Two dimensional canonical correlation analysis (2DCCA) is a data driven method that has been used to preserve the local spatial structure of functional magnetic resonance (fMR) images and to detect brain activation patterns. 2DCCA finds pairs of left and right linear transforms by directly operating on two dimensional data (i.e., image data) such that the correlation between their projections is maximized without neglecting the local spatial structure of the data. However, in the context of high dimensional data, the performance of 2DCCA suffers from interpretability of learned projection variables. In this study, to improve the interpretability of projection variables while preserving the local spatial structure of fMR images, we propose two new 2DCCA approaches, sparse 2DCCA and regularized 2DCCA. The proposed algorithms aim at improving the activation detection performance in terms of specificity of activated voxels by directly operating on image data without rearranging fMRI slices in 1D-vectors. The validity of the proposed algorithms has been evaluated on synthetic and real fMRI datasets and it has been shown that the proposed algorithms produce activation maps with higher specificity of activated voxels compared with CCA, 2DCCA, and existing sparse 2DCCA (S2DCCA).