Affine invariant descriptors using Fourier series

doi:10.1016/S0167-8655(02)00027-2

Pattern Recognition Letters

Volume 23, Issue 10, August 2002, Pages 1109-1118

https://doi.org/10.1016/S0167-8655(02)00027-2 Get rights and content

Abstract

In this paper we use Fourier series (FS) to produce normalized descriptors for shape. These descriptors are invariant under any affine transformation. A motion estimation algorithm using FS capable of estimating general two-dimensional affine motion is also proposed.

In the case of parameterized shape we use a pseudo affine arc length to compute FS coefficients.

Introduction

The determination of invariant characteristics has been an important problem in pattern recognition since the origin of the field in the 1960s. Invariant descriptors are properties of geometric configurations, which remain unchanged under an appropriate class of transformations. In many cases, one is only interested in those properties that are invariant under a particular class of transformations. The fundamental difficulty in recognizing an object from its image is that the appearance of shape depends on the viewpoint. Invariant descriptions can be measured directly from objects without detailed prior knowledge of the actual affine, projective, or other transformations (Rothe et al., 1996; Weiss, 1993).

Recently, several authors have proposed the use of invariant descriptors for tracking contours in a sequence of images (Huang, 1981; Blake et al., 1993; Daoudi et al., 1999). Some of them have proposed to use Fourier Descriptors for motion estimation and invariant description (Daoudi et al., 1999; Wang and Clarke, 1990; Pauwels et al., 1995; Zahn and Roskies, 1972; Persoon and Fu, 1977) in the case of similarity group (rotation, scaling). Abter et al. (1990) has proposed invariant descriptors based on Fourier Descriptors for the case of Affine group.

A great amount of work in invariant descriptors field deals with differential ones. For example, the affine curvature proposed by Spivac (1970) and Pauwels et al. (1995) is invariant under affine transformations except for its starting point.

In this paper we combine Fourier series (FS) and differential invariant descriptors to produce normalized ones that remain unchanged under any affine transformation. In the same time, a motion estimation algorithm using FS is proposed which allows estimating general two-dimensional affine motions (i.e. affinity matrix and translation vector).

Deriving invariant descriptors from FS is motivated by the fact that this method has some very important advantages over known ones:

•
The main advantage of the proposed method is that, contrarily to the Fourier descriptors, no knowledge on the relation between starting points on a shape and its transformed is supposed.
•
The method works very well on scenes with multiple objects.
•
The method is easily extended to three-dimensional space.

The remainder of this paper is organized as follows. In Section 2 we will present the FS decomposition and the FS coefficients. The investigated affine invariant descriptors and motion estimation algorithm are described in Section 3. Section 4 gives a set of affine invariant descriptors for scene with multiple objects. In Section 5 the complexity of the FS method is discussed.

Section snippets

Fourier series decomposition

Given a periodic function f, with a period T=2π/w, the trigonometric series $∑_{n=0}^{∞} (a_{n} cos (nwt)+b_{n} sin (nwt)),$ where $c_{0} =a_{0} = 1 T ∫f(t) d t, a_{n} = 2 T ∫f(t) cos (nwt) d t, b_{n} = 2 T ∫f(t) cos (nwt) d t$ are called FS of the function f.

Another equivalent decomposition into FS is $f(t)=a_{0} +∑_{n=1}^{∞} (c_{n} e^{jnwt} +c_{−n} e^{−jnwt}),$ where c_n=(1/2)(a_n−jb_n).

In the following the coefficients $a_{n}, b_{n}$ and c_n are called FS coefficients.

The proposed method is called FS method because it uses these FS coefficients as we will see in the remainder of the paper.

Representation

Any set of points can be described in the real space $R^{2}$ or in the complex plane. Using these two representations, the set of points X̃ will be said to be similar to the set of points X, if it can be mapped into X by a composition of an affine transformation A, and a translation B: $X ̃ =AX+B,$ where X and $X ̃ ∈ R^{2}$ , A is a 2×2 matrix, and B is a 2-vector representing translation.

Parameterization

It is well known that there are different parameterizations to represent a given set of points. The familiar arc length one

Multiple objects

The invariant descriptors given in (3) cannot handle the possibility that some curves constituting the whole shape remain unchanged or are subject to different affine transformations.

Solution, to that problem, is presented through the use of a new set of affine invariants given by $I_{n} (X)=c_{n}^{|X(τ),X_{t}(τ)||X_{t}(τ),X_{tt}(τ)|} J_{n} (X)=c_{n}^{|X(τ),X_{tt}(τ)||X_{t}(τ),X_{tt}(τ)|} for all n≠0,$ where $c_{n}^{f(τ)} = 12 (a_{n}^{f(τ)} −jb_{n}^{f(τ)}).$

In fact, if a part of X undergoes an affine transform A₁ and another part undergoes another affine transform

Complexity of the FS method

The algorithm contains at least two steps:

1.
The parameterization which needs the computation of first and second derivatives that are approximated by convolution with a Gaussian kernel of width σ=1 and then the computation of $τ(t)= |X_{t} (t),X_{tt} (t)| 3 L_{a}$ and X(τ) for all t. This step is not specific to the algorithm.
2.
The computation of the needed Fourier coefficients:
- ◦
  We use a normal formula to compute Fourier transform: We have to compute first the cosine and sine functions for each sample if the sampling

Conclusion and future work

In this paper, we have proposed the use of one-dimensional Fourier Series coefficients to derive affine invariant descriptors. We have also discussed a motion estimation algorithm based on the minimization of a distance defined between FS coefficients. The obtained experimental results from synthetic and real shapes demonstrate the validity of our approach and its applicability.

In future, and since the vector form for the coefficients a_n,b_n could easily be extended to more than two dimensions,

References (11)

M. Daoudi et al.
Shape distance for contour tracking and motion estimation
Pattern Recognition
(1999)
K. Abter et al.
Application of affine-invariant Fourier descriptors to recognition of 3-D objects
IEEE, PAMI
(1990)
A. Blake et al.
Affine invariant contour tracking with automatic control of spatio temporal scale
T.S. Huang
Image sequence analysis
(1981)
E.J. Pauwels
Recognition of planar shapes under affine distortion
Internat. J. Comput. Vision
(1995)

There are more references available in the full text version of this article.

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Affine invariant descriptors using Fourier series

Abstract

Introduction

Section snippets

Fourier series decomposition

Representation

Parameterization

Multiple objects

Complexity of the FS method

Conclusion and future work

Pattern Recognition

Application of affine-invariant Fourier descriptors to recognition of 3-D objects

IEEE, PAMI

Affine invariant contour tracking with automatic control of spatio temporal scale

Image sequence analysis

Recognition of planar shapes under affine distortion

Internat. J. Comput. Vision