Estimation of general 2D affine motion using Fourier descriptors☆
Introduction
Motion estimation (i.e. the computation of the motion parameters of objects) plays an important and central role in image processing and communication. Indeed, it provides for the analysis modules, segmentation and temporal tracking, the ability to compute their own criteria. Temporal tracking consists of matching primitives extracted from two images at two different instants (generally consecutive in an image sequence). These primitives, may be segments of straight lines or curves, characteristic points, pixels, blocks or regions according to the context. They are defined by certain attributes: brightness, local contrast, shape, relative position with respect to the neighboring primitives. Matching these primitives consists of minimizing a cost function, or maximizing a correlation function. Usually, motion estimation methods differ depending on the used primitives, correlation function and optimization method [1]. Contours have been used in this paper. The contours of an object in consecutive frames are first extracted, and an algorithm using Fourier descriptors (FDs) is developed to estimate the motion parameters given the ones for the global object.
In the following developments, we will assume that the object of interest has been extracted from the background.
In Refs. [2], [3], [4], [5], [6] the motion estimation algorithm based on FDs was developed in the case of similarity group (translation, rotation and zoom).
In this paper, a motion estimation algorithm using FDs capable of estimating general two-dimensional affine motions is proposed (i.e. 2D motion consisting of translation, rotation, scaling and stretching).
This algorithm, in addition to the similarity group parameters, allows the estimation of the stretching ones. Under stretching, the shape of the object will no longer be preserved. Such shape distortion can typically arise if a planar object is observed by a camera under arbitrary orientation with respect to the plane.
The relative positions of the camera and the objects are arbitrary, but the viewing conditions are supposed to be such that orthogonal projection combined with a scaling factor allows a good approximation of the perspective projection.
Under these conditions, two views of the contours of the same object are known to be related to each other by a two-dimensional affine transformation [7]. These transformations constitute the special affine motion group SA(2).
The rest of the paper is organized as follows:
In Section 2, we describe the used parameterization procedure. The affine motion algorithm based on FDs is presented in Section 3. Section 4 is dedicated to the algorithm evaluation using a synthetic curve and the Claire sequence. This sequence was selected by International Expert Groups, and was provided by the “Centre National d’Etudes des Télécommunications” (CNET), France.
Section snippets
Parameterization
It is well known that a given curve can be represented using different parameterizations. The normalized arc length l is required when considering invariance under similarities. It is the same for the estimation of the global movement of objects assumed to be rigid. Pointwise correspondence between two equivalent curves can then be achieved efficiently.
In the case of affine motion group, as shown in Refs. [7], [8], the reparametrization can be formulated in terms of the action of affine motion
Motion estimation
In this section, our goal is to show how to compute an affine transformation using the FDs. More particularly, let us suppose that we are looking for an affine transformation between two curves.
Synthetic example
We describe here the experiments carried out on synthetic shapes. Fig. 2a shows a “peanut” shape, created asFig. 2b shows an affine transformation of our peanut with a matrix A.
Table 1 and the reconstructed curve illustrated in Fig. 3 demonstrate that the estimated motion parameters are close to the actual motion algorithm.
Claire sequences
As typically available standard image sequences, for example Claire (Fig. 4), contain small and irregular frame-to-frame motions. A
Conclusion
In this paper, a general affine motion estimation algorithm using the Fourier descriptors is proposed. The normalization of the contours based on the affine arc length is indispensable when the movement is assumed affine. Under this hypothesis, we recover the shift, the scale and the four affine parameters. The evaluation of the first two parameters is made independently. The estimation of the affine matrix uses the first two parameters. The experimental results obtained on synthetic data and
Summary
Motion estimation (i.e. the computation of the motion parameters of an object) plays an important and central role in image processing and communication. Indeed, it provides for the analysis and segmentation modules and temporal tracking, the ability to compute their own criteria.
Recently, in the case of similarity group (translation, scaling and rotation), we have developed a method for motion estimation using Fourier descriptors (FDs).
In the present paper, a general affine motion estimation
About the Author—DRISS ABOUTAJDINE received the Doctorat de 3rd Cycle and the Doctorat d’Etat-és-Sciences degrees in Signal Processing from the Mohammed V University, Rabat, Morocco, in 1980 and 1985, respectively.
He joined Mohamed V University, Rabat, Morocco, in 1978, first as an assistant professor, then as an associate professor in 1985, and he is presently a Professor with the “Faculté des sciences,” since 1990. He was a visiting scholar with the Southeastern Massachusetts University (SMU)
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About the Author—DRISS ABOUTAJDINE received the Doctorat de 3rd Cycle and the Doctorat d’Etat-és-Sciences degrees in Signal Processing from the Mohammed V University, Rabat, Morocco, in 1980 and 1985, respectively.
He joined Mohamed V University, Rabat, Morocco, in 1978, first as an assistant professor, then as an associate professor in 1985, and he is presently a Professor with the “Faculté des sciences,” since 1990. He was a visiting scholar with the Southeastern Massachusetts University (SMU) and the ISL Laboratory, Stanford University, during the summers of 1981 and 1990, Guest Scientist during the summers of 1986 and 1995 with ENST Paris, France, and the Polytechnic University of Catalunya (UPC). From 1986 to 1988, he was on leave at the ENSERB, Bordeaux I University. Since October 1999 he is senior IEEE member. He has contributed to more than 70 articles to journals and conference proceedings. He organized the international symposium ISIVC 2000 in Rabat, Morocco. His research interests include statistical and adaptive signal and image processing, pattern recognition and their applications.
About the Author—MOHAMED DAOUDI received his Ph.D. Degree and Habilatation to Manage Research (HDR) in Computer Science Engineering in 1993 and 2000 from the University of Lille and the University of Littoral, respectively. Since 1994 he has been an Associate Professor at the ENIC/INT (Institut National des Télécommunications) in the computer science department. His research interests include digital pattern recognition, image processing, invariant representation of images and shapes, neural network and more recently content-based image retrieval. He has contributed to more than 40 articles to journals and conference proceedings.
About the Author—AHMED EL OIRRAK received the CEUS and 3rd Cycle thesis both in Computer Science from the university Mohamed V Faculty of the Sciences of Rabat, Morocco in 1996 and 1999, respectively. He is actually preparing his Ph.D. thesis. His research interests include invariance, motion estimation, image processing.
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This work has been supported by The “comite mixte Inter-Universitaire franco-marocain AI 74/SI/97/RI” and The “Moroccan CNCPRST, Protars P4T1/14”.