Motion correction is a challenging problem in free breathing under sampled cardiac cine MRI and cardiac perfusion MRI. Respiratory motion during cine MRI acquisition can cause strong blurring artifacts in the reconstructed image. These artifacts become more prominent when use with the fast imaging reconstruction techniques like compressed sensing (CS). CS has been developed as an MRI reconstruction technique, to provide good quality sparse images from the under sampled measurements. To exploit the CS, it is needed to use CS with the motion corrected samples in cardiac cine MRI. In perfusion MRI, the under-sampling artifacts and the rapid contrast changes can cause adverse effect in the quality of reconstructed perfusion MRI. In addition to the recovery limitations, many registration techniques underperform in the presence of strong intensity changes in the cardiac perfusion MRI. In this dissertation, new reconstruction algorithms have been proposed to obtain the motion corrected cardiac cine MRI and cardiac perfusion MRI. In the first part of the dissertation, we propose two novel motion correction based CS reconstruction technique to obtain good quality images. In first technique, reconstructed cine images with the highly under sampled k-space data are achieved using motion correction based CS framework. First, image registration based similarity measure is used to bin the data in different respiratory states. Then the motion correction based CS framework is used to obtain the good quality motion free cine images. The proposed method is simple to implement in clinical settings as compared to existing motion corrected methods. The performance of the developed technique is examined using simulated data and clinical data. Results show that this method performs better reconstruction of cardiac cine images as compared to the CS reconstruction method. In the second technique, first, k-space data has been assigned to different respiratory state with the help of frequency domain phase correlation method. Then, multiple sparsity constraints have been used to provide good quality reconstructed cardiac cine images with the highly undersampled k-space measurements. The proposed method exploits the multiple sparsity constraints, in combination with demon based registration technique and a novel reconstruction technique to provide the final motion free images. The performance of the method is examined using simulated data and clinical data with different acceleration rates. In the second part of the dissertation, two novel motion correction techniques are proposed to reconstruct the motion corrected images from under sampled cardiac perfusion MRI. First technique utilizes the robust principal component analysis along with the periodic decomposition to separate the respiratory motion component that can be registered, from the unchanged contrast intensity variations. It is tested on synthetic data, simulated data and the clinically acquired data. The performance of the method is qualitatively assessed and validated by comparing manually acquired time-intensity curves of the myocardial sectors to automatically obtained curves before and after registration. In second technique, a new algorithm for robust principal component analysis is developed to separate contrast agent from the perfusion images in the presence of the acquisition noise. Then the periodic decomposition in combination with the image registration is used to remove the respiratory motion artifacts from the perfusion images. It is tested on simulated data and the clinically acquired data. The performance of the technique is stage wise compared with the existing motion correction methods.