From 852aa65a9c8be2b36c781fbc13a2108918a5b4ff Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Thu, 25 Apr 2024 11:52:41 +0200 Subject: [PATCH] Add IPCV2 Zhang's method and warping --- .../module2/img/_backward_warping.pdf | Bin 0 -> 44166 bytes .../module2/img/_forward_warping.pdf | Bin 0 -> 41166 bytes .../module2/img/_homography_refinement.pdf | Bin 0 -> 149268 bytes .../module2/sections/_image_formation.tex | 295 ++++++++++++++++-- 4 files changed, 277 insertions(+), 18 deletions(-) create mode 100644 src/year1/image-processing-and-computer-vision/module2/img/_backward_warping.pdf create mode 100644 src/year1/image-processing-and-computer-vision/module2/img/_forward_warping.pdf create mode 100644 src/year1/image-processing-and-computer-vision/module2/img/_homography_refinement.pdf diff --git a/src/year1/image-processing-and-computer-vision/module2/img/_backward_warping.pdf 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@@ -34,7 +34,7 @@ The conversion from the camera reference frame to the image reference frame is done in two steps: \begin{descriptionlist} - \item[Discetization] \marginnote{Discetization} + \item[Discretization] \marginnote{Discretization} Given the sizes (in mm) $\Delta u$ and $\Delta v$ of the pixels, it is sufficient to modify the perspective projection to map CRF coordinates into a discrete grid: \[ @@ -346,7 +346,7 @@ The PPM is based on the pinhole model and is unable to capture distortions that \caption{Example of distortions w.r.t. a perfect rectangle} \end{figure} - \item[Tangental distortion] + \item[Tangential 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} @@ -430,9 +430,14 @@ Therefore, the complete workflow for image formation becomes the following: \begin{remark} In practice, it is easier to get multiple images of the same pattern. \end{remark} -\end{description} -\begin{description} + \item[Algebraic error] \marginnote{Algebraic error} + Error minimized to estimate an initial guess for a subsequent refinement step. + It should be cheap to compute. + + \item[Geometric error] \marginnote{Geometric error} + Error minimized to match the actual geometrical location of a problem. + \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. @@ -477,21 +482,21 @@ Therefore, the complete workflow for image formation becomes the following: Due to the choice of the $z$-axis position, the perspective projection matrix and the WRF points can be simplified: \[ \begin{split} - k \tilde{\vec{m}} &\equiv - k \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} \equiv - \matr{P} \tilde{\vec{M}}_W \equiv + k \tilde{\vec{m}} &= + k \begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = + \matr{P} \tilde{\vec{M}}_W = \begin{bmatrix} p_{1,1} & p_{1,2} & \cancel{p_{1,3}} & p_{1,4} \\ p_{2,1} & p_{2,2} & \cancel{p_{2,3}} & p_{2,4} \\ p_{3,1} & p_{3,2} & \cancel{p_{3,3}} & p_{3,4} \end{bmatrix} \begin{bmatrix} x \\ y \\ \cancel{0} \\ 1 \end{bmatrix} \\ - &\equiv \begin{bmatrix} + &= \begin{bmatrix} p_{1,1} & p_{1,2} & p_{1,4} \\ p_{2,1} & p_{2,2} & p_{2,4} \\ p_{3,1} & p_{3,2} & p_{3,4} \end{bmatrix} - \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \equiv + \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = \matr{H}\tilde{\vec{w}} \end{split} \] @@ -533,7 +538,7 @@ Therefore, the complete workflow for image formation becomes the following: \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 $\nullvec$: + It can be shown that two vectors have the same direction if their cross product is $\nullvec$: \begin{align*} \tilde{\vec{m}}_{i,j} \equiv \matr{H}_i \tilde{\vec{w}}_j &\iff @@ -556,9 +561,9 @@ Therefore, the complete workflow for image formation becomes the following: \\ &\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} + \nullvec_{1 \times 3} & -\tilde{\vec{w}}_j^T & v_{i,j} \tilde{\vec{w}}_j^T \\ + \tilde{\vec{w}}_j^T & \nullvec_{1\times 3} & -u_{i,j} \tilde{\vec{w}}_j^T \\ + -v_{i,j} \tilde{\vec{w}}_j^T & u_{i,j} \tilde{\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} @@ -568,8 +573,8 @@ Therefore, the complete workflow for image formation becomes the following: & \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 \\ + \nullvec_{1 \times 3} & -\tilde{\vec{w}}_j^T & v_{i,j} \tilde{\vec{w}}_j^T \\ + \tilde{\vec{w}}_j^T & \nullvec_{1\times 3} & -u_{i,j} \tilde{\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} @@ -580,9 +585,263 @@ Therefore, the complete workflow for image formation becomes the following: \end{align*} \end{description} - \item[Homographies refinement] + Given $c$ corners, a homogeneous overdetermined linear system of $2c$ equations to estimate the (vectorized) homography $\vec{h}_i$ is defined as follows: + \[ + \underset{\mathbb{R}^{2c \times 9}}{ + \begin{bmatrix} + \nullvec_{1 \times 3} & -\tilde{\vec{w}}_1^T & v_{i,1} \tilde{\vec{w}}_1^T \\ + \tilde{\vec{w}}_1^T & \nullvec_{1 \times 3} & -u_{i,1} \tilde{\vec{w}}_1^T \\ + \vdots & \vdots & \vdots \\ + \nullvec_{1 \times 3} & -\tilde{\vec{w}}_c^T & v_{i,c} \tilde{\vec{w}}_c^T \\ + \tilde{\vec{w}}_c^T & \nullvec_{1 \times 3} & -u_{i,c} \tilde{\vec{w}}_c^T \\ + \end{bmatrix} + } + \underset{\mathbb{R}^{9 \times 1}}{ + \begin{bmatrix} + \vec{h}_{i, 1} \\ \vec{h}_{i, 2} \\ \vec{h}_{i, 3} + \end{bmatrix} + } + = + \nullvec_{2c \times 1} + \Rightarrow + \matr{L}_i \vec{h}_i = \nullvec + \] + With the constraint $\Vert \vec{h}_i \Vert = 1$ to avoid the trivial solution $\vec{h}_i = \nullvec$. + + The solution $\vec{h}_i^*$ is found by minimizing the norm of $\matr{L}_i \vec{h}_i$: + \[ + \vec{h}_i^* = \arg\min_{\vec{h_i} \in \mathbb{R}^{9}} \Vert \matr{L}_i \vec{h}_i \Vert \text{ subject to } \Vert \vec{h}_i \Vert = 1 + \] + $\vec{h}_i^*$ can be found using the singular value decomposition of $\matr{L}_i = \matr{U}_i \matr{D}_i \matr{V}_i^T$. + It can be shown that $\vec{h}_i^* = \vec{v}_9$ where $\vec{v}_9$ is the last column of $\matr{V}_i$, associated with the smallest singular value. + + \begin{remark} + This step minimizes an algebraic error. + \end{remark} + + \item[Homographies non-linear refinement] + The homographies $\matr{H}_i$ estimated at the previous step are obtained using a linear method and need to be refined as, for each image $i$, + the IRF coordinates $\matr{H}_i\vec{w}_j = (\frac{h_{i, 1}^T \tilde{\vec{w}}_j}{h_{i, 3}^T \tilde{\vec{w}}_j}, \frac{h_{i, 2}^T \tilde{\vec{w}}_j}{h_{i, 3}^T \tilde{\vec{w}}_j})$ + of the world point $\vec{w}_j$ are still not matching the known IRF coordinates $\vec{m}_{i,j}$ of the $j$-corner in the $i$-image. + \begin{figure}[H] + \centering + \includegraphics[width=0.7\linewidth]{./img/_homography_refinement.pdf} + \end{figure} + + Given an initial guess for the homography $\matr{H}_i$, we can refine it through a non-linear minimization problem: + \[ \matr{H}_i^* = \arg\min_{\matr{H}_i} \sum_{j=1}^{c} \Vert \vec{m}_{i,j} - \matr{H}_i \vec{w}_j \Vert^2 \] + + This can be solved using an iterative algorithm (e.g. Levenberg-Marquardt algorithm). + + \begin{remark} + This step minimizes a geometric error. + \end{remark} + \item[Initial intrinsic parameters guess] + From the PPM, the following relationship between intrinsic and extrinsic parameters can be established: + \[ + \begin{aligned} + \matr{P}_i &\equiv \matr{A}[\matr{R}_i | \vec{t}_i] = \matr{A} \begin{bmatrix} \vec{r}_{i,1} & \vec{r}_{i,2} & \vec{r}_{i,3} & \vec{t}_{i} \end{bmatrix} \\ + &\Rightarrow \matr{H}_i = \begin{bmatrix} \vec{h}_{i,1} & \vec{h}_{i,2} & \vec{h}_{i,3} \end{bmatrix} = + \begin{bmatrix} k\matr{A}\vec{r}_{i,1} & k\matr{A}\vec{r}_{i,2} & k\matr{A}\vec{t}_{i} \end{bmatrix} + & \text{By definition of $\matr{H}_i$} \\ + &\Rightarrow (k\vec{r}_{i,1} = \matr{A}^{-1} \vec{h}_{i,1}) \land (k\vec{r}_{i,2} = \matr{A}^{-1}\vec{h}_{i,2}) + \end{aligned} + \] + Moreover, as $\matr{R}_i$ is an orthogonal matrix, the following two constraints must hold: + \[ + \begin{split} + \langle \vec{r}_{i,1}, \vec{r}_{i,2} \rangle = \nullvec &\Rightarrow + \langle \matr{A}^{-1} \vec{h}_{i,1}, \matr{A}^{-1} \vec{h}_{i,2} \rangle = \nullvec \\ + &\Rightarrow \vec{h}_{i,1}^T (\matr{A}^{-1})^T \matr{A}^{-1} \vec{h}_{i,2} = \nullvec + \end{split} + \] + \[ + \begin{split} + \langle \vec{r}_{i,1}, \vec{r}_{i,1} \rangle = \langle \vec{r}_{i,2}, \vec{r}_{i,2} \rangle &\Rightarrow + \langle \matr{A}^{-1} \vec{h}_{i,1}, \matr{A}^{-1} \vec{h}_{i,1} \rangle = \langle \matr{A}^{-1} \vec{h}_{i,2}, \matr{A}^{-1} \vec{h}_{i,2} \rangle \\ + &\Rightarrow \vec{h}_{i,1}^T (\matr{A}^{-1})^T \matr{A}^{-1} \vec{h}_{i,1} = \vec{h}_{i,2}^T (\matr{A}^{-1})^T \matr{A}^{-1} \vec{h}_{i,2} + \end{split} + \] + where $\langle \cdot, \cdot \rangle$ is the dot product. + + If at least 3 images have been collected, by stacking the two constraints for each image, + we obtain a homogeneous system of equations that can be solved with SVD over the unknown $(\matr{A}^{-1})^T \matr{A}^{-1}$. + + Note that $(\matr{A}^{-1})^T \matr{A}^{-1}$ is symmetric, therefore reducing the number of independent parameters to 5 (6 with skew). + + Once $(\matr{A}^{-1})^T \matr{A}^{-1}$ has been estimated, the actual values of $\matr{A}$ can be found by solving a + traditional system of equations using the structure and results in $(\matr{A}^{-1})^T \matr{A}^{-1}$. + + \begin{remark} + This step minimizes an algebraic error. + \end{remark} + \item[Initial extrinsic parameters guess] + For each image, given the estimated intrinsic matrix $\matr{A}$ and the homography $\matr{H}_i$, + it holds that: + \[ + \begin{split} + \matr{H}_i &= \begin{bmatrix} \vec{h}_{i,1} & \vec{h}_{i,2} & \vec{h}_{i,3} \end{bmatrix} = + \begin{bmatrix} k\matr{A}\vec{r}_{i,1} & k\matr{A}\vec{r}_{i,2} & k\matr{A}\vec{t}_{i} \end{bmatrix} \\ + &\Rightarrow \vec{r}_{i,1} = \frac{\matr{A}^{-1} \vec{h}_{i,1}}{k} + \end{split} + \] + Then, as $\vec{r}_{i,1}$ is a unit vector, it must be that $k = \Vert \matr{A}^{-1} \vec{h}_{i,1} \Vert$. + + Now, with $k$ estimated, $\vec{r}_{i,2}$ and $\vec{t}_{i}$ can be computed: + \[ \vec{r}_{i,2} = \frac{\matr{A}^{-1}\vec{h}_{i,2}}{k} \hspace{3em} \vec{t}_{i} = \frac{\matr{A}^{-1}\vec{h}_{i,3}}{k} \] + + Finally, $\vec{r}_{i,3}$ can be computed as: + \[ \vec{r}_{i,3} = \vec{r}_{i,1} \times \vec{r}_{i,2} \] + where $\times$ is the cross-product. It holds that: + \begin{itemize} + \item $\vec{r}_{i,3}$ is orthogonal to $\vec{r}_{i,1}$ and $\vec{r}_{i,2}$. + \item $\Vert \vec{r}_{i,3} \Vert = 1$ as the cross-product computes the area of the square defined by $\vec{r}_{i,1}$ and $\vec{r}_{i,2}$ (both unit vectors). + \end{itemize} + + Note that the resulting rotation matrix $\matr{R}_i$ is not exactly orthogonal as: + \begin{itemize} + \item $\vec{r}_{i,1}$ and $\vec{r}_{i,2}$ are not necessarily orthogonal. + \item $\vec{r}_{i,2}$ does not necessarily have unit length as $k$ was computed considering $\vec{r}_{i,1}$. + \end{itemize} + SVD for $\matr{R}_i$ can be used to find the closest orthogonal matrix by substituting the singular value matrix $\matr{D}$ with the identity $\matr{I}$. + + \begin{remark} + This step minimizes an algebraic error. + \end{remark} + \item[Initial distortion parameters guess] - \item[Parameters refinement] -\end{description} \ No newline at end of file + The current estimate of the homographies $\matr{H}_i$ project WRF points into ideal (undistorted) IRF coordinates $\vec{m}_\text{undist}$. + On the other hand, the coordinates $\vec{m}$ of the corners in the actual image are distorted. + + The original algorithm estimates the parameters of the radial distortion function defined as: + \[ + \begin{bmatrix} x \\ y \end{bmatrix} = L(r) \begin{bmatrix} x_\text{undist} \\ y_\text{undist} \end{bmatrix} = + (1 + k_1r^2 + k_2r^4) \begin{bmatrix} x_\text{undist} \\ y_\text{undist} \end{bmatrix} + \] + where $k_1$ and $k_2$ are parameters. + + \begin{remark} + OpenCV uses a different method to estimate: + \begin{itemize} + \item 3 parameters $k_1$, $k_2$, $k_3$ for radial distortion. + \item 2 parameters $p_1$, $p_2$ for tangential distortion. + \end{itemize} + \end{remark} + + Using the estimated intrinsic matrix $\matr{A}$, it is possible to obtain the CRF coordinates $(x, y)$ from the IRF coordinates $(u, v)$ + of $\vec{m}$ or $\vec{m}_\text{undist}$: + \[ + \begin{bmatrix} ku \\ kv \\ k \end{bmatrix} \equiv \matr{A} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} \,\,\Rightarrow\,\, + \begin{bmatrix} ku \\ kv \\ k \end{bmatrix} \equiv \begin{bmatrix} f_u x + u_0 \\ f_v y + v_0 \\ 1 \end{bmatrix} \,\,\Rightarrow\,\, + \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{u-u_0}{f_u} \\ \frac{v-v_0}{f_v} \end{bmatrix} + \] + Then, the distortion equation can be rewritten in IRF coordinates as: + \[ + \begin{split} + \begin{bmatrix} \frac{u-u_0}{f_u} \\ \frac{v-v_0}{f_v} \end{bmatrix} &= + (1 + k_1r^2 + k_2r^4) \begin{bmatrix} \frac{u_\text{undist}-u_0}{f_u} \\ \frac{v_\text{undist}-v_0}{f_v} \end{bmatrix} \\ + &\Rightarrow \begin{bmatrix} u-u_0 \\ v-v_0 \end{bmatrix} = (1 + k_1r^2 + k_2r^4) \begin{bmatrix} u_\text{undist}-u_0 \\ v_\text{undist}-v_0 \end{bmatrix} \\ + &\Rightarrow \begin{bmatrix} u-u_0 \\ v-v_0 \end{bmatrix} - \begin{bmatrix} u_\text{undist}-u_0 \\ v_\text{undist}-v_0 \end{bmatrix} = + (k_1r^2 + k_2r^4) \begin{bmatrix} u_\text{undist}-u_0 \\ v_\text{undist}-v_0 \end{bmatrix} \\ + &\Rightarrow \begin{bmatrix} u-u_\text{undist} \\ v-v_\text{undist} \end{bmatrix} = + \begin{bmatrix} + (u_\text{undist} - u_0)r^2 & (u_\text{undist} - u_0)r^4 \\ + (v_\text{undist} - v_0)r^2 & (v_\text{undist} - v_0)r^4 + \end{bmatrix} + \begin{bmatrix} k_1 \\ k_2 \end{bmatrix} + \end{split} + \] + With $n$ images with $c$ corners each, we obtain $2nc$ equations + to form a system $\vec{d} = \matr{D} \vec{k}$ in 2 unknowns $\vec{k} = \begin{bmatrix} k_1 & k_2\end{bmatrix}^T$. + This can be solved in a least squares approach as: + \[ \vec{k}^* = \min_k \Vert \matr{D}\vec{k} - \vec{d} \Vert_2 = \matr{D}^\dagger \vec{d} = (\matr{D}^T\matr{D})^{-1} \matr{D}^T \vec{d} \] + where $\matr{D}^\dagger$ is the pseudo-inverse of $\matr{D}$. + + \begin{remark} + This step minimizes an algebraic error. + \end{remark} + + \item[Parameters non-linear refinement] + A final non-linear refinement of all the estimated parameters is done to obtain a solution closer to their physical meaning. + + Assuming i.i.d. noise, this is done through the maximum likelihood estimate (MLE) using the estimated parameters as starting point: + \[ + \matr{A}^*, \vec{k}^*, \matr{R}_i^*, \vec{t}_i^* = + \arg\min_{\matr{A}, \vec{k}, \matr{R}_i, \vec{t}_i} \sum_{i=1}^n \sum_{j=1}^c + \Vert \tilde{\vec{m}}_{i,j} - \hat{\vec{m}}(\matr{A}, \vec{k}, \matr{R}_i, \vec{t}_i, \tilde{\vec{w}}_j) \Vert^2 + \] + where $\tilde{\vec{m}}_{i,j}$ are the known IRF coordinates in projective space of the $j$-th corner in the $i$-th image and + $\hat{\vec{m}}(\cdot)$ is the projection from WRF to IRF coordinates using the estimated parameters. + + This can be solved using iterative algorithms. + + \begin{remark} + This step minimizes a geometric error. + \end{remark} +\end{description} + + + +\section{Warping} + +\begin{description} + \item[Warp] \marginnote{Warp} + Transformation of an image on the spatial domain. + + Given an image $I$, warping can be seen as a function $w$ that computes the new coordinates of each pixel: + \[ u' = w_u(u, v) \hspace{2em} v' = w_v(u, v) \] + + The transformation can be: + \begin{descriptionlist} + \item[Rotation] + $\begin{bmatrix} u' \\ v' \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\begin{bmatrix} u \\ v \end{bmatrix}$ + \item[Full homography] + $k\begin{bmatrix} u' \\ v' \\ 1 \end{bmatrix} = + \begin{bmatrix} h_{1,1} & h_{1,2} & h_{1,3} \\ h_{2,1} & h_{2,2} & h_{2,3} \\ h_{3,1} & h_{3,2} & h_{3,3} \end{bmatrix}\begin{bmatrix} u \\ v \\ 1 \end{bmatrix}$ + + \end{descriptionlist} + + \begin{remark} + Differently from warping, filtering transforms the pixel intensities of an image. + \end{remark} +\end{description} + + +\subsection{Forward mapping} +\marginnote{Forward mapping} + +Starting from the input image coordinates, apply the warping function to obtain the output image. + +Output coordinates might be continuous and need to be discretized (e.g. truncated or rounded). +This might give rise to two problems: +\begin{descriptionlist} + \item[Fold] More than one input pixel ends up in the same output pixel. + \item[Hole] An output pixel does not have a corresponding input pixel. +\end{descriptionlist} + +\begin{figure}[H] + \centering + \includegraphics[width=0.55\linewidth]{./img/_forward_warping.pdf} + \caption{Example of fold and hole} +\end{figure} + + +\subsection{Backward mapping} +\marginnote{Backward mapping} + +Starting from the output image coordinates, use the inverse of the warping function $w^{-1}$ to find its corresponding input coordinates. +\[ u = w_u^{-1}(u', v') \hspace{2em} v = w_v^{-1}(u', v') \] + +\begin{figure}[H] + \centering + \includegraphics[width=0.55\linewidth]{./img/_backward_warping.pdf} +\end{figure} + +The computed input coordinates might be continuous. Possible discretization strategies are: +\begin{itemize} + \item Truncation. + \item Nearest neighbor (i.e. rounding). + \item Interpolation between the 4 closest points (e.g. bilinear, bicubic, \dots). +\end{itemize} \ No newline at end of file