1. (10 pts) The MRI signal from a specific tissue type (e.g. gray matter or white matter) decays at an exponential rate, referred to as the 72 relaxation rate. Because of lower resolution and the noisy nature of the acquisition, the measured signal at a specific pixel is often the combination of two different tissue types. y(t) = Ce exp (- 7. ) + co exp (-75 ) + n(t). (1) Here, cy and cu are the concentration of gray matter and white matter, respectively. Also assume that relaxation constants of the tissue types are known (T, = 1 seconds, Tw = 1/2 seconds). The noise signal n(t) ~ N(0,o') is white Gaussian distributed measurement noise at time t. Assume that the noise process at two different time points are independent. Assume that we have two signal measurements at t = 0 and t = 0.693, which can be concatenated to a 2 x 1 vector: y = wo.6931 = 0.33 (2) 1.1. Express the relation between the unknown concentration vector c Cg and y in a matrix form. 1.2. Using the above signal equations, determine the conditional probability density p(y|c).1.3. Using the above conditional density (likelihood), determine the maximum likelihood estimate of the tissue concentrations, denoted by CML-1.4. Assume that the tissue concentrations are independent and are distributed as exponential distri- butions P(x) = Aexp (-7) (3) with Ag = 10 and Aw = 10. Determine the maximum aposteriori estimate of the concentrations, specified by CMAP