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In distributed estimation fusion, processed data from each sensor is sent to the fusion center. By taking linear transformation of the raw measurements of each sensor, two optimal distributed fusion algorithms are proposed in this paper. Compared with existing fusion algorithms, they have three nice properties. First, they are optimal in the sense that they are equivalent to the optimal centralized fusion. Second, their communication requirements from each sensor to the fusion center are equal to or less than those of the centralized and most existing distributed fusion algorithms. Third, they do not need the inverses of estimation error covariance matrices, which are assumed to exist in most existing algorithms but can not be guaranteed to exist. So the proposed algorithms can be applied in more cases. Pros and cons of these two new algorithms are analyzed. A possible way to reduce the computational complexity of the new algorithms, an extension to the case of a singular covariance matrix of measurement noise, and an extension to the reduced-rate communication case for some simple systems are also discussed.