Wavelet transforms have been successfully applied in many fields of image processing. Yet, to our knowledge, they have never been directly incorporated to the objective function in Emission Computed Tomography (ECT) image reconstruction. Our aim has been to investigate if the ℓ1-norm of non-decimated discrete cosine transform (DCT) coefficients of the estimated radiotracer distribution could be effectively used as the regularization term for the penalized-likelihood (PL) reconstruction, where a regularizer is used to enforce the image smoothness in the reconstruction. In this study, the ℓ1-norm of 2D DCT wavelet decomposition was used as a regularization term. The Preconditioned Alternating Projection Algorithm (PAPA), which we proposed in earlier work to solve penalized likelihood (PL) reconstruction with non-differentiable regularizers, was used to solve this optimization problem. The DCT wavelet decompositions were performed on the transaxial reconstructed images. We reconstructed Monte Carlo simulated SPECT data obtained for a numerical phantom with Gaussian blobs as hot lesions and with a warm random lumpy background. Reconstructed images using the proposed method exhibited better noise suppression and improved lesion conspicuity, compared with images reconstructed using expectation maximization (EM) algorithm with Gaussian post filter (GPF). Also, the mean square error (MSE) was smaller, compared with EM-GPF. A critical and challenging aspect of this method was selection of optimal parameters. In summary, our numerical experiments demonstrated that the ℓ1-norm of discrete cosine transform (DCT) wavelet frame transform DCT regularizer shows promise for SPECT image reconstruction using PAPA method.
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.