Polynomial model

[riaanvddool]

I would like to add a 2nd order polynomial model:

x' = a0 + a1.x + a2.y + a3.x.y + a4.x^2 + a5.y^2 y' = b0 + b1.x + b2.y + b3.x.y + b4.x^2 + b5.y^2

We use this quite a lot and it should be faster that the spline model because is only has 12 parameters.

[nfaggian]

Looks like this will be easy to derive the warp jacobian for - if you need help, let me know.

[riaanvddool]

Maybe you can check my attempt:

    n = self.coordinates.homogenous.shape[1]
    x = self.coordinates.homogenous[0,:]    
    y = self.coordinates.homogenous[1,:]

    dx = np.zeros((n, 12))
    dy = np.zeros((n, 12))

    dx[:,0] = 1.0
    dx[:,1] = x
    dx[:,2] = y
    dx[:,3] = x*y
    dx[:,4] = x**2
    dx[:,5] = y**2

    dy[:,6] = 1.0
    dy[:,7] = x
    dy[:,8] = y
    dy[:,9] = x*y
    dy[:,10] = x**2
    dy[:,11] = y**2

    return (dx, dy)

[riaanvddool]

I am getting the following error while trying out the Poly2 model:

Traceback (most recent call last): File "nonlinreg.py", line 60, in plotCB=plot.gridPlot File "/usr/local/lib/python2.6/dist-packages/python_register-0.1-py2.6-linux-i686.egg/register/register.py", line 225, in register p=p File "/usr/local/lib/python2.6/dist-packages/python_register-0.1-py2.6-linux-i686.egg/register/register.py", line 111, in __deltaP return np.dot( np.linalg.inv(H), np.dot(J.T, e)) File "/usr/lib/python2.6/dist-packages/numpy/linalg/linalg.py", line 355, in inv return wrap(solve(a, identity(a.shape[0], dtype=a.dtype))) File "/usr/lib/python2.6/dist-packages/numpy/linalg/linalg.py", line 254, in solve raise LinAlgError, 'Singular matrix' numpy.linalg.linalg.LinAlgError: Singular matrix

[nfaggian]

I am having a look at this now - if the polynomial model is easy to invert it will be a great addition

[nfaggian]

Ok.

Riaan please have a look at how I have added the polynomial model you mentioned. I realized that there were a few problems.

1) in the optimization context identitiy should really map to no deformation - look at how the shift and affine warps add nothing to the warp. This parameterization was in Bakers paper.

2) The inverse warp is required for sampling - what I did in the spline model is a rough (and probably wrong approximation) :

 (grid coordinates minus warp) = inverse

We should clean up the examples before we merge this stuff - also I want to be sure that what I did is correct so please check the output of non-linreg.

While doing this I realized the comments are pretty out of date for some of the code - my fault :-(

[riaanvddool]

1) I though that with the polynomial model as i defined it, the parameters a1 and b2 should be =1 to get x= x and y = y?

2) Mm, not sure i understand.

[nfaggian]

1) For the optimization to work a starting point needs to be defined which is what I called "identity" in the code. The identity transform is the parameter set which yields no displacement in coordinates.

The deformation model (in all cases) is something like this:

x = x0 + dx y = y0 + dy

where x0, y0 are the grid coordinates and dx, dy are estimated using some sort of displacement model, beit affine or non-linear.

So if dx = f(p), the optimization relies on the fact that if p = 0, dx = 0.. same applies to dy.

2) For non-linear warps you have a nice equation for displacement (refer to the Kybic paper) but what is really computed (for sampling) is an inverse warp - otherwise the image will have holes in it after re-sampling. So I made an approximation (which is probably not great) and say that the inverse warp is:

x = x0 - dx y = y0 - dy

Hope this helps...

-N

[riaanvddool]

Please remember to try warping an image using the likeSpline model and look at the result...

[nfaggian]

Do we still want this model?

[nfaggian]

Made a branch for this work, ticket_0014