Matthijs van EedeUniversity of Toronto

August 22nd, 2006

Joint work with Diego Macrini, Alex Telea, Cristian Sminchisescu, and Sven Dickinson

The skeleton of a shape yields a symmetry-based parts decomposition (e.g., a shock graph) which can support effective object indexing and recognition, e.g., Siddiqi et al. (1999), Sebastian et al. (2004).

But, they suffer from two forms of instability…

Ligature branch

Ligature segment

Blum (1973)

• “Smooth” these structural instabilities while retaining the object’s salient shape structure.

• Two exemplar shapes drawn from the same category should therefore yield two graphs with the same structure.

• Prune skeletal branches that don’t contribute to the salient shape structure of the object.

• Simpler graphs with fewer unstable nodes lead to more efficient and more effective indexing and matching.

• But how do we measure branch saliency and when do we stop pruning?

Reconstructionerror

Saliency favors elongated and thick parts

External branches rank-ordered by saliency:

1

2

3

4

5

6

7

8

910

11

12

Strong SmoothingMild SmoothingNo Smoothing

The cost of external branch smoothing: increased reconstruction error

Intuitively: create similar topologies in the skeletons by pruning short (low saliency) ligature

segments and branches

Ligature branch

Ligature segment

Fit piecewise linear skeleton fragments subject to endpoint and tangent constraints

No Smoothing

Strong SmoothingMild Smoothing

The cost of internal branch smoothing: altering the shape’s appearance

• Fact the medial axis transform of a shape is unique; skeleton changes introduce reconstruction error

• Goal minimize a cost function that promotes simpler skeletons with low reconstruction error

# branches Reconstruction error

p

spR(sp)

1. Rank-order external branches by saliency2. Iteratively prune low-saliency external

branches until cost function is minimized3. For internal branches, identify the

ligature branches as candidates for pruning, and rank-order them by saliency

4. Iteratively prune low-saliency candidate internal branches until cost function is minimized

Three hand shapes and their skeletons using no simplifications

Three hand shapes and their skeletons using both externalas well as internal simplifications

Notice the isomorphic graph structure

• Shock graphs are computed for 15 views of 8 three dimensional CAD models. A total of 120 shapes in the database.

• Each object view is removed from the database and used as a query

• Successful object recognition best ranked view belongs to the same object as query view

• Successful pose estimation neighbouring view of query is among top ranked views

• Noise is simulated by adding random “bumps” and “notches” to the query.

Results when using simplifications

- Object recognition performance increased up to 16%- Pose estimation performance increased up to 20%- (r5) = having a radius of 5 pixels

Results without usingsimplifications

• Skeletal descriptions of a shape offer a powerful shape representation for object recognition, yet their structural instability has long been an obstacle to their widespread use.

• Our structural simplification framework isolates this instability at both external and internal branches, and removes non-salient branches.

• The removal of internal branches requires a proper smoothing of neighboring branches so that the resulting skeleton is a MAT and reconstruction error is minimized.

• Results on a shock graph recognition experiment indicate a significant improvement in recognition and pose estimation performance when both query and database are structurally simplified prior to recognition.