Scaling and rotation invariant analysis approach to object recognition based on Radon and Fourier–Mellin transforms☆
Introduction
Description of objects invariant to geometric transformation including translation, scaling and rotation is useful in image analysis, object recognition and classification [1], [2]. The simplest rotationally invariant feature is the Fourier transform of the boundary curve, which is invariant with regard to translation and rotation [1]. A popular class of the invariant features is based on the moment techniques including orthogonal moments and nonorthogonal moments. Nonorthogonal moments such as geometric moments [3] and complex moments [5], [6], [7] are components of the projection of the image onto monomial functions, and present a low computational cost, but are highly sensitive to noise; furthermore, reconstruction is extremely difficult. The orthogonal moments including the Zernike moments (ZM) [4], [8], [9], the pseudo-Zernike moments [4], the Legendre moments [4], [10], the orthogonal Fourier–Mellin moments (OFM) [8] and the Tchebichef moments (TM) [11] are the projection of the image onto a set of orthogonal basis. They have proven less sensitive to noise and very accurate in image reconstruction, but the major drawback is the lack of native scaling invariance, image binarization and normalization should be used prior to moments’ extraction [see Fig. 1(a)] and lead to inaccuracy of object recognition and classification since the normalization of the image generates error of resampling and requantifying and the binarization of the image destroys much useful information. This paper proposes a set of scaling and rotation invariant descriptors for image recognition (RFM) [see Fig. 1(b)]. The Radon transform is utilized in the proposed method to project the image onto projection space. In the space, a rotation of the original image results in a translation in the angle variable and a scaling of the original image leads to a scaling in the spatial variable together with an amplitude scaling. Then, the Fourier–Mellin transform is applied to convert the translation in the angle variable to a phase shift and the scaling in the spatial variable together with the amplitude scaling to an amplitude scaling. Based on the result, a rotation and scaling invariant function is constructed to achieve a set of completely invariant descriptors. A -nearest neighbors’ classifier is employed to implement classification. Theoretical and experimental results show the superiority of this approach compared with orthogonal moment-based analysis methods. The outline of this paper is as follows: In Section 2, we briefly review Radon and Fourier–Mellin transforms. The proposed approach is presented in Section 3. In Section 4, noise robustness has been proven. Experimental results are described in Section 5, and conclusions are presented in Section 6.
Section snippets
Radon transform and some of its properties
The Radon transform of a two-dimensional (2-D) function is defined aswhere is the perpendicular distance of a straight line from the origin [see Fig. 2], is the angle between the distance vector and the -axis, i.e., [12].
The Radon transform has useful properties about translation, rotation and scaling as outlined in (2), (3), (4).
Translation:
Rotation by :
The proposed approach
Let be the scaled and rotated version of an image function with the scale factor and the rotation angle , according to (2), (3), (4). The Radon transform of is given bywhere is the Radon transform of . Then, the Fourier–Mellin transform of is
Let , , we have , , , , Eq. (8) can be rewritten as
Noise robustness of this method
Suppose the image is corrupted by white noise with zero mean and variance .
Then
Since the Radon transform is line integrals of the image, for the continuous case, the Radon transform of noise is constant for all of the points and directions and is equal to the mean value of the noise, which is assumed to be zero. Therefore
This means white noise with zero mean has no effect on the
Simulation results and performance analysis
The proposed method has been implemented using Matlab. Two image sets as shown in Fig. 4 were considered in the computer implementation, the image set 1 consists of eight gray-level images of airplane with size and the image set 2 consists of eight gray-level images of butterfly with size . The experiments were conducted to test the classification accuracy of this approach. The second objective was to verify the robustness of the proposed method. A comparison of the performance
Conclusion
We have presented an approach to scaling and rotation invariant analysis for images. Unlike conventional orthogonal moment-based analysis methods in which the image needs to be binarized and normalized, this approach extracted invariant features from the Fourier–Mellin transforms of the original image's Radon projection. Experimental results show that this approach has higher classification accuracy and noise robustness to white noise compared with orthogonal moment-based analysis methods.
About the Author—XUAN WANG was born in 1966. He received the B.S. and M.S. degrees in Electrical Engineering from Shaanxi Normal University Xi’An, China in 1983 and 1987. He is currently vice professor and head of School of Physics and Information Technology at Shaanxi Normal University. He is currently pursuing the Ph.D. degree at Xidian University. His research interests include image processing and pattern recognition.
References (15)
On the independence of rotation moment invariants
Pattern Recognition
(2000)On the inverse problem of rotation moment invariants
Pattern Recognition
(2002)- et al.
Invariant character recognition with Zernike and orthogonal Fourier–Mellin moments
Pattern Recognition
(2002) - et al.
Robust and efficient Fourier–Mellin transform approximations for gray-level image reconstruction and complete invariant description
Comput. Vision Image Understanding
(2001) - et al.
Shape discrimination using Fourier descriptors
IEEE Trans. Syst. Man Cybern.
(1977) Syntactic Pattern Recognition and Application
(1982)Visual pattern recognition by moment invariants
IEEE Trans. Inf. Theory
(1962)
Cited by (0)
About the Author—XUAN WANG was born in 1966. He received the B.S. and M.S. degrees in Electrical Engineering from Shaanxi Normal University Xi’An, China in 1983 and 1987. He is currently vice professor and head of School of Physics and Information Technology at Shaanxi Normal University. He is currently pursuing the Ph.D. degree at Xidian University. His research interests include image processing and pattern recognition.
About the Author—BIN XIAO was born in 1982. He received the B.S. degree in Electrical Engineering from Shaanxi Normal University Xi’An, China in 2000, He is currently pursuing the M.S. degree at Shaanxi Normal University. His research interests include image processing and pattern recognition.
About the Author—JIAN-FENG MA was born in 1961. He received the Ph.D. degree in communication and electronic systems from Xidian University. He is currently a professor and the Dean of School of Computer in Xidian University. He is also a member of IEEE. His research interests include image processing and information and network security.
About the Author—XIU-LI BI, was born in 1982. She received the B.S. degree in Electrical Engineering from Shaanxi Normal University Xi’An, China in 2000. She is currently pursuing the M.S. degree at Shaanxi Normal University. Her research interests include image processing and pattern recognition.
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This work was supported by the National Natural Science Foundation of China under Grant No. 60573036 and by the National High-Tech Research and Development Plan of China under Grant No. 60633020.