Fast global SA(2,R) shape registration based on invertible invariant descriptor

Highlights

• A global affine shape registration is presented.

• A numerical reconstruction from a complete and stable affine invariant descriptor.

• Experiments are conducted on several known datasets for shape retrieval.

Abstract

Here, we intend to introduce a fast and non-rigid global registration for simple planar closed curves relatively to the planar Special Affine group SA(2,R). In previous work, Ghorbel (1998) has been introduced a complete and stable set of invariant for simple closed planar curves which is invariant jointly under its original parametrization and special planar affine transformations. Such property allows us a robust reconstruction of the considered object up to a special affine transformation. In this paper, several numerical difficulties of the computation of the proposed reconstruction are considered. The robustness of this inverse problem with respect to noise and reasonable deformations of non-rigid shape is demonstrated experimentally. The proposed new registration is based, on the one hand, on the shift theorem relating to the group SA(2,R) and, on the other hand, on the invertibility of the set of invariant. Since this shift theorem allows the extraction of a pose parameters exciting between reference and target objects. The low algorithmic complexity is due to the fact that the computation of the inverse descriptors are based essentially on the Fast Fourier Transformation (FFT) algorithm. Experiments are conducted on different known datasets such as MPEG-7, MCD, Kimia-99, Kimia216, ETH-80 and Swedish leaf datasets. Promising results on the sense of shape retrieval and shape recognition rates will also be demonstrated.

Introduction

It is well known that it is difficult to compare images when they are acquired at different times, sensors and poses. The obtained images need to be aligned in such a way that differences can be detected. A similar problem appears when looking for a prototype or a model in an image. These problems and several variations can be solved by image registration techniques. The image registration algorithm consists on the determination of the geometric transformation in order that the different points in one image find their correspondence in another image. An optimal transformation describing such alignment is determined according to a given criterion. The most registration methods proposed in literature is related to a rigid deformation which is modeled by a rotation followed by a translation. Such transformation is often referred to as Euclidean. However, in the case of three-dimensional object reconstruction from two of its projections, the transformation linking these two projections is often modeled by planar homography. The matching of curves up to a planar homography, needs planar equi-projective re-parametrization curves which requires at least five numerical derivations [1]. This induces errors of approximation equivalent to the value order of the quantities to estimate. For this reason, it is often proposed to replace the homography by the associated affine transformation. In addition to 3D reconstruction, the need for image registration occurs in several practical problems with various computer vision and image analysis applications such as robot navigation [2], medical image matching [3] and face recognition [4], [5]...

In this work, we intend to introduce a fast global planar curve registration algorithm relatively to planar special affine group SA(2,R) and independent to the original parametrization. This result becomes possible thanks to the existence of an invertible and stable invariant set of descriptors which allows us to reconstruct the target curve from the set of invariant descriptors of a reference contour. The proposed algorithm consists on three steps. First, an affine curve re-parametrization is applied [6]. The second step consists on the computation of the invertible set of descriptors for the reference and target objects. Using these two sets of descriptors, pose parameters are estimated by applying geometric shift theorem relative to SA(2,R). The invariant of the reference curve and the pose parameters, allows us the alignment.

The remainder of the paper is organized as follows: Section 2 introduces the related work to our approach. Then, we recall the affine invariant descriptor and evaluate the shape reconstruction by solving many numeric problems in Section 3. In the next section, we give a description of the proposed method for shape Registration. In Section 5, we study the accuracy of the proposed approach using MPEG-7, MCD Kimia-99, Kimia216, ETH-80 and Swedish leaf database in the task of shape retrieval. Finally, the last section submits the conclusion.