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Uncovering Interactions and Interactors: Joint Estimation of Head, Body Orientation and F-formations from Surveillance Videos Elisa Ricci1,3 , Jagannadan Varadarajan2 , Ramanathan Subramanian2 , Samuel Rota Bul`o1 , Narendra Ahuja2,4 , Oswald Lanz1 1 Fondazione Bruno Kessler, Trento, Italy 2 Advanced Digital Sciences Center, University of Illinois at Urbana-Champaign, Singapore 3 Department of Engineering, University of Perugia, Italy 4 University of Illinois at Urbana-Champaign, IL USA {eliricci,rotabulo,lanz}@fbk.eu, {vjagan,subramanian.r}@adsc.com.sg, {n-ahuja}@illinois.edu

Supplementary Material

In this document we provide the derivations of the update rules for ΘH and ΘB .

A. Update rules for ΘH and ΘB Consider the body and head regressors defined in Subsection 3.2.3 of our ICCV contribution. The update rules for ΘH and ΘB that we provide in Subsection 3.2.4 are obtained by setting to zero the partial derivative of the objective function in (2) with respect to Θ⋄ with ⋄ ∈ {B, H}, and by solving the resulting equations, which are given by ∂ ∂ LH (fH (·; ΘH ); T , S) + LC (fB (·; ΘB ), fH (·; ΘH ); S) = 0 ∂ΘH ∂ΘH ∂ ∂ ∂ LB (fB (·; ΘB ); T , S) + LC (fB (·; ΘB ), fH (·; ΘH ); S) + LF (fB (·; ΘB ), C; S) = 0 , ∂ΘB ∂ΘB ∂ΘB where we have replaced LP in (2) with its definition in (3). The L⋄ term is given by L⋄ (f⋄ (·; Θ⋄ ); T⋄ , S) =

N⋄ X

kΘ⋄ v ⋄i − y ⋄i k2M + λR kΘ⋄ k2F + λU

i=1

X

⋄ ωij kΘ⋄ (v ⋄i − v ⋄j )k2M ,

(i,j)∈E⋄

and its derivative with respect to Θ⋄ is ∂ ˆ ⊤ + 2λR Θ⋄ + 2λU M Θ⋄ V ⋄ L⋄ V ⊤ L⋄ (f⋄ (·; Θ⋄ ); T , S) = 2M (Θ⋄ Xˆ ⋄ − Y ⋄ )X ⋄ ⋄ ∂Θ⋄ ⊤ ˆ ⋄X ˆ + λU V ⋄ L⋄ V ⊤ ) + 2λR Θ⋄ − 2M Y ⋄ X ˆ⊤. = 2M Θ⋄ (X ⋄



The LC term is given by LC (fB (·; ΘB ), fH (·; ΘH ); S) = λC

NK X NT X

kΘB xBkt − ΘH xHkt k2M ,

k=1 t=1

and its derivative with respect to Θ⋄ is ∂ LC (fB (·; ΘB ), fH (·; ΘH ); S) = 2λC M (Θ⋄ X ⋄ − Θ⋆ X ⋆ )X ⊤ ⋄ ∂Θ⋄ ⊤ = 2λC M Θ⋄ X ⋄ X ⋄ − 2λC M Θ⋆ X ⋆ X ⊤ ⋄ , 1



(A) (B)

where (⋄, ⋆) ∈ {(H, B), (B, H)}. The LF term is given by NT NK X X

LF (fB (·; ΘB ), C; S) = λF

kckt − (pkt + DAΘB xBkt )k22 + const ,

k=1 t=1

where “const” indicates terms not depending on ΘB , and its derivative with respect to ΘB is ∂ LF (fB (·; ΘB ), C; S) = 2λF DA⊤ (DAΘB X B + P − C)X ⊤ B ∂ΘB ⊤ ⊤ = 2λF D2 A⊤ AΘB X B X ⊤ B + 2λF DA (P − C)X B . By replacing the computed gradient terms in (A), and after few algebraic manipulations, we obtain ⊤

ˆ + λU V H L H V ⊤ + λC X H X ⊤ ) + λR Θ H − F H = 0 , ˆ HX M ΘH ( X H H H and by vectorizing both sides we get E H vec(ΘH ) = vec(F H )

=⇒

vec(ΘH ) = E −1 H vec(F H ) .

By replacing the computed gradient terms in (B), and after few algebraic manipulations, we obtain ˆ BX ˆ ⊤ + λU V B LB V ⊤ + λC X B X ⊤ ) + λR ΘB + λF D2 A⊤ AΘB X B X ⊤ − G = 0 , M ΘB ( X B B B B and by vectorizing both sides we get ⊤ (E B + λF D2 X B X ⊤ B ⊗ A A)vec(ΘB ) = vec(G)

=⇒

⊤ −1 vec(ΘB ) = (E B + λF D2 X B X ⊤ vec(G) . B ⊗ A A)

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