We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights.
Abstract
The prescribed algorithm for removing impulse noise effectively even under high noise densities without causing any loss of image details. Hence a cascaded section of median filters that, involves an Decision-based Median Filter followed by a Recursive Weighted Median (RWM) Filter employing exponential weights are used. The median controlled algorithm is employed to calculate the exponential weights. In the algorithms that where proposed in earlier which involves a cascaded section of the median with the RWM filters provided lesser Peak Signal/Noise Ratio (PSNR) and greater Mean Square Error(MSE) values. Hence the output appeared to be distorted for higher noise levels. These drawbacks have been eliminated in this proposed algorithm.
In the recent years, many objective image quality assessment methods have been proposed by different researchers, leading to a significant increase in their correlation with subjective quality evaluations. Although many recently proposed image quality assessment methods, particularly full-reference metrics, are in some cases highly correlated with the perception of individual distortions, there is still a need for their verification and adjustment for the case when images are affected by multiple distortions. Since one of the possible approaches is the application of combined metrics, their analysis and optimization are discussed in this paper. Two approaches to metrics’ combination have been analyzed that are based on the weighted product and the proposed weighted sum with additional exponential weights. The validation of the proposed approach, carried out using four currently available image datasets, containing multiply distorted images together with the gathered subjective quality scores, indicates a meaningful increase of correlations of the optimized combined metrics with subjective opinions for all datasets.
Let Ux=∏i=1rx−tipi, 0<p<∞, −1=tr<tr−1<⋯<t2<t1=1, r≥2, pi>−1/p, i=1,2,…,r, and W=e−Qx where Q:−1,1⟶0,∞. We give the estimates of the zeros of orthogonal polynomials for the Jacobi-Exponential weight WU on −1,1. In addition, Markov–Bernstein inequalities for the weight WU are also obtained.
Heat Kernels Estimates for Hermitian Line Bundles on Manifolds of Bounded Geometry
