public class AlgorithmNoise
extends AlgorithmBase

Algorithm used to add Gaussian, Poisson, Uniform, Rayleigh, or Rician noise to an image. The additive noise is clamped to the lowest or highest value is the source image type. For example a byte image where the source pixel = 120 + noise = 15 would be clamped to 127 the maximum pixel value for a byte image. For Rayleigh noise the formula is simply sigma * sqrt(-2.0 * ln(U)), where U is a uniform 0 to 1 distribution not including 0. The Rayleigh cumulative distribution function F(x) = 1 - exp(-(x**2)/(2*sigma**2)) is simply set equal to U and then the inverse function is obtained by solving for x in terms of U, giving x = sigma * sqrt(-2.0 * ln(1 - U)) and finally noting that 1 - U has the same distribution as U. -----Original Message----- From: hakim.achterberg@gmail.com [mailto:hakim.achterberg@gmail.com] On Behalf Of Hakim Achterberg Sent: Wednesday, April 18, 2012 9:30 AM To: Gandler, William (NIH/CIT) [E] Cc: McAuliffe, Matthew (NIH/CIT) [E]; Senseney, Justin (NIH/CIT) [E] Subject: Re: Which Rician noise generator is the best? Dear William, All methods come down to the same equations, however there are some differences in the assumption of the signal. Adding Rician noise to the magnitude of a complex signal, is equivalent to adding Gaussian noise to both the real and imaginary part of the signal and then taking the magnitude. However, in our case the phase of this signal was unknown, so some assumption had to be made about the phase in the simulation. The assumption I can deduct from the methods mentioned: In equation (14) in "A Nonlocal Maximum Likelihood Estimation Method for Rician Noise Reduction in MR Images" the phase of the signal is assumed to be zero (or 0.5 pi, pi, 1.5 pi) so that the magnitude is completely in one of the two components. In equation (4) in "Rician noise removal by non-Local Means filtering for low signal-to-noise ratio MRI: applications to DT-MRI" the phase of the signal is assumed to be 0.25 pi (or an equivalent rotation) so that the magnitude is split into two equal parts (a + bi, where a=b=M/sqrt(2)). In equation (6) in "Development of computer-generated phantoms for FMRI software evaluation", I have the feeling they make an assumption which is incorrect and that is that adding the squared noises separately to the magnitude it is similar as to adding the noise to the real and imaginary parts of the signal in computation of the gradient magnitude. My feeling says this one is not correct, but I haven't extensively checked this. In our work ( Algorithm 2.1 in "Optimal acquisition schemes in high angular resolution diffusion weighted imaging" ), we assume the phase of the signal to be random. So we create a random phase, change the signal from a magnitude and phase into real and imaginary parts, add noise to these parts and calculate the gradient magnitude. To be honest, from discussion with our MR physicist we thought our assumption to be (more or less) valid, but I do not recall that this assumption has been carefully checked. To really establish which algorithm is the best, the assumptions made in each model should be checked against the data you are trying to simulate. I hope that this helps and does not only add confusion. Kind regards, Hakim Email from MRI manufacturer neusoft supports using random phase for Rician noise generation: FW: NIH questions FW: MRI phase characterization Dear Keith There are several key points about the questions as below: 1) About Quadrature coils, the two coils are orthogonally orientated having 90 degree phase difference to maximizing the SNR. We do not adjust the phase difference between the 2 coils to 45 degrees just before combining them into a magnitude signal. 2) About a complex signal, the phases of a set of actual data of MRI not simulation is usually not the only 45 degree. 3) About noise, it be added to the real and imaginary parts of the signal respectively is benefit to getting Rician noise. FYI. Best Regards, Polly Yang Software Team Leader, MR R&D Department Philips and Neusoft Medical Systems Co.,Ltd. No.16 Century Road, Hun Nan New District Shenyang 110179, PRC Tel: (86 24)8366 0778 Fax: (86 24)8366 1915 Mobile: (86)13940069636 Email: yangp@neusoft.com Dear William, Thanks for your interest in my paper. If X and Y are independent Gaussian random variables, then R=sqrt(X^2+Y^2) is Rician distributed. In particular, if X had mean mu*cos(theta) and variance sigma^2 and Y has mean mu*sin(theta) and variance sigma^2, then R is Rician distributed with parameters mu and sigma. If mu is much larger than sigma, then R is approximately Gaussian distributed with mean mu and variance sigma^2. The motivation for using this form in MRI is the fact that the raw (received) signals have real and imaginary component (the component in-phase with the RF excitation and the component out-of-phase with the excitation). The value of theta, above, reflects how much is in and out of phase. However, the raw signals are measured with noise (due to many factors) and the noise in each component is independent. This gives rise to the Rician model that is generally accepted in MRI. This paper by Prof. Makovski is gives an excellent explanation of the physics that lead to this model: www.ncbi.nlm.nih.gov/pubmed/8875425 I hope this is helpful. Best wishes, Rob Gandler, William (NIH/CIT) [E] wrote: > > Dear Robert Nowak: > > > > In Noise Removal Methods for High Resolution MRI you present a model > for Rician noise generation in equation (1). Is there a reference or > motivation for using this particular form? Some researchers use an > equation for Rician noise generation where the phase is zero and all > of the signal is in the real part. Other researchers split the signal > into equal real and imaginary parts. > > > > > > Sincerely, > > > > > William Gandler > > > -- Robert Nowak University of Wisconsin-Madison www.ece.wisc.edu/~nowak Dear William, To generate Rician noise, it doesn't really matter if the signal is in real part only or if it is split into real and imaginary parts. The important thing is to add Gaussian noise to both channels and then make the magnitude image. You will always get exactly the same noise distribution independently of the phase angle. Equation 2 represents the signal in a real MR image, and there is phase angle neither zero or 45 degrees. Best regards Adnan Bibic, MSc Research Engineer Preclinical MRI systems, LBIC Faculty of Medicine, Lund University Klinikgatan 32, building D11 SE-221 84 Sweden Phone: +46 46 222 42 15 homepage (LBIC): www.med.lu.se/bioimaging_center ________________________________________ From: Gandler, William (NIH/CIT) [E] [ilb@mail.nih.gov] Sent: 29 May 2012 20:51 To: Adnan Bibic Cc: Senseney, Justin (NIH/CIT) [E]; McAuliffe, Matthew (NIH/CIT) [E] Subject: Rician noise generation equation Dear Adnan Bibic: In your master of science thesis you present a model for Rician noise generation in equation (2). Is there a reference or motivation for using this particular form? Some researchers use an equation for Rician noise generation where the phase is zero and all of the signal is in the real part. Other researchers split the signal into equal real and imaginary parts. Sincerely, William Gandler Rician noise is not additive, but is instead data dependent. Let a be the original noiseless data value, and x and y be gaussian random variables with zero mean and identical standard deviations sigma. Then with a uniformly distributed theta the resulting value m = sqrt((a*cos(theta) + x)**2 + (a*sin(theta) + y)**2) is Rician distributed. MR images have Rician noise. References for Rician noise generation: Optimal acquisition schemes in high angular resolution diffusion weighted imaging" MSC Thesis by Hakim Achterberg. Denoising of Complex MRI Data by Wiener-like Filtering in the Wavelet Domain: Application to High b-value Diffusion Weighted Imaging by Adnan Bibic. Noise Removal Methods for High Resolution MRI by R. L. Gregg and R. D. Nowak Wavelet-Based Rician Noise Removal for Magnetic Resonance Imaging by Robert D. Nowak MRI denoising via phase error estimation by Dylan Tisdall and M. Stella Atkins Confirmation that the zero angle phase noise generator leads to a Rician noise distribution is found in "Analytically exact correct scheme for signal extraction from noisy magnitude MR signals" by Cheng Guan Koay and Peter J. Basser, Journal of Magnetic Resonance, 179, 2006, pp. 317-322, Equations 2 - 4.