Camera calibration is an important prerequisite towards the solution of 3D computer vision problems. Traditional methods rely on static images of a calibration pattern. This raises interesting challenges towards the practical usage of event cameras,
which notably require image change to produce sufficient measurements. The current standard for event camera calibration therefore consists of using flashing patterns. They have the advantage of simultaneously triggering events in all reprojected pattern feature locations, but it is difficult to construct or use such patterns in the field. We present the first dynamic event camera calibration algorithm. It calibrates directly from events captured during relative motion between camera and calibration pattern. The method is propelled by a novel feature extraction mechanism for calibration patterns, and leverages existing calibration tools before optimizing all parameters through a multi-segment continuous-time formulation. As demonstrated through our results on real data, the obtained calibration method is highly convenient and reliably calibrates from data sequences spanning less than 10 seconds.
We present a new solution to tracking and mapping with an event camera. The motion of the camera contains both rotation and translation, and the displacements happen in an arbitrarily structured environment. As a result, the image matching may no lon
ger be represented by a low-dimensional homographic warping, thus complicating an application of the commonly used Image of Warped Events (IWE). We introduce a new solution to this problem by performing contrast maximization in 3D. The 3D location of the rays cast for each event is smoothly varied as a function of a continuous-time motion parametrization, and the optimal parameters are found by maximizing the contrast in a volumetric ray density field. Our method thus performs joint optimization over motion and structure. The practical validity of our approach is supported by an application to AGV motion estimation and 3D reconstruction with a single vehicle-mounted event camera. The method approaches the performance obtained with regular cameras, and eventually outperforms in challenging visual conditions.
The chemotaxis--Navier--Stokes system begin{equation*}label{0.1} left{begin{array}{ll} n_t+ucdot abla n=triangle n-chi ablacdotp left(displaystylefrac n {c} abla cright)+n(r-mu n), c_t+ucdot abla c=triangle c-nc, u_t+ (ucdot abla) u=Delta
u+ abla P+n ablaphi, ablacdot u=0, end{array}right. end{equation*} is considered in a bounded smooth domain $Omega subset mathbb{R}^2$, where $phiin W^{1,infty}(Omega)$, $chi>0$, $rin mathbb{R}$ and $mu> 0$ are given parameters. It is shown that there exists a value $mu_*(Omega,chi, r)geq 0$ such that whenever $ mu>mu_*(Omega,chi, r)$, the global-in-time classical solution to the system is uniformly bounded with respect to $xin Omega$. Moreover, for the case $r>0$, $(n,c,frac {| abla c|}c,u)$ converges to $(frac r mu,0,0,0)$ in $L^infty(Omega)times L^infty(Omega)times L^p(Omega)times L^infty(Omega)$ for any $p>1$ exponentially as $trightarrow infty$, while in the case $r=0$, $(n,c,frac {| abla c|}c,u)$ converges to $(0,0,0,0)$ in $(L^infty(Omega))^4$ algebraically. To the best of our knowledge, these results provide the first precise information on the asymptotic profile of solutions in two dimensions.
We are concerned with the Keller--Segel--Navier--Stokes system begin{equation*} left{ begin{array}{ll} rho_t+ucdot ablarho=Deltarho- ablacdot(rho mathcal{S}(x,rho,c) abla c)-rho m, &!! (x,t)in Omegatimes (0,T), m_t+ucdot abla m=Delta m-rho m, &!! (x
,t)in Omegatimes (0,T), c_t+ucdot abla c=Delta c-c+m, & !! (x,t)in Omegatimes (0,T), u_t+ (ucdot abla) u=Delta u- abla P+(rho+m) ablaphi,quad ablacdot u=0, &!! (x,t)in Omegatimes (0,T) end{array}right. end{equation*} subject to the boundary condition $( ablarho-rho mathcal{S}(x,rho,c) abla c)cdot u!!=! abla mcdot u= abla ccdot u=0, u=0$ in a bounded smooth domain $Omegasubsetmathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $mathcal{S}in C^2(overlineOmegatimes [0,infty)^2)^{3times 3}$ satisfies $|mathcal{S}(x,rho,c)|leq C_S $ for some $C_S>0$, and the initial data satisfy certain smallness conditions.
This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain $Omega subset mathbb{R}^N$ ($N=1,2$): $$label{0} left{begin{array}{ll} p_t=Delta p- ablacdotp p(displaystylefrac alpha {1+c}
abla c+rho abla w)+lambda p(1-p),,& xin Omega, t>0, c_t=Delta c-c-mu pc,, &xin Omega, t>0, w_t= gamma p(1-w),,& xin Omega, t>0, end{array}right. $$ where $alpha, rho, lambda, mu$ and $gamma$ are positive parameters. For any reasonably regular initial data $(p_0, c_0, w_0)$, we prove the global boundedness ($L^infty$-norm) of $p$ via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution $(p,c,w)$ converges to $(1,0,1)$ with an explicit exponential rate as time tends to infinity.
