Arc length and Affine length

Consider a parametric vector equation of a curve:

where u is an arbitrary parameter. In many shape descriptors, the curve is reparametrised by normalised arc length, s which is defined as follows.

Arc length is preserved under similarity transforms ie the corresponding points between the original and the transformed curve have the same normalised arc length under change in orientation, translation, and differ by a scale factor under scaling.

Arc length is not preserved under general affine transforms:

where and represent the coordinates of the transformed curve.

In some applications, where curves are subject to general affine transforms, arc length is replaced by affine length which is defined as follows and is proved to be affine-invariant.

The corresponding points differ by a scale factor which is , and A is called the transformation matrix, .

Curvature and Affine curvature

Curvature has been proved to be a very useful shape descriptor which can be computed as:

If arc length, s, is used to parameterize the curve, the formula for curvature will be as follows.

Curvature is invariant under similarity transforms. It remains unchanged under rotation and translation; and its value under uniform scaling is inversely proportional to the scale factor. However, under general affine transforms, the change in curvature is not a linear function of the transformation matrix. Affine curvature has been defined as an alternative for curvature which changes linearly under affine transforms.

The definition of affine curvature is based on affine length , as follows:

Note the similarity of equations (4) and (3). It is much easier to compute affine curvature if it is expressed as a function of an arbitrary parameter.

where , , and denote the first, second, third, and fourth derivatives of x with respect to u, respectively. Derivatives of y are defined similarly.

The following facts can be observed from this equation.

• The formula is rather complicated and involves up to fourth derivatives of x and y. This can be problematic in digital implementation of the formula, where the precise approximation of high order derivatives is a difficult task. These derivatives tend to cause instability and involve more errors.
• Unlike conventional curvature, affine curvature does not have a straight forward physical interpretation.
• Affine curvature has not been defined at inflection points. The denominator in equation (5), is the same as the numerator of equation (2) without a power. As a result, whenever conventional curvature approaches to zero, affine curvature goes to infinity.
• The main advantage of this descriptor is expected to be its invariance under general affine transforms. In fact, it is not absolute invariant, ie the value of affine curvature does not remain constant under affine transformation. However, it is only a function of the determinant of the matrix A. More precisely, by substituting x and y in equation (5) by and from equation (1), we obtain the following result:

  

  where and denote affine curvature on the original and affine transformed curve respectively and det denotes determinant. The difference between the affine curvature of the original and affine transformed shape is represented by a scale factor.

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