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Barcelona June 2013 1
II- Ground truth energies and Saliency Transform
B. Ravi Kiran Jean Serra
LIMG ESIEE Université Paris-Est
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UPC / Pompeu Fabra Barcelona, June 13 2013
Problem context
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A Problem: Inputs
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Ground truth: Evaluation of Hierarchies
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Problems
1. Given a hierarchy H and ground truth partition G find the partition in H closest to G.
1. Closest from H -> G
2. Closest from G -> H
2. Compare any hierarchy H with multiple ground truth partitions of the same image
3. Compare any two hierarchies H1, H2, with respect to a common ground truth partition G
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Hausdorff distance and associated problems
Y
X
i.e. smallest disc dilation of X that contains Y and of X to contain Y
- Global Measure - Large variations when object are asymmetric w.r.t each other
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Local Hausdorff distances
• Local measures: Each class S in H is assigned 2 radii: ,
• Both are h-increasing energies
• Local optimization to obtain a globally optimal solution
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minimum radius of dilation of ground truth contour that covers the contour of S.
minimum radius of dilation of the contour of S to cover GT within S. Barcelona June 2013
Example
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Energy at various levels of H
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Energy at various levels of H
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Problems
1. Given a hierarchy H and ground truth partition G find the partition in H closest to G.
1. Closest from H -> G
2. Closest from G -> H
2. Compare any hierarchy H with multiple ground truth partitions of the same image
3. Compare any two hierarchies H1, H2, with respect to a common ground truth partition G
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Optimal Cuts
Initial Image
GT2 13 Barcelona June 2013
Optimal Cuts
Initial Image
GT7 14 Barcelona June 2013
Problems
1. Given a hierarchy H and ground truth partition G find the partition in H closest to G.
1. Closest from H -> G
2. Closest from G -> H
2. Compare any hierarchy H with multiple ground truth partitions of the same image
3. Compare any two hierarchies H1, H2, with respect to a common ground truth partition G
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Composition of ground truths:
GT5_1 GT5_2 GT5 = GT5_1 + GT5_2
The distance function of the union(sum) is the inf of the distance functions 16 Barcelona June 2013
Composition of ground truths:
GT5_1 GT5_2 GT5 = GT5_1 + GT5_2
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Problems
1. Given a hierarchy H and ground truth partition G find the partition in H closest to G.
1. Closest from H -> G
2. Closest from G -> H
2. Compare any hierarchy H with multiple ground truth partitions of the same image
3. Compare any two hierarchies H1, H2, with respect to a common ground truth partition G
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Global Precision-Recall Energies
Local dissimilarity measure Counterpart Global similarity measures
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The two half distances yield two local and then two global energies: • Precision (P) : How close is on average the ground truth to the class (G->S) • Recall (R) : How close is on average the Class contour to the Ground truth (S->G)
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Comparing Hierarchies (saliencies) with Precision-recall similarity measures
UCM Cousty (floodings of watershed)
UCM random hierarchy Cousty random hierarchy
GT6 GT5 GT4 GT3 GT2 GT7 GT1 20 Barcelona June 2013
Comparing Hierarchies (saliencies) with Precision-recall similarity measures
Image 25098
UCM UCM random
Cousty Cousty random
Precision energy
4.4 0.27 0.13 0.09
Recall energy
3.9 0.28 0.16 0.10
Integrals from PR equations expressed per 1000 pixels in the image
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Problem context
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Ground truth: Evaluation of Hierarchies
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A problem: Transforming hierarchies
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Representations of Hierarchies: Saliency function
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Saliency Ultrametric contour Map (UCM)
Introducing an external function
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Introducing an external function
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Introducing an external function
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Binary Class Opening
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Numerical (grayscale) class opening
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Example: Class Opening
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Properties of class opening I
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Properties of class opening II
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Saliency degeneracy
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1 2 3 4 5 6 7 8 9 100
2
4
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linear
power law x > 1 (compression)
power law x 0 < < 1(expansion)
Logarithmic
Evaluating Hierarchies w.r.t Ground Truth
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Evaluating Hierarchies w.r.t Ground Truth
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Composing two external functions
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Composing two external functions
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Generating Random Hierarchies
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Hierarchy Fusion: Matching hierarchies
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Hierarchy Fusion: Matching hierarchies
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Hierarchy Fusion: Matching hierarchies
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Hierarchy Fusion: Matching hierarchies
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Conclusion
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http://www.esiee.fr/~kiranr/HierarchEvalGT.html
Future work
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