1. Kiera Henning, Jiwon Kim,
Lost and Found
The Distance Transform
Kiera Henning, Jiwon Kim,

2. Pictorial Structures Represent the ‘lost’ item with a tree G = (V, E).
V = {parts of the object}
E = {deformations between parts}
[Felzenszwalb, Huttenlocher 00]

3. An example template tree

4. The Problem
Given an object, find the best location of the object in the image while respecting the structure of the object defined by the tree.

5. The Problem (cont) Define
For a given part, j, if we know the location of our parent, i, then the best location for j is that which minimizes

6. The Solution
Beginning with the leaves and continuing toward the root one depth at a time, determine B. If all nodes at depth at least d have already determined B, then all nodes at depth d-1 can determine B.

7. Deformation cost
Suppose

8. Transformed Grid
Let location , and define
, where

9. Transformed Grid (cont)
Then
Run DT on grid defined by T.

10. Match cost
Using PCA, can initialize m to measure the distance from space spanned by eigenvectors.
Using simple color scheme, m measures how well the area matches the average color and variance of the part.
Using sum of squared differences to match a part expected to change very little.

11. Examples Using PCA
Success!
Failed…

12. Examples Using PCA

13. Example Using Sum Sq Diff

14. Document Recognition
Documents on Desk
Document Image

15. Document Template
SIFT features [Lowe 04]

16. Example

17. Feature Points

18. Distance Transform

19. chamf(A,B) = ∑a∈A minb∈B ||a-b||
Approach 1
L22 distance defined in 2D feature space [px,py] (2D location x, y)
Assymetric Chamfer distance
chamf(A,B) = ∑a∈A minb∈B ||a-b||
Brute-force search for best matching translation and rotation (x,y,θ) between A and B

20. Result
Sparse document incorrectly matched to more dense document

21. Approach 2
L22 distance defined in 2D feature space [px,py] (2D location x, y)
Symmetric Chamfer distance
chamf(A,B) + chamf(B,A)
Brute-force search for best matching translation and rotation (x,y,θ) between A and B

22. Result
H = 268.7
H = 239.3
Non-existing document scores only slightly lower than existing document

23. Approach 3
L22 distance defined in 3D feature space [px,py,αpo] (x, y + edge orientation)
Symmetric Chamfer distance
chamf(A,B) + chamf(B,A)
Brute-force search for best matching translation and rotation (x,y,θ) between A and B
 Better at answering existence query

24. Result
H = 673.1
H = 267.0
Non-existing document scores clearly lower than existing document

25. Approach 4
L2 distance defined in 3D feature space [px,py,αpo] (x, y + edge orientation)
Symmetric, partial Hausdorff distance
h(A,B) = K-tha∈A minb∈B ||a-b||
H(A,B) = max(h(A,B), h(B,A))
Efficient search for best matching translation, rotation and scale (x,y,θ) between A and B under partial occlusion

26. Partial Hausdorff Distance
h(A,B) = K-tha∈A minb∈B ||a-b||
Rank a’s according to distance to closest b
K=|A| equivalent to h(A,B) = max…
1-K/|A| dictates amount of partial occlusion allowed

27. Fast Hausdorff Search (1)
H(A,B) decreases linearly
 can compute lower bound for a cell
Subdivide and prune in transformation space
[tx,ty,sx,sy]
[Huttenlocher et al. 92]

28. Fast Hausdorff Search (2)
Additional acceleration tricks
Early rejection
N = # of points in A whose dtrans value > thresh
If |A| > N > K, skip rest of points for the given cell
Early acceptance
Probe a randomly chosen fraction of A (e.g., 20%)
If fraction of points < thresh larger than K/|A|, accept cell
Skipping forward
[Huttenlocher et al. 92]

29. Result Speedup Test image: Template: Xform space: 1600x1200
~10,000 features
Template:
~600x500
~1,000 features
Xform space:
36 rotation bins
100 scales between 0.5 and 1.0
(tx,ty,θ)
(tx,ty,θ,sx,sy)
Exhaustive search
> 10 minutes ?
Fast Hausdorff search
< 1 minute
4-5 minutes

30. Result
Partial occlusion (success, failure)
works up to ~20% occlusion

31. Discussion Comparison w/ SIFT-based recognition
Accuracy (error rate, occlusion handling)
Inherently lower, because full feature descriptor information not used
Efficiency (speed)
Fast multi-resolution search performs quite well
Potentially could be made faster by using fewer features
However, SIFT-based method also has room for speedup

32. References
Daniel Huttenlocher, William Rucklidge. A Multi-Resolution Technique for Comparing Images Using the Hausdorff Distance. Technical Report TR , 1992.
Pedro Felzenszwalb, Daniel Huttenlocher. Efficient Matching of Pictorial Structures. In CVPR, 2000.
Martin Handford. Where’s Waldo Now?