Similarity transformation parameters recovery based on Radon transform. Application in image registration and object recognition
Introduction
The estimation of the affine transformation is very useful for the building of invariant vision systems, on which are based many computer vision problematics, such as, image registration and object recognition. These disciplines play an important role in computer vision applications (e.g. medical imaging, automatic target recognition, and industrial inspection) where the determination of the spatial transformations between two images is often the keystone of some existing methods. This spatial transformation has as role, for image registration, to bring homologous points in registered images into correspondence [1], whilst for object recognition, it permits to match two objects. Among other spatial transformations, the similarity transforms, i.e. rotation, scaling and translation (RST) are often involved in many applications, where the object shape should be preserved, such as (1) analysis of pictures captured by camera in a common reference frame, (2) state change quantification of a moving target and (3) weld defect identification [2] on radiograms in nondestructive testing (NDT).
Although on many decades, methods and techniques such as invariant moments and invariant correlation have carried out noticeable progress for RST parameters estimation, it is only very recently that the methods based on the Radon transform [3] have received gain of interest. In fact, working directly with the data provided by the Radon projections allows us to avoid image reconstruction techniques like filtered back projection, which are computationally expensive and prone to reconstruction artifacts [4]. Computed tomography (CT) is an essential imaging technology in medicine, NDT and materials research [5], examples of application where the determination of the internal structure of an object is closely associated with the Radon transform; and only the projection data (often represented as sinograms) are available to the imaging system manipulator. The techniques designed to recover spatial transformations linking two patterns or two images using directly their Radon transforms have received recently a lot of interest and are increasingly investigated. For image registration tasks [6], the authors in [7] estimate the RST parameters between a reference image and a distorted image. Here, only the rotation angle is estimated using Radon transform. Knowing that the sinograms of both images are the same, except for the circular shift along the direction of the angular component (ϕ), they use the cross-correlation on the rows of the Radon matrix to recover the rotation angle. However, results remain fairly weak on some images. For the tomographic registration purpose, the algorithm in [8] uses some properties of the Radon and Fourier transforms, such as the central slice and Fourier shift theorems to identify translational and angular offsets and then, the fitting is obtained by using the cross-correlation method. Related to the NDT community, the work in [9] is devoted to the application of the Fourier phase matching method in the registration of radiography and computed tomography projections which has the potential to allow automated metrology and defect detection. Still using image projections, the authors in [5] develop a method which incorporates all degrees of freedom of an affine transformation. The parameters of the latter are estimated as an optimization problem where a trust region-based optimizer is used [10]. However, the mean relative magnitude error rely high around 10% and 5° for the mean angular error, for both cases of two-dimensional parallel and fan beam geometries. In [11], the authors present a method which consists to estimate transformation parameters for a binary object subject to reflection, scaling, translation and rotation using only the Radon projections. Nevertheless, the objective function, as it is formulated by the authors, cannot be generalized for any rotation angle in the range . In [12], the authors use the results obtained in [11] to identify the transform parameters for a fast matching algorithm. However, the rotation angle estimation is not very accurate compared to [11] and needs two running steps to distinguish the rotation angle ϕ0 from .
In this paper, we develop a method in order to estimate the RST transforms relating two unknown images, directly from their projection data. A preliminary study of the work reported in this paper has already been published in [13]. Here, we propose a deep analysis on theoretical and experimental aspects. As first step, we define the 2π-based Radon transform to deal with the problem of rotation by any angle belonging in , unlike [11]. The resulting algorithm may be used on sinograms obtained from spatial images as well on projections considered as raw data. Consequently, this method may be introduced in an intensity-based image registration working in the space domain and where the mapping model consists in a similarity transform. Moreover, this method is originally applied in object recognition where the objective function used for the rotation angle estimation is taken as a similarity measure. In addition to recognize object through a pattern classification task, the main advantage of the proposed method is that the RST transform parameters between two matched patterns are implicitly recovered without any additional computing. However, the major part of methods using Radon transform devoted to invariant pattern recognition, e.g. [14], [15], [16], bring together similar patterns in the same class but are not able to provide the estimation of the geometric transforms giving the best similarity between two patterns.
The remainder of the paper is organized as follows. Section 2 gives the basic material to understand the main properties of the Radon transform. Section 3 is devoted to the problem statement whereas in Section 4, the 2π-based Radon transform is defined and its utilization is motivated. In Section 5, the proposed RST parameters recovery method is detailed and summarized. The experimental results are discussed in Section 6. Finally, the conclusion and some future directions of this work are drawn in Section 7.
Section snippets
Basic material
To define the Radon transform (RT) [3], let L be a straight line in the plane and be an increment of length along L. Then, the Radon transform of a real valued function f, denoted , is defined by its integral as is determined by integration of all lines , in the plane, , . From Fig. 1, the equation of L is given by . If we rotate the axes by an angle ϕ, and label the new axes p and s, we obtainThen,
Problem statement
We consider the case of 2D mono-channel images. Assume f as an image in plane, subjected to a sequence of affine transformations , , and , where ϕ0 (in ), α (in ) and (in ) are, respectively, the rotation angle, the scaling factor and the translation vector components. The transform image, noted g with coordinates is a composition of three functions formulated asor as a matrix product using homogeneous coordinates
The 2π-based Radon transform
Let us examine in details the rotation property for which modifications will be brought to make the proposed algorithm reliable in image registration and object recognition tasks. Let fr be the function representing the rotated object by an angle ϕ0 (), i.e. . For the sake of simplicity in notation, the centroid of the object is set to the origin of the plane. This assumption is set to avoid the translation effect introduced by the rotation, since, in this section, only
The proposed algorithm for RT-based RST parameters recovery
To be useful, a shape recognition framework should allow explicit RST operations invariance. When a given shape is subjected to a sequence of RST transformations, it will be difficult to recover all the RST parameters using the RT properties. In the aim to identify the underlying transformations, we will use the 2π-based RT defined in the previous section, some results given in [11] and our own strategy to build the parameter recovery algorithm.
The principle of this algorithm is to deduce one
Experiment results and discussion
In order to show the effectiveness of the proposed algorithm, two experiments have been carried out. The first experiment consists in the recovery, in the Radon space, of the RST transforms allowing the comparison between two gray level images corrupted or not by additive noise in case of image registration purposes. The second experiment is based on the results obtained from the first experiment for object recognition (shapes, logos and graphical symbols) where the robustness to additive noise
Conclusion
In this paper, we have developed a method to find the RST parameters from Radon projections. This is of particular interest in domains where only image projections are provided, as in medical or industrial computed tomography. We have described in details and motivated the extended version of Radon transform on to deal with image rotation in . Experiments are conducted for image registration and object recognition purposes where binary and gray level images with or without additive
Conflict of interest
None declared.
Nafaa Nacereddine was graduated in Electrical Engineering from Ecole Nationale Polytechnique (ENP) of Algiers in 1992. From 1994, when he obtained the specialized post-graduation degree at the Centre de Recherche en Soudage et Contrôle (CSC), he was a researcher in signal and image processing laboratory, where he heads since 2004, the date of the receiving of the Magister degree, a research team on pattern recognition applied on industrial radiography imaging. From 2008 to 2010, he was a
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Nafaa Nacereddine was graduated in Electrical Engineering from Ecole Nationale Polytechnique (ENP) of Algiers in 1992. From 1994, when he obtained the specialized post-graduation degree at the Centre de Recherche en Soudage et Contrôle (CSC), he was a researcher in signal and image processing laboratory, where he heads since 2004, the date of the receiving of the Magister degree, a research team on pattern recognition applied on industrial radiography imaging. From 2008 to 2010, he was a visiting scientist as Ph.D. student at QGAR research team at LORIA research center in Nancy, France. He received Doctorate degree in Electronics from ENP in 2011. His research interests include industrial radiography, NDT, image processing and analysis, computer vision, content-based image retrieval and pattern recognition. He has many contributions in these fields through scientific publications.
Salvatore Tabbone is a professor in Computer Science at University of Université de Lorraine (France). Since 2007, he is the scientific leader of the QGAR research team at LORIA research center. His research topics include image and document indexing, content-based image retrieval, and pattern recognition. He is the (co-)author of more than 100 articles in refereed journals and conferences. He serves as PC members for numerous national and international conferences.
Djemel Ziou received the B.Eng. degree in Computer Science from the University of Annaba, Algeria, in 1984 and the Ph.D. degree in Computer Science from the Institut National Polytechnique de Lorraine (INPL), France, in 1991. From 1987 to 1993, he served as a lecturer in several universities in France. During the same period, he was a researcher at the Centre de Recherche en Informatique de Nancy (CRIN) and the Institut National de Recherche en Informatique et Automatique (INRIA) in France. Presently, he is a full professor in the Department of Computer Science, Universite de Sherbrooke, Quebec, Canada. He is the holder of the Natural Sciences and Engineering Research Council (NSERC)/Bell Canada Research chair in personal imaging. He has served on numerous conference committees as a member or chair. He heads the laboratory MOIVRE and the consortium CoRIMedia, which he founded. His research interests include image processing, information retrieval, computer vision and pattern recognition.