## 1 Introduction  
Coronary artery disease (CAD) is the leading cause of death in the Western world and developing countries [1]. The gold standard imaging modality for detection of CAD is X-ray angiography, which is invasive, exposes patients to potentially harmful radiation and requires iodinated contrast agents. Multi-Detector Computed Tomography (MDCT) has been shown to be a reliable technique for non-invasive detection of coronary artery stenosis [2]. However, MDCT, like X-ray, utilizes ionizing radiation. Furthermore, the diagnostic accuracy of MDCT is reduced in patients with heavily calcified plaques. In recent decades coronary magnetic resonance angiography (CMRA) has emerged as a non-invasive alternative for the diagnosis of CAD without exposing patients to potentially harmful radiation or nephrotoxic contrast agents. But despite advancements in CMRA hardware and software techniques, artifacts due to residual coronary artery motion remain an impediment to diagnostic CMRA in a small but significant amount of patients. Several studies have investigated respiration induced coronary motion along the foot-head (FH), left-right (LR), and anterior-posterior (AP) directions throughout the respiratory cycle [3; 4; 5; 6]. Although the principal motion component is predominately in the FH direction and is often one order of magnitude larger than the CMRA spatial resolution, the displacement magnitude in the LR or AP direction has high subject specific variability and can contribute to motion artifacts in CMRA. Furthermore, rotation and non-rigid motion such as affine or even non-linear deformation may occur between the end-expiratory and end-inspiratory respiratory phase [7; 8; 9; 10]. As a result, CMRA is particularly susceptible to respiratory motion artifacts due to the relatively slow data acquisition speed of MRI and the high spatial resolution required to visualize the coronary artery lumen. The main coronary arteries, which consist of the right coronary artery (RCA) and the left main (LM) artery which branches into the left anterior descending (LAD) artery and left circumflex (LCX) artery, have a diameter of approximately 3 mm in the proximal coronary segments which reduces to 1 mm in more distal segments. A respiratory motion corrupted CMRA dataset is shown in Figure 1(a), along with the same dataset after respiratory motion correction in Figure 1(b), reformatted to visualize the RCA and LAD. Figure 1: CMRA dataset without respiratory motion correction (**a**) and with respiratory motion correction (**b**). The 3D dataset is reformatted to visualize the right coronary artery (RCA) and left anterior descending artery (LAD). Ao = ascending aorta; RV = right ventricle; LV = left ventricle. Adapted from [11]. For diagnostic purposes it is desirable to detect obstructive coronary stenosis of at least 50% reduction of the coronary lumen. Thus, CMRA spatial resolution is typically on the order of 0.7 to 1.3 mm. This extends the CMRA scan time beyond what can be comfortably acquired within a breath-hold. In comparison, due to superior acquisition speed, X-ray angiography and MDCT allows for data acquisition with sub-millimeter resolution within a single breath-hold, which effectively minimizes the problem or respiratory motion. Although respiratory motion compensation in the early days of CMRA involved breath-holding [12], image acquisition typically only employed two-dimensional (2D) encoding and therefore provided limited coverage of the coronary arteries [13, 14, 15]. As the coronary arteries are long and tortuous, three-dimensional (3D) coverage is important to visualize one or more coronary arteries in a single scan. Developments in MRI acceleration techniques including partial Fourier [16] and parallel imaging [17, 18] have enabled 3D CMRA data acquisitions within a single breath-hold [19, 20, 21]. Further acceleration may be achieved with compressed sensing [22, 23, 24] although its efficacy in conjunction with breath-hold CMRA has yet to be established.