1. 1 Manifold Clustering of Shapes Dragomir Yankov, Eamonn Keogh Dept. of Computer Science & Eng. University of California Riverside

2. 2 Outline Problem formulation Shape space representation. Similarity metric. Manifold clustering of shapes Handling noisy and bridged clusters Experimental evaluation

3. 3 Problem formulation Object recognition systems dependent heavily on the accurate identification of shapes Learning the shapes without supervision is essential when large image collections are available In this work we propose a robust approach for clustering of 2D shapes *The malaria images are part of the Hoslink medical databank, and the diatoms images are part of the collection used in the ADIAC project.

4. 4 Data representation Requirements –invariant to basic geometric transformations –handle limited rotations –low dimensionality for meaningful clustering Centroid-based “time series” representation All extracted time series are further standardised and resampled to the same length

5. 5 Measuring shape similarity The Euclidean distance does not capture the real similarities Rotationally invariant distance rd approximate rotations as: and define: Metric properties of rd

6. 6 Manifold clustering of shapes Vision data often reside on a nonlinear embedding that linear projections fail to reconstruct We apply Isomap to detect the intrinsic dimensionality of the shapes data. Isomap moves further apart different clusters, preserving their convexity

7. 7 Handling noisy and bridged clusters Instability of the Isomap projection The degree-k-bounded minimum spanning tree (k-MST) problem The b-Isomap algorithm short circuits disconnected components

8. 8 Experimental evaluation Diatom dataset –4classes –2 classes (Stauroneis and Flagilaria) DatasetMethodDimkAcc(%)Std(%) 4-class diatoms MDS3DN/A62.35.2 Isomap3D1686.23.0 b-Isomap3D483.03.6 2-class diatoms MDS3DN/A90.21.3 Isomap3D592.71.3 b-Isomap3D398.30.9

9. 9 Experimental evaluation Marine creatures DataMethodDimkAcc Marine creatures MDS2DN/A61.0 Isomap3D477.6 b-Isomap3D480.0 Arrowheads DataMethodDimkAcc Arrow heads MDS3DN/A75.6 Isomap3D1485.2 b-Isomap2D685.1

10. 10 Conclusions and future work We presented a method for clustering of shapes data invariantly to basic geometric transformations We demonstrated that the Isomap projection built on top of a rotationally invariant distance metric can reconstruct correctly the intrinsic nonlinear embedding in which the shape examples reside. The degree-bounded MST modification of the Isomap algorithm can decreases the effect of bridging elements and noise in the data. Our future efforts are targeted towards an automatic adaptive approach for combining the features of Isomap and b-Isomap Thank you!