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

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