cool new equivariance result

let Φ\Phi be an U, U equivariant registration network.

Ψ0:=ΦΨ1:=TwoStep{Ψ0,Φ}Ψ2:=TwoStep{Ψ1,Φ}Ψ3:=TwoStep{Ψ2,Φ}...\begin{aligned} \Psi_0 := \Phi \\ \Psi_1 := TwoStep\{\Psi_0, \Phi\} \\ \Psi_2 := TwoStep\{\Psi_1, \Phi\} \\ \Psi_3 := TwoStep\{\Psi_2, \Phi\} \\ ... \end{aligned}

If the sequence Ψi\Psi_i is convergent, call its limit Ψ\Psi

using usual fixed point argument,

Ψ=TwoStep{Ψ,Φ} \Psi = TwoStep\{\Psi, \Phi\}

Apply to images A, B

Ψ[A,B]=Ψ[A,B]Φ[AΨ[A,B],B]\Psi[A, B] = \Psi[A, B] \circ \Phi[A \circ \Psi[A, B], B] Φ[AΨ[A,B],B]=id\Phi[A \circ \Psi[A, B], B] = id

Substitute A=MW,B=FU A = M \circ W, B = F \circ U

Φ[MWΨ[MW,FU],FU]=id\Phi[M \circ W \circ \Psi[M \circ W, F \circ U], F \circ U] = id

Use that Φ\Phi is U, U equivariant

U1Φ[MWΨ[MW,FU]U1,F]U=idU^{-1} \circ \Phi[M \circ W \circ \Psi[M \circ W, F \circ U] \circ U^{-1}, F] \circ U = id Φ[MWΨ[MW,FU]U1,F]=id\Phi[M \circ W \circ \Psi[M \circ W, F \circ U] \circ U^{-1}, F] = id

Substitute A = M, B = F, use that both sides are identity

Φ[MWΨ[MW,FU]U1,F]=Φ[MΨ[M,F],F]\Phi[M \circ W \circ \Psi[M \circ W, F \circ U] \circ U^{-1}, F] = \Phi[M \circ \Psi[M , F], F]

This is not quite the same thing as W, U equivariance, which can be stated as

WΨ[MW,FU]U1=Ψ[M,F] W \circ \Psi[M \circ W, F \circ U] \circ U^{-1} = \Psi[M, F]

But it's really close. Informally, we showed, "If Psi is not W, U equivariant, it has to be either in a way that the resulting transform warps M identically, or in a way that the differently warped M causes Phi to have the same output"

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