From 412c906a4236b9a6c0204352093d147cd950e436 Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Sat, 20 Apr 2024 15:46:47 +0200 Subject: [PATCH] Add IPCV2 lenses and Zhang's method --- .../module2/img/_zhang_corner_homography.pdf | Bin 0 -> 45642 bytes .../module2/img/_zhang_image_acquistion.pdf | Bin 0 -> 254821 bytes .../module2/img/radial_distortion.png | Bin 0 -> 18560 bytes .../module2/sections/_image_formation.tex | 401 ++++++++++++++++-- 4 files changed, 368 insertions(+), 33 deletions(-) create mode 100644 src/image-processing-and-computer-vision/module2/img/_zhang_corner_homography.pdf create mode 100644 src/image-processing-and-computer-vision/module2/img/_zhang_image_acquistion.pdf create mode 100644 src/image-processing-and-computer-vision/module2/img/radial_distortion.png diff --git a/src/image-processing-and-computer-vision/module2/img/_zhang_corner_homography.pdf 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zZutSsV{h=6ta-<0X)u=alJGn4{mf_U@0l8KnzeW(pO;t+B>Emo4G}v{Nx7u={W{7v zzCR(ki?#SHn?vnFs)wd9c<=W$=;*1Sq_qXUy`G0^C#rzDD4@?ouP?4!!K9yEgkDF7 zN!s2Qr&IlU+X4JVJc%R%_5dc7JZ~6lv-a!4;@A&qaGm%)5GtcDNB{YsX}g(>Svma}CKs z!rECCECTb)5;Ko#RkfgLnyuK{rbA-)f>o^}YYo1u(Q9WS_Jkkqli3|dPu!5} zcwO1{i^}o(_~1{r<0Rq<^47-93*@~9xy@fM1BI-}sNIUaFgUey2}6#xF^o!1?QH1kOK VQ-#>WDPS0onv#}c>4PWF{tq)s_h|qC literal 0 HcmV?d00001 diff --git a/src/image-processing-and-computer-vision/module2/sections/_image_formation.tex b/src/image-processing-and-computer-vision/module2/sections/_image_formation.tex index 2915b91..c839731 100644 --- a/src/image-processing-and-computer-vision/module2/sections/_image_formation.tex +++ b/src/image-processing-and-computer-vision/module2/sections/_image_formation.tex @@ -59,8 +59,6 @@ is done in two steps: \end{figure} \item[Intrinsic parameters] \marginnote{Intrinsic parameters} - Parameters needed to convert from CRF to IRF. - By fixing $f_u = \frac{f}{\Delta u}$ and $f_v = \frac{f}{\Delta v}$, the projection can be rewritten as: \[ u = f_u\frac{x_C}{z_C} + u_0 @@ -68,6 +66,17 @@ is done in two steps: v = f_v\frac{y_C}{z_C} +v_0 \] Therefore, there is a total of 4 parameters: $f_u$, $f_v$, $u_0$ and $v_0$. + + \begin{remark} + A more general model includes a further parameter (skew) + to account for non-orthogonality between the axes of the image sensor such as: + \begin{itemize} + \item Misplacement of the sensor so that it is not perpendicular to the optical axis. + \item Manufacturing issues. + \end{itemize} + + Nevertheless, in practice skew is always 0. + \end{remark} \end{descriptionlist} @@ -85,20 +94,20 @@ Given: \end{itemize} the coordinates $\vec{M}_C$ in CRF corresponding to $\vec{M}_W$ are given by: \[ - \vec{M}_C = \begin{pmatrix} x_C \\ y_C \\ z_C \end{pmatrix} = + \vec{M}_C = \begin{bmatrix} x_C \\ y_C \\ z_C \end{bmatrix} = \matr{R}\vec{M}_W + \vec{t} = - \begin{pmatrix} - r_{11} & r_{12} & r_{13} \\ - r_{21} & r_{22} & r_{23} \\ - r_{31} & r_{32} & r_{33} \\ - \end{pmatrix} - \begin{pmatrix} + \begin{bmatrix} + r_{1,1} & r_{1,2} & r_{1,3} \\ + r_{2,1} & r_{2,2} & r_{2,3} \\ + r_{3,1} & r_{3,2} & r_{3,3} \\ + \end{bmatrix} + \begin{bmatrix} x_W \\ y_W \\ z_W - \end{pmatrix} + \end{bmatrix} + - \begin{pmatrix} + \begin{bmatrix} t_1 \\ t_2 \\ t_3 - \end{pmatrix} + \end{bmatrix} \] \begin{remark} @@ -127,9 +136,9 @@ the coordinates $\vec{M}_C$ in CRF corresponding to $\vec{M}_W$ are given by: It is not possible to combine the intrinsic camera model and the extrinsic roto-translation to create a linear model for the forward imaging model. \[ - u = f_u \frac{r_{11}x_W + r_{12}y_W + r_{13}z_W + t_1}{r_{31}x_W + r_{32}y_W + r_{33}z_W + t_3} + u_0 + u = f_u \frac{r_{1,1}x_W + r_{1,2}y_W + r_{1,3}z_W + t_1}{r_{3,1}x_W + r_{3,2}y_W + r_{3,3}z_W + t_3} + u_0 \hspace{1.5em} - v = f_v \frac{r_{21}x_W + r_{22}y_W + r_{23}z_W + t_2}{r_{31}x_W + r_{32}y_W + r_{33}z_W + t_3} + v_0 + v = f_v \frac{r_{2,1}x_W + r_{2,2}y_W + r_{2,3}z_W + t_2}{r_{3,1}x_W + r_{3,2}y_W + r_{3,3}z_W + t_3} + v_0 \] \end{remark} @@ -187,19 +196,19 @@ the coordinates $\vec{M}_C$ in CRF corresponding to $\vec{M}_W$ are given by: Given the parametric equation of a 2D line defined as: \[ \vec{m} = \vec{m}_0 + \lambda \vec{d} = - \begin{pmatrix} u_0 \\ v_0 \end{pmatrix} + \lambda \begin{pmatrix} a \\ b \end{pmatrix} = - \begin{pmatrix} u_0 + \lambda a \\ v_0 + \lambda b \end{pmatrix} + \begin{bmatrix} u_0 \\ v_0 \end{bmatrix} + \lambda \begin{bmatrix} a \\ b \end{bmatrix} = + \begin{bmatrix} u_0 + \lambda a \\ v_0 + \lambda b \end{bmatrix} \] It is possible to define a generic point in the projective space along the line $m$ as: \[ \tilde{\vec{m}} \equiv - \begin{pmatrix} \vec{m} \\ 1 \end{pmatrix} \equiv - \begin{pmatrix} u_0 + \lambda a \\ v_0 + \lambda b \\ 1 \end{pmatrix} \equiv - \begin{pmatrix} \frac{u_0}{\lambda} + a \\ \frac{v_0}{\lambda} + b \\ \frac{1}{\lambda} \end{pmatrix} + \begin{bmatrix} \vec{m} \\ 1 \end{bmatrix} \equiv + \begin{bmatrix} u_0 + \lambda a \\ v_0 + \lambda b \\ 1 \end{bmatrix} \equiv + \begin{bmatrix} \frac{u_0}{\lambda} + a \\ \frac{v_0}{\lambda} + b \\ \frac{1}{\lambda} \end{bmatrix} \] The projective coordinates $\tilde{\vec{m}}_\infty$ of the point at infinity of a line $m$ is given by: - \[ \tilde{\vec{m}}_\infty = \lim_{\lambda \rightarrow \infty} \tilde{\vec{m}} \equiv \begin{pmatrix} a \\ b \\ 0 \end{pmatrix} \] + \[ \tilde{\vec{m}}_\infty = \lim_{\lambda \rightarrow \infty} \tilde{\vec{m}} \equiv \begin{bmatrix} a \\ b \\ 0 \end{bmatrix} \] \begin{figure}[H] \centering @@ -210,8 +219,8 @@ the coordinates $\vec{M}_C$ in CRF corresponding to $\vec{M}_W$ are given by: In 3D, the definition is trivially extended as: \[ \tilde{\vec{M}}_\infty = - \lim_{\lambda \rightarrow \infty} \begin{pmatrix} \frac{x_0}{\lambda} + a \\ \frac{y_0}{\lambda} + b \\ \frac{z_0}{\lambda} + c \\ \frac{1}{\lambda} \end{pmatrix} \equiv - \begin{pmatrix} a \\ b \\ c \\ 0 \end{pmatrix} + \lim_{\lambda \rightarrow \infty} \begin{bmatrix} \frac{x_0}{\lambda} + a \\ \frac{y_0}{\lambda} + b \\ \frac{z_0}{\lambda} + c \\ \frac{1}{\lambda} \end{bmatrix} \equiv + \begin{bmatrix} a \\ b \\ c \\ 0 \end{bmatrix} \] \item[Perspective projection] \marginnote{Perspective projection in projective space} @@ -220,11 +229,11 @@ the coordinates $\vec{M}_C$ in CRF corresponding to $\vec{M}_W$ are given by: \[ \begin{split} \tilde{\vec{m}} &\equiv - \begin{pmatrix} u \\ v \\ 1 \end{pmatrix} \equiv - \begin{pmatrix} f_u\frac{x_C}{z_C} + u_0 \\ f_v\frac{y_C}{z_C} +v_0 \\ 1 \end{pmatrix} \equiv - z_C \begin{pmatrix} f_u\frac{x_C}{z_C} + u_0 \\ f_v\frac{y_C}{z_C} +v_0 \\ 1 \end{pmatrix} \\ - &\equiv \begin{pmatrix} f_u x_C + z_C u_0 \\ f_v y_C + z_C v_0 \\ z_C \end{pmatrix} \equiv - \begin{pmatrix} f_u & 0 & u_0 & 0 \\ 0 & f_v & v_0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} x_C \\ y_C \\ z_C \\ 1 \end{pmatrix} \equiv + \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} \equiv + \begin{bmatrix} f_u\frac{x_C}{z_C} + u_0 \\ f_v\frac{y_C}{z_C} +v_0 \\ 1 \end{bmatrix} \equiv + z_C \begin{bmatrix} f_u\frac{x_C}{z_C} + u_0 \\ f_v\frac{y_C}{z_C} +v_0 \\ 1 \end{bmatrix} \\ + &\equiv \begin{bmatrix} f_u x_C + z_C u_0 \\ f_v y_C + z_C v_0 \\ z_C \end{bmatrix} \equiv + \begin{bmatrix} f_u & 0 & u_0 & 0 \\ 0 & f_v & v_0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_C \\ y_C \\ z_C \\ 1 \end{bmatrix} \equiv \matr{P}_\text{int} \tilde{\vec{M}}_C \end{split} \] @@ -240,14 +249,340 @@ the coordinates $\vec{M}_C$ in CRF corresponding to $\vec{M}_W$ are given by: In projective space, we can also project in Euclidean space the point at infinity of parallel 3D lines in CRF with direction $(a, b, c)$: \[ \tilde{\vec{m}}_\infty \equiv - \matr{P}_\text{int} \begin{pmatrix} a \\ b \\ c \\ 0 \end{pmatrix} \equiv - \begin{pmatrix} f_u & 0 & u_0 & 0 \\ 0 & f_v & v_0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \\ 0 \end{pmatrix} \equiv - \begin{pmatrix} f_u a + c u_0 \\ f_v b + c v_0 \\ c \end{pmatrix} \equiv - c\begin{pmatrix} f_u \frac{a}{c} + u_0 \\ f_v \frac{b}{c} + v_0 \\ 1 \end{pmatrix} + \matr{P}_\text{int} \begin{bmatrix} a \\ b \\ c \\ 0 \end{bmatrix} \equiv + \begin{bmatrix} f_u & 0 & u_0 & 0 \\ 0 & f_v & v_0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \\ 0 \end{bmatrix} \equiv + \begin{bmatrix} f_u a + c u_0 \\ f_v b + c v_0 \\ c \end{bmatrix} \equiv + c\begin{bmatrix} f_u \frac{a}{c} + u_0 \\ f_v \frac{b}{c} + v_0 \\ 1 \end{bmatrix} \] Therefore, the Euclidean coordinates are: - \[ \vec{m}_\infty = \begin{pmatrix} f_u \frac{a}{c} + u_0 \\ f_v \frac{b}{c} + v_0 \end{pmatrix} \] + \[ \vec{m}_\infty = \begin{bmatrix} f_u \frac{a}{c} + u_0 \\ f_v \frac{b}{c} + v_0 \end{bmatrix} \] - Note that this is not possible when $c = 0$. + Note that this is not possible when $c = 0$ (i.e. the line is parallel to the image plane). \end{remark} + + \item[Intrinsic parameter matrix] \marginnote{Intrinsic parameter matrix} + The intrinsic transformation can be expressed through a matrix: + \[ + \matr{A} = + \begin{bmatrix} + f_u & 0 & u_0 \\ 0 & f_v & v_0 \\ 0 & 0 & 1 + \end{bmatrix} + \] + $\matr{A}$ is always upper right triangular and models the characteristics of the imaging device. + + \begin{remark} + If skew is considered, it would be at position $(1, 2)$. + \end{remark} + + \item[Extrinsic parameter matrix] \marginnote{Extrinsic parameter matrix} + The extrinsic transformation can be expressed through a matrix: + \[ + \matr{G} = + \begin{bmatrix} \matr{R} & \vec{t} \\ \nullvec & 1 \end{bmatrix} = + \begin{bmatrix} + r_{1,1} & r_{1,2} & r_{1,3} & t_1 \\ + r_{2,1} & r_{2,2} & r_{2,3} & t_2 \\ + r_{3,1} & r_{3,2} & r_{3,3} & t_3 \\ + 0 & 0 & 0 & 1 + \end{bmatrix} + \] + + \item[Perspective projection matrix (PPM)] \marginnote{Perspective projection matrix} + As the following hold: + \[ + \matr{P}_\text{int} = [ \matr{A} | \nullvec ] \hspace{3em} \tilde{\vec{M}}_C \equiv \matr{G} \tilde{\vec{M}}_W + \] + The perspective projection can be represented in matrix form as: + \[ \tilde{\vec{m}} \equiv \matr{P}_\text{int} \tilde{\vec{M}}_C \equiv \matr{P}_\text{int} \matr{G} \tilde{\vec{M}}_W \equiv \matr{P} \tilde{\vec{M}}_W \] + where $\matr{P} = \matr{P}_\text{int} \matr{G}$ is the perspective projection matrix. It is full-rank and has shape $3 \times 4$. + + \begin{remark} + Every full-rank $3 \times 4$ matrix is a PPM. + \end{remark} + + \begin{description} + \item[Canonical perspective projection] \marginnote{Canonical perspective projection} + PPM of form: + \[ \matr{P} \equiv [\matr{I} | \nullvec] \] + + It is useful to represent the core operations carried out by a perspective projection as any general PPM can be factorized as: + \[ \matr{P} \equiv \matr{A} [\matr{I} | \nullvec] \matr{G} \] + where: + \begin{itemize} + \item $\matr{G}$ converts from WRT to CRF. + \item $[\matr{I} | \nullvec]$ performs the canonical perspective projection (i.e. divide by the third coordinate). + \item $\matr{A}$ applies camera specific transformations. + \end{itemize} + + A further factorization is: + \[ + \matr{P} \equiv \matr{A} [\matr{I} | \nullvec] \matr{G} \equiv + \matr{A}[\matr{I} | \nullvec] \begin{bmatrix} \matr{R} & \vec{t} \\ \nullvec & 1 \end{bmatrix} \equiv + \matr{A} [ \matr{R} | \vec{t} ] + \] + \end{description} +\end{description} + + + +\section{Lens distortion} + +The PPM is based on the pinhole model and is unable to capture distortions that a lens introduces. + +\begin{description} + \item[Radial distortion] \marginnote{Radial distortion} + Deviation from the ideal pinhole caused by the lens curvature. + + \begin{descriptionlist} + \item[Barrel distortion] \marginnote{Barrel distortion} + Defect associated with wide-angle lenses that causes straight lines to bend outwards. + \item[Pincushion distortion] \marginnote{Pincushion distortion} + Defect associated with telephoto lenses that causes straight lines to bend inwards. + \end{descriptionlist} + + \begin{figure}[H] + \centering + \includegraphics[width=0.25\linewidth]{./img/radial_distortion.png} + \caption{Example of distortions w.r.t. a perfect rectangle} + \end{figure} + + \item[Tangental distortion] + Second-order effects caused by misalignment or defects of the lens (i.e. capture distortions that are not considered in radial distortion). +\end{description} + + +\subsection{Modeling lens distortion} +\marginnote{Modeling lens distortion} + +Lens distortion can be modeled using a non-linear transformation that maps ideal (undistorted) image coordinates $(x_\text{undist}, y_\text{undist})$ into +the observed (distorted) coordinates $(x, y)$: +\[ + \begin{bmatrix} x \\ y \end{bmatrix} = + \underbrace{ L(r) \begin{bmatrix} x_\text{undist} \\ y_\text{undist} \end{bmatrix} }_{\text{Radial distortion}} + + \underbrace{ \begin{bmatrix} dx(x_\text{undist}, y_\text{undist}, r) \\ dy(x_\text{undist}, y_\text{undist}, r) \end{bmatrix} }_{\text{Tangential distortion}} +\] +where: +\begin{itemize} + \item $r$ is the distance from the distortion center which is usually assumed to be the piercing point $c = (0, 0)$. + Therefore, $r = \sqrt{ (x_\text{undist})^2 + (y_\text{undist})^2 }$. + \item $L(r)$ is the radial distortion function which is a linear operator defined for positive $r$ only and is approximated using the Taylor series: + \[ L(0) = 1 \hspace{2em} L(r) = 1 + k_1 r^2 + k_2 r^4 + k_3 r^6 + \dots \] + where $k_i$ are additional intrinsic parameters. + \item The tangential distortion is approximated as: + \[ + \begin{bmatrix} dx(x_\text{undist}, y_\text{undist}, r) \\ dy(x_\text{undist}, y_\text{undist}, r) \end{bmatrix} = + \begin{bmatrix} + 2 p_1 x_\text{undist} y_\text{undist} + p_2 (r^2 + 2(x_\text{undist})^2) \\ + 2 p_1 y_\text{undist} x_\text{undist} + p_2 (r^2 + 2(y_\text{undist})^2) \\ + \end{bmatrix} + \] + where $p_1$ and $p_2$ are additional intrinsic parameters. + \begin{remark} + This approximation has empirically been shown to work. + \end{remark} +\end{itemize} + +\begin{remark} + The additivity of the two distortions in an assumption. + Other models might add arbitrary complexity. +\end{remark} + + +\subsection{Image formation with lens distortion} +\marginnote{Image formation with lens distortion} + +Lens distortion is applied after the canonical perspective projection. +Therefore, the complete workflow for image formation becomes the following: +\begin{enumerate} + \item Transform points from WRF to CRF: + \[ \matr{G} \tilde{\vec{M}}_W \equiv \begin{bmatrix} x_C & y_C & z_C & 1 \end{bmatrix}^T \] + \item Apply the canonical perspective projection: + \[ + \begin{bmatrix} \frac{x_C}{z_C} & \frac{y_C}{z_C} \end{bmatrix}^T = + \begin{bmatrix} x_\text{undist} & y_\text{undist} \end{bmatrix}^T + \] + \item Apply the lens distortion non-linear mapping: + \[ + L(r) \begin{bmatrix} x_\text{undist} \\ y_\text{undist} \end{bmatrix} + + \begin{bmatrix} dx(x_\text{undist}, y_\text{undist}, r) \\ dy(x_\text{undist}, y_\text{undist}, r) \end{bmatrix} = + \begin{bmatrix} x \\ y \end{bmatrix} + \] + \item Transform points from CRF to IRF: + \[ + \matr{A} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \equiv + \begin{bmatrix} ku \\ kv \\ k \end{bmatrix} \mapsto + \begin{bmatrix} u \\ v \end{bmatrix} + \] +\end{enumerate} + + + +\section{Zhang's method} + +\begin{description} + \item[Calibration patterns] \marginnote{Calibration patterns} + There are two approaches to camera calibration: + \begin{itemize} + \item Use a single image of a 3D calibration object (i.e. image with at least 2 planes with a known pattern). + \item Use multiple (at least 3) images of the same planar pattern (e.g. a chessboard). + \end{itemize} + + \begin{remark} + In practice, it is easier to get multiple images of the same pattern. + \end{remark} +\end{description} + +\begin{description} + \item[Zhang's method] \marginnote{Zhang's method} + Algorithm to determine the intrinsic and extrinsic parameters of a camera setup given multiple images of a pattern. + + \begin{description} + \item[Image acquisition] + Acquire $n$ images of a planar pattern with $c$ internal corners. + + Consider a chessboard for which we have prior knowledge of: + \begin{itemize} + \item The number of internal corners, + \item The size of the squares. + \end{itemize} + + \begin{remark} + To avoid ambiguity, the number of internal corners should be odd along one axis and even along the other + (otherwise, a $180^\circ$ rotation of the board would be indistinguishable). + \end{remark} + + The WRF can be defined such that: + \begin{itemize} + \item The origin is always at the same corner of the chessboard. + \item The $z$-axis is at the same level of the pattern so that $z=0$ when referring to points of the chessboard. + \item The $x$ and $y$ axes are aligned to the grid of the chessboard. $x$ is aligned along the short axis and $y$ to the long axis. + \end{itemize} + + \begin{remark} + As each image has its own extrinsic parameters, + during the execution of the algorithm, for each image $i$ will be computed + an estimate of its own extrinsic parameters $\matr{R}_i$ and $\vec{t}_i$. + \end{remark} + + \begin{figure}[H] + \centering + \includegraphics[width=0.45\linewidth]{./img/_zhang_image_acquistion.pdf} + \caption{Example of two acquired images} + \end{figure} + \end{description} + + \item[Initial homographies guess] + For each image $i$, compute an initial guess of its homography $H_i$. + + Due to the choice of the $z$-axis position, the perspective projection matrix and the WRF points can be simplified: + \[ + \begin{split} + k \tilde{\vec{m}} &\equiv + k \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} \equiv + \matr{P} \tilde{\vec{M}}_W \equiv + \begin{bmatrix} + p_{1,1} & p_{1,2} & \cancel{p_{1,3}} & p_{1,4} \\ + p_{2,1} & p_{2,2} & \cancel{p_{2,3}} & p_{2,4} \\ + p_{3,1} & p_{3,2} & \cancel{p_{3,3}} & p_{3,4} + \end{bmatrix} + \begin{bmatrix} x \\ y \\ \cancel{0} \\ 1 \end{bmatrix} \\ + &\equiv \begin{bmatrix} + p_{1,1} & p_{1,2} & p_{1,4} \\ + p_{2,1} & p_{2,2} & p_{2,4} \\ + p_{3,1} & p_{3,2} & p_{3,4} + \end{bmatrix} + \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \equiv + \matr{H}\tilde{\vec{w}} + \end{split} + \] + where $\matr{H}$ is a homography and represents a general transformation between projective planes. + + \begin{description} + \item[DLT algorithm] + Consider the $i$-th image with its $c$ corners. + For each corner $j$, we have prior knowledge of: + \begin{itemize} + \item Its 3D coordinates in the WRF. + \item Its 2D coordinates in the IRF. + \end{itemize} + Then, for each corner $j$, we can define 3 linear equations where the homography $\matr{H}_i$ of the $i$-th image is the unknown: + \[ + \tilde{\vec{m}}_{i,j} \equiv + \begin{bmatrix} u_{i,j} \\ v_{i,j} \\ 1 \end{bmatrix} \equiv + \begin{bmatrix} + p_{i,1,1} & p_{i,1,2} & p_{i,1,4} \\ + p_{i,2,1} & p_{i,2,2} & p_{i,2,4} \\ + p_{i,3,1} & p_{i,3,2} & p_{i,3,4} \\ + \end{bmatrix} + \begin{bmatrix} x_j \\ y_j \\ 1 \end{bmatrix} \equiv + \matr{H}_i \tilde{\vec{w}}_j \equiv + \underset{\mathbb{R}^{3 \times 3}}{\begin{bmatrix} + \vec{h}_{i, 1}^T \\ \vec{h}_{i, 2}^T \\ \vec{h}_{i, 3}^T + \end{bmatrix}} + \tilde{\vec{w}}_j \equiv + \underset{\mathbb{R}^{3 \times 1}}{\begin{bmatrix} + \vec{h}_{i, 1}^T \tilde{\vec{w}}_j \\ + \vec{h}_{i, 2}^T \tilde{\vec{w}}_j \\ + \vec{h}_{i, 3}^T \tilde{\vec{w}}_j + \end{bmatrix}} + \] + Geometrically, we can interpret $\matr{H}_i \tilde{\vec{w}}_j$ as a point in $\mathbb{P}^2$ + that we want to align to the projection of $(u_{i,j}, v_{i,j})$ by tweaking $\matr{H}_i$ + (i.e. find $\matr{H}_i^*$ such that $\matr{H}_i^* \tilde{\vec{w}}_j \equiv k \begin{bmatrix} u_{i,j} & v_{i,j} & 1 \end{bmatrix}^T$). + \begin{center} + \includegraphics[width=0.7\linewidth]{./img/_zhang_corner_homography.pdf} + \end{center} + + It can be shown that two points lay on the same line if their cross product is $\nullvec$: + \begin{align*} + \tilde{\vec{m}}_{i,j} \equiv \matr{H}_i \tilde{\vec{w}}_j + &\iff + \tilde{\vec{m}}_{i,j} \times \matr{H}_i \tilde{\vec{w}}_j = \nullvec \iff + \begin{bmatrix} u_{i, j} \\ v_{i, j} \\ 1 \end{bmatrix} \times + \begin{bmatrix} + \vec{h}_{i, 1}^T \tilde{\vec{w}}_j \\ + \vec{h}_{i, 2}^T \tilde{\vec{w}}_j \\ + \vec{h}_{i, 3}^T \tilde{\vec{w}}_j + \end{bmatrix} = + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + \\ + &\iff + \begin{bmatrix} + v_{i,j} \vec{h}_{i,3}^T \tilde{\vec{w}}_j - \vec{h}_{i, 2}^T \tilde{\vec{w}}_j \\ + \vec{h}_{i, 1}^T \tilde{\vec{w}}_j - u_{i, j} \vec{h}_{i, 3}^T \tilde{\vec{w}}_j \\ + u_{i, j} \vec{h}_{i, 2}^T \tilde{\vec{w}}_j - v_{i, j} \vec{h}_{i, 1}^T \tilde{\vec{w}}_j + \end{bmatrix} = + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + \\ + &\iff + \underset{\mathbb{R}^{3 \times 9}}{\begin{bmatrix} + \nullvec_{1\times 3} & -\vec{w}_j^T & v_{i,j}\vec{w}_j^T \\ + \vec{w}_j^T & \nullvec_{1\times 3} & -u_{i,j} \vec{w}_j^T \\ + -v_{i,j} \vec{w}_j^T & u_{i,j} \vec{w}_j^T & \nullvec_{1\times 3} + \end{bmatrix}} + \underset{\mathbb{R}^{9 \times 1}}{\begin{bmatrix} + \vec{h}_{i,1} \\ \vec{h}_{i,2} \\ \vec{h}_{i,3} + \end{bmatrix}} + = + \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + & \text{\parbox[t]{5cm}{$\vec{h}_{*}^T \tilde{\vec{w}}_j = \tilde{\vec{w}}_j^T \vec{h}_{*}$\\and factorization}} \\ + &\iff + \underset{\mathbb{R}^{2 \times 9}}{\begin{bmatrix} + \nullvec_{1\times 3} & -\vec{w}_j^T & v_{i,j}\vec{w}_j^T \\ + \vec{w}_j^T & \nullvec_{1\times 3} & -u_{i,j} \vec{w}_j^T \\ + \end{bmatrix}} + \underset{\mathbb{R}^{9 \times 1}}{\begin{bmatrix} + \vec{h}_{i,1} \\ \vec{h}_{i,2} \\ \vec{h}_{i,3} + \end{bmatrix}} + = + \begin{bmatrix} 0 \\ 0 \end{bmatrix} + & \text{\parbox{5cm}{only the first two\\equations are\\linearly independent}} \\ + \end{align*} + \end{description} + + \item[Homographies refinement] + \item[Initial intrinsic parameters guess] + \item[Initial extrinsic parameters guess] + \item[Initial distortion parameters guess] + \item[Parameters refinement] \end{description} \ No newline at end of file