We arrived at the fixed-point formulation of the total variation maximum a posteriori (MAP) regularized emission computed tomography (ECT) reconstruction problem and we proposed an iterative alternating scheme to numerically calculate the fixed point. We theoretically proved that our algorithm converges to unique solutions. Because the obtained algorithm exhibits slow convergence speed, we further developed the proximity algorithm in the transformed image space, i.e. the preconditioned proximity algorithm. We used the bias-noise curve method to select optimal regularization hyperparameters for both our algorithm and expectation maximization with total variation regularization (EM-TV). We showed in the numerical experiments that our proposed algorithms, with an appropriately selected preconditioner, outperformed conventional EM-TV algorithm in many critical aspects, such as comparatively very low noise and bias for Shepp-Logan phantom. This has major ramification for nuclear medicine because clinical implementation of our preconditioned fixed-point algorithms might result in very significant radiation dose reduction in the medical applications of emission tomography.