Target Response Adaptaton for Correlaton Flter Tracng Adel Bb, Matthas Mueller, and Bernard Ghanem Image and Vdeo Understandng Laboratory IVUL, Kng Abdullah Unversty of Scence and Technology KAUST, Saud Araba {adel.bb,matthas.mueller.,bernard.ghanem}@aust.edu.sa In ths supplementary materal, we followup our dscusson on the soluton to the followng problem: mn w,y Aw y + λ w + λ y y 0 The problem s convex quadratc and a statonary pont s necessary and suffcent for global optmalty. The followng sectons wll dscuss how Problem s solved n the prmal doman, dual doman, and for both sngle and multple templates along wth the formula used to generate the response map, whose maxmum value determnes the current detecton. Lastly, we dscuss a one way of ncorporatng SRDCF wth our proposed target adaptve framewor. Soluton to Problem n the Prmal Doman. Usng a Sngle Template Problem can be rewrtten n terms of z wth z T w T y T : fz A I z + λ I 0 z + λ 0 I z y 0, where w R n, A R n n, y R n, y 0 R n, z R n, then: A z fz T A A T I 0 0 0 0 z + λ A I z + λ 0 0 z λ 0 I y I 0 0 A z fz T A + λ I A T 0 z λ A + λ I y I 0 F 0 dagâ â + λ dagâ F H 0 0 0 F dagâ dag + λ 0 F H z λ y I 0 dagâ â + λ dagâ ŵ 0 dagâ dag + λ ŷ λ F H y 0 ŵ dagâ â ŷ λ + λ dagâ 0 y dagâ dag + λ 0 F H
Adel Bb, Matthas Mueller, and Bernard Ghanem Note that the nverse lemma states: B N B NC V B NC V NC V C C VB NC V C VB NC V NC + C 3 B NC V dagâ â + λ dagâ dag + λ dagâ B NC V + λ dag λ â â + λ + λ B NC V NC â dag λ â â + λ + λ Snce: ŵ ŵ λ B NC V NC F H y o 4 λ â ŷo λ â â + λ + λ ŵ λ â ŷ o λ â â + λ + λ 5 Detecton Formula wth a Sngle Template As for the detecton formula n the prmal doman, we consder a new test sample u. For detecton, we construct all the crcular shfts of u n matrx U. Therefore, the response map on the test sample s: Tu Uw FdagûF H w ˆT u û ŵ ˆTu û ŵ 6. Usng Multple Templates Followng s the dervaton of the soluton for the multple template case n prmal doman,.e. when à Rn n, s the total number of templates, and à T A T A T... A T and Ĩ T I I... I. fz à Ĩ z + λ I 0 z + λ 0 I z y 0 fz ÃT à ÃT Ĩ I 0 0 0 0 z + λ z + λ ĨT à ĨT Ĩ 0 0 z λ 0 I y I 0 0 ÃT à + λ fz I ÃT Ĩ 0 z λ y ĨT à + λ I I 0 }{{} Γ
Target Response Adaptaton for Correlaton Flter Tracng 3 Now, note the followng: à T à AT A, ĨT Ĩ IT I I, and à T Ĩ AT, and that ĨT à A. It s clear that ÃT à and ĨT Ĩ are crculant whch s a sum of crculant matrces. Therefore, the matrx Γ s bloc wse crculant such that Γ R n n. Smlar to the sngle template case, we have: w ỹ λ dagâ â + λ dagâ 0 dagâ dag + λ F H y 0 7 Usng the nverse lemma n Eq 3 on Γ, we get: B NC V dag â â + λ dag â dag â + λ 8 B NC V NC dag â + λ â aˆ + + λ λ â â ŵ λ ˆ a ŷ o + λ â aˆ + + λ λ â â 9 0 It s to be noted that when, then Eq 0 reduces to the sngle template case n Eq 5, snce â â â â for. Detecton Formula wth Multple Templates Here, the detecton formula s smlar to Eq 6. Soluton to Problem n the Dual Doman. Usng a Sngle Template The optmzaton problem becomes: fz A I z + λ I 0 z + λ 0 I z y 0 For smpler notaton, let G A I, also let E I 0, and D 0 I fz Gz + λ Ez + λ Dz y 0 λ z T d y 0 + λ Ez + Gz
4 Adel Bb, Matthas Mueller, and Bernard Ghanem z fz λ z T d y 0 d + λ E T Ez + G T Gz 0 λ z T d y 0 d λ E T E + G T G z z λ λ E T E + G T G z T d y 0 d Let α λ z T d y 0. Then z λ E T E + G T G. By substtutng the dual varables α nto Eq, we obtan: fα λ λ E T E + G T G D T α. Let K α T DK d y 0 + λ EK D T α + GK D T α Then, the new dual objectve s gven by: 3 fα λ DK D T α y 0 + λ EK D T α + GK D T α 4 By settng the gradent to zero, the soluton to Problem 4 s obtaned by solvng the followng lnear system: λ DK D T DK D T + λ DK E T EK D T + DK G T GK D T α λ DK D T y 0 DK λ D T D + λ E T E + G T G K D T α λ DK D T y 0 }{{} 5 Ψ A D T A + λ I A T A T A + λ I A T A T A + λ I A T A I A + λ I A I }{{} D T A α λ D T A + λ I A T D T y A I 0 6 By usng the nverse lemma and the dagonalzaton propertes of crculant matrces smlar to nvertng the crculant dagonal matrx n the sngle template Ψ
Target Response Adaptaton for Correlaton Flter Tracng 5 case, one can show the followng: A K T A + λ I A T A I K λ I λ A T 7 λ A λ AA T + I K ΨK λ I λ A T I 0 λ λ AA T λ λ A T + + λ I λ A λ AA T + I λ I + λ A T +λ A λ λ A T + λ A T AA T λ λ AA T A + +λ λ λ A λ AA T AA T + +λ λ λ AA T + + λ I 8 Therefore, by substtutng Eq 8 nto Eq 6, we get the lnear system: λ λ AA T AA T + + λ AA T + + λ I α λ AA T + I y 0 9 λ λ The prevous problem can be effcently dagonalzed and solved as follows: λ F λ dagâ â â â + + λ dagâ â λ + + λ I ˆα λ F dagâ â λ + λ I ŷo λ ˆα λ â â + λ ŷo λ â λ â â â + +λ λ â â + + λ λ λ â â + λ ŷ o ˆα λ â λ â â â + +λ λ â â + + λ 0 Detecton Formula wth a Sngle Template Tu Uw UEK D T α U I 0 λ I λ A T 0 λ A λ AA T + I UA T α α λ λ Fdagû â ˆα ˆTu λ û â ˆα
6 Adel Bb, Matthas Mueller, and Bernard Ghanem. Usng Multple Templates For multple templates, the only dfference s that G T Therefore, the new dual objectve s gven as follows: where A T A T... A T I I... I D K λ D T D + λ E T E + G T G K D T α λ D K D T y 0 K AT A + λ I A T A I F 0 K dagâ â + λ I F dagâ 0 F H 0 dagâ I 0 F H 3 4 To fnd K, we use the nverse lemma n Eq 3 agan. Frst, we need to fnd: B NC V: B NC V dagâ â + λ I dagâ dagâ dag â â + λ I dag â dag â dag â â + λ dag â dag â â + λ â â â Then, usng the nverse lemma 3 and smlar trcs as used for tranng the flter n the sngle template case n the prmal doman, the nverse of K s gven as follows: K F 0 F Λ H 0 0 F 0 F H where dag Λ dag â â+λ â â â â â+λ â â â dag dag â â+λ â â â â + â â+λ â â 5
Target Response Adaptaton for Correlaton Flter Tracng 7 And snce: Ψ λ D T D + λ E T E + G T G AT A + λ I A T A + λ I F 0 0 F Ω F H 0 0 F H where Ω dagâ â + λ I dagâ dagâ + λ I Then, K Ψ K dag ΛΩ dag0 F 0 F ΛΩΛ H 0 0 F 0 F H dag dag λ â â â+λ â â â â+ +λ â â+ λ +λ â â+λ â â dag ΛΩΛ dag0 dag dag λ â â â+λ â â â â+ +λ â â+ λ +λ â â+λ â â dag dag â â+λ â â â â â+λ â â â dag dag â â+λ â â â â + â â+λ â â Υ Υ Υ Υ
8 Adel Bb, Matthas Mueller, and Bernard Ghanem Note that to to compute D K Ψ K D T, only the last bloc Υ s relevant. So, D K Ψ K D T FdagΥ F H, where λ Υ â + â â â â â â â + λ â â â â â â + λ â â + +λ â â â â + λ â â + + λ â â â â + λ â â + 6 Wth some mathematcal manpulaton Eq 6 s exactly equvalent to Eq 7 n the paper. As for the rght hand sde of Eq, one need to fnd: D K D T Fdag â â â â + λ â â + Therefore, soluton to Problem s computed as follows: FdagΥ F H α λ F â â â â + λ â â + F H 7 F H y o 8 ˆα λ dag Υ â â â â + λ ŷ 0 â ŷ0 + â 9 Detecton Formula wth Multple Templates Tu Uw UE K D T û α F â ˆα â â + λ â â ˆTu û â ˆα â â + λ â â 30
Target Response Adaptaton for Correlaton Flter Tracng 9 3 Integratng SRDCF Instead of re-dervng a closed form soluton to SRDCF s formulaton, one can use alternatng optmzaton, where the flter w and the target response y are updated n an teratve fashon eepng one of them fxed at any gven tme. The target adapted SRDCF objectve becomes: mn w,y Aw y + λ Xw + λ y y 0, 3 where X s the weght mas over the flter. Usng alternatng optmzaton, t can be mnmzed by solvng the next two optmzaton problems at teraton j. w j+ arg mn Aw y j + λ Xw 3 w y j+ arg mn Aw j+ y + λ y y 0 33 y Problem 3 can be solved usng the standard SRDCF soluton, whle Problem 33 has a closed form soluton as follows: y j+ + λ Aw j+ + λ y o 34 To compute y j+ effcently n the Fourer doman, we can use the followng strategy: y j+ Fdagâ F H w j+ + λ y o + λ 35 Fdagâ ŵ j+ + λ y o + λ 36 ŷ dagâ ŵ j+ + λ ŷ + λ o 37 â ŵ j+ + λ ŷ + λ o 38 Acnowledgments. Research n ths publcaton was supported by the Kng Abdullah Unversty of Scence and Technology KAUST Offce of Sponsored Research.
0 Adel Bb, Matthas Mueller, and Bernard Ghanem References. Bb, A., Mueller, M., Ghanem, B.: Target response adaptaton for correlaton flter tracng. In: European Conference on Computer Vson, ECCV, 06.. Danelljan, M., Hager, G., Shahbaz Khan, F., Felsberg, M.: Learnng spatally regularzed correlaton flters for vsual tracng. In: IEEE Internatonal Conference on Computer Vson, ICCV, 05.