Abstruct-A new expression for the output moments of weighted mcdian filtered data is derived in this paper. The noise attenuation capability of a weighted median filter can now be assessed using the L-vector and AI-vector parameters in the new expression.The second major contribution of the paper is the development of a new optimality theory for weighted median filters. This theory is based on the new expression for the output moments, and combines the noise attenuation and some structural constraints on the filter's behavior.In certain special cases, the optimal weighted median filter can be obtained by merely solving a set of linear inequalities. This leads in some cases to closed form solutions for optimal weighted median filters. Some applications of the theory developed in this paper, in 1-D signal processing and image processing are discussed.Throughout the analysis, some striking similarities are pointed out between linear FIR filters and weighted median filters.

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In today's data-driven world, it is critical that we use appropriate datasets for analysis and decision-making. Datasets could be biased because they reflect existing inequalities in the world, due to the data scientists' biased world view, or due to the data scientists' limited control over the data collection process. For these reasons, it is important to ensure adequate data coverage across different groups over the intersection of multiple attributes. Often, the dataset to be analyzed is obtained through complex joins and predicate combinations over multiple relational tables in a database. Due to the sheer data volume we often have to deal with, determining adequate coverage can require an unacceptably long execution time. In this paper, we provide an efficient approach for coverage analysis, given a set of attributes across multiple tables. To identify regions with insufficient coverage in the combinatorially large set of value combinations, we design an index scheme to avoid explicit table joins, achieve efficient memory usage, and support predicate combination at a high level of parallelism. We also propose P-WALK , a priority-based search algorithm, to traverse the lattice space. Since in practice, coverage assessment typically does not require precise COUNT aggregation results, we further present approximate methods to reduce computation time. Experimental evaluation using three real-world datasets shows the effectiveness, efficiency, and accuracy of proposed methods.

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