Automatically estimating noise-induced signal variance in magnitude reconstructed MRI using kernel estimator
Gustavo Kunde Rohde, Peter J. Basser, Carlo M. Pierpaoli · 2005
2 2 σ = σnoise . Note that the value of M for which p(M) in (2) is maximum is equal to σ . This finding can be easily checked by differentiating (2) with respect to M, setting the result equal to zero, and solving for M (4). This information can be used to extract the standard deviation of the intensity σ in the images by simply identifying the peak of the noise distribution. We use the kernel or the Parzen density estimator (5) for this purpose, the most popular technique for nonparametric density estimation. The choice of basis function is not very important so long as it is smooth and bell-shaped (5). We chose the Gaussian kernel because it is easy to manipulate and derive. The kernel size or window width is very important and sometimes is adapted to the application. There is a trade-off between too much variability on one hand (if the window width is too small) and increased bias on the other (if it is too large). The window width can be computed by minimizing the mean square error between the true and estimated density. In our simulation, we set the window width equal 1.06*(sample standard deviation)*n -1/5 where n is the sample size as suggested in (5). We created synthetic images containing different size objects and added Gaussian distributed noise in quadrature to simulate images with different S/N. Our objective is to test the accuracy of the proposed method. RESULTS Figure 1 shows the simulation result on the amount of background required for the proposed method to properly estimate the peak of the noise distribution. If the error of the estimated standard deviation is set within 10%, 65% of the background in an image is required when S/N =3, 22% is required when S/N=4, and only a small amount of background is needed when S/N 5. In general, more background would provide a better result. If the background is less than required, for example, less than 60% when S/N=3, the estimated standard deviation is somewhat over-estimated. This is understandable because the noise distribution is contaminated by the signal and the mixture of noise and object will always cause the estimated peak to be shift to the right. We use echo-planar diffusion weighted images as a test application. We carefully chose 30 regions manually from the background and compute the average of variance from those regions. Our preliminary result shows that the estimated variance using the Parzen window approach has similar results when compared with this conventional estimation (has the error rate within 10% in our testing data sets). Simulations with artifacts will be conducted to further verify our preliminary findings. CONCLUSION An automatic method for estimating the variance in magnitude reconstructed MRI is presented. This method needs only one image, does not require any user interaction as no background pixels need to be selected, and does not require prior brain segmentation. The result is promising when compared with the conventional manual object-free background selection.