\form#0:\[ COV = H M H^\top \]
\form#1:$ [x y \phi]^T $
\form#2:$ [x y z yaw pitch roll]^T $
\form#3:$ [x y]^T $
\form#4:$ [x y z]^T $
\form#5:$ (r,\mathbf{u}) $
\form#6:$ \mathbf{u} = (x,y,z) $
\form#7:$ p = f(\cdot) = q_{this} \times r $
\form#8:$ \frac{\partial f}{\partial q_{this} } $
\form#9:$ D_\mathrm{KL}(\mathcal{N}_0 \| \mathcal{N}_1) = { 1 \over 2 } ( \log_e ( { \det \Sigma_1 \over \det \Sigma_0 } ) + \mathrm{tr} ( \Sigma_1^{-1} \Sigma_0 ) + ( \mu_1 - \mu_0 )^\top \Sigma_1^{-1} ( \mu_1 - \mu_0 ) - N ) $
\form#10:\[ v_out = \left( \begin{array}{c c c} \hat{i} ~ \hat{j} ~ \hat{k} \\ x0 ~ y0 ~ z0 \\ x1 ~ y1 ~ z1 \\ \end{array} \right) \]
\form#11:\[ M([x ~ y ~ z]^\top) = \left( \begin{array}{c c c} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{array} \right) \]
\form#12:\[ -M([x ~ y ~ z]^\top) = \left( \begin{array}{c c c} 0 & z & -y \\ -z & 0 & x \\ y & -x & 0 \end{array} \right) \]
\form#13:$ d = [ dx ~ dy ~ dz ] $
\form#14:$ M $
\form#15:\[ M = \left( \begin{array}{c c c} v^1_x ~ v^2_x ~ v^3_x \\ v^1_y ~ v^2_y ~ v^3_y \\ v^1_z ~ v^2_z ~ v^3_z \end{array} \right) \]
\form#16:\[ v^1 = \frac{d}{|d|} \]
\form#17:\[ v^2 = \frac{[-dy ~ dx ~ 0 ]}{\sqrt{dx^2+dy^2}} \]
\form#18:\[ v^2 = [1 ~ 0 ~ 0] \]
\form#19:\[ v^3 = v^1 \times v^2 \]
\form#20:$\vec{w}$
\form#21:$ \theta = |\vec{w}|$
\form#22:$\frac{\sin \theta}{\theta}$
\form#23:$\frac{1 - \cos \theta}{\theta^2}$
\form#24:\[ \mathbf{q} = \left( \begin{array}{c} \cos (\phi /2) \cos (\theta /2) \cos (\psi /2) + \sin (\phi /2) \sin (\theta /2) \sin (\psi /2) \\ \sin (\phi /2) \cos (\theta /2) \cos (\psi /2) - \cos (\phi /2) \sin (\theta /2) \sin (\psi /2) \\ \cos (\phi /2) \sin (\theta /2) \cos (\psi /2) + \sin (\phi /2) \cos (\theta /2) \sin (\psi /2) \\ \cos (\phi /2) \cos (\theta /2) \sin (\psi /2) - \sin (\phi /2) \sin (\theta /2) \cos (\psi /2) \\ \end{array}\right) \]
\form#25:$ \phi = roll $
\form#26:$ \theta = pitch $
\form#27:$ \psi = yaw $
\form#28:$ f(x,u) = x \oplus u $
\form#29:$ \frac{\partial f}{\partial x} $
\form#30:$ \frac{\partial f}{\partial u} $
\form#31:$Ax+By+C=0$
\form#32:$Ax+By+Cz+D=0$
\form#33:$\left[A,B,C\right]$
\form#34:$\left[A,B,C,D\right]$
\form#35:\[ return = - \log N + \log \sum_{i=1}^N e^{ll_i-ll_{max}} + ll_{max} \]
\form#36:$ ]-\pi,\pi [ $
\form#37:$ \pi $
\form#38:\[ return = \log \left( \frac{1}{\sum_i e^{lw_i}} \sum_i e^{lw_i} e^{ll_i} \right) \]
\form#39:\[ d^2 = (X-MU)^\top \Sigma^{-1} (X-MU) \]
\form#40:\[ d = \sqrt{ (X-MU)^\top \Sigma^{-1} (X-MU) } \]
\form#41:\[ d^2 = \Delta_\mu^\top (\Sigma_1 + \Sigma_2 - 2 \Sigma_12 )^{-1} \Delta_\mu \]
\form#42:\[ d = \sqrt{ \Delta_\mu^\top (\Sigma_1 + \Sigma_2 - 2 \Sigma_12 )^{-1} \Delta_\mu } \]
\form#43:\[ d^2 = \Delta_\mu^\top \Sigma^{-1} \Delta_\mu \]
\form#44:\[ d^2 = \sqrt( \Delta_\mu^\top \Sigma^{-1} \Delta_\mu ) \]
\form#45:\[ D = \frac{1}{(2 \pi)^{0.5 N} \sqrt{} } \exp( \Delta_\mu^\top (\Sigma_1 + \Sigma_2 - 2 \Sigma_12)^{-1} \Delta_\mu) \]
\form#46:\[ D = \frac{1}{(2 \pi)^{0.5 N} \sqrt{} } \exp( \Delta_\mu^\top (\Sigma_1 + \Sigma_2)^{-1} \Delta_\mu) \]
\form#47:\[ exp( -\frac{1}{2} D^2 ) \]
\form#48:$ D^2 $
\form#49:$ p(\mathbf{x}) = [x ~ y ~ z ]^t $
\form#50:$ p(\mathbf{x}) = \sum\limits_{i=1}^N \omega^i \mathcal{N}( \mathbf{x} ; \bar{\mathbf{x}}^i, \mathbf{\Sigma}^i ) $
\form#51:$ a \ominus b $
\form#52:$ a = this \oplus D $
\form#53:$ this = A \oplus B $
\form#54:$ u' = this \oplus u $
\form#55:$ G = P \oplus L $
\form#56:$ this = A \ominus B $
\form#57:$ RET = this \oplus b $
\form#58:$ this = this \oplus b $
\form#59:$ a \oplus b $
\form#60:$ L = G \ominus this $
\form#61:$ p(\mathbf{x}) = [x ~ y ~ z ~ yaw ~ pitch ~ roll]^t $
\form#62:$ y = x \oplus u $
\form#63:$ G = this \oplus L $
\form#64:$ ret = this \oplus p $
\form#65:$ this = this \ominus b $
\form#66:$ ret = this \ominus p $
\form#67:$ p(\mathbf{x}) = [x ~ y ~ z ~ qr ~ qx ~ qy ~ qz]^\top $
\form#68:$ \oplus $
\form#69:$ \ominus $
\form#70:$ ||\mathbf{x}|| = \sqrt{x^2+y^2+z^2} $
\form#71:$ p(\mathbf{x}) = [x ~ y ~ \phi ]^t $
\form#72:$ \mathbf{C} $
\form#73:$ \mathbf{C} = \mathbf{A} \oplus \mathbf{B} $
\form#74:$ \mathbf{R}~\mathbf{COV}~\mathbf{R}^t $
\form#75:$ \mathbf{R} = \left[ \begin{array}{ccc} \cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{array}\right] $
\form#76:$ this = x1 \ominus x0 $
\form#77:$ [ x ~ y ~ \phi ] $
\form#78:$ [ x ~ y ~ z ~ yaw ~ pitch ~ roll ] $
\form#79:\[ \frac{\partial pseudoLn(P_1 D P_2^{-1}) }{\partial \epsilon_1} \]
\form#80:\[ \frac{\partial pseudoLn(P_1 D P_2^{-1}) }{\partial \epsilon_2} \]
\form#81:$ \epsilon_1 $
\form#82:$ \epsilon_2 $
\form#83:\[ \sum\limits_i e_i \]
\form#84:$ e_i $
\form#85:\[ e_i = | x_{this} - q \oplus x_{other} |^2 \]
\form#86:$ \hat{x}_{k|k-1} = f( \hat{x}_{k-1|k-1}, u_k ) $
\form#87:\[ \hat{x}_{k-1|k-1} \]
\form#88:$ \hat{x}_{k|k-1} $
\form#89:$V \times V$
\form#90:$ Q_k $
\form#91:$ h_i(x) $
\form#92:$ \frac{\partial h_i}{\partial x} $
\form#93:$ \frac{\partial h_i}{\partial y_i} $
\form#94:$ y_n=y(x,z_n) $
\form#95:$F \times V$
\form#96:$ \frac{\partial y_n}{\partial x_v} $
\form#97:$F \times O$
\form#98:$ \frac{\partial y_n}{\partial h_n} $
\form#99:$ \frac{\partial y_n}{\partial h_n} R \frac{\partial y_n}{\partial h_n}^\top $
\form#100:$ p_{ij} $
\form#101:$ \Delta_i^j = j \ominus i $
\form#102:$ \Delta_i^j = i \ominus j $
\form#103:$ p(m_{xy}) = \frac{1}{1+exp(-log_odd)} $
\form#104:$ p_i $
\form#105:$ p'_i = b \oplus p_i $
\form#106:$ \sum{\frac{wR}{w}} $
\form#107:$\frac\pi N$
\form#108:$ point_this = q \oplus point_other $
\form#109:$\sigma^2_p$
\form#110:$x$
\form#111:$y$
\form#112:\[ K(x^2) = \frac{x^2}{x^2+\rho^2} \]
\form#113:$ \sigma^2_p $
\form#114:$ (0.02)^2 $
\form#115:$ \ominus F $
\form#116:$ F $
\form#117:$ \frac{\partial h}{\partial y} $
\form#118:$ h = (h_x ~ h_y) $
\form#119:$ y=( y_x ~ y_y ~ y_z ) $
\form#120:\[ \frac{\partial h}{\partial y} = \frac{\partial h}{\partial u} \frac{\partial u}{\partial y} \]
\form#121:\[ \frac{\partial u}{\partial y} = \left( \begin{array}{ccc} \frac{f_x}{y_z} & 0 & - y \frac{f_x}{y_z^2} \\ 0 & \frac{f_y}{y_z} & - y \frac{f_y}{y_z^2} \\ \end{array} \right) \]
\form#122:$ f_x, f_y $
\form#123:\[ f = 1+ 2 k_1 (u_x^2+u_y^2) \]
\form#124:\[ \frac{\partial h}{\partial u} = \left( \begin{array}{cc} \frac{ 1+2 k_1 u_y^2 }{f^{3/2}} & -\frac{2 u_x u_y k_1 }{f^{3/2}} \\ -\frac{2 u_x u_y k_1 }{f^{3/2}} & \frac{ 1+2 k_1 u_x^2 }{f^{3/2}} \end{array} \right) \]
\form#125:$ \frac{\partial y}{\partial h} $
\form#126:$ P \ominus F $
\form#127:$ F \oplus P $
\form#128:$ A $
\form#129:$ A = P^TLDL^*P $
\form#130:$ A x = b $
\form#131:$ \ell^p $
\form#132:$ \ell^\infty $
\form#133:$ s \Vert \frac{*this}{s} \Vert $
\form#134:$ \sigma $
\form#135:$ P_\sigma $
\form#136:$ (e_1,\ldots,e_p) $
\form#137:\[ P_\sigma(e_i) = e_{\sigma(i)}. \]
\form#138:$ \sigma, \tau $
\form#139:\[ P_{\sigma\circ\tau} = P_\sigma P_\tau. \]
\form#140:$[T_{n-1} \ldots T_{i} \ldots T_{0}]$
\form#141:$ T_{i} $
\form#142:$ T_{i} M$
\form#143:$ M T_{i}$
\form#144:$ v $
\form#145:$ w $
\form#146:$ p $
\form#147:\[ \Vert v - w \Vert \leqslant p\,\min(\Vert v\Vert, \Vert w\Vert). \]
\form#148:$ x $
\form#149:\[ \Vert v \Vert \leqslant p\,\vert x\vert. \]
\form#150:\[ \Vert v \Vert \leqslant p\,\Vert w\Vert. \]
\form#151:$ Ax = \lambda B x $
\form#152:$ ABx = \lambda x $
\form#153:$ BAx = \lambda x $
\form#154:$(ij)$
\form#155:$ this = this + \alpha u v^* + conj(\alpha) v u^* $
\form#156:$ this = this + \alpha ( u u^* ) $
\form#157:$ this = this + \alpha ( u^* u ) $
\form#158:$ A = Q T Q^* $
\form#159:$ Q $
\form#160:$ T $
\form#161:$ H $
\form#162:$ A = Q H Q^T $
\form#163:$ Q^{-1} = Q^T $
\form#164:$ A = Q H Q^* $
\form#165:$ Q^{-1} = Q^* $
\form#166:$ A = U T U^T $
\form#167:$ U^{-1} = U^T $
\form#168:$ \lambda $
\form#169:$ Av = \lambda v $
\form#170:$ D $
\form#171:$ V $
\form#172:$ A V = V D $
\form#173:$ A = V D V^{-1} $
\form#174:\[ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} \]
\form#175:$ u $
\form#176:$ u \pm iv $
\form#177:$ Av = \lambda Bv $
\form#178:$ B $
\form#179:$ A = U T U^*$
\form#180:\[ \|A\|_2 = \max_x \frac{\|Ax\|_2}{\|x\|_2} \]
\form#181:$ A^*A $
\form#182:$ 4n^3/3 $
\form#183:$ n $
\form#184:$ Q = H_{N-1} \ldots H_1 H_0 $
\form#185:$ H_i $
\form#186:$ H_i = (I - h_i v_i v_i^T) $
\form#187:$ h_i $
\form#188:$ i $
\form#189:$ v_i $
\form#190:$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T $
\form#191:$ mat = Q T Q^* $
\form#192:$ 10n^3/3 $
\form#193:$25n^3$
\form#194:$10n^3$
\form#195:$ k $
\form#196:$ AV = VD $
\form#197:$ \begin{bmatrix} u & v \\ -v & u \end{bmatrix} $
\form#198:$ 25n^3 $
\form#199:$ 10n^3 $
\form#200:$ 9n^3 $
\form#201:$ A^{1/2} = V D^{1/2} V^{-1} $
\form#202:$ V D^{-1/2} V^{-1} $
\form#203:$ x^* B x = 1 $
\form#204:$ B = LL^* $
\form#205:$ L^{-1} A (L^*)^{-1} $
\form#206:$ L^{*} A L $
\form#207:$ L^{-1} A (L^*)^{-1} (L^* x) = \lambda (L^* x) $
\form#208:$ O(n^3) $
\form#209:$ w+xi+yj+zk $
\form#210:$ \left( \begin{array}{cc} linear & translation\\ 0 ... 0 & 1 \end{array} \right) $
\form#211:$ \left( \begin{array}{cc} I & t \\ 0\,...\,0 & 1 \end{array} \right) $
\form#212:$ \left( \begin{array}{cc} R & 0\\ 0\,...\,0 & 1 \end{array} \right) $
\form#213:$ \left( \begin{array}{cc} L & 0\\ 0\,...\,0 & 1 \end{array} \right) $
\form#214:$ \left( \begin{array}{c} A\\ 0\,...\,0\,1 \end{array} \right) $
\form#215:$ \left( \begin{array}{c} v\\ 1 \end{array} \right) $
\form#216:$ \left( \begin{array}{ccc} v_1 & ... & v_n\\ 1 & ... & 1 \end{array} \right) $
\form#217:$ n \cdot x + d = 0 $
\form#218:$ d $
\form#219:$ \mathbf{o} $
\form#220:$ \mathbf{d} $
\form#221:$ l(t) = \mathbf{o} + t \mathbf{d} $
\form#222:$ t \in \mathbf{R} $
\form#223:$ c, \mathbf{R}, $
\form#224:$ \mathbf{t} $
\form#225:\begin{align*} \frac{1}{n} \sum_{i=1}^n \vert\vert y_i - (c\mathbf{R}x_i + \mathbf{t}) \vert\vert_2^2 \end{align*}
\form#226:$ \Sigma_{\mathbf{x}\mathbf{y}} \in \mathbb{R}^{d \times d} $
\form#227:$ \mathbf{x} $
\form#228:$ \mathbf{y} $
\form#229:$d$
\form#230:$O(d^3)$
\form#231:$O(dm)$
\form#232:$d \times m$
\form#233:$ \mathbf{x} = \left( x_1, \hdots, x_n \right) $
\form#234:$ \mathbf{y} = \left( y_1, \hdots, y_n \right) $
\form#235:$ c=1 $
\form#236:\begin{align*} T = \begin{bmatrix} c\mathbf{R} & \mathbf{t} \\ \mathbf{0} & 1 \end{bmatrix} \end{align*}
\form#237:$ c_0*x_0 + ... + c_{d-1}*x_{d-1} + c_d = 0 $
\form#238:$ H *this = [ beta 0 ... 0]^T $
\form#239:$ H = I - tau v v^*$
\form#240:$ v^T = [1 essential^T] $
\form#241:$ n \times n $
\form#242:$ H = \prod_{i=0}^{n-1} H_i $
\form#243:$ H_i = I - h_i v_i v_i^* $
\form#244:\[ v_i = [\underbrace{0, \ldots, 0}_{i-1\mbox{ zeros}}, 1, \underbrace{*, \ldots,*}_{n-i\mbox{ arbitrary entries}} ]. \]
\form#245:$ n-i $
\form#246:$ MH $
\form#247:$ HM $
\form#248:$ H = H_0 H_1 \ldots H_{n-1} $
\form#249:$ H = H_{\mathrm{shift}} H_{\mathrm{shift}+1} \ldots H_{n-1} $
\form#250:$ H^T = H_{n-1}^T \ldots H_1^T H_0^T $
\form#251:$ \theta $
\form#252:$ J = \left ( \begin{array}{cc} c & \overline s \\ -s & \overline c \end{array} \right ) $
\form#253:$ v = J^* v $
\form#254:$ B = \left ( \begin{array}{cc} x & y \\ \overline y & z \end{array} \right )$
\form#255:$ A = J^* B J $
\form#256:$ B = \left ( \begin{array}{cc} \text{this}_{pp} & \text{this}_{pq} \\ (\text{this}_{pq})^* & \text{this}_{qq} \end{array} \right )$
\form#257:$ G^* $
\form#258:$ V = \left ( \begin{array}{c} p \\ q \end{array} \right )$
\form#259:$ G^* V = \left ( \begin{array}{c} r \\ 0 \end{array} \right )$
\form#260:$ \left ( \begin{array}{cc} x \\ y \end{array} \right ) = J \left ( \begin{array}{cc} x \\ y \end{array} \right ) $
\form#261:$ B = \left ( \begin{array}{cc} \text{*this.row}(p) \\ \text{*this.row}(q) \end{array} \right ) $
\form#262:$ B = \left ( \begin{array}{cc} \text{*this.col}(p) & \text{*this.col}(q) \end{array} \right ) $
\form#263:$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert $
\form#264:\[ \mathbf{A} = \mathbf{Q} \, \mathbf{R} \]
\form#265:\[ \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R} \]
\form#266:\[ A = U S V^* \]
\form#267:$ O(n^2p) $
\form#268:$ \Vert A x - b \Vert $