Vision-based ACC with a Single Camera
单目相机测距的方法通常有两个：y = f H Z
y=\frac{f H}{Z}
y = Z f H Z e r r Z_{err} Z e r r Z e r r = Z n − Z = f H y + H n − Z = f H f H Z + n = n Z 2 f H + n Z
Z_{e r r}=Z_{n}-Z=\frac{f H}{y+\frac{H}{n}}-Z=\frac{f H}{\frac{f H}{Z}+n}=\frac{n Z^{2}}{f H+n Z}
Z e r r = Z n − Z = y + n H f H − Z = Z f H + n f H = f H + n Z n Z 2 f H > > n Z fH>>nZ f H > > n Z Z e r r ≈ n Z 2 f H
Z_{e r r} \approx \frac{n Z^{2}}{f H}
Z e r r ≈ f H n Z 2 Z = Z e r r Z f H = 0.05 ∗ 740 ∗ 1.2 = 44 m
Z=\frac{Z_{e r r}}{Z} f H=0.05 * 740 * 1.2=44 m
Z = Z Z e r r f H = 0 . 0 5 ∗ 7 4 0 ∗ 1 . 2 = 4 4 m
离散系统通过使用离散积分v = Δ Z Δ t v = \frac{\Delta Z}{\Delta t} v = Δ t Δ Z
令w和w’分别为目标车辆在位置Z和Z’的高度，方程可以写为：w = f W Z w ′ = f W Z ′
w =\frac{f W}{Z} \\
w^{\prime} =\frac{f W}{Z^{\prime}}
w = Z f W w ′ = Z ′ f W v = Δ Z Δ t = Z ′ − Z Δ t = f H W ′ − f H W Δ t = f H w − w ′ w ′ w Δ t = Z w − w ′ w ′ Δ t
v=\frac{\Delta Z}{\Delta t}=\frac{Z^{\prime}-Z}{\Delta t}=\frac{\frac{f H}{W^{\prime}}-\frac{f H}{W}}{\Delta t}=\frac{f H \frac{w-w^{\prime}}{w^{\prime} w}}{\Delta t}=\frac{Z \frac{w-w^{\prime}}{w^{\prime}}}{\Delta t}
v = Δ t Δ Z = Δ t Z ′ − Z = Δ t W ′ f H − W f H = Δ t f H w ′ w w − w ′ = Δ t Z w ′ w − w ′ s = w − w ′ w ′
s=\frac{w-w^{\prime}}{w^{\prime}}
s = w ′ w − w ′ v = Z s Δ t
v=\frac{Z s}{\Delta t}
v = Δ t Z s
这里引入了一个对齐误差S e r r S_{err} S e r r S a c c S_{acc} S a c c s a c c = s e r r w = s e r r Z f W Δ t
s_{a c c} =\frac{s_{e r r}}{w} =\frac{s_{e r r} Z}{fW\Delta t}
s a c c = w s e r r = f W Δ t s e r r Z v e r r c = Z s a c c Δ t = Z 2 s e r r f W Δ t
v_{e r r}^{c} =\frac{Z s_{a c c}}{\Delta t}
=\frac{Z^{2} s_{e r r}}{f W \Delta t}
v e r r c = Δ t Z s a c c = f W Δ t Z 2 s e r r
相对速度误差与相对速度无关
相对速度误差与距离的平方成正比
相对速度误差与时间间隔成反比，因此时间越长越好
有一个窄的FOV（增大f）可以提高精度
此时将测距的误差项添加上，v z e r r v_{zerr} v z e r r v z e r r = Z e r r S Δ t = n Z 2 f H s Δ t = n Z v f H
v_{z e r r}=\frac{Z_{e r r} S}{\Delta t}=\frac{n Z^{2}}{f H} \frac{s}{\Delta t}=\frac{n Z v}{f H}
v z e r r = Δ t Z e r r S = f H n Z 2 Δ t s = f H n Z v v e r r = Z 2 s e r r f W Δ t + n Z v f H
v_{e r r}=\frac{{Z^{2} s_{e r r}}}{f W \Delta t}+ \frac{n Z v}{f H}
v e r r = f W Δ t Z 2 s e r r + f H n Z v Δ t \Delta t Δ t
Z ( Δ t ) = 1 2 a Δ t 2 + v Δ t + Z 0 Z(\Delta t) =\frac{1}{2} a \Delta t^{2}+v \Delta t+Z_{0} Z ( Δ t ) = 2 1 a Δ t 2 + v Δ t + Z 0 v ( Δ t ) = a Δ t + v 0 v(\Delta t) =a \Delta t+v_{0} v ( Δ t ) = a Δ t + v 0 Δ Z = Z ( Δ t ) − Z 0 = 1 2 a Δ t 2 + v 0 Δ t
\Delta Z=Z(\Delta t)-Z_{0}=\frac{1}{2} a \Delta t^{2}+v_{0} \Delta t
Δ Z = Z ( Δ t ) − Z 0 = 2 1 a Δ t 2 + v 0 Δ t Δ t \Delta t Δ t v 0 v_0 v 0 Δ Z Δ t = 1 2 a Δ t + v 0
\frac{\Delta Z}{\Delta t}=\frac{1}{2} a \Delta t + v_0
Δ t Δ Z = 2 1 a Δ t + v 0 v e r r v_{err} v e r r v e r r = Z 2 S e r r f W Δ t + n Z v f H + 1 2 a Δ t
v_{e r r}=\frac{Z^{2} S_{e r r}}{f W \Delta t}+\frac{n Z v}{f H}+\frac{1}{2} a \Delta t
v e r r = f W Δ t Z 2 S e r r + f H n Z v + 2 1 a Δ t − Z 2 s e r r f W Δ t 2 + 1 2 a = 0
-\frac{Z^{2} s_{e r r}}{f W \Delta t^{2}}+\frac{1}{2} a=0
− f W Δ t 2 Z 2 s e r r + 2 1 a = 0 Δ t = 2 Z 2 s e r r f W a
\Delta t=\sqrt{\frac{2 Z^{2} s_{e r r}}{f W a}}
Δ t = f W a 2 Z 2 s e r r v e r r = Z 2 a s e r r f W + n Z v f H
v_{err} = Z \sqrt{\frac{2as_{err}}{fW}} + \frac{nZv}{fH}
v e r r = Z f W 2 a s e r r + f H n Z v Δ t = 2 s \Delta t = 2s Δ t = 2 s