 When FOV is not very large and the camera motion has a small rotation, the 2D displacement (u,v) of an image point (x,y) in the image plane can be expressed:
 where




Estimate the 2D parameters of a single motion
Register two successive frames according to the computed
2D parameters. This alignment was proved to cancel the rotation motion for the entire scene, resulting in a new sequence which only contains 3D translation, and looks as if taken from a stabilized platform with no jitter.
Further option: Compute FOE (translation) from the residual epipolar displacement field between the two registered frames and recover the 3D rotation parameters from the 2D parameters and 3D translation parameters.

 For small motions, assume gray level constancy:

- For translation: 2 parameters, u(x,y) = a, v(x,y) = b
- For Affine: 6 parameters, u(x,y) = a+bx+cy, v(x,y) = d+ex+fy.
- For a pseudo projective transformation: u(x,y)=a+bx+cy+gx^2+hxy, v(x,y)=c+dx+fy+gxy+hy^2
 First order Taylor expansion and neglecting all nonlinear terms:
(

,
 v t
 
( , , )
 uI x
 vI y

I t where
I x


I x y t
 x
, I y


, I t


 y
 t
Minimize the error function at Frame t in the region of R:
Err ( , )
 
( , )

R
( uI x
 vI y

I t
)
2
(Its derivatives with respect to the parameters are set to zero.)




Multiresolution iterative framework
Construct a Gaussian pyramid
Start at the lowest resolution level, motion parameters are estimated by solving the set of linear equation to minimize the error function according to the appropriate model.


Incremental motion estimation can be performed at each resolution level.
Coarse-fine alignment: at each successive iteration, the shift and estimates steps are performed on the next higher resolution pyramid level, that is, the shift estimated at level k+1 of frame t is applied to level k of frame t to form a new level k of frame t, then the residual can be computed between the new level k of frame t and the level k of frame t+1.





Warping two successive frames according to the computed 2D parameters.
Theoretically, this step should be able to cancel the rotation motion for the entire scene, resulting in a new sequence which only contains 3D translation, and looks as if taken from a stabilized platform with no jitter.
Stabilized GoCart (not so good)
Problems: Didn’t work for other test videos ( Got zero gradient, thus NaN …. Need debug and further improvements)