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

• 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 (I)

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)

• After splitting (time t+1)

• Update centers▫ Select a direction -

eigenvectors▫ Move by a step -

eigenvalue

▫ 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

Code available at: www.cs.uoi.gr/~dgerogia

The proposed algorithm

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