By deploying a large number of antennas with sub-half-wavelength spacing in a compact space, dense array systems (DASs) can fully unleash the multiplexing and diversity gains of limited apertures. To acquire these gains, accurate channel state information acquisition is necessary but challenging due to the large antenna numbers. To overcome this obstacle, this paper reveals that designing the observation matrix to exploit the high spatial correlation of DAS channels is crucial for realizing near-optimal Bayesian channel estimation. Specifically, we prove that the observation matrix design for channel estimation is equivalent to a time-domain duality of point-to-point multiple-input multiple-output precoding, except for the change in the total power constraint on the precoding matrix to the pilot-wise discrete power constraint on the observation matrix. Inspired by Bayesian regression, a novel ice-filling algorithm is proposed to design amplitude-and-phase controllable observation matrices, and a majorization-minimization algorithm is proposed to address the phase-only controllable case. Particularly, we prove that the ice-filling algorithm can be interpreted as a ``quantized" water-filling algorithm, wherein the latter's continuous power-allocation process is converted into the former's discrete pilot-assignment process. To support the near-optimality of the proposed designs, we provide comprehensive analyses on the achievable mean square errors and their asymptotic expressions. Finally, numerical results confirm that our proposed designs achieve the near-optimal channel estimation performance and outperform existing approaches significantly.