## Abstract

In this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue
*Rc* ≥ *−K*, by using Li-Yau’s gradient estimate for the heat equation.

In this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue
*Rc* ≥ *−K*, by using Li-Yau’s gradient estimate for the heat equation.

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A new impulse detection and filtering algorithm is proposed for restoration of images that are highly corrupted by impulse noise. It is based on the average absolute value of four convolutions obtained by one-dimensional Laplacian operators. The proposed algorithm can effectively remove the impulse noise with a wide range of noise density and produce better results in terms of the qualitative and quantitative measures of the images even at noise density as high as 90%. Extensive simulations show that the proposed algorithm provides better performance than many of the existing switching median filters in terms of noise suppression and detail preservation.