IEEE 2015 Conference on Computer Vision and Pattern Recognition C ORRELATION F ILTERS WITH L IMITED B OUNDARIES H AMED K IANI ,T ERENCE S IM AND S IMON L UCEY C ORRELATION F ILTERS Boundary Effects: Synthetic training patches ∆ ℎ * E (h)= 1 2 N X i=1 ||y i - x i ? h|| 2 2 + λ 2 ||h|| 2 2 1- Frequency domain: O (ND log D ) E ( ˆ h)= 1 2 N X i=1 || ˆ y i - diag(ˆ x i ) > ˆ h|| 2 2 + λ 2 || ˆ h|| 2 2 ( ˆ y denotes the DFT of vector y ) 2- Spatial domain: O (D 3 + ND 2 ) E (h)= 1 2 N X i=1 D X j =1 ||y i (j ) - h > x i [Δτ j ] | {z } circular shift || 2 2 + λ 2 ||h|| 2 2 C ONTRIBUTIONS 1. A new correlation filter objective to drastically reduce the number of synthetic patches. 2. Optimizing the new objective using ADMM with very effi- cient complexity and memory usage. K EY I DEA * ℎ T≫D E (h)= 1 2 N X i=1 T X j =1 ||y i (j ) - h > Px i [Δτ j ]|| 2 2 + λ 2 ||h|| 2 2 ∆ 1. # of training patches: T vs. D (T D ) 2. # of patches affected by circular shift (synthetic): D -1 T vs. D -1 D 3. Complexity: O (D 3 + NDT ) E (h, ˆ g ) = 1 2 N X i=1 || ˆ y i - diag(ˆ x i ) > ˆ g || 2 2 + λ 2 ||h|| 2 2 s.t. ˆ g = √ D FP > h (F: D × D discrete Fourier transform matrix) Augmented Lagrangian L(ˆ g , h, ˆ ζ ) = 1 2 N X i=1 || ˆ y i - diag(ˆ x i ) > ˆ g || 2 2 + λ 2 ||h|| 2 2 + ˆ ζ > (ˆ g - √ D FP > h)+ μ 2 || ˆ g - √ D FP > h|| 2 2 Augmented Lagrangian is solved using Alternating Direction Method of Multipliers (ADMM) with a time complexity of O ([N + K ]T log T ) and memory usage of O (T ) . RESULTS (2) MOSSE KMOSSE MILTrack STRUCK OAB(1) SemiBoost FragTrack Our method mean {0.80, 11} {0.84, 12} {0.72, 16} {0.91, 12} {0.53, 31} {0.62, 29} {0.51, 37} {0.97, 8} fps 600 100 25 11 25 25 2 100 R ESULTS (1) Runtime and Convergence Performance 1 100 200 300 400 500 600 0 1 2 3 4 5 6 7 8 x 10 4 Number of training images Time to converge (s) Spatial steepest descent (32x32) Our method (32x32) Spatial steepest descent (64x64) Our method (64x64) 0 500 1000 1500 10 0 10 2 10 4 10 6 10 8 10 10 10 12 # of iteration Objective value Spatial steepest descent Our method Localization Performance MOSSE ASEF UMACE MACE OTF Our method 1 50 100 150 200 250 300 350 400 0 0.5 1 Number of training images Localization rate at d < 0.10 0.05 0.10 0.15 0.20 0 0.5 1 Threshold (fraction of interocular distance) Localization rate Tracking Performance: 100 fps girl clifbar 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 Threshold Precision MOSSE K_MOSSE MILTrack STRUCK FragTrack IVTs Our method 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 Threshold Precision MOSSE K_MOSSE MILTrack STRUCK FragTrack IVTs Our method 100 200 300 400 0 50 100 150 200 Frame # Position Error (pixel) MOSSE K_MOSSE STRUCK FragTrack Our method 50 100 150 200 250 300 0 20 40 60 80 100 120 140 Frame # Position Error (pixel) MOSSE K_MOSSE STRUCK FragTrack Our method REFERENCES [1] D. Bolme, J. Beveridge, B. Draper, and Y. Lui. Visual Object Tracking using Adaptive Correlation Filters CVPR’10 [2] Code and Demo: http://www.hamedkiani.com/cfwlb.html