Image up-sampling in the discrete cosine transform (DCT) domain is a challenging problem because DCT coefficients are de-correlated, such that it is nontrivial to estimate directly high-frequency DCT coefficients from observed low-frequency DCT coefficients. In the literature, DCT-based up-sampling algorithms usually pad zeros as high-frequency DCT coefficients or estimate such coefficients with limited success mainly due to the nonadaptive estimator and restricted information from a single observed image. In this paper, we tackle the problem of estimating high-frequency DCT coefficients in the spatial domain by proposing a learning-based scheme using an adaptive $k$ -nearest neighbor weighted minimum mean squares error (MMSE) estimation framework. Our proposed scheme makes use of the information from precomputed dictionaries to formulate an adaptive linear MMSE estimator for each DCT block. The scheme is able to estimate high-frequency DCT coefficients with very successful results. Experimental results show that the proposed up-sampling scheme produces the minimal ringing and blocking effects, and significantly better results compared with the state-of-the-art algorithms in terms of peak signal-to-noise ratio (more than 1 dB), structural similarity, and subjective quality measurements.