It is often desirable to separate voxels that contain signal from tissue along with measurement noise from those that contain purely measurement noise. Generally, this separation called thresholding utilizes only the magnitude portion of the images. Recently, methods have been developed that utilize both the magnitude and phase for thresholding voxels. This manuscript is an extension previous work and uses the bivariate normality of the real and imaginary values with phase coupled means. A likelihood ratio statistic is derived that simplifies to a more familiar form that is F-distributed in large samples. It is shown that in small samples, critical values from Monte Carlo simulation can be used to threshold this statistic with the proper Type I and Type II error rates. This method is applied to susceptibility weighted magnetic resonance images and shown to produce increased tissue contrast.