The purpose of this study is to assess the safety and effectiveness (usefulness) of an investigational technique used to repair the mitral valve for patients with mitral regurgitation due to a defective valve leaflet. This procedure is called trans-ventricular off-pump mitral valve repair (TRANSOM) and it repairs the mitral valve by replacing some of the cords that support the mitral valve. This type of repair is minimally-invasive since it is performed through a small incision and avoids making a breast-bone-splitting incision.