Motion Model

The general process of image acquisition (e.g., taking an image by a camera) can be modeled by

$$g(x,y)=\int_0^T \int \int_{-\infty}^{\infty} h(x,y,x',y',t) f(x',y',t)dx'dy'dt +n(x,y)$$

where $T$ is the exposure time, $n(x,y)$ is some additive noise, and $h(x,y,x',y',t)$ is a function characterizing the distortion introduced by the imaging system, caused by, for example, limited aperture, out of focus, random atmospheric turbulence, and/or relative motion. If the imaging system is ideal, spatial and time invariant, and noise-free, i.e.,

$$h(x,y,x',y',t)=\delta(x-x',y-y')$$

then the equation above for the imaging process becomes

$$g(x,y)=\int_0^T f(x,y,t) dt$$

If the signal is also time invariant (a stationary scene), i.e., $f(x,y,t)=f(x,y)$, the image obtained is simply

$$g(x,y)=T\;f(x,y)$$

However, if there exists some relative planar motion (only in the x-y plane) between the object and the camera system, i.e., the signal $f(x,y,t)$ is no longer time invariant. This planar motion can be described by its two components in $x$ and $y$ directions $\{x_d(t), y_d(t)\}$, and the image of this moving object becomes

$$g(x,y)=\int_0^T f(x,y,t) dt=\int_0^T f(x-x_d(t),y-y_d(t)) dt$$

For simplicity, we assume a 1-D linear motion of velocity $v$ in $x$ direction only:

$$x_d(t)=vt,\;\;\;\;\;\;\;\;\;y_d(t)=0$$

We introduce a new variable $x'=vt$, we have $dt=dx'/v$, the integral from $0$ to $T$ with respect to $t$ becomes integral from $0$ to $L=vT$ with respect to $x'$, the equation for the imaging process becomes

$$g(x,y) = \int_0^T f(x-x_d(t),y-y_d(t),t) dt =\int_0^T f(x-vt,y) dt=\frac{1}{v}\int_0^Lf(x-x',y) dx'$$

$$= \int_{-\infty}^{\infty} f(x-x',y) h(x') dx'=f(x,y) * h(x)$$

where $h(x)$ is the impulse response function of the imaging system

$$h(x)\stackrel{\triangle}{=} \left\{ \begin{array}{ll} 1/v & ... ...x{if $0 \leq x \leq L$} 0 & \mbox{else} \end{array} \right.$$

which is a square impulse. Note that variable $y$ can be treated as a parameter and in this case motion restoration is essentially a 1-D problem.

If the linear motion is not in either $x$ or $y$ direction, we can rotate the image so that the motion direction is aligned with one of the axes after the rotation and modeled by the equation above.