Integral Image Integral Tree

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Integral Image / Integral Tree

• Werner Kloihofer

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Overview

• Integral Image
• Fast Face Detection Algorithm
• Integral Tree
• Finding a Spanning Tree
• Distance Transform on Graphs
• Integral Tree of Depths
• Integral Tree of Diameters
• Decomposition

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Face Detection Algorithm using the Integal Image

• Implemented in the OpenCV library
• Uses many simple features which can be computed
  in constant time using the pre-calculated
  integral image
• AdaBoost to form a classifier
• Paul Viola and Michael J. Jones. Robust Real-Time
  Face Detection. International Journal of Computer
  Vision 57(2), 137-154, 2004

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Features

• Haar-Like rectangular features
• 1 value per feature
• Thresholding
• Windows placed over the whole image
• Leads to 160000 features for a 24x24 resolution

Haar-Wavelet
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Integral Image

• Calculated once per image
• In linear time
• line-by-line

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Fast Calculation of the Haar-Features

• D 4 2 3 1

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Boosting

• KN weak classifiers
• K number of features
• N number of examples (and therefore thresholds)
• Perceptron-Like Calculate weights for the
  classifiers

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Attentional Cascade

• Increases Performance
• Negatives rejected early in detection process
• First 2 features reject 50 of non-faces

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Application of the Detector

• Different scales achieved by scaling the
  detector, not the image
• Same computation time for every scale
• Search-Window shifted over the image

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Normalization

• Per Search-Window
• Goal Minimize effect of lighting conditions
• Using the Integral Image and the Integral Image
  of the squared image

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Conclusion

• Invariants
• - 15 rotation-invariant in plane
• to some degree lighting-invariant
• Problems
• with occlusions, not perfect as shown on the
  image here
• Performance
• 384x288 image with 20 fps (P3 700)
• Training needs of course more time

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Overview

• Integral Image
• Fast Face Detection Algorithm
• Integral Tree
• Finding a Spanning Tree
• Distance Transform on Graphs
• Integral Tree of Depths
• Integral Tree of Diameters
• Decomposition

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Integral Tree

• Integral features on trees
• How do you get those trees?
• shape -gt graph -gt spanning tree
• Approaches
• Minimal Spanning Tree
• Dual Graph Contraction

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Distance Transform on Graphs

• First integral property

• Algorithm
• Initialize boundary vertexes with 1, others with
  8
• Repeat for all Vertices in parallel

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Minimum Spanning Tree

• Simple Algorithm
• compute distance transform
• use edge weights -dmin(u)dmin(v)
• find the MST using Kruskal
• why -dmin(u)dmin(v)?
• ab -gt max for ab
• highest when far away from theboundary

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Dual Graph Contraction

• thinning
• achieved by merging faces with low connectivity
  to the background with the background
• Iterative until no face remains
• The equivalent contraction kernel of the apex is
  then the spanning tree

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Integral Tree of Depths

• Subtree Depth
• Labels each vertex with the length of the longest
  branch from the center
• Algorithm
• Initialize leafs with 0, others with 8
• Repeat for all vertices

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Integral Tree of Depths and Diameters
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Integral Tree of Depths

• Center
• Out of this algorithm the center of the graph is
  determined
• It is the edge or vertex with highest subtree
  depth dmax
• Edge Even diameter
• Vertex Odd diameter

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Integral Tree of Diameters

• Diameter
• Longest path through the tree
• Labels each vertex with the diameter of the
  subtree
• Algorithm
• Initialize diameters d(v) dmax(v)
• repeat for all vertices

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Integral Tree of Depths and Diameters
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Decomposition

• Decomposition
• top-down approach

• Allows to decompose complex structure-gt 2
  balls
• Achieved by cutting the tree at specific
  positions
• For example Depth of outer tree smaller than a
  threshold

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Other Applications

• k-TSP
• TSP with multiple salespersons
• Each visits n/k cities
• Goal for a heuristic find a spanning forest with
  balanced diameters

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Integral Properties

• Pre-Processing
• Global properties locally stored
• Allows decisions based on global information in
  constant time

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The End

• Presentation mainly based upon
• Paul Viola and Michael J. Jones. Robust Real-Time
  Face Detection. International Journal of Computer
  Vision 57(2), 137-154, 2004
• Walter G. Kropatsch, Yll Haxhimusa and Zygmunt
  Pizlo. Integral Trees Subtree Depth and
  Diameter. Technical Report PRIP-TR-92, 2004