Commit 5b8ea19d by Victor Poughon

DOC: fix math markup

parent bca81855
 ... ... @@ -149,94 +149,94 @@ For each option parameter, the list below gives the formula used. #. msinclairtocoherency (SinclairToReciprocalCoherencyMatrixFunctor) #. :math: 0.5 . (S_{hh}+S_{vv}).(S_{hh}+S_{vv})^{*}  #. :math:0.5 . (S_{hh}+S_{vv}).(S_{hh}+S_{vv})^{*} #. :math: 0.5 . (S_{hh}+S_{vv}).(S_{hh}-S_{vv})^{*}  #. :math:0.5 . (S_{hh}+S_{vv}).(S_{hh}-S_{vv})^{*} #. :math: 0.5 . (S_{hh}+S_{vv}).(2 S_{hv})^{*}  #. :math:0.5 . (S_{hh}+S_{vv}).(2 S_{hv})^{*} #. :math: 0.5 . (S_{hh}-S_{vv}).(S_{hh}-S_{vv})^{*}  #. :math:0.5 . (S_{hh}-S_{vv}).(S_{hh}-S_{vv})^{*} #. :math: 0.5 . (S_{hh}-S_{vv}).(2 S_{hv})^{*}  #. :math:0.5 . (S_{hh}-S_{vv}).(2 S_{hv})^{*} #. :math: 0.5 . (2 S_{hv}).(2 S_{hv})^{*}  #. :math:0.5 . (2 S_{hv}).(2 S_{hv})^{*} #. msinclairtocovariance (SinclairToReciprocalCovarianceMatrixFunctor) #. :math: S_{hh}.S_{hh}^{*}  #. :math:S_{hh}.S_{hh}^{*} #. :math: \sqrt{2}.S_{hh}.S_{hv}^{*}  #. :math:\sqrt{2}.S_{hh}.S_{hv}^{*} #. :math: S_{hh}.S_{vv}^{*}  #. :math:S_{hh}.S_{vv}^{*} #. :math: 2.S_{hv}.S_{hv}^{*}  #. :math:2.S_{hv}.S_{hv}^{*} #. :math: \sqrt{2}.S_{hv}.S_{vv}^{*}  #. :math:\sqrt{2}.S_{hv}.S_{vv}^{*} #. :math: S_{vv}.S_{vv}^{*}  #. :math:S_{vv}.S_{vv}^{*} #. msinclairtocircovariance (SinclairToReciprocalCircularCovarianceMatrixFunctor) #. :math: S_{ll}.S_{ll}^{*}  #. :math:S_{ll}.S_{ll}^{*} #. :math: S_{ll}.S_{lr}^{*}  #. :math:S_{ll}.S_{lr}^{*} #. :math: S_{ll}.S_{rr}^{*}  #. :math:S_{ll}.S_{rr}^{*} #. :math: S_{lr}.S_{lr}^{*}  #. :math:S_{lr}.S_{lr}^{*} #. :math: S_{lr}.S_{rr}^{*}  #. :math:S_{lr}.S_{rr}^{*} #. :math: S_{rr}.S_{rr}^{*}  #. :math:S_{rr}.S_{rr}^{*} With: - :math: S_{ll} = 0.5(S_{hh}+2j S_{hv}-S_{vv})  - :math:S_{ll} = 0.5(S_{hh}+2j S_{hv}-S_{vv}) - :math: S_{lr} = 0.5(j S_{hh}+j S_{vv})  - :math:S_{lr} = 0.5(j S_{hh}+j S_{vv}) - :math: S_{rr} = 0.5(-S_{hh}+2j S_{hv}+S_{vv})  - :math:S_{rr} = 0.5(-S_{hh}+2j S_{hv}+S_{vv}) #. mcoherencytomueller (ReciprocalCoherencyToReciprocalMuellerFunctor) #. :math: 0.5*( C_{11}+C_{22}+C_{33} )  #. :math:0.5*( C_{11}+C_{22}+C_{33} ) #. :math: Re(C_{12}) + Im(C_{22})  #. :math:Re(C_{12}) + Im(C_{22}) #. :math: Re(C_{13})  #. :math:Re(C_{13}) #. :math: Im(C_{23})  #. :math:Im(C_{23}) #. :math: Re(C_{12})  #. :math:Re(C_{12}) #. :math: 0.5*( C_{11}+C_{22}-C_{33} )  #. :math:0.5*( C_{11}+C_{22}-C_{33} ) #. :math: Re(C_{23})  #. :math:Re(C_{23}) #. :math: Im(C_{13})  #. :math:Im(C_{13}) #. :math: -Re(C_{13})  #. :math:-Re(C_{13}) #. :math: -Re(C_{23})  #. :math:-Re(C_{23}) #. :math: 0.5.Re(VAL1)  #. :math:0.5.Re(VAL1) #. :math: 0.5.Im(VAL0)  #. :math:0.5.Im(VAL0) #. :math: Im(C_{23})  #. :math:Im(C_{23}) #. :math: Im(C_{13})  #. :math:Im(C_{13}) #. :math: 0.5.Im(VAL1^{*})  #. :math:0.5.Im(VAL1^{*}) #. :math: 0.5.Re(VAL0)  #. :math:0.5.Re(VAL0) With: - :math: VAL0 = C_{33}+C_{12}-C_{11}-(C_{12}-C_{22})^{*}  - :math:VAL0 = C_{33}+C_{12}-C_{11}-(C_{12}-C_{22})^{*} - :math: VAL1 = -C_{33}+C_{12}-C_{11}-(C_{12}-C_{22})^{*}  - :math:VAL1 = -C_{33}+C_{12}-C_{11}-(C_{12}-C_{22})^{*} Where :math:C_{ij} are related to the elements of the reciprocal coherence matrix. ... ... @@ -244,26 +244,26 @@ For each option parameter, the list below gives the formula used. #. mcovariancetocoherencydegree (ReciprocalCovarianceToCoherencyDegreeFunctor) #. :math: abs(S_{hh}.S_{vv}^{*}) / sqrt(S_{hh}.S_{hh}^{*}) / sqrt(S_{vv}.S_{vv}^{*})  #. :math:abs(S_{hh}.S_{vv}^{*}) / sqrt(S_{hh}.S_{hh}^{*}) / sqrt(S_{vv}.S_{vv}^{*}) #. :math: abs(S_{hv}.S_{vv}^{*}) / sqrt(S_{hv}.S_{hv}^{*}) / sqrt(S_{vv}.S_{vv}^{*})  #. :math:abs(S_{hv}.S_{vv}^{*}) / sqrt(S_{hv}.S_{hv}^{*}) / sqrt(S_{vv}.S_{vv}^{*}) #. :math: abs(S_{hh}.S_{hv}^{*}) / sqrt(S_{hh}.S_{hh}^{*}) / sqrt(S_{hv}.S_{hv}^{*})  #. :math:abs(S_{hh}.S_{hv}^{*}) / sqrt(S_{hh}.S_{hh}^{*}) / sqrt(S_{hv}.S_{hv}^{*}) #. mcovariancetocoherency (ReciprocalCovarianceToReciprocalCoherencyFunctor) #. :math: 0.5 . ( C_{33} + C_{13} + C_{13}^{*} + C_{11} )  #. :math:0.5 . ( C_{33} + C_{13} + C_{13}^{*} + C_{11} ) #. :math: 0.5 . ( -C_{33} - C_{13} + C_{13}^{*} + C_{11} )  #. :math:0.5 . ( -C_{33} - C_{13} + C_{13}^{*} + C_{11} ) #. :math: 0.5 . ( \sqrt{2}.C_{12} + \sqrt{2}.C_{23}^{*} )  #. :math:0.5 . ( \sqrt{2}.C_{12} + \sqrt{2}.C_{23}^{*} ) #. :math: 0.5 . ( C_{33} - C_{13} - C_{13}^{*} + C_{11} )  #. :math:0.5 . ( C_{33} - C_{13} - C_{13}^{*} + C_{11} ) #. :math: 0.5 . ( \sqrt{2}.C_{12} - \sqrt{2}.C_{23}^{*} )  #. :math:0.5 . ( \sqrt{2}.C_{12} - \sqrt{2}.C_{23}^{*} ) #. :math: 0.5 . ( 2 . C_{22} )  #. :math:0.5 . ( 2 . C_{22} ) Where :math:C_{ij} are related to the elements of the reciprocal linear covariance matrix. ... ... @@ -271,168 +271,168 @@ For each option parameter, the list below gives the formula used. #. mlinearcovariancetocircularcovariance (ReciprocalLinearCovarianceToReciprocalCircularCovarianceFunctor) #. :math: 0.25 . ( C_{33}-i.\sqrt{2}.C_{23}-C_{13}+i.\sqrt{2}.C_{23}^{*}-C_{13}^{*}+2.C_{22}-i.\sqrt{2}.C_{12}+i.\sqrt{2}.C_{12}^{*}+C_{11} )  #. :math:0.25 . ( C_{33}-i.\sqrt{2}.C_{23}-C_{13}+i.\sqrt{2}.C_{23}^{*}-C_{13}^{*}+2.C_{22}-i.\sqrt{2}.C_{12}+i.\sqrt{2}.C_{12}^{*}+C_{11} ) #. :math: 0.25 . ( i.\sqrt{2}.C_{33}+2.C_{23}-i.\sqrt{2}.C_{13}+i.\sqrt{2}.C_{13}^{*}+2.C_{12}^{*}-i.\sqrt{2}.C_{11} )  #. :math:0.25 . ( i.\sqrt{2}.C_{33}+2.C_{23}-i.\sqrt{2}.C_{13}+i.\sqrt{2}.C_{13}^{*}+2.C_{12}^{*}-i.\sqrt{2}.C_{11} ) #. :math: 0.25 . ( -C_{33}+i.\sqrt{2}.C_{23}+C_{13}+i.\sqrt{2}.C_{23}^{*}+C_{13}^{*}+2.C_{22}-i.\sqrt{2}.C_{12}-i.\sqrt{2}.C_{12}^{*}-C_{11} )  #. :math:0.25 . ( -C_{33}+i.\sqrt{2}.C_{23}+C_{13}+i.\sqrt{2}.C_{23}^{*}+C_{13}^{*}+2.C_{22}-i.\sqrt{2}.C_{12}-i.\sqrt{2}.C_{12}^{*}-C_{11} ) #. :math: 0.25 . ( 2.C_{33}+2.C_{13}+2.C_{13}^{*}+2.C_{11} )  #. :math:0.25 . ( 2.C_{33}+2.C_{13}+2.C_{13}^{*}+2.C_{11} ) #. :math: 0.25 . ( i.\sqrt{2}.C_{33}+i.\sqrt{2}.C_{13}+2.C_{23}^{*}-i.\sqrt{2}.C_{13}^{*}+2.C_{12}-i.\sqrt{2}.C_{11} )  #. :math:0.25 . ( i.\sqrt{2}.C_{33}+i.\sqrt{2}.C_{13}+2.C_{23}^{*}-i.\sqrt{2}.C_{13}^{*}+2.C_{12}-i.\sqrt{2}.C_{11} ) #. :math: 0.25 . ( C_{33}+i.\sqrt{2}.C_{23}-C_{13}-i.\sqrt{2}.C_{23}^{*}-C_{13}^{*}+2.C_{22}+i.\sqrt{2}.C_{12}-i.\sqrt{2}.C_{12}^{*}+C_{11} )  #. :math:0.25 . ( C_{33}+i.\sqrt{2}.C_{23}-C_{13}-i.\sqrt{2}.C_{23}^{*}-C_{13}^{*}+2.C_{22}+i.\sqrt{2}.C_{12}-i.\sqrt{2}.C_{12}^{*}+C_{11} ) Where :math:C_{ij} are related to the elements of the reciprocal linear covariance matrix. #. muellertomcovariance (MuellerToReciprocalCovarianceFunctor) #. :math: 0.5.(M_{11}+M_{22}+2.M_{12})  #. :math:0.5.(M_{11}+M_{22}+2.M_{12}) #. :math: 0.5.\sqrt{2}.[(M_{13}+M_{23}) + j.(M_{14}+M_{24})]  #. :math:0.5.\sqrt{2}.[(M_{13}+M_{23}) + j.(M_{14}+M_{24})] #. :math: -0.5.(M_{33}+M_{44}) - j.M_{34}  #. :math:-0.5.(M_{33}+M_{44}) - j.M_{34} #. :math: M_{11}-M_{22}  #. :math:M_{11}-M_{22} #. :math: 0.5.\sqrt{2}.[(M_{13}-M_{23}) + j.(M_{14}-M_{24})]  #. :math:0.5.\sqrt{2}.[(M_{13}-M_{23}) + j.(M_{14}-M_{24})] #. :math: 0.5.(M_{11}+M_{22}-2.M_{12})  #. :math:0.5.(M_{11}+M_{22}-2.M_{12}) — Bistatic case — #. bsinclairtocoherency (SinclairToCoherencyMatrixFunctor) #. :math: (S_{hh}+S_{vv}).(S_{hh}+S_{vv})^{*}  #. :math:(S_{hh}+S_{vv}).(S_{hh}+S_{vv})^{*} #. :math: (S_{hh}+S_{vv}).(S_{hh}-S_{vv})^{*}  #. :math:(S_{hh}+S_{vv}).(S_{hh}-S_{vv})^{*} #. :math: (S_{hh}+S_{vv}).(S_{hv}+S_{vh})^{*}  #. :math:(S_{hh}+S_{vv}).(S_{hv}+S_{vh})^{*} #. :math: (S_{hh}+S_{vv}).( j (S_{hv}-S_{vh}))^{*}  #. :math:(S_{hh}+S_{vv}).( j (S_{hv}-S_{vh}))^{*} #. :math: (S_{hh}-S_{vv}).(S_{hh}-S_{vv})^{*}  #. :math:(S_{hh}-S_{vv}).(S_{hh}-S_{vv})^{*} #. :math: (S_{hh}-S_{vv}).(S_{hv}+S_{vh})^{*}  #. :math:(S_{hh}-S_{vv}).(S_{hv}+S_{vh})^{*} #. :math: (S_{hh}-S_{vv}).( j (S_{hv}-S_{vh}))^{*}  #. :math:(S_{hh}-S_{vv}).( j (S_{hv}-S_{vh}))^{*} #. :math: (S_{hv}+S_{vh}).(S_{hv}+S_{vh})^{*}  #. :math:(S_{hv}+S_{vh}).(S_{hv}+S_{vh})^{*} #. :math: (S_{hv}+S_{vh}).( j (S_{hv}-S_{vh}))^{*}  #. :math:(S_{hv}+S_{vh}).( j (S_{hv}-S_{vh}))^{*} #. :math: j (S_{hv}-S_{vh}).( j (S_{hv}-S_{vh}))^{*}  #. :math:j (S_{hv}-S_{vh}).( j (S_{hv}-S_{vh}))^{*} #. bsinclairtocovariance (SinclairToCovarianceMatrixFunctor) #. :math: S_{hh}.S_{hh}^{*}  #. :math:S_{hh}.S_{hh}^{*} #. :math: S_{hh}.S_{hv}^{*}  #. :math:S_{hh}.S_{hv}^{*} #. :math: S_{hh}.S_{vh}^{*}  #. :math:S_{hh}.S_{vh}^{*} #. :math: S_{hh}.S_{vv}^{*}  #. :math:S_{hh}.S_{vv}^{*} #. :math: S_{hv}.S_{hv}^{*}  #. :math:S_{hv}.S_{hv}^{*} #. :math: S_{hv}.S_{vh}^{*}  #. :math:S_{hv}.S_{vh}^{*} #. :math: S_{hv}.S_{vv}^{*}  #. :math:S_{hv}.S_{vv}^{*} #. :math: S_{vh}.S_{vh}^{*}  #. :math:S_{vh}.S_{vh}^{*} #. :math: S_{vh}.S_{vv}^{*}  #. :math:S_{vh}.S_{vv}^{*} #. :math: S_{vv}.S_{vv}^{*}  #. :math:S_{vv}.S_{vv}^{*} #. bsinclairtocircovariance (SinclairToCircularCovarianceMatrixFunctor) #. :math: S_{ll}.S_{ll}^{*}  #. :math:S_{ll}.S_{ll}^{*} #. :math: S_{ll}.S_{lr}^{*}  #. :math:S_{ll}.S_{lr}^{*} #. :math: S_{ll}.S_{rl}^{*}  #. :math:S_{ll}.S_{rl}^{*} #. :math: S_{ll}.S_{rr}^{*}  #. :math:S_{ll}.S_{rr}^{*} #. :math: S_{lr}.S_{lr}^{*}  #. :math:S_{lr}.S_{lr}^{*} #. :math: S_{lr}.S_{rl}^{*}  #. :math:S_{lr}.S_{rl}^{*} #. :math: S_{lr}.S_{rr}^{*}  #. :math:S_{lr}.S_{rr}^{*} #. :math: S_{rl}.S_{rl}^{*}  #. :math:S_{rl}.S_{rl}^{*} #. :math: S_{rl}.S_{rr}^{*}  #. :math:S_{rl}.S_{rr}^{*} #. :math: S_{rr}.S_{rr}^{*}  #. :math:S_{rr}.S_{rr}^{*} With: - :math: S_{ll} = 0.5(S_{hh}+j S_{hv}+j S_{vh}-S_{vv})  - :math:S_{ll} = 0.5(S_{hh}+j S_{hv}+j S_{vh}-S_{vv}) - :math: S_{lr} = 0.5(j S_{hh}+S_{hv}-S_{vh}+j S_{vv})  - :math:S_{lr} = 0.5(j S_{hh}+S_{hv}-S_{vh}+j S_{vv}) - :math: S_{rl} = 0.5(j S_{hh}-S_{hv}+ S_{vh}+j S_{vv})  - :math:S_{rl} = 0.5(j S_{hh}-S_{hv}+ S_{vh}+j S_{vv}) - :math: S_{rr} = 0.5(-S_{hh}+j S_{hv}+j S_{vh}+S_{vv})  - :math:S_{rr} = 0.5(-S_{hh}+j S_{hv}+j S_{vh}+S_{vv}) — Both cases — #. sinclairtomueller (SinclairToMueller) #. :math: 0.5 Re( T_{xx}.T_{xx}^{*} + T_{xy}.T_{xy}^{*} + T_{yx}.T_{yx}^{*} + T_{yy}.T_{yy}^{*} )  #. :math:0.5 Re( T_{xx}.T_{xx}^{*} + T_{xy}.T_{xy}^{*} + T_{yx}.T_{yx}^{*} + T_{yy}.T_{yy}^{*} ) #. :math: 0.5 Re( T_{xx}.T_{xx}^{*} - T_{xy}.T_{xy}^{*} + T_{yx}.T_{yx}^{*} - T_{yy}.T_{yy}^{*} )  #. :math:0.5 Re( T_{xx}.T_{xx}^{*} - T_{xy}.T_{xy}^{*} + T_{yx}.T_{yx}^{*} - T_{yy}.T_{yy}^{*} ) #. :math: Re( T_{xx}.T_{xy}^{*} + T_{yx}.T_{yy}^{*} )  #. :math:Re( T_{xx}.T_{xy}^{*} + T_{yx}.T_{yy}^{*} ) #. :math: Im( T_{xx}.T_{xy}^{*} + T_{yx}.T_{yy}^{*} )  #. :math:Im( T_{xx}.T_{xy}^{*} + T_{yx}.T_{yy}^{*} ) #. :math: 0.5 Re( T_{xx}.T_{xx}^{*} + T_{xy}.T_{xy}^{*} - T_{yx}.T_{yx}^{*} - T_{yy}.T_{yy}^{*} )  #. :math:0.5 Re( T_{xx}.T_{xx}^{*} + T_{xy}.T_{xy}^{*} - T_{yx}.T_{yx}^{*} - T_{yy}.T_{yy}^{*} ) #. :math: 0.5 Re( T_{xx}.T_{xx}^{*} - T_{xy}.T_{xy}^{*} - T_{yx}.T_{yx}^{*} + T_{yy}.T_{yy}^{*} )  #. :math:0.5 Re( T_{xx}.T_{xx}^{*} - T_{xy}.T_{xy}^{*} - T_{yx}.T_{yx}^{*} + T_{yy}.T_{yy}^{*} ) #. :math: Re( T_{xx}.T_{xy}^{*} - T_{yx}.T_{yy}^{*} )  #. :math:Re( T_{xx}.T_{xy}^{*} - T_{yx}.T_{yy}^{*} ) #. :math: Im( T_{xx}.T_{xy}^{*} - T_{yx}.T_{yy}^{*} )  #. :math:Im( T_{xx}.T_{xy}^{*} - T_{yx}.T_{yy}^{*} ) #. :math: Re( T_{xx}.T_{yx}^{*} + T_{xy}.T_{yy}^{*} )  #. :math:Re( T_{xx}.T_{yx}^{*} + T_{xy}.T_{yy}^{*} ) #. :math: Im( T_{xx}.T_{yx}^{*} - T_{xy}.T_{yy}^{*} )  #. :math:Im( T_{xx}.T_{yx}^{*} - T_{xy}.T_{yy}^{*} ) #. :math: Re( T_{xx}.T_{yy}^{*} + T_{xy}.T_{yx}^{*} )  #. :math:Re( T_{xx}.T_{yy}^{*} + T_{xy}.T_{yx}^{*} ) #. :math: Im( T_{xx}.T_{yy}^{*} - T_{xy}.T_{yx}^{*} )  #. :math:Im( T_{xx}.T_{yy}^{*} - T_{xy}.T_{yx}^{*} ) #. :math: Re( T_{xx}.T_{yx}^{*} + T_{xy}.T_{yy}^{*} )  #. :math:Re( T_{xx}.T_{yx}^{*} + T_{xy}.T_{yy}^{*} ) #. :math: Im( T_{xx}.T_{yx}^{*} - T_{xy}.T_{yy}^{*} )  #. :math:Im( T_{xx}.T_{yx}^{*} - T_{xy}.T_{yy}^{*} ) #. :math: Re( T_{xx}.T_{yy}^{*} + T_{xy}.T_{yx}^{*} )  #. :math:Re( T_{xx}.T_{yy}^{*} + T_{xy}.T_{yx}^{*} ) #. :math: Im( T_{xx}.T_{yy}^{*} - T_{xy}.T_{yx}^{*} )  #. :math:Im( T_{xx}.T_{yy}^{*} - T_{xy}.T_{yx}^{*} ) With : - :math: T_{xx} = -S_{hh}  - :math:T_{xx} = -S_{hh} - :math: T_{xy} = -S_{hv}  - :math:T_{xy} = -S_{hv} - :math: T_{yx} = S_{vh}  - :math:T_{yx} = S_{vh} - :math: T_{yy} = S_{vv}  - :math:T_{yy} = S_{vv} #. muellertopoldegandpower (MuellerToPolarisationDegreeAndPowerFunctor) #. :math: P_{min}  #. :math:P_{min} #. :math: P_{max}  #. :math:P_{max} #. :math: DegP_{min}  #. :math:DegP_{min} #. :math: DegP_{max}  #. :math:DegP_{max} Examples : ... ... @@ -549,17 +549,17 @@ Alpha, A(Anisotropy). Here are the formula used (refer to the previous section about how the coherence matrix is obtained from the Sinclair one): #. :math: entropy = -\sum_{i=0}^{2} \frac{p[i].\log{p[i]}}{\log{3}}  #. :math:entropy = -\sum_{i=0}^{2} \frac{p[i].\log{p[i]}}{\log{3}} #. :math: \alpha = \sum_{i=0}^{2} p[i].\alpha_{i}  #. :math:\alpha = \sum_{i=0}^{2} p[i].\alpha_{i} #. :math: anisotropy = \frac {SortedEigenValues[1] - SortedEigenValues[2]}{SortedEigenValues[1] + SortedEigenValues[2]}  #. :math:anisotropy = \frac {SortedEigenValues[1] - SortedEigenValues[2]}{SortedEigenValues[1] + SortedEigenValues[2]} Where: - :math: p[i] = max(SortedEigenValues[i], 0) / \sum_{i=0}^{2, SortedEigenValues[i]>0} SortedEigenValues[i]  - :math:p[i] = max(SortedEigenValues[i], 0) / \sum_{i=0}^{2, SortedEigenValues[i]>0} SortedEigenValues[i] - :math: \alpha_{i} = \left| SortedEigenVector[i] \right|* \frac{180}{\pi} - :math:\alpha_{i} = \left| SortedEigenVector[i] \right|* \frac{180}{\pi} Example : ... ...
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