Various works have used 2D motion models to describe the change between two images. [LINKS]
These methods approximate the flow field between the images with some restricted function. A hierarchy of such models has evolved and was used extensibely, each model includes the previous ones.
From the table below we can see that only the last two models are directly related to real 3D motion. They fit any motion of the 3D surface + any motion of the cameras + any re-parametrization on the surface.

The hierarchy of motion models

The hierarchy of motion models (MORE DETAILS)

	Model	No. of parameters	The formula	Physical meaning
1	Translation	2	u=c, v=f	2D translation
2	Euclidean	3	u=c + xcosa - ysina v=f + xsina + ycosa	2D translation + rotation
3	Affine	6	u = ax + by + c v = ex + fy + f	above + skew
4	Pseudo Projective	8		(none)
5	Homography	8	x'=(ax + by + c)/(gx + hy + 1) y'=(dx + ey + f)/(gx + hy + 1)	motion of a 3D planar surface
6	Q-warping	17		motion of a 3D quadratic surface

What is this method good for?

All the previous methods listed above are limited to planar images or images where the size of the object is negligible relative to it's distance from the camera. For example, the Affine model was used extensively to stabilize aerial images. Common 'in-door' images are usually far from being planar and thus these methods will not perform well on them. A human face is an example for such image.

The Algorithm

The algorithm sets the parameters as if there is no motion between the images and then tries to improve the result iteratively. All this is incorporated into a coarse-to-fine scheme.

• Build image pyramids from the two images.
• Staring from the top level to the bottom one:
  • Repeat until convergence:
    • Update the motion model (starting with zero)
    • Warp the images
  • Scale the computed flow fields to the next level
• Output the flow-field and the warped image

The motion estimation

Each pixel contributes one equation:
u' and v' are the numerators in the Q-warping formula above. Ix, Iy and It are the spatial and temporal derivatives of the images. This give a highly over-constrained system of linear equation for the 17 parameters which can be solved in least-squares fasion.

Figure

Q-Warping is the process of transferring one image to the other. A ray from the center of the first camera hits the quadric in one point. This point is then projected onto the second image. It turns out that this transformation can described using 17 parameters.

The goal of our algorithm is to find the best combination of the parameters for the given images. Here, we define 'Best' by the SSD (sum of squared difference) of the resulting image and the second image.
 
 
 
 
 
 

Description	Original	Q-Warped
Cylinder
General Shape
Planar scene
Rigid face

Here is the abstract from the paper:

We consider the problem of wrapping around an object, of which two views are available, a reference surface and recovering the resulting parametric flow using direct computations (via spatio-temporal derivatives). The well known examples are affine flow models and 8-parameter flow models --- both describing a flow field of a planar reference surface. We extend those classic flow models to deal with a quadric reference surface and work out the explicit parametric form of the flow field. As a result we derive a simple warping algorithm that maps between two views and leaves a residual flow proportional to the 3D deviation of the surface from a virtual quadric surface. The applications include image morphing, model building, image stabilization, and disparate view correspondence.

The change between two images of the same world (such as the left and right eyes) is a function of many parameters. Among them are the motion of the camera, the motion of the objects in the world, the shape of the world, changes in illumination and more. Not all these parameters can be computed from the images (only) and some of them are coupled in a way that makes it hard to find them. For many tasks in computer vision, it is not necessary though.

This work models the change between two images as if they were taken around a quadratic surface. The method is performs a search for the best quadric+motion pair that minimizes the change between the images. Each set of parameters defines a warping function between the images (see figure).

This work extends a hierarchy of motion models that were used extensively in the past for various applications. It's novelty is in that the reference surface is not limited to be planar. For this reason, this method can work on images with large perspective distortion, such as a person sitting close to a camera.

Contact

The paper "Q-warping: Direct Computation of Quadratic Reference Surfaces" by Yoni Wexler & Amnon Shashua, to appear in CVPR June 1999. Retrieve a copy here.