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Melanoma can be cured if it is detected early, so early diagnosis is very important in dermatological practice today. Early and non-invasive diagnosis of melanomas can be done by accurate image segmentation of skin lesions. The medical images, while acquisition are generally bound to contain noise. This paper proposes a robust and efficient image segmentation algorithm using LOG edge detector to extract the true border of noisy clinical skin images containing lesions, that reveals the global structure irregularity, which may suggest excessive cell growth or regression of a melanoma. The proposed segmentation algorithm is applied to a RGB image containing the lesion, where the RGB image is converted to grayscale intensity image and adds salt and pepper noise to the image and uses background noise reduction techniques to filter noise. The algorithm converts an image to a binary image, based on threshold, and finds the xor image and then finds the edges in this image using log edge detector. The black and white image is used to find the row co-ordinate of the pixel on the border of the object to be traced and the edge detected image is used to find the column co-ordinate of the pixel on the border of the object to be traced and using this pixel found on the border of the object as the starting pixel, the border of the lesion is traced using the proposed robust segmentation algorithm successfully. In this paper, we have developed a robust segmentation method using LOG detector for border detection of real skin lesions for noisy skin lesion images and compared with variational formulation for geometric active contours for detecting, the desired image features, such as object boundaries by Chunming Li et al. The algorithm was applied on many noisy clinical skin images containing lesions. The proposed segmentation algorithm successfully detects the border of noisy clinical skin images.