This paper presents an adaptive regularized image interpolation algorithm, which can restore high frequency details in the original high resolution image. In order to apply the regularization approach to the interpolation procedure, we first present a two-dimensional separable image degradation model for a low resolution imaging system. Based on the image degradation model, we can have an interpolated image which minimizes both residual between the high resolution and the interpolated images with a prior constraints. In addition, by using spatially adaptive constraints and regularization parameters, directional high frequency components are preserved with efficiently suppressed noise. We also analyze convergence of the proposed adaptive iterative algorithm. As a result, step length of the adaptive algorithm should be less than the non-adaptive algorithm, and the ratio of two quantities is proportional to the number of different constraints used in the adaptive algorithm. In the experimental results, interpolated images using the conventional algorithms are shown to compare the conventional algorithms with the proposed adaptive algorithm. Moreover, we provide experimental results which are classified into non- adaptive and adaptive algorithms. Based on the experimental results, the proposed algorithm provides a better interpolated image than the conventional non-adaptive interpolation algorithms in the sense of both subjective and objective criteria. More specifically, the proposed algorithm has the advantage of preserving directional high frequency components and suppressing undesirable artifacts such as noise.