vopra lecture02
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\section{Определение параметров объекта}
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Удалённость от камеры, размеры объекта, кинематические характеристики (скорость, направление движения).
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Метод пропорций -- должны быть априорные данные об объекте, для которого мы хотим определять характеристики. Если нет данных об объекте -- должны быть размеры объектов в сцене (дорожные знаки, разметка, и так далее), на основе данных о сцене и изображения объекта на сцене можем вычислить нужные параметры.
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\subsection{Метод пропорций}
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должны быть априорные данные об объекте, для которого мы хотим определять характеристики. Если нет данных об объекте -- должны быть размеры объектов в сцене (дорожные знаки, разметка, и так далее), на основе данных о сцене и изображения объекта на сцене можем вычислить нужные параметры.
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Исходные данные:
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\begin{itemize}
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\item $H_{\text{объекта}}$ -- например, высота объекта в пикселях $h$ -- априорная высота;
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\item $\alpha_{\text{кадр}}, \beta_{\text{кадр}}$ -- характеристики камеры -- углы обзора по вертикали и горизонтали, соответственно.
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\item $H_{\text{кадр}}$, $W_{\text{кадр}}$ -- высота и ширина кадра
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\item $H_{o}$ -- высота объекта в пикселях $h$ -- априорная высота (в физическом мире);
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\item $\alpha_{k}, \beta_{k}$ -- характеристики камеры -- углы обзора кадра по вертикали и горизонтали, соответственно.
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\item $H_{k}$, $W_{k}$ -- высота и ширина кадра
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\end{itemize}
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найти $l$ -- расстояние до объекта, $v$ -- скорость.
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(1)
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для вычисления скорости нужно взять два кадра с известным временем между ними.
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(2)
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\begin{figure}[H]
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\centering
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\fontsize{14}{1}\selectfont
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\includesvg[scale=1.01]{pics/04-vora-00-obj-height.svg}
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\end{figure}
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Высота объекта находится по формулам
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\begin{equation*}
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\begin{gathered}
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\tg(\alpha) = \frac{h}{l} \approx \alpha_{o}\\
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\frac{\alpha_o}{\alpha_k} = \frac{H_o}{H_k}\Rightarrow \alpha_o = \frac{\alpha_k \cdot H_o}{H_k}\\
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l = \frac{h \cdot H_k}{\alpha_k \cdot H_o}
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\end{gathered}
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\end{equation*}
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Для вычисления скорости нужно взять два кадра с известным временем между ними.
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\begin{equation*}
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\begin{gathered}
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v=\sqrt{v_x^2, v_y^2, v_z^2}\\
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\frac{\Delta\alpha_o}{\alpha_k} = \frac{\Delta Y_o}{H_k}\Rightarrow \Delta\alpha_o = \frac{\alpha_k \cdot \Delta Y_o}{H_k}\\
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\tg\Delta\alpha_o = \frac{\Delta y}{l} \approx \Delta\alpha_o\\
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\Delta y = \frac{\alpha_k\cdot\Delta Y_o\dot l}{H_k}\\
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v_y = \frac{\Delta y}{N\cdot\tau} = \frac{\alpha_k\cdot\Delta y_o\cdot l}{H_k\cdot N\tau}
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\end{gathered}
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\end{equation*}
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где $N$ -- число кадров между замерами, а $\tau$ -- длительность одного кадра (из информации о кадре (fps, frames pre second, кадров в секунду)).
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\begin{figure}[H]
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\centering
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\fontsize{12}{1}\selectfont
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\includesvg[scale=1.01]{pics/04-vora-00-obj-moving.svg}
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\end{figure}
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$v_x$ тоже касательный считается по аналогии
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\[ v_x = \frac{\beta_k * \Delta_o * l}{W_k * N \tau} \]
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(3)
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Недостаток в том, что нам нужны априорные знания об объектах.
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\[ v_x = \frac{\beta_k\cdot\Delta_o\cdot l}{W_k \cdot N\tau} \]
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Метод pinhole
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(4)
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мы знаем, что все лучи проходят через одну точку, тогда стоит задача по координатам $(X,Y,Z)$ получить двумерные координаты $(u, v)$.
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(5)
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Для $v_z$ формула отличается, так как движение радиальное и мы фактически считаем расстояние до объекта
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(6) - матрица поворота
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\[ v_z = \frac{\Delta z}{N\tau} = \frac{\Delta l(t)}{N\tau} = \frac{h\cdot H_k}{N\tau\alpha_k}\cdot\left(\frac{1}{H_o(t+N\tau)} - \frac{1}{H_o(t)}\right) \]
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вектор $T$ отвечает за центр масс объекта.
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Координаты $(X, Y, Z)$ приводятся к двумерным $x', y'$, масштабируются $f(x)$ и делаем сдвиг $c(x)$.
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Основной недостаток метода в том, что нам нужны априорные знания об объектах.
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\[x' = x/Z; y' = y/Z\]
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\subsection{Метод pinhole}
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\[u = f_x*x' + c_x; v = f_y*y'+c_y\]
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(7)
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\begin{figure}[H]
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\centering
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\fontsize{14}{1}\selectfont
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\includesvg[scale=1.01]{pics/04-vora-00-pinhole.svg}
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\end{figure}
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$P$ -- проекционная матрица.
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Мы знаем, что все лучи проходят через одну точку, тогда стоит задача по координатам $(X, Y, Z)$ получить двумерные координаты $(u, v)$.
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\begin{figure}[H]
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\centering
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\fontsize{14}{1}\selectfont
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\includesvg[scale=1.01]{pics/04-vora-00-pinhole-iso.svg}
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\end{figure}
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\[ \begin{pmatrix} X\\Y\\Z \end{pmatrix} = R \begin{pmatrix} X_0\\Y_0\\Z_0 \end{pmatrix} + T \]
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Матрица поворота, вектор $T$ отвечает за центр масс объекта. Координаты $(X, Y, Z)$ приводятся к двумерным $x', y'$, масштабируются $f(x)$ и сдвигаются $c(x)$.
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\begin{equation*}
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\begin{gathered}
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x' = \frac{x}{Z}; y' = \frac{y}{Z} \\
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u = f_x\cdot x' + c_x\\
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v = f_y\cdot y' + c_y\\
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\end{gathered}
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\end{equation*}
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\[ \begin{pmatrix} u \\ v \end{pmatrix} = P \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix} \]
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где $P$ -- проекционная матрица.
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\[ P = \begin{pmatrix} f(x) & 0 & c(x) \\ 0 & f(y) & c(y) \\ 0 & 0 & 1 \end{pmatrix} \]
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В данной задаче возникает проблема искажений (аберрации, дисторсия).
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\[x'' = x'(1+k_1*r^2 + k_2*r^4 + k_3*r^6) + 2p_1x'y' + p_2(r^2+2x'^2)\]
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\[r^2 = x'^2 + y'^2\]
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аналошгично y'
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аналошгично $y'$
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\[y'' = y'(1+k_1*r^2 + k_2*r^4 + k_3*r^6) + p_1(r^2+2y'^2) + 2p_2x'y'\]
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По изображению можем получить все коэффициенты и посчитать координаты $u, v$. Коэффициенты находятся путём калибровки камеры. И используются для обратного вычисления координат.
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(8) цель минимизировать ошибку, видеале = 0
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(uia via) = P(xi,yi,zi)
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\begin{figure}[H]
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\centering
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\fontsize{12}{1}\selectfont
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\includesvg[scale=1.01]{pics/04-vora-00-blurring.svg}
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\end{figure}
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$A$ -- не чёткое изображение, на рисунке -- границы размытия $\sigma$. Цель минимизировать ошибку, в идеале, получить ошибку, равную нулю.
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Зная, что матрица P - это проекционная матрица мы можем варьировать матрицы поворота и сдвига(R, T), которые входят в её состав. Perspective Points Problem - проблема того что реальная точка может восстановиться в две и нужно понять у какой коэффициент ошибки меньше.
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\[error(A) = \sum_i\left( \begin{pmatrix} u_i\\v_i \end{pmatrix} - \begin{pmatrix} u_i^A\\v_i^A \end{pmatrix} \right)^2 \to \min(R, T)\]
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Определение на изображении планарных (плоских) объектов -- гомография.
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В иделаьном случае матрицы будут равны, а их разность равняться нулю.Ошибка возводится в квадрат для увеличения чувствительности и удобства распознавания.
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(9)
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Как понять, что объект плоский. Все точки объекта связаны определёнными геометрическими преобразованиями и возможно построить между ними зависимостями.
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\[ \begin{pmatrix} u_i^A\\v_i^A \end{pmatrix} = P \begin{pmatrix} x_i\\y_i\\z_i \end{pmatrix} \]
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объект = u,v
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изображение = \tilde{u}, \tilde{v}
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Зная, что матрица $P$ -- это проекционная матрица, мы можем варьировать матрицы поворота и сдвига $(R, T)$, которые входят в её состав. \textbf{Perspective Points Problem} -- проблема того что реальная точка может восстановиться в две и нужно понять у какой коэффициент ошибки меньше.
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\tu = h_11u+h_12v+h13/h31u+h32v+h_33
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\tv = h_21u+h_22v+h13/h31u+h32v+h_33
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H = h11 h12 h13\\h21 h22 h23\\h31 h32 h33 - матрица гомографии
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(tu tv 1) = H(u v 1)
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\subsection{Определение на изображении планарных (плоских) объектов}
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Гомография.
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задача - поиск точек, подверженных гомографии. Такой поиск называется схема RANSAC.
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\begin{figure}[H]
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\centering
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\fontsize{12}{1}\selectfont
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\includesvg[scale=1.01]{pics/04-vora-00-homographia.svg}
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\end{figure}
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Как понять, что объект плоский? Все точки объекта связаны определёнными геометрическими преобразованиями и возможно построить между ними зависимостями. Координаты объекта -- $u,v$; координаты объекта на изображении -- $\tilde{u}, \tilde{v}$
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\begin{equation*}
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\begin{gathered}
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\tilde{u} = \frac{h_{11}u + h_{12}v + h_{13}}{h_{31}u + h_{32}v + h_{33}}\\
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\tilde{v} = \frac{h_{21}u + h_{22}v + h_{13}}{h_{31}u + h_{32}v + h_{33}}\\
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\begin{pmatrix} \tilde{u}\\\tilde{v}\\1 \end{pmatrix} = H \cdot \begin{pmatrix} u\\ v\\ 1 \end{pmatrix}
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\end{gathered}
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\end{equation*}
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Матрица гомографии
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\[ H = \begin{pmatrix} h_{11}&h_{12}&h_{13}\\h_{21}&h_{22}&h_{23}\\h_{31}&h_{32}&h_{33} \end{pmatrix} \]
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Основная задача -- поиск точек, подверженных гомографии. Такой поиск называется схема RANSAC.
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\end{document}
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After Width: | Height: | Size: 7.9 KiB |
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Reference in New Issue