Question about motion blur My goal is to model motion blur for a supposedly simple case:
Consider a 1 dimensions image  I of a scene. Also consider that the scence is in relative motion to the camera.
This means that at time t, the pixel  of coordinates x,  I(x)=I(x-v*t) where v is a constant.
Using the previous hypothesis how can one model this phenomenom using convolutions?
Now to be honest with you, this a homework question.
Being mostly in computer sciences, I honestly don't know how to solve it.
 A: Since this is homework, I'll leave the conversion of continuous functions to discrete functions to you.
Motion blur occurs because a camera shutter is open for a finite, non-zero amount of time. If you imagine that the in-game camera is one with a shutter, then the intensity of a pixel at position $x$ and time $t$ (when the shutter opens) is given by
$$I(x,t) = \int_t^{t+s}P(x,t')\,dt'$$
where $s$ is the time the shutter is open (perhaps the reciprocal of the frame rate) and $P(x,t)$ is the power of the light emanating from the first surface that intersects the line going from your viewpoint through the pixel at $x$ on the screen. This formula will give the correct motion blur for all cases of motion, even where different parts of the scene move around at different velocities. However, it can be very slow to calculate for large scenes and long exposures.
If your scene is static and moving past the camera at velocity $v$, then you can write the first integral as
$$I(x,t) = \frac{1}{v}\int_{x-vs}^xP(x',t)\,dx'$$
In other words, the pixel is a sum of all of the light sources that will cross $x$ while the shutter is open. Notice that the integral is over space instead of time since the scene is static. This gives you the result you want, but it can be slow for large scenes and long exposures. From now on, I'll ignore the $1/v$ factor to keep the equations clear.
The integral above is equivalent to the following that uses a windowing function:
$$I(x,t) = \int_{-\infty}^\infty P(x',t)\,w(x'; x-vs, x)\,dx'$$
where $w(x;a,b)$ is a window function given by
$$w(x;a,b) = \left\{\begin{matrix}1 & a \leq x \leq b \\ 0 & \textrm{elsewhere}\end{matrix}\right..$$
Since the window is symmetrical, we can put the integral into standard form like so:
$$I(x,t) = \int_{-\infty}^\infty P(x',t)\,w(x-x'; 0, vs)\,dx'$$
This is a convolution of the light intensity of the scene $P$ and the window function $w$:
$$I(x,t) = P(x,t)\star w(x,0,vs).$$
This can be computed quickly using the Fast Fourier Transform since
$$\mathscr{F}_x[I(x,t)] = \mathscr{F}_x[P(x,t)]\mathscr{F}_x[w(x,0,vs)]$$
where $\mathscr{F}_x$ indicates the Fourier transform over position and the right side is just a multiplication.
One last note: the discrete Fourier transform and the FFT assume the functions are periodic, so you may get motion blurs on one edge of the screen appearing on the other side. You may need to add zero-padding to the off-screen parts of your scene.
