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Image Segmentation based on multilevel thresholding using non-extensive (non-additive) entropy based techniques is challenging, and the optimal choice of thresholds is an effective approach to solve this problem. In this paper, we propose a novel optimization technique based on the Particle Swarm Optimization (PSO) called Fibonacci Particle Swarm Optimization (FPSO) that helps decide the optimal thresholds by maximizing the objective function of Tsallis entropy. The superiority of our proposed method has been demonstrated by comparing the results with some of the contemporary algorithms like Genetic Algorithm (GA), Bacterial Foraging Optimization (BFO), the Standard Particle Swarm Optimization (PSO) and the Golden Ratio Particle Swarm Optimization (GRPSO). The quality of the segmented images has been evaluated using Peak Signal to Noise Ratio (PSNR) and Compression Ratios of the original images and reconstructed images. The results obtained by the proposed method have been found to be significantly better than those obtained by the above mentioned algorithms.