ORB_SLAM3_IMU预积分理论推导(噪声分析)

噪声分析

1. δϕ⃗ij\\delta \\vec{\\phi}_{i j}δϕ​ij​：
   Exp⁡(−δϕ⃗ij)=∏k=ij−1Exp⁡(−ΔR~k+1jT⋅Jrk⋅ηkgdΔt)\\operatorname{Exp}\\left(-\\delta \\vec{\\phi}_{i j}\\right)=\\prod_{k=i}^{j-1} \\operatorname{Exp}\\left(-\\Delta \\tilde{\\mathbf{R}}_{k+1 j}^{T} \\cdot \\mathbf{J}_{r}^{k} \\cdot \\boldsymbol{\\eta}_{k}^{g d} \\Delta t\\right) Exp(−δϕ​ij​)=k=i∏j−1​Exp(−ΔR~k+1jT​⋅Jrk​⋅ηkgd​Δt)
   其中，Jrk=Jr((ω~k−big)Δt)\\mathbf{J}_{r}^{k}=\\mathbf{J}_{r}\\left(\\left(\\tilde{\\boldsymbol{\\omega}}_{k}-\\mathbf{b}_{i}^{g}\\right) \\Delta t\\right)Jrk​=Jr​((ω~k​−big​)Δt)

对上式两边取对数：
δϕ⃗ij=−log⁡(∏k=ij−1Exp⁡(−ΔR~k+1jT⋅Jrk⋅ηkgdΔt))\\delta \\vec{\\phi}_{i j}=-\\log \\left(\\prod_{k=i}^{j-1} \\operatorname{Exp}\\left(-\\Delta \\tilde{\\mathbf{R}}_{k+1 j}^{T} \\cdot \\mathbf{J}_{r}^{k} \\cdot \\boldsymbol{\\eta}_{k}^{g d} \\Delta t\\right)\\right) δϕ​ij​=−log(k=i∏j−1​Exp(−ΔR~k+1jT​⋅Jrk​⋅ηkgd​Δt))
利用性质：当δϕ⃗\\delta \\vec{\\phi}δϕ​是小量时，log⁡(Exp⁡(ϕ⃗)⋅Exp⁡(δϕ⃗))=ϕ⃗+J⁡r−1(ϕ⃗)⋅δϕ⃗\\log{\\left(\\operatorname{Exp}\\left(\\vec{\\phi} \\right) \\cdot \\operatorname{Exp}\\left(\\delta\\vec{\\phi}\\right)\\right)}=\\vec{\\phi }+\\operatorname{J}^{-1}_{r}\\left (\\vec{\\phi } \\right ) \\cdot \\delta \\vec{\\phi }log(Exp(ϕ​)⋅Exp(δϕ​))=ϕ​+Jr−1​(ϕ​)⋅δϕ​, 其中ηkgd\\boldsymbol{\\eta}_{k}^{g d}ηkgd​是小量，ξ=ΔR~k+1jT⋅Jrk⋅ηkgdΔt\\xi =\\Delta \\tilde{\\mathbf{R}}_{k+1 j}^{T} \\cdot \\mathbf{J}_{r}^{k} \\cdot \\boldsymbol{\\eta}_{k}^{g d} \\Delta tξ=ΔR~k+1jT​⋅Jrk​⋅ηkgd​Δt是小量，于是J⁡r−1(ξk)≈I\\operatorname{J}^{-1}_{r}\\left (\\xi_{k} \\right )\\approx IJr−1​(ξk​)≈I

δϕ⃗ij=−log⁡(∏k=ij−1Exp⁡(−ξk))=−log⁡(Exp⁡(−ξi)∏k=i+1j−1Exp⁡(−ξk))≈−(−ξi+I⋅log⁡(∏k=i+1j−1Exp⁡(−ξk)))=ξi−log⁡(∏k=i+1j−1Exp⁡(−ξk))=ξi−log⁡(Exp⁡(−ξi+1)∏k=i+2j−1Exp⁡(−ξk))≈ξi+ξi+1−log⁡(∏k=i+2j−1Exp⁡(−ξk))≈⋯≈∑k=ij−1ξk\\begin{aligned} \\delta \\vec{\\phi}_{i j} & =-\\log \\left(\\prod_{k=i}^{j-1} \\operatorname{Exp}\\left(-\\xi_{k}\\right)\\right) \\\\ & =-\\log \\left(\\operatorname{Exp}\\left(-\\xi_{i}\\right) \\prod_{k=i+1}^{j-1} \\operatorname{Exp}\\left(-\\xi_{k}\\right)\\right) \\\\ & \\approx-\\left(-\\xi_{i}+\\mathbf{I} \\cdot \\log \\left(\\prod_{k=i+1}^{j-1} \\operatorname{Exp}\\left(-\\xi_{k}\\right)\\right)\\right)=\\xi_{i}-\\log \\left(\\prod_{k=i+1}^{j-1} \\operatorname{Exp}\\left(-\\xi_{k}\\right)\\right) \\\\ & =\\xi_{i}-\\log \\left(\\operatorname{Exp}\\left(-\\xi_{i+1}\\right) \\prod_{k=i+2}^{j-1} \\operatorname{Exp}\\left(-\\xi_{k}\\right)\\right) \\\\ & \\approx \\xi_{i}+\\xi_{i+1}-\\log \\left(\\prod_{k=i+2}^{j-1} \\operatorname{Exp}\\left(-\\xi_{k}\\right)\\right) \\\\ & \\approx \\cdots \\\\ & \\approx \\sum_{k=i}^{j-1} \\xi_{k} \\end{aligned} δϕ​ij​​=−log(k=i∏j−1​Exp(−ξk​))=−log(Exp(−ξi​)k=i+1∏j−1​Exp(−ξk​))≈−(−ξi​+I⋅log(k=i+1∏j−1​Exp(−ξk​)))=ξi​−log(k=i+1∏j−1​Exp(−ξk​))=ξi​−log(Exp(−ξi+1​)k=i+2∏j−1​Exp(−ξk​))≈ξi​+ξi+1​−log(k=i+2∏j−1​Exp(−ξk​))≈⋯≈k=i∑j−1​ξk​​
即：
δϕ⃗ij≈∑k=ij−1ΔR~k+1jTJ⁡rkηkgdΔt\\delta\\vec{\\phi}_{ij}\\approx \\sum_{k=i}^{j-1} {\\Delta\\tilde{R}_{k+1 j}^{T}\\operatorname{J}_{r}^{k}\\eta_{k}^{gd}\\Delta t } δϕ​ij​≈k=i∑j−1​ΔR~k+1jT​Jrk​ηkgd​Δt
由于ΔR~k+1jT\\Delta\\tilde{R}_{k+1 j}^{T}ΔR~k+1jT​、J⁡rk\\operatorname{J}_{r}^{k}Jrk​、$\Delta t$都是已知量，而ηkgd\\eta_{k}^{gd}ηkgd​为零均值高斯噪声，因此δϕ⃗ij\\delta \\vec{\\phi}_{i j}δϕ​ij​也为零均值高斯噪声
2. δvij\\delta v_{ij}δvij​
δvij=∑k=ij−1[ΔR~ikηkadΔt−ΔR~ik⋅(f~k−bia)∧⋅δϕ⃗ik⋅Δt]\\delta \\mathbf{v}_{i j}=\\sum_{k=i}^{j-1}\\left[\\Delta \\tilde{\\mathbf{R}}_{i k} \\mathbf{\\eta}_{k}^{a d} \\Delta t-\\Delta \\tilde{\\mathbf{R}}_{i k} \\cdot\\left(\\tilde{\\mathbf{f}}_{k}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\cdot \\delta \\vec{\\phi}_{i k} \\cdot \\Delta t\\right] δvij​=k=i∑j−1​[ΔR~ik​ηkad​Δt−ΔR~ik​⋅(f~k​−bia​)∧⋅δϕ​ik​⋅Δt]
3. δpij\\delta p_{ij}δpij​
δpij=∑k=ij−1[δvikΔt−12ΔR~ik⋅(f~k−bia)∧δϕ⃗ikΔt2+12ΔR~ikηkadΔt2]\\delta \\mathbf{p}_{i j}=\\sum_{k=i}^{j-1}\\left[\\delta \\mathbf{v}_{i k} \\Delta t-\\frac{1}{2} \\Delta \\tilde{\\mathbf{R}}_{i k} \\cdot\\left(\\tilde{\\mathbf{f}}_{k}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\delta \\vec{\\phi}_{i k} \\Delta t^{2}+\\frac{1}{2} \\Delta \\tilde{\\mathbf{R}}_{i k} \\mathbf{\\eta}_{k}^{a d} \\Delta t^{2}\\right] δpij​=k=i∑j−1​[δvik​Δt−21​ΔR~ik​⋅(f~k​−bia​)∧δϕ​ik​Δt2+21​ΔR~ik​ηkad​Δt2]

噪声更新

1. δϕ→ij−1→δϕ→ij\\delta\\overrightarrow{\\phi}_{ij-1}\\to \\delta\\overrightarrow{\\phi}_{ij}δϕ​ij−1​→δϕ​ij​
   δϕ⃗ij=∑k=ij−1ΔR~k+1jTJrkηkgdΔt=∑k=ij−2ΔR~k+1jTJrkηkgdΔt+ΔR~jjT⏟IJrj−1ηj−1gdΔt=∑j−2k=i(ΔR~k+1j−1ΔR~j−1j)TJrkηkgdΔt+Jrj−1ηj−1gdΔt=ΔR~jj−1∑k=ij−2ΔR~k+1j−1TJrkηkgdΔt+Jrj−1ηj−1gdΔt=ΔR~jj−1δϕ⃗ij−1+Jrj−1ηj−1gdΔt\\begin{aligned} \\delta \\vec{\\phi}_{i j} & =\\sum_{k=i}^{j-1} \\Delta \\tilde{\\mathbf{R}}_{k+1 j}^{T} \\mathbf{J}_{r}^{k} \\boldsymbol{\\eta}_{k}^{g d} \\Delta t \\\\ & =\\sum_{k=i}^{j-2} {\\color{Red} \\Delta \\tilde{\\mathbf{R}}_{k+1 j}^{T}} \\mathbf{J}_{r}^{k} \\boldsymbol{\\eta}_{k}^{g d} \\Delta t+\\underbrace{\\Delta \\tilde{\\mathbf{R}}_{j j}^{T}}_{I} \\mathbf{J}_{r}^{j-1} \\boldsymbol{\\eta}_{j-1}^{g d} \\Delta t \\\\ &= \\sum_{j-2}^{k=i}{\\color{Red} \\left(\\Delta \\tilde{\\mathbf{R}}_{k+1 j-1} \\Delta \\tilde{\\mathbf{R}}_{j-1 j}\\right)^{T}} \\mathbf{J}_{r}^{k} \\boldsymbol{\\eta}_{k}^{g d} \\Delta t+\\mathbf{J}_{r}^{j-1} \\boldsymbol{\\eta}_{j-1}^{g d} \\Delta t \\\\ & =\\Delta \\tilde{\\mathbf{R}}_{j j-1} \\sum_{k=i}^{j-2} \\Delta \\tilde{\\mathbf{R}}_{k+1 j-1}^{T} \\mathbf{J}_{r}^{k} \\boldsymbol{\\eta}_{k}^{g d} \\Delta t+\\mathbf{J}_{r}^{j-1} \\boldsymbol{\\eta}_{j-1}^{g d} \\Delta t \\\\ & =\\Delta \\tilde{\\mathbf{R}}_{j j-1} \\delta \\vec{\\phi}_{i j-1}+\\mathbf{J}_{r}^{j-1} \\boldsymbol{\\eta}_{j-1}^{g d} \\Delta t \\end{aligned} δϕ​ij​​=k=i∑j−1​ΔR~k+1jT​Jrk​ηkgd​Δt=k=i∑j−2​ΔR~k+1jT​Jrk​ηkgd​Δt+IΔR~jjT​​​Jrj−1​ηj−1gd​Δt=j−2∑k=i​(ΔR~k+1j−1​ΔR~j−1j​)TJrk​ηkgd​Δt+Jrj−1​ηj−1gd​Δt=ΔR~jj−1​k=i∑j−2​ΔR~k+1j−1T​Jrk​ηkgd​Δt+Jrj−1​ηj−1gd​Δt=ΔR~jj−1​δϕ​ij−1​+Jrj−1​ηj−1gd​Δt​
2. δvij−1→δvij\\delta v_{ij-1} \\to \\delta v_{ij}δvij−1​→δvij​
   δvij=∑k=ij−1[ΔR~ikηkadΔt−ΔR~ik⋅(f~k−bia)∧⋅δϕ⃗ik⋅Δt]=∑k=ij−2[ΔR~ikηkadΔt−ΔR~ik⋅(f~k−bia)∧⋅δϕ⃗ik⋅Δt]…+ΔR~ij−1ηj−1adΔt−ΔR~ij−1⋅(f~j−1−bia)∧⋅δϕ⃗ij−1⋅Δt=δvij−1+ΔR~ij−1ηj−1adΔt−ΔR~ij−1⋅(f~j−1−bia)∧⋅δϕ⃗ij−1⋅Δt\\begin{aligned} \\delta \\mathbf{v}_{i j}= & \\sum_{k=i}^{j-1}\\left[\\Delta \\tilde{\\mathbf{R}}_{i k} \\mathbf{\\eta}_{k}^{a d} \\Delta t-\\Delta \\tilde{\\mathbf{R}}_{i k} \\cdot\\left(\\tilde{\\mathbf{f}}_{k}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\cdot \\delta \\vec{\\phi}_{i k} \\cdot \\Delta t\\right] \\\\ = & \\sum_{k=i}^{j-2}\\left[\\Delta \\tilde{\\mathbf{R}}_{i k} \\boldsymbol{\\eta}_{k}^{a d} \\Delta t-\\Delta \\tilde{\\mathbf{R}}_{i k} \\cdot\\left(\\tilde{\\mathbf{f}}_{k}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\cdot \\delta \\vec{\\phi}_{i k} \\cdot \\Delta t\\right] \\ldots \\\\ & +\\Delta \\tilde{\\mathbf{R}}_{i j-1} \\boldsymbol{\\eta}_{j-1}^{a d} \\Delta t-\\Delta \\tilde{\\mathbf{R}}_{i j-1} \\cdot\\left(\\tilde{\\mathbf{f}}_{j-1}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\cdot \\delta \\vec{\\phi}_{i j-1} \\cdot \\Delta t \\\\ = & \\delta \\mathbf{v}_{i j-1}+\\Delta \\tilde{\\mathbf{R}}_{i j-1} \\boldsymbol{\\eta}_{j-1}^{a d} \\Delta t-\\Delta \\tilde{\\mathbf{R}}_{i j-1} \\cdot\\left(\\tilde{\\mathbf{f}}_{j-1}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\cdot \\delta \\vec{\\phi}_{i j-1} \\cdot \\Delta t \\end{aligned} δvij​===​k=i∑j−1​[ΔR~ik​ηkad​Δt−ΔR~ik​⋅(f~k​−bia​)∧⋅δϕ​ik​⋅Δt]k=i∑j−2​[ΔR~ik​ηkad​Δt−ΔR~ik​⋅(f~k​−bia​)∧⋅δϕ​ik​⋅Δt]…+ΔR~ij−1​ηj−1ad​Δt−ΔR~ij−1​⋅(f~j−1​−bia​)∧⋅δϕ​ij−1​⋅Δtδvij−1​+ΔR~ij−1​ηj−1ad​Δt−ΔR~ij−1​⋅(f~j−1​−bia​)∧⋅δϕ​ij−1​⋅Δt​
3. δpij−1→δpij\\delta p_{ij-1} \\to \\delta p_{ij}δpij−1​→δpij​
   δpij=∑k=ij−1[δvikΔt−12ΔR~ik⋅(f~k−bia)∧δϕ⃗ikΔt2+12ΔR~ikηkadΔt2]=δpij−1+δvij−1Δt−12ΔR~ij−1⋅(f~j−1−bia)∧δϕ⃗ij−1Δt2+12ΔR~ij−1ηj−1adΔt2\\begin{aligned} \\delta \\mathbf{p}_{i j} & =\\sum_{k=i}^{j-1}\\left[\\delta \\mathbf{v}_{i k} \\Delta t-\\frac{1}{2} \\Delta \\tilde{\\mathbf{R}}_{i k} \\cdot\\left(\\tilde{\\mathbf{f}}_{k}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\delta \\vec{\\phi}_{i k} \\Delta t^{2}+\\frac{1}{2} \\Delta \\tilde{\\mathbf{R}}_{i k} \\boldsymbol{\\eta}_{k}^{a d} \\Delta t^{2}\\right] \\\\ & =\\delta \\mathbf{p}_{i j-1}+\\delta \\mathbf{v}_{i j-1} \\Delta t-\\frac{1}{2} \\Delta \\tilde{\\mathbf{R}}_{i j-1} \\cdot\\left(\\tilde{\\mathbf{f}}_{j-1}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\delta \\vec{\\phi}_{i j-1} \\Delta t^{2}+\\frac{1}{2} \\Delta \\tilde{\\mathbf{R}}_{i j-1} \\boldsymbol{\\eta}_{j-1}^{a d} \\Delta t^{2} \\end{aligned} δpij​​=k=i∑j−1​[δvik​Δt−21​ΔR~ik​⋅(f~k​−bia​)∧δϕ​ik​Δt2+21​ΔR~ik​ηkad​Δt2]=δpij−1​+δvij−1​Δt−21​ΔR~ij−1​⋅(f~j−1​−bia​)∧δϕ​ij−1​Δt2+21​ΔR~ij−1​ηj−1ad​Δt2​
   矩阵C:
   χij=[ηijT,γijT]T∼N(015×1,Cij)ηij=[δϕijT,δvijT,δpijT]T∼N(09×1,Σijη)γij=[δbijaT,δbijgT]T∼N(06×1,Σijγ)\\begin{aligned} \\chi_{i j} & =\\left[\\eta_{i j}^{T}, \\gamma_{i j}^{T}\\right]^{T} \\sim N\\left(0_{15 \\times 1}, C_{i j}\\right) \\\\ \\eta_{i j} & =\\left[\\delta \\phi_{i j}^{T}, \\delta v_{i j}^{T}, \\delta p_{i j}^{T}\\right]^{T} \\sim N\\left(0_{9 \\times 1}, \\Sigma_{i j}^{\\eta}\\right) \\\\ \\gamma_{i j} & =\\left[\\delta b_{i j}^{a T}, \\delta b_{i j}^{g T}\\right]^{T} \\sim N\\left(0_{6 \\times 1}, \\Sigma_{i j}^{\\gamma}\\right) \\end{aligned} χij​ηij​γij​​=[ηijT​,γijT​]T∼N(015×1​,Cij​)=[δϕijT​,δvijT​,δpijT​]T∼N(09×1​,Σijη​)=[δbijaT​,δbijgT​]T∼N(06×1​,Σijγ​)​
   根据$\delta \varphi$、$\delta v$、$\delta p$、δba\\delta b_{a}δba​、δbg\\delta b_{g}δbg​，可以得到：
   Cij=(Σijη09×606×9Σijγ)15×15C_{i j}=\\left(\\begin{array}{cc} \\Sigma_{i j}^{\\eta} & 0_{9 \\times 6} \\\\ 0_{6 \\times 9} & \\Sigma_{i j}^{\\gamma} \\end{array}\\right)_{15 \\times 15} Cij​=(Σijη​06×9​​09×6​Σijγ​​)15×15​

ηijΔ≜[δϕ⃗ijTδvijTδpijT]T\\boldsymbol{\\eta}_{i j}^{\\Delta} \\triangleq\\left[\\begin{array}{lll}\\delta \\vec{\\phi}_{i j}^{T} & \\delta \\mathbf{v}_{i j}^{T} & \\delta \\mathbf{p}_{i j}^{T}\\end{array}\\right]^{T}ηijΔ​≜[δϕ​ijT​​δvijT​​δpijT​​]T的递推形式如下
ηijΔ=[ΔR~jj−100−ΔR~ij−1⋅(f~j−1−bia)∧ΔtI0−12ΔR~ij−1⋅(f~j−1−bia)∧Δt2ΔtII]⏟Aj−1ηij−1Δ⋯+[Jrj−1Δt00ΔR~ij−1Δt012ΔR~ij−1Δt2]⏟Bj−1[(ηj−1gd)T(ηj−1ad)T]⏟ηj−1d\\begin{aligned} \\boldsymbol{\\eta}_{i j}^{\\Delta}= &{\\underbrace{ {\\left[\\begin{array}{ccc} \\Delta \\tilde{\\mathbf{R}}_{j j-1} & \\mathbf{0} & \\mathbf{0} \\\\ -\\Delta \\tilde{\\mathbf{R}}_{i j-1} \\cdot\\left(\\tilde{\\mathbf{f}}_{j-1}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\Delta t & \\mathbf{I} & \\mathbf{0} \\\\ -\\frac{1}{2} \\Delta \\tilde{\\mathbf{R}}_{i j-1} \\cdot\\left(\\tilde{\\mathbf{f}}_{j-1}-\\mathbf{b}_{i}^{a}\\right)^{\\wedge} \\Delta t^{2} & \\Delta t \\mathbf{I} & \\mathbf{I} \\end{array}\\right]}} _{A_{j-1} }\\boldsymbol{\\eta}_{i j-1}^{\\Delta} \\cdots } \\\\ & +\\underbrace{\\left[\\begin{array}{cc} \\mathbf{J}_{r}^{j-1} \\Delta t & \\mathbf{0} \\\\ \\mathbf{0} & \\Delta \\tilde{\\mathbf{R}}_{i j-1} \\Delta t \\\\ \\mathbf{0} & \\frac{1}{2} \\Delta \\tilde{\\mathbf{R}}_{i j-1} \\Delta t^{2} \\end{array}\\right]}_{B_{j-1}}\\underbrace{ \\begin{bmatrix} \\left(\\eta_{j-1}^{gd}\\right)^T\\\\ \\left(\\eta_{j-1}^{ad}\\right)^T \\end{bmatrix} } _{\\boldsymbol{\\eta}_{j-1}^{d}} \\end{aligned} ηijΔ​=​Aj−1​​ΔR~jj−1​−ΔR~ij−1​⋅(f~j−1​−bia​)∧Δt−21​ΔR~ij−1​⋅(f~j−1​−bia​)∧Δt2​0IΔtI​00I​​​​ηij−1Δ​⋯+Bj−1​​Jrj−1​Δt00​0ΔR~ij−1​Δt21​ΔR~ij−1​Δt2​​​​ηj−1d​​(ηj−1gd​)T(ηj−1ad​)T​​​​​
Σijη\\Sigma_{ij}^{\\eta}Σijη​的递推形式如下：
Σijη=Aj−1Σij−1ηAj−1T+Bj−1ΣηBj−1T\\boldsymbol{\\Sigma}_{i j}^{\\eta}=\\mathbf{A}_{j-1} \\boldsymbol{\\Sigma}_{i j-1}^{\\eta} \\mathbf{A}_{j-1}^{T}+\\mathbf{B}_{j-1} \\boldsymbol{\\Sigma}_{\\boldsymbol{\\eta}} \\mathbf{B}_{j-1}^{T} Σijη​=Aj−1​Σij−1η​Aj−1T​+Bj−1​Ση​Bj−1T​
其中：Ση\\Sigma_{\\eta}Ση​为IMU噪声的协方差矩阵

γij=[δbijaT,δbijgT]T\\gamma_{i j} =\\left[\\delta b_{i j}^{a T}, \\delta b_{i j}^{g T}\\right]^{T}γij​=[δbijaT​,δbijgT​]T的递推形式如下
γij=γij−1+σj−1d\\gamma_{i j}=\\gamma_{i j-1}+\\sigma_{j-1}^{d} γij​=γij−1​+σj−1d​
其中：
σkd=[(σkbgd)T(σkbad)T]T\\sigma_{k}^{d}=\\left[\\left(\\sigma_{k}^{b g d}\\right)^{T} \\left(\\sigma_{k}^{b a d}\\right)^{T}\\right]^{T} σkd​=[(σkbgd​)T(σkbad​)T]T
Σijγ\\Sigma_{i j}^{\\gamma}Σijγ​的递推形式如下：
Σijγ=Σij−1γ+Σγ\\begin{array}{l} \\Sigma_{i j}^{\\gamma}=\\Sigma_{i j-1}^{\\gamma}+\\Sigma_{\\gamma} \\end{array} Σijγ​=Σij−1γ​+Σγ​​