Elastography, which is based on applying pressure and estimating the resulting deformation, involves the forward problem to obtain the strain distributions and inverse problem to construct the elastic distributions consistent with the obtained strains on observation points. This thesis focuses on the former problem whose solution is used as an input to the latter problem. The aim is to provide the inverse problem community with accurate strain estimates of a coronary artery vessel wall. In doing so, a new ultrasonic image-based elastography approach is developed. Because the accuracy and quality of the estimated strain fields depend on the resolution level of the ultrasound image and to date best resolution levels obtained in the literature are not enough to clearly see all boundaries of the artery, one of the main goals is to acquire high-resolution coronary vessel wall ultrasound images at different pressures. For this purpose, first an experimental setup is designed to collect radio frequency (RF) signals, and then image formation algorithm is developed to obtain ultrasound images from the collected signals. To segment the noisy ultrasound images formed, a geodesic active contour-based segmentation algorithm with a novel stopping function that includes local phase of the image is developed. Then, region-based information is added to make the segmentation more robust to noise. Finally, elliptical deformable template is applied so that a priori information regarding the shape of the arteries could be taken into account, resulting in more stable and accurate results. The use of this template also implicitly provides boundary point correspondences from which high-resolution, size-independent, non-rigid and local strain fields of the coronary vessel wall are obtained.