Figure 2

Definition of energy functional for the synthetic image estimation . Here I(x, y, t) is the acquired MOLLI image, and S(x, y, t) is the MOLLI signals which is calculated from the initial T1 parameter fitting. Note for the inversion recovery sequence, the polarity of acquired magnitude image has to be corrected. The weight function w(x, y) is added to maintain the edge sharpness in the estimated synthetic image. The first term constrains the distance between synthetic images and original MOLLI images. The second term is the regularizer using first-order derivative. It is added to penalize occasional error in the original T1 estimation and keep sufficient SNR of synthetic images. This term does not constrain the smoothness of the termporal behavior. The last term is added to minimize the distance between estimated images and the MOLLI signal recovery curve. As the MOLLI signal recovery curve is essentially smoothing, the term actually implicitly constrain the temporal smoothness of estimated synthetic images. α and β are weights to balance the contribution across three terms. This energy minimization problem can be solved by evaluating the resulting Euler equation.