We propose signal reconstruction algorithms which utilize a guiding subspace that represents desired properties of reconstructed signals. Optimal reconstructed signals are shown to belong to a convex bounded set, called the ``reconstruction'' set. Iterative reconstruction algorithms, based on conjugate gradient methods, are developed to approximate optimal reconstructions with low memory and computational costs. Effectiveness of the proposed method is demonstrated with an application to image magnification. [http://arxiv.org/abs/1509.02465]