Customized TRS invariants for 2D vector fields via moment normalization☆
Introduction
Moment invariants are one of the fundamental techniques to describe and compare real-valued objects because they are robust and easy to use. They are a number of values representing a function that do not change under certain transformations. Their invariance property allows to compare objects in one single step instead of having to compare every possible transformed version of it.
Two-dimensional invariants with respect to translation, rotation, and scaling (TRS) were introduced to the pattern recognition community by Hu [1]. The use of complex moments [2], [3] simplified the construction of rotation invariants because of the easy way to describe rotations by means of complex exponentials. In the last decade, Flusser [4], [5], [6] structured the theory of complex moment invariants into a clear framework. That paved the way for a generalization to vector-valued data. In [7], a comprehensive treatment of their work can be found.
Recently, Schlemmer et al. [8], [9] applied their results to flow fields. They constructed a basis of flow field moment invariants, developed an algorithm that calculates them efficiently, and successfully used it to detect features in real-world data.
In contrast to the use of an independent bases [4], [9], there is a different approach for the construction of moment invariants, called normalization [1], [10], [7]. First, the function is brought into a standard position by setting certain moments to given values. Then, all the remaining moments are used as the discriminating invariants. The transformation of the first step can take various forms even in the case where it is only the combination of translation, rotation, and scaling.
We will show how invariants with respect to all of these forms can be constructed by means of normalization. As an example, we will calculate the set of moment invariants that are customized to the problem of finding patterns in flow fields.
For a function and , the moments are defined byFor the analysis of functions over the plane, we can make use of the isomorphism between the Euclidean and the complex plane [11], [12], [13], interpret them as functionsand use the complex moments . For , they are defined byAnalogously, two-dimensional vector fieldswith , can be interpreted as complex functionsFor , the definition of complex moments (3) can easily be generalized. We will work with complex functions during the calculations and keep in mind that the results are also valid for vector fields.
In order to customize the results to practical applications, we assume the functions to vanish outside an area with characteristic functionAlthough the functions with infinite support are easier to deal with, they will not appear very often in real-world applications. For the sake of completeness, the case is not excluded.
Section snippets
Translation, rotation, and scaling on vector fields
Vector fields can have very different properties under affine transformations. The specific behavior depends on the interpretation of the field. When working with vector fields, one has to distinguish at least three cases. In this paper, we show how moment invariants can be constructed that satisfy the different requirements.
In contrast to scalar fields, the term rotational misalignment is ambiguous for vector fields. A simple example rotated by can be found visualized in Fig. 1. Let be
Moment invariants of scalar functions
In this section, we state the classical method of the normalization of moments of real-valued functions. Even though the results are commonly known, we show how they can be achieved in order to pave the way to the following sections. In the classical case the transforms are always inner transforms. It is possible to show the results without the use of the characteristic function. They are valid for all integrable functions. But since we will definitely need the characteristic function to
Moment invariants of complex functions
Now, we have the case that the affine transforms can not only be applied to the arguments but also to the values of the functions. That means that we have far more degrees of freedom. In order to normalize with respect to outer and inner transformations, we analyze the relation of the moments of a functionto the ones of its transformed copywith the inner and outer scaling factors , translational differences , rotation angles
Finding flow field patterns
Now, we want to focus on pattern matching and feature extraction [16], [17] of flow fields. The problem is as follows. We have a relatively small pattern and want to decide where it appears in a larger vector field independent from its orientation, size, or position. As we depicted in the Fig. 1, Fig. 2, Fig. 3 this application is a special case of the general one treated in Theorem 2. We have to treat some parameters different than others.
In this application, the calculation of the inner
Conclusions and outlook
The requirements for invariants of vector fields or complex functions differ from the ones of the well analyzed real-valued functions depending on their meaning and application.
In Theorem 2, we showed how moments have to be normalized such that they are invariant with respect to inner and outer translation, rotation and scaling. This general result can be customized to specific problems by leaving out the superfluous parameters. Representative for all possible applications, Corollary 3 presents
Acknowledgments
We thank Prof. Kollmann from the University of California at Davis for producing the swirling jet dataset. We would further like to thank the FAnToM development group from the Leipzig University for providing the environment for the visualization of the presented work, especially Wieland Reich, Jens Kasten and Stefan Koch. This work was partially supported by the European Social Fund (Application No. 100098251).
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This paper has been recommended for acceptance by L. Heutte.