update notations

content/CSE559A/CSE559A_L26.md

@@ -17,16 +17,16 @@ If we set the config for the first camera as the world origin and $[I|0]\begin{p
 Notice that $x'\cdot [t\times (Ry)]=0$
 
 $$
-x'^T E x_1 = 0
+x'^\top E x_1 = 0
 $$
 
 We denote the constraint defined by the Essential Matrix as $E$.
 
 $E x$ is the epipolar line associated with $x$ ($l'=Ex$)
 
-$E^T x'$ is the epipolar line associated with $x'$ ($l=E^T x'$)
+$E^\top x'$ is the epipolar line associated with $x'$ ($l=E^\top x'$)
 
-$E e=0$ and $E^T e'=0$ ($x$ and $x'$ don't matter)
+$E e=0$ and $E^\top e'=0$ ($x$ and $x'$ don't matter)
 
 $E$ is singular (rank 2) and have five degrees of freedom.
 
@@ -35,13 +35,13 @@ $E$ is singular (rank 2) and have five degrees of freedom.
 If the calibration matrices $K$ and $K'$ are unknown, we can write the epipolar constraint in terms of unknown normalized coordinates:
 
 $$
-x'^T_{norm} E x_{norm} = 0
+x'^\top_{norm} E x_{norm} = 0
 $$
 
 where $x_{norm}=K^{-1} x$, $x'_{norm}=K'^{-1} x'$
 
 $$
-x'^T_{norm} E x_{norm} = 0\implies x'^T_{norm} Fx=0
+x'^\top_{norm} E x_{norm} = 0\implies x'^\top_{norm} Fx=0
 $$
 
 where $F=K'^{-1}EK^{-1}$ is the **Fundamental Matrix**.
@@ -60,17 +60,17 @@ Properties of $F$:
 
 $F x$ is the epipolar line associated with $x$ ($l'=F x$)
 
-$F^T x'$ is the epipolar line associated with $x'$ ($l=F^T x'$)
+$F^\top x'$ is the epipolar line associated with $x'$ ($l=F^\top x'$)
 
-$F e=0$ and $F^T e'=0$
+$F e=0$ and $F^\top e'=0$
 
 $F$ is singular (rank two) and has seven degrees of freedom
 
 #### Estimating the fundamental matrix
 
-Given: correspondences $x=(x,y,1)^T$ and $x'=(x',y',1)^T$
+Given: correspondences $x=(x,y,1)^\top$ and $x'=(x',y',1)^\top$
 
-Constraint: $x'^T F x=0$
+Constraint: $x'^\top F x=0$
 
 $$
 (x',y',1)\begin{bmatrix}
@@ -95,7 +95,7 @@ F=U\begin{bmatrix}
 \sigma_1 & 0 \\
 0 & \sigma_2 \\
 0 & 0
-\end{bmatrix}V^T
+\end{bmatrix}V^\top
 $$
 
 ## Structure from Motion
@@ -126,7 +126,7 @@ a_{21} & a_{22} & a_{23} & t_2 \\
 0 & 0 & 0 & 1
 \end{bmatrix}=\begin{bmatrix}
 A & t \\
-0^T & 1
+0^\top & 1
 \end{bmatrix}
 $$
 
@@ -160,10 +160,10 @@ The reconstruction is defined up to an arbitrary affine transformation $Q$ (12 d
 $$
 \begin{bmatrix}
 A & t \\
-0^T & 1
+0^\top & 1
 \end{bmatrix}\rightarrow\begin{bmatrix}
 A & t \\
-0^T & 1
+0^\top & 1
 \end{bmatrix}Q^{-1}, \quad \begin{pmatrix}X_j\\1\end{pmatrix}\rightarrow Q\begin{pmatrix}X_j\\1\end{pmatrix}
 $$