1834 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 8, NO. 12, DECEMBER 1999 (a) (b) Fig. 5. (a) Real image of a rotationally symmetric shape. (b) Recovered pattern by using the proposed method. demonstrate the robustness of this method to noise and digitization errors. The output images of the last experiment are imperfect RSS. This is due to the nonlinear distortion caused by the camera lens. In such situation, orthographic projection assumption is no longer valid. However, fairly well results are provided and this indicates the usability of our method to imperfect rotational symmetry. To achieve better results , a camera calibration procedure may be placed before our system. It is beyond the scope of this research. V. CONCLUSION In this work, we propose an algorithm to recover the skewed RSS. Our algorithm needs no numeric method and no information about the number of folds. Since this method is based on the moments, it does not rely on smooth or continuous contours and is robust to noise or digitization errors but assumes there is no occlusion. We do not intend to finding the axis of symmetry, because a given RSS may not have any axis of reflective symmetry. However, it does have axes, several kinds of axes are proposed by previous researchers [1]–[4]. The experimental results confirm our derivations of constraints and show availability of our algorithm. Shapes with and without reflective symmetry are all presented. The algorithm gives accurate estimation of the skew parameters and After applying our algorithm, any of the algorithms proposed in [1]–[5] can be used to find the axes and normalize the deskewed RSS. Thus, the whole normalization procedure is completed. REFERENCES [1] W.-H. Tsai and S.-L. Chou, “Detection of generalized principal axes in rotationally symmetric shapes,” Pattern Recognit., vol. 24, pp. 95–104, 1991. [2] S.-C. Pei and C.-N. Lin, “Normalization of rotationally symmetric shapes for pattern recognition,” Pattern Recognit., vol. 25, pp. 913–920, 1992. [3] J.-C. Lin, “Universal principal axes: An easy-to-construct tool useful in defining shape orientations for almost every kind of shape,” Pattern Recognit., vol. 26, pp. 485–493, 1993. [4] J.-C. Lin, W.-H. Tsai, and J.-A. Chen, “Detecting number of folds by a simple mathematical property,” Pattern Recognit. Lett., vol. 15, pp. 1081–1088, 1994. [5] R. K. K. Yip, W. C. Y. Lam, P. K. S. Tam, and D. N. K. Leung, “A Hough transform technique for the detection of rotational symmetry,” Pattern Recognit. Lett., vol. 15, pp. 919–928, 1994. [6] T. Kanade, “Recovery of the three-dimensional shape of an object from a single view,” Artif. Intell. vol. 17, pp. 409–460, 1981. [7] S. A. Friedberg, “Finding axes of skewed symmetry,” Comput. Vis., Graph., Image Process., vol. 34, pp. 138–155, 1986. [8] A. D. Gross and T. E. Boult, “Analyzing skewed symmetries,” Int. J. Comput. Vis., vol. 13, pp. 91–111, 1994. [9] M.-K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inform. Theory, vol. IT-8, pp. 179–187, 1962. Tri-State Median Filter for Image Denoising Tao Chen, Kai-Kuang Ma, and Li-Hui Chen Abstract—In this work, a novel nonlinear filter, called tri-state median (TSM) filter, is proposed for preserving image details while effectively suppressing impulse noise. We incorporate the standard median (SM) filter and the center weighted median (CWM) filter into a noise detection framework to determine whether a pixel is corrupted, before applying filtering unconditionally. Extensive simulation results demonstrate that the proposed filter consistently outperforms other median filters by balancing the tradeoff between noise reduction and detail preservation. Index Terms—Impulse noise, median filter, noise detection. I. INTRODUCTION Digital images are often corrupted by impulse noise during the acquisition or transmission through communication channels. Con- sequently, some pixel intensities are inevitably altered while others remain noise-free. The image model containing impulse noise with probability of occurrence can be described as follows: with probability ; with probability (1) where denotes the noiseless image pixel and the noise substituting for the original pixel. In order to remove impulse noise and enhance image quality, the median filter has been extensively studied and presented in the literature (e.g., [1] and [2]). Median filtering being a nonlinear filtering technique, it is generally superior to linear filtering (e.g., Manuscript received June 30, 1998; revised April 27, 1999. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Henri Maitre. T. Chen is with the School of Computer Science and Software Engi- neering, Monash University, Clayton Campus, Vic. 3168, Australia (e-mail: tchen@cs.monash.edu.au). K.-K. Ma and L.-H. Chen are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, Republic of Sin- gapore, 639798 (e-mail: elhchen@ntu.edu.sg). Publisher Item Identifier S 1057-7149(99)09349-5. 1057–7149/99$10.00  1999 IEEE