Customized TRS invariants for 2D vector fields via moment normalization

Highlights

• We extend the theory of moment normalization from real functions to 2D vector fields.

• We calculate the most general TRS normalized moments.

• A customized normalization can be derived from the general results.

• We explicitly show this for the example of pattern recognition for flow fields.

Abstract

The behavior of vector fields under translation, rotation and scaling differs with respect to the underlying application. Moment invariants that are customized to the specific problem can be constructed by means of normalization.

In this paper, we calculate general TRS (translation, rotation, and scaling) moment invariants for two-dimensional vector fields. As an example, we show explicitly how to customize the result for the detection of flow field patterns.

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 $f:{\mathbb{R}}^2\to \mathbb{R}$ and $p\text{,}q\in \mathbb{N}$, the moments $m_{p\text{,}q}$ are defined by

$$m_{p\text{,}q}={\int }_{{\mathbb{R}}^2}{x}^p{y}^{q}f(x\text{,}y)\text{d}x\text{d}y\text{.}$$

For 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 functions

$$f^{\prime }(x_1\text{,}{x}_2)=f(x_1+{\mathit{ix}}_2)=f(z):\mathbb{C}\to \mathbb{R}$$

and use the complex moments $c_{p\text{,}q}$. For $f:\mathbb{C}\to \mathbb{R}$, they are defined by

$$c_{p\text{,}q}={\int }_{\mathbb{C}}{z}^p{\bar{z}}^{q}f(z)\text{d}z\text{.}$$

Analogously, two-dimensional vector fields

$$\mathit{v}(\mathit{x})=v_1(x_1\text{,}{x}_2){\mathit{e}}_1+v_2(x_1\text{,}{x}_2){\mathit{e}}_2:{\mathbb{R}}^2\to {\mathbb{R}}^2$$

with $v_1\text{,}{v}_2:{\mathbb{R}}^2\to \mathbb{R}$, can be interpreted as complex functions

$$f(z)=f(x_1+{\mathit{ix}}_2)=f^{\prime }(x_1\text{,}{x}_2)=v_1(x_1\text{,}{x}_2)+{\mathit{iv}}_2(x_1\text{,}{x}_2):\mathbb{C}\to \mathbb{C}\text{.}$$

For $f:\mathbb{C}\to \mathbb{C}$, 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 $A\subseteq \mathbb{C}$ with characteristic function

$${\chi }_A(z)=\left\{\begin{array}{cc}1\text{,}\hfill & \text{if}z\in A\text{,}\hfill \\ 0\text{,}\hfill & \text{else.}\hfill \end{array}\right.$$

Although 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 $A=\mathbb{C}$ is not excluded.