Why do we normalize homography or fundamental matrix?-CodePudding

I want to know about why do we normalize the homography or fundamental matrix? Here is the code in particular.

H = H * (1.0 / H[2, 2]) # Normalization step. H is [3, 3] matrix.

I can understand that we have to normalize the data before computing SVD because of instability caused by linear least squares but why do we normalize it in end?

CodePudding user response：

A homography in 3D space has 8 degrees of freedom by definition, mapping from one plane to another using perspective. Such a homography can be defined by giving four points, which makes eight coordinates (scalars).

A 3x3 matrix has 9 elements, so it has 9 degrees of freedom. That is one degree more than needed for a homography.

The homography doesn't change when the matrix is scaled (multiplied by a scalar). All the math works the same. You don't need to normalize your homography matrix.

It is a good idea to normalize.

For one, it makes the arithmetic somewhat tamer. Have some wikipedia links to fields of study because weaving all these into a coherent sentence... doesn't add anything: Numerical analysis, Condition number, Floating-point arithmetic, Numerical error, Numerical stability, ...

Also, normalization makes the matrix easier for humans to interpret. The most common normalization is to scale the matrix such that the last element becomes 1. That is convenient because this whole math happens in a projective space, where the projection causes points to be mapped to the w=1 plane, making vectors have a 1 for the last element.

CodePudding user response：

How is the homography matrix provided to you?

For example, in the scene that some library function calculates and provides the homography matrix to you, if the function specification doesn't mention about the scale...

In an extreme case, the function can be implemented as:

Matrix3x3 CalculateHomographyMatrix( some arguments )
{
   Matrix3x3 H = ...;  //Homogoraphy Calculation
   return Non_Zero_Random_Value * H;  //Wow!
}

Element values may become very large or very small and using such values to your process may cause problems (floating point computation errors).