M ≃ (x, y, z, t) x^{2} + y^{2} + z^{2} = 0 = t

( \begin{matrix}fx \\ fy \\ z \end{matrix} ) = [ \begin{matrix}f & 0 & 0 & 0 \\ 0 & f & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ] ( \begin{matrix}x \\ y \\ z \\ 1 \end{matrix} )

zm = P M

G = [ \begin{matrix}R & t \\ 0 & 1 \end{matrix} ]

K = [ \begin{matrix}f/s_{x} & f/s_{x} cot θ & o_{x} \\ 0 & f/s_{y} & o_{y} \\ 0 & 0 & 1 \end{matrix} ]

P = K[I|0]G = K[R|t] ζm = P M m ≃ P M P = λK[R|t], P C = P ( C̃ 1 ) = 0 ˜ C ˜ C C̃ = −P_{3×3}^{−1}p_{4} ζm = P M = P M − P C = P (M − C) = P_{3×3}(M̃ − C̃).

˜ M ˜ C ˜ M ˜ C

M = ( \begin{matrix}−P_{3×3}^{−1}p_{4} \\ 1 \end{matrix} ) + ζ ( \begin{matrix}P_{3×3}^{−1}m \\ 0 \end{matrix} ) , ζ ∈ ℝ.

˜ M m ≃ K[R | t](M̃^{T} , 0)^{T} = KRM̃.

˜ M ˜ M m^{T} (K^{−T} K^{−1})m = 0.

1 / z

˜ M

cos θ = \frac{M̃_{1}^{T} M̃_{2}}{||M̃_{1}||||M̃_{2}||} = \frac{m_{1}^{T} ωm_{2}}{\sqrt{m_{1}^{T} ωm_{1}}\sqrt{m_{2}^{T} ωm_{2}}}

m_{i} ≃ P M_{i} for all i. m_{i} × P M_{i} = 0. m_{i} × P M_{i} = 0 ⟺ [m_{i}]_{×}P M_{i} = 0 ⟺ vec([m_{i}]_{×}P M_{i}) = 0 ⟺ ⟺ (M_{i}^{T} ⊗ [m_{i}]_{×}) vec P = 0 ⟺ ([m_{i}]_{×} ⊗ M_{i}^{T}) vec P^{T} = 0

[\begin{matrix}0^{T} & −M_{i}^{T} & vM_{i}^{T} \\ M_{i}^{T} & 0^{T} & −uM_{i}^{T} \\ −vM_{i}^{T} & uM_{i}^{T} & 0^{T} \end{matrix} ] vec P^{T} = 0

ζ_{ℓ}m_{ℓ} = P_{ℓ}M ζ_{r}m_{r} = P_{r}M.

ζ_{r}m_{r} = P_{r}M = \underbrace{P_{r} ( \begin{matrix}−P_{3×3,ℓ}^{−1}p_{4,ℓ} \\ 1 \end{matrix} )}_{e_{r}} + ζ_{ℓ}P_{r} ( \begin{matrix}P_{3×3,ℓ}^{−1}m_{ℓ} \\ 0 \end{matrix} )

ζ_{r}m_{r} = e_{r} + ζ_{ℓ}P_{3×3,r}P_{3×3,l}^{−1}m_{ℓ} P_{ℓ} = [I|0] and P_{r} = [R|t]. ζ_{r}m_{r} = t + ζ_{ℓ}Rm_{ℓ}; m_{r}^{T} (t × Rm_{ℓ}) = 0, m_{r}^{T} [t]_{×}Rm_{ℓ} = 0, E ≜ [t]_{×}R, m_{r}^{T} Em_{ℓ} = 0.

P_{ℓ} = K_{ℓ}[I|0] and P_{r} = K_{r}[R|t]. ζ_{r}m_{r} = e_{r} + ζ_{ℓ}K_{r}RK_{ℓ}^{−1}m_{ℓ} with e_{r} = K_{r}t, m_{r}^{T} [e_{r}]_{×}K_{r}RK_{ℓ}^{−1}m_{ℓ} = 0.

F = [e_{r}]_{×}K_{r}RK_{ℓ}^{−1} m_{r}^{T} F m_{ℓ} = 0. l_{r} = F m_{ℓ}. l_{ℓ} = F^{T} m_{r}.

F e_{ℓ} = 0  (38)
e_{r}^{T} F = 0  (39)

F = K_{r}^{−T} EK_{ℓ}^{−1}. ζ_{r}F^{T} m_{r} = ζ_{ℓ}(e_{ℓ} × m_{ℓ}). m_{r}^{T} F m_{ℓ} = 0 ⟺ vec(m_{r}^{T} F m_{ℓ}) = 0 ⟺ (m_{r}^{T} ⊗ m_{ℓ}^{T}) vec F = 0. e_{r} = ζ_{r}m_{r} − ζ_{ℓ}m′_{ℓ}

ζ_{r} = \frac{(e_{r} × m′_{ℓ}) · (m_{r} × m′_{ℓ})}{||m_{r} × m′_{ℓ}||^{2}}

ζ [ \begin{matrix}u \\ v \\ 1 \end{matrix} ] = [\begin{matrix}P_{1,1} & P_{1,2} & P_{1,3} & P_{1,4} \\ P_{2,1} & P_{2,2} & P_{2,3} & P_{2,4} \\ P_{3,1} & P_{3,2} & P_{3,3} & P_{3,4} \end{matrix}] [ \begin{matrix}x \\ y \\ 0 \\ 1 \end{matrix} ] = [\begin{matrix}P_{1,1} & P_{1,2} & P_{1,4} \\ P_{2,1} & P_{2,2} & P_{2,4} \\ P_{3,1} & P_{3,2} & P_{3,4} \end{matrix}] [ \begin{matrix}x \\ y \\ 1 \end{matrix} ] .

m_{r} ≃ H_{Π}m_{ℓ} if M ∈ Π.

(H_{Π}m_{ℓ})^{T} F m_{ℓ} = 0 H_{Π}^{T}F + F^{T} H_{Π} = 0 F ≃ [e_{r}]_{×}H H_{Π}e_{ℓ} ≃ e_{r}

\frac{ζ_{r}}{ζ_{ℓ}} m_{r} = K_{r} (R + \frac{t n^{T}}{d} ) K_{ℓ}^{−1}m_{ℓ}.

H_{Π} = K_{r} (R + \frac{t n^{T}}{d} ) K_{ℓ}^{−1}

H_{∞} = K_{r}RK_{ℓ}^{−1}

\frac{ζ_{r}}{ζ_{ℓ}} m_{r} = H_{∞}m_{ℓ} + \frac{1}{ζ_{ℓ}} e_{r},

1 / (ζ_{ℓ})

H_{∞} = H_{Π} − K_{r} (\frac{t n^{T}}{d} ) K_{ℓ}^{−1}

\frac{ζ_{r}}{ζ_{ℓ}} m_{r} = H_{Π}m_{ℓ} + γe_{r}

(\frac{a}{d ζ_{ℓ}} )

F ≃ [e_{r}]_{×}H_{Π} m_{r}^{i} ≃ Hm_{ℓ}^{i} for all i m_{r} × Hm_{ℓ} = 0 m_{r} × Hm_{ℓ} = 0 ⟺ [m_{r}]_{×}Hm_{ℓ} = 0 ⟺ vec([m_{r}]_{×}Hm_{ℓ}) = 0 ⟺ (m_{ℓ}^{T} ⊗ [m_{r}]_{×}) vec H = 0 ⟺ ([m_{r}]_{×} ⊗ m_{ℓ}^{T}) vec H^{T} = 0

[\begin{matrix}0^{T} & −m_{ℓ}^{T} & vm_{ℓ}^{T} \\ m_{ℓ}^{T} & 0^{T} & −um_{ℓ}^{T} \\ −vm_{ℓ}^{T} & um_{ℓ}^{T} & 0^{T} \end{matrix} ] vec H^{T} = 0

e_{r} ≃ (H_{Π}m_{ℓ}^{0} × m_{r}^{0}) × (H_{Π}m_{ℓ}^{1} × m_{r}^{1})

H_{Π}m_{ℓ} = −γe_{r} + \frac{ζ_{r}}{ζ_{ℓ}} m_{r}

γ = \frac{(H_{Π}m_{ℓ} × m_{r})^{T} (m_{r} × e_{r})}{||m_{r} × e_{r}||^{2}}

m_{ℓ}^{i} = P_{ℓ}M^{i} and m_{r}^{i} = P_{r}M^{i}, ∀i F = [e_{r}]_{×}A P_{ℓ} = [I|0] and P_{r} = [A|e_{r}], F_{3,1}e_{3,2} ≃ e_{1,3} × e_{1,2} F_{1,2}e_{1,3} ≃ e_{2,1} × e_{2,3} F_{2,3}e_{2,1} ≃ e_{3,2} × e_{3,1} m_{3} ≃ F_{1,3}m_{1} × F_{2,3}m_{2} P_{1} = [I|0], P_{2} = [A_{2}|e_{2,1}], and P_{3} = [A_{3}|e_{3,1}].

0 = s_{2}^{T} e_{2,1} + ζ_{1}s_{2}^{T} A_{2}m_{1} 0 = s_{3}^{T} e_{3,1} + ζ_{1}s_{3}^{T} A_{3}m_{1} 0 = (s_{2}^{T} e_{2,1})(s_{3}^{T} A_{3}m_{1}) − (s_{3}^{T} e_{3,1})(s_{2}^{T} A_{2}m_{1})

0 = s_{2}^{T} (e_{2,1}m_{1}^{T} A_{3}^{T} − A_{2}m_{1}e_{3,1}^{T}) s_{3} 0 = (s_{3}^{T} ⊗ s_{2}^{T}) ((A_{3} ⊗ e_{2,1}) − (e_{3,1} ⊗ A_{2})) m_{1} = (m_{1}^{T} ⊗ s_{3}^{T} ⊗ s_{2}^{T}) vec(T)

T = (A_{3} ⊗ e_{2,1}) − (e_{3,1} ⊗ A_{2}) s_{2}^{T} (T m_{1})^{(3)}s_{3} = 0 [m_{2}]_{×}(T m_{1})^{(3)}[m_{3}]_{×} = 0.

(m_{1}^{T} ⊗ [m_{3}]_{×} ⊗ [m_{2}]_{×}) vec(T) = 0 (s_{2}^{T} (T m_{1})^{(3)}) [m_{3}]_{×} = 0 m_{3} ≃ (T m_{1})^{(3)}^{T} s_{2} (T m_{1})^{(3)} = ([t_{1}|t_{2}|t_{3}]m_{1})^{(3)} = (ut_{1} + vt_{2} + t_{3})^{(3)} = ut_{1}^{(3)} + vt_{2}^{(3)} + t_{3}^{(3)} ˆ t ˆ t ˆ t t_{13} = R_{23}t_{12} + t_{23} t̂_{13} = μ_{1}R_{23}t̂_{12} + μ_{2}t̂_{23} ˆ t ˆ t ˆ t

\frac{||t_{12}||}{||t_{13}||} = μ_{1} = \frac{(t̂_{13} × t̂_{23}) · (R_{23}t̂_{12} × t̂_{23})}{||R_{23}t̂_{12} × t̂_{23}||^{2}}

ζ_{i}^{j}m_{i}^{j} = P_{i}M^{j} i = 1 … m, j = 1 … n.

\underbrace{[\begin{matrix}ζ_{1}^{1}m_{1}^{1}, & ζ_{1}^{2}m_{1}^{2}, & … & ζ_{1}^{n}m_{1}^{n} \\ ζ_{2}^{1}m_{2}^{1}, & ζ_{2}^{2}m_{2}^{2}, & … & ζ_{2}^{n}m_{2}^{n} \\ ⋮ & ⋮ & ... & ⋮ \\ ζ_{m}^{1}m_{m}^{1}, & ζ_{m}^{2}m_{m}^{2}, & … & ζ_{m}^{n}m_{m}^{n} \end{matrix}]}_{measurements W} = \underbrace{[P_{1}, ; P_{2}, ; ⋮ ; P_{m}]}_{P}\underbrace{[M^{1}, M^{2}, … M^{n}]}_{structure S}.

W = UDV^{T} .

W = U_{3m×4} diag(σ_{1}, σ_{2}, σ_{3}, σ_{4}) V_{n×4}^{T}.

P = U_{3m×4} diag(σ_{1}, σ_{2}, σ_{3}, σ_{4}) and S = V_{n×4}^{T} [m_{i}^{1}, m_{i}^{2}, … m_{i}^{n}] Z_{i} = P_{i}S ζ_{i}m_{i} = P_{i}M i = 1 … m

[\begin{matrix}P_{1} & m_{1} & 0 & … & 0 \\ P_{2} & 0 & m_{2} & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ P_{m} & 0 & 0 & … & m_{m} \end{matrix}] [ M ; −ζ_{1} ; −ζ_{2} ; ⋮ ; −ζ_{m}] = [0 0 ⋮ 0]

5m − 8 ≥ (m − 1)(5 − m_{k} − m_{c}) + 5 − m_{k}.

E_{ij} = K_{i}^{T} F_{ij}K_{j},

C(K_{i}, i = 1 … n) = ∑_{i=1}^{n} ∑_{j>n}^{n} w_{ij} \frac{^{1}σ_{ij} −^{2}σ_{ij}}{^{1}σ_{ij} +^{2}σ_{ij}} ,

m_{i}^{j} ≃ P_{i}M^{j} i = 0 … m, j = 0 … n.

P_{0} = [I | 0]; P_{i} = [A_{i} | e_{i}] for i = 1 … m

m_{i}^{j} ≃ \underbrace{P_{i}T}_{P_{i}^{E}}\underbrace{T^{−1}M^{j}}_{structure},

P_{0}^{E} = K_{0}[I | 0] P_{i}^{E} = K_{i}[R_{i} | t_{i}] for i = 1 … m.

T = [\begin{matrix}K_{0} & 0 \\ r^{T} & s \end{matrix}]

P_{i}^{E} = [K_{i}R_{i} | K_{i}t_{i}] ≃ P_{i}T = [A_{i}K_{0} + e_{i}r^{T} | e_{i}] for i > 0

K_{i}R_{i} ≃ A_{i}K_{0} + e_{i}r^{T} = P_{i} [ \begin{matrix}K_{0} \\ r^{T} \end{matrix} ]

K_{i}K_{i}^{T} ≃ P_{i} [ \begin{matrix}K_{0}K_{0}^{T} & K_{0}r \\ r^{T} K_{0}^{T} & r^{T} r \end{matrix} ] P_{i}^{T}

C^{∗} ≃ P Q^{∗}P^{T} Ω^{∗} = diag(1, 1, 1, 0). ω^{∗} = (KK^{T}) ≃ P^{E}Ω^{∗}P^{E}^{T} Ω^{∗} = T diag(1, 1, 1, 0)T^{T} .

Ω^{∗} = [ \begin{matrix}K_{0}K_{0}^{T} & K_{0}r \\ r^{T} K_{0}^{T} & r^{T} r \end{matrix} ]

ω_{i}^{∗} ≃ P_{i}Ω^{∗}P_{i}^{T} ω_{i}^{∗} − λ_{i}P_{i}Ω^{∗}P_{i}^{T} = 0 vec ω_{i}^{∗} ≃ vec(P_{i}Ω^{∗}P_{i}^{T}) ⟺ rank [vec ω_{i}^{∗}| vec(P_{i}Ω^{∗}P_{i}^{T})] = 1

\frac{ω_{i}^{∗}}{||ω_{i}^{∗}||_{F}} − \frac{P_{i}Ω^{∗}P_{i}^{T}}{||P_{i}Ω^{∗}P_{i}^{T} ||_{F}} = 0

ω_{i}^{∗}(k, ℓ) = 0 ⇒ p_{i,k}^{T}Ω^{∗}p_{i,ℓ} = 0

KK^{T} ≃ P_{i} [ \begin{matrix}KK^{T} & Kr \\ r^{T} K^{T} & r^{T} r \end{matrix} ] P_{i}^{T}

\sqrt{2}

ˆ P

ˆ P

min_{P̂} ∑_{j} d(P̂M^{j}, m^{j})^{2}

ˆ M ˆ M

min_{M̂} ∑_{i} d(P_{i}M̂, m_{i})^{2}

ˆ F

min_{F̂} ∑_{j} d(F̂m_{ℓ}^{j}, m_{r}^{j})^{2} + d(F̂^{T} m_{r}^{j}, m_{ℓ}^{j})^{2}

ˆ H

min_{Ĥ} ∑_{j} d(Ĥm_{ℓ}^{j}, m_{r}^{j})^{2} + d(Ĥ^{−1}m_{r}^{j}, m_{ℓ}^{j})^{2}

ˆ P

ˆ M

min_{P̂_{i},M̂^{j}} ∑_{i,j} d(P̂_{i}M̂^{j}, m_{i}^{j})^{2}

ˆ P

ˆ M

min_{P̂_{i},M̂^{j}} ∑_{i,j} d(P̂_{i}M̂^{j}, m_{i}^{j})^{2}

ˆ P

ˆ K

ˆ R

ˆ t

min_{θ} ∑_{i}(r_{i}/σ_{i})^{2}

min_{θ} ∑_{i} ρ(r_{i}/σ_{i})

∑_{i} \frac{1}{σ_{i}} ρ′(r_{i}/σ_{i})\frac{dr_{i}}{dθ} = 0

P = 1 − (1 − (1 − ϵ)^{p})^{N}

min_{θ} med_{i}r_{i}

H ≃ K[r_{1}, r_{2}, t] r_{1} = λK^{−1}h_{1} and r_{2} = λK^{−1}h_{2} λ^{2}h_{1}^{T} (KK^{T})^{−1}h_{2} = 0 h_{1}^{T} ωh_{2} = 0 h_{1}^{T} ωh_{1} = h_{2}^{T} ωh_{2} ((h_{1}^{T} ⊗ h_{1}^{T}) − (h_{2}^{T} ⊗ h_{2}^{T}))

[r_{1}, r_{2}, t] = \frac{1}{||K^{−1}h_{1}||}K^{−1}[h_{1}, h_{2}, h_{3}] r_{3} = r_{1} × r_{2}

min_{P̂_{i}} ∑_{i=1}^{n} ∑_{j=1}^{m} d(P̂_{i}M^{j}, m_{i}^{j})^{2}

ˆ P

ˆ K

ˆ R

ˆ t

ˆ v

\{ \begin{matrix}û & = (u − u_{0})(1 + k_{1}r_{d}^{2}) + u_{0} \\ v̂ & = (v − v_{0})(1 + k_{1}r_{d}^{2}) + v_{0} \end{matrix} .

(\frac{(u−u_{0})}{a_{u}})

(\frac{(v−v_{0})}{a_{v}})

ˆ m

ˆ v

\{ \begin{matrix}(u − u_{0}) ((\frac{(u − u_{0})}{a_{u}})^{2} + (\frac{(v − v_{0})}{a_{v}})^{2}) k_{1} & = û − u \\ (v − u_{0}) ((\frac{(u − u_{0})}{a_{u}})^{2} + (\frac{(v − v_{0})}{a_{v}})^{2}) k_{1} & = v̂ − v \end{matrix}

P_{nℓ} = K[R | − R C̃_{ℓ}], P_{nr} = K[R | − R C̃_{r}].

˜ C

˜ C

R = [\begin{matrix}r_{1}^{T} \\ r_{2}^{T} \\ r_{3}^{T} \end{matrix} ]

˜ C

˜ C

˜ C

˜ C

\{ \begin{matrix}M̃ = C̃_{ℓ} + ζ_{o}P_{3×3,oℓ}^{−1}m_{oℓ} \\ M̃ = C̃_{ℓ} + ζ_{n}P_{3×3,nℓ}^{−1}m_{nℓ} \end{matrix}

m_{nℓ} = \frac{ζ_{o}}{ζ_{n}} P_{3×3,nℓ}P_{3×3,oℓ}^{−1}m_{oℓ}

(ζ_{o}) / (ζ_{n})

Π

Π

σ ρ π μ λ θ ζ ϵ γ

ℓ

∈ ∞ ≃ ↔ → ← ⊗

∞

𝕏 ℝ ℙ Ω ω