Covariance of reconstruction images are useful to analyze the magnitude and correlation of noise in the evaluation of systems and reconstruction algorithms. The covariance estimation requires a big number of image samples that are hard to acquire in reality. A covariance propagation method from projection by a few noisy realizations is studied in this work. Based on the property of convergent points of cost funtions, the proposed method is composed of three steps, 1) construct a relationship between the covariance of projection and corresponding reconstruction from cost functions at its convergent point, 2) simplify the covariance relationship constructed in 1) by introducing an approximate gradient of penalties, and 3) obtain an analytical covariance estimation according to the simplified relationship in 2). Three approximation methods for step 2) are studied: the linear approximation of the gradient of penalties (LAM), the Taylor apprximation (TAM), and the mixture of LAM and TAM (MAM). TV and qGGMRF penalized weighted least square methods are experimented on. Results from statistical methods are used as reference. Under the condition of unstable 2nd derivative of penalties such as TV, the covariance image estimated by LAM accords to reference well but of smaller values, while the covarianc estimation by TAM is quite off. Under the conditon of relatively stable 2nd derivative of penalties such as qGGMRF, TAM performs well and LAM is again with a negative bias in magnitude. MAM gives a best performance under both conditions by combining LAM and TAM. Results also show that only one noise realization is enough to obtain reasonable covariance estimation analytically, which is important for practical usage. This work suggests the necessity and a new way to estimate the covariance for non-quadratically penalized reconstructions. Currently, the proposed method is computationally expensive for large size reconstructions.Computational efficiency is our future work to focus.
Image mean and covariance required for a model observer are usually calculated by the statistical method using image samples, which is hard to acquire in reality. Although some analytical methods are proposed to estimate image covariance from a single projection, these methods are of high computational cost for large-dimensional images (e.g., 512×512), and images of large dimension are commonly required. Considering the covariance used for a model observer is the covariance of the channel response, whose dimension is much smaller than the image covariance, we aim to obtain the covariance of small-dimensional channel response directly from its projection. Channel filters are applied to the analytical projection to image (Prj2Img) covariance estimation method to derive the analytical projection to channel response (Prj2CR) covariance estimation method, which successfully reduces the computational cost and connects the covariance of projection and channel response. In addition, a transition matrix is introduced in Prj2CR method to stabilize the connections. The transition matrix mainly depends on channel filters, not the system, phantom, and reconstruction algorithm, which means it can be calibrated by small-dimensional reconstructions and then applied to any situation with a same channel filter. We validate the feasibility and utility of the proposed Prj2CR method by simulations. 128×128 reconstructions from qGGMRF-WLS are adopted for calibration, while 512×512 reconstructions are used for validation. SNR of CHO is chosen as the figure of merit for performance evaluation, and the covariance estimated by 290 image samples are used as the reference. Results show that the SNR by the Prj2CR method is within 95% confidence interval of the SNR * by 290 image samples, indicating that the proposed method accords with statistical method. The Prj2CR method may be beneficial for subjective image quality assessment since it only needs a single sample of projection and has low computational cost.
Gain from more energy channels could be significant with the increase of energy channel number. Introduction of contrast agents in scanned objects will increase overall error in spectral CT imaging. Energy thresholding optimization is beneficial for information recovery. Moreover, the choice of basis materials could also be important to obtain low noise results. With these studies of the effect from various configurations for PCD-SCT, one may optimize the configuration of PCD-SCT accordingly.
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