Geometric Hashing techinque based Alignment Algorithm

This approach was originally developed for object recognition in Computer Vision (Lamdan, Schwartz, Wolfson, 1988-rigid [7], Wolfson, 1991-flexible [8]).

Definition:

A 3D reference frame is a triplet of orthogonal vectors emanating from a common arbitrary origin. It can be uniquely defined by the ordered vertices of a non-degenerate triangle.

  

Figure 13.9: Example of possible Reference Frame choice

There are many ways to define it. For example we can select the origin of the reference frame to be p₁. The x-axis will be in the direction of a vector p₂-p₁. The y-axis is on the triangle plane orthogonal to x-axis in the counterclockwise direction and the z-axis is orthogonal to the triangle plane, its direction defined by right hand rule. Since we are dealing with rigid objects, we can pick a unit length vectors (see figure [13.9]).
Suppose e_x,e_y,e_z are the relevant unit vectors, than each point v in the 3D space can be represented in the above reference frame as .
The lengths of the triangle sides are translation/rotation invariant, and therefore define a valid shape signature of the reference frame.