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, Greece2 Institute of Computer Science, FORTH, Crete, Greece

3 Computer Science Department, University of Crete, Greece

WebPage: https://sites.google.com/site/costaspanagiotakis/research/EFA

https://sites.google.com/site/costaspanagiotakis/research/EFA


• 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


• 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


• 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


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:


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.


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.


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.


• 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


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.


AEFA results (qualitative)

Representative success (top) and failure (bottom) examples of AEFA method. Captions show the estimated values of shape coverage.


DEFA results (qualitative)

Representative success (top) and failure (bottom) examples of DEFA method. Captions show the estimated values of shape coverage .


• 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


• 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

https://sites.google.com/site/costaspanagiotakis/research/EFA