A split-and-merge framework for
2D shape summarization
D. Gerogiannis, C. Nikou and A. Likas
Department of Computer Science,
University of Ioannina,
Greece
Presentation Outline
• Problem definition
▫ Multiple line fitting on 2D unordered set of points
• Description of the proposed methodology
▫ Line segments are represented by major axis of
ellipses
▫ An iterative split and merge algorithm
• Experimental results
• Conclusion and Future Work
Problem Definition
• Input: 2D unordered set of points describing a contour
• Output: A set of lines that describe the contour
• Point set may contain joints and inner structures or
scattered data
Iterative split and merge algorithm (I)
• Let
be the points of the unordered set
• Let
be the set of line segments of the
model
• Then we can define the distortion Δ as:
is the distance of from
is 1 if corresponds to
Iterative split and merge algorithm (II)
Iterative split and merge algorithm (III)
• The problem is augmented to a multiple lines
fitting problem
▫ Minimize distortion Δ
• Each line segment is modeled with an ellipse
▫ Mean and covariance computed from the
corresponding points
• A two step optimization method
▫ Step1: Split (iterative)
▫ Step2: Merge (iterative)
Split process (I)
• Split Step (time t)
▫ The ellipse modeling the line
segment (blue line) in time t
▫ Split (split criterion)?
▫ If yes then compute the new
centers else keep the current
ellipse
• After splitting (time t+1)
▫ Update centers
▫ The resulting ellipse is more
eccentric (elongated)
Split process (II)
• Split Step (time t)
• Update centers
▫ Select a direction - eigenvectors
▫ Move by a step - eigenvalue
• After splitting (time t+1)
▫ Nearest neighbor classification
▫ Final centers and covariances
of new clusters
When to split: Linearity
• Linearity: estimated by the minimum eigenvalue
of the corresponding covariance matrix
(Threshold T1)
• Trivial calculation
When to split: Connectivity
• Connectivity: the maximum distance (gap)
between two successive points (Threshold T2)
• Demands ordering of points
Connectivity Computation
• Determines the split direction
▫ one of the axis of the ellipse
• Defined for each axis and keep the maximum
• Computation
▫ Project on the corresponding axis (scalar values)
▫ Sort the projections
▫ Find the maximum difference between two successive
projections
When to split: The split criterion
• A set of points correspond to a line segment:
▫ if the corresponding ellipse is eccentric (linearity)
▫ if it is tightly connected (connectivity)
▫ Thresholds T1 (for linearity), T2 (for connectivity)
Black lines: tightly connected points
Merge process
•
•
•
•
•
Iterative process
Merge collinear neighboring ellipses.
Aims to reduce the complexity of the model.
The split criterion is employeed.
Merge two ellipses if the resulting ellipse DOES
NOT satisfy the split criterion.
Demonstration of the method steps
The proposed algorithm
Code available at: www.cs.uoi.gr/~dgerogia
Thresholds computation
• Computed from the data
Experiments show that the final result is not
strongly dependant on the value of α.
Experimental Results(I)
• Comparison with Gaussian Mixture Models and
Hough Transform
• Silhouettes of objects (MPEG7) and fishes
(GatorBait 100)
• Edge points extracted by Canny Edge Detector
• α = 0.6
Experimental Results(II)
Our method
GMM
Experimental Results(III)
Hough Transform
Our method
Conclusions
• A low level methodology.
▫ Extracting shape features (oriented line
segments)
• Does not necessitate ordering of points.
• It handles joints and multiple structures.
Future Work
• Improve the split criterion
• Improve the definition of the thresholds
• Extend to 3D sets of points
A split-and-merge framework for
2D shape summarization
THANK YOU