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Ignas Budvytis*, Tae-Kyun Kim*, Roberto Cipolla * - indicates equal contribution Making a Shallow Network Deep: Growing a Tree from Decision Regions of a Boosting Classifier

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Introduction Aim – improved classification time of a learnt boosting classifier Shallow network of boosting classifier converted into a “deep” decision tree based structure Applications Real time detection and tracking Object segmentation Design goals Significant speed up Similar accuracy BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge2/22

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Speeding up a boosting classifier Creating a cascade of boosting classifiers Robust Real-time Object Detection [Viola & Jones 02] Single path of varying length “Fast exit” [Zhou 05] Sequential probability ratio test [Sochman et. al. 05] Multiple paths of different lengths A binary decision tree implementation of a boosted strong classifier [Zhou 05] Feature sharing between multiple classifiers Sharing visual features [Torralba et. al 07] VectorBoost [Huang et. al 05] Boosted trees AdaTree [Grossmann 05] BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge3/22 Weak classifier Strong classifier

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Brief review of boosting classifier Aggregation of weak learners yields a strong classifier Many variations of learning method and weak classifier functions. Anyboost [Mason et al 00] implementation with discrete decision stumps Weak classifiers: Haar-basis like functions (45,396 in total) BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge4/22 Weak classifier Strong classifier

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Brief review of boosting classifier Smooth decision regions BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge5/22

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Brief review of decision tree classifier BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge6/22 1 2 3 674 9 5 8 category c split nodes leaf nodes v 10111213 14151617 feature vector v split functions f n ( v ) thresholds t n Classifications P n ( c ) feature vector v split functions f n ( v ) thresholds t n Classifications P n ( c ) ≥ < < ≥ Slide taken and modified from Shotton et. al (2008)

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Brief review of decision tree classifier Short classification time BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge7/22 1 2 3 674 9 5 8 category c v 10111213 14151617 ≥ < < ≥

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Boosting Classifier vs Decision Tree Preserving (smooth) decision regions for good generalisation Short classification time BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge8/22 Decision treeBoosting

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Converting boosting classifier to a decision tree – Super Tree BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge9/22 Boosting 6 8 11 16 13 2 2 3 2 1 4 7 Preserving (smooth) decision regions for good generalisation Short classification time Super tree

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Boolean optimisation formulation For a learnt boosting classifier split a data space into 2 m primitive regions by m binary weak-learners. Code regions R i i=1,..., 2 m by boolean expressions. BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge10/22BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge W2W2 R3 R5 R6 R1 R2 R4 R7 W1W1 0 0 0 1 1 1 W3W3 W1W1 W2W2 W3W3 C R1000F R2001F R3010F R4011T R5100T R6101T R7110T R8111X Data space Data space as a boolean table

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Boolean optimisation formulation Boolean expression minimisation by optimally joining the regions of the same class label or don’t care label. A short tree built from the minimised boolean expression by placing more frequent variables at the top. BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge11/22 W2W2 R3 R5 R6 R1 R2 R4 R7 W1W1 0 0 0 1 1 1 W3W3 W1W1 W2W2 W3W3 C R1000F R2001F R3010F R4011T R5100T R6101T R7110T R8111X R1,R2 01 0 0 1 1 W1W1 W2W2 W3W3 TF F T R4 R5,R6,R7,R8 R3 Data space Data space as a boolean table Data space as a tree don’t care

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Boolean optimisation formulation Optimally short tree is defined in terms of average expected path length of data points as where region prior p(R i )=M i /M. Constraint: tree must duplicate the decision regions of the boosting classifier BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge12/22

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Growing a Super Tree Regions of data points R i taken as input s.t. p(R i )>0 A tree grown by maximising the region information gain Where Key ideas – Growing a tree from the decision regions – Using the region prior (data distribution). BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge13/22 Region prior p Entropy H Weak learner w j Region set R n at node n Region prior p Entropy H Weak learner w j Region set R n at node n

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Synthetic data exp1 BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge14/22 Examples generated from GMMs

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Synthetic data exp2 BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge15/22 Imbalanced cases

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Growing a Super Tree BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge16/22 W1W2W3W4W5SumC Weight1.00.80.70.50.23.2 Region101101.21 Boundary region101000.21 Extended region1x1xx0.2-3.21 When number of weak learners is relatively large, too many regions of no data points maybe assigned to different class labels from the original ones Solution: Extending regions Modifying information gain: “dont’ care” variable

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Face detection experiment Training set: MPEG-7 face data set (11,845 faces) Validation set (for boostrapping): BANCA face set (520 faces) + Caltech background dataset (900 images) Total number: 50128 Testing set: MIT+CMU face test set (130 images of 507 faces) 21,780 Harr-like features BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge17/22

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Face detection experiment The proposed solution is about 3 to 5 times faster than boosting and 1.5 to 2.8 times faster than [Zhou 05], at the similar accuracy. BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge18/22 BoostingFast Exit [Zhou 05]Super Tree No. of weak learners False positive False negative Average path length False positive False negative Average path length False positive False negative Average path length 205011202050112011.704761227.51 402641264026412623.2623112712.23 602221436022214337.2421214214.38 Total test data points = 57507

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Face detection experiment For more than 60 weak-learners a boosting cascade is considered. BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge19/22 Total test data points = 57507 BoostingFast Exit [Zhou 05]Super Tree No. of weak learners False positive False negative Average path length False positive False negative Average path length False positive False negative Average path length 10014814610014814669.2814515215.1 200120143200120143146.1912814615.8 Fast Exit Cascade No. of weak learners False positive rate False negative rate Average path length 10014414937.4 20014614838.1 Class A Super Tree “Fast Exit” Class A Class B

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Experiments with tracking and segmentation by ST BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge20/22

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Summary Speeded up boosting classifier without sacrificing accuracy Formalized the problem as a boolean optimization task Proposed a boolean optimisation method for a large number of binary variables (~60) Proposed a 2 stage cascade to handle almost any number of weak learners (binary variables) BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge21/22

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Questions? BMVC 2010, Budvytis, Kim, Cipolla, University of Cambridge22/22

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