperforms the two others
under the coverage α.

11. Parameter-free Modelling of 2D Shapes with Ellipses
AEFA results (qualitative)
Representative success (top) and failure (bottom) examples of AEFA
method. Captions show the estimated values of shape coverage.

12. Parameter-free Modelling of 2D Shapes with Ellipses
DEFA results (qualitative)
Representative success (top) and failure (bottom) examples of DEFA
method. Captions show the estimated values of shape coverage .

13. Parameter-free Modelling of 2D Shapes with Ellipses
• A parameter-free methodology for estimating automatically the
number and the parameters of ellipses under the Equal Area
constraint.
• Experiments on more than 4,000 2D shapes assess the
effectiveness of AEFA and DEFA on a variety of shapes, shape
transformations, noise models and noise contamination levels.
• DEFA slightly outperforms AEFA especially for shapes of middle
and high complexity.
• The solutions proposed by AEFA and DEFA seem to agree with
human intuition.
Summary

14. Parameter-free Modelling of 2D Shapes with Ellipses
• Application of the proposed approach on the problem of
recovering automatically the unknown kinematic structure of an
unmodelled articulated object based on several, temporally
ordered views of it.
• Extensions of DEFA/AEFA towards handling shape primitives
other than ellipses.
Next steps/future work
Acknowledgments:
This work was partially supported by the EU FP7-ICT-2011-9-601165 project WEARHAP.
WebPage: https://sites.google.com/site/costaspanagiotakis/research/EFA