Rician Noise Removal in

Diffusion Tensor MRI

Saurav Basu, Tom Fletcher, Ross WhitakerSchool of ComputingUniversity of Utah

• DT-MRI plagued by low SNR

‣ Multiple scans needed to increase SNR

‣ Issues: long acquisition time, patient comfort system throughput

‣ Noise in DT-MRI adversely affects tensor measurements used in clinical studies

Why DT-MRI filtering?

Rician noise in DT-MRI

•DW images are magnitudes of complex valued signals.

•If the real and imaginary components of the signal are assumed to have a Gaussian noise, the resulting magnitude image will have Rician distributed noise.

gaussian

magnitude

where is zero mean, stationary Gaussian noise with standard deviation

Rician Noise

Unlike the normal distribution the pdf is not symmetric about the true signal value A

A signal is said to be corrupted with Rician noise if the pdf of the noisy signal has a Rice distribution

p(x)

ARice Distribution

How does Rician noise affect estimated tensors?

Tensor splitting gradient direction

Tensor aligned with gradient direction

We performed Monte Carlo simulations with two cases:

Previous filtering approaches2 categorie

s

DWI spaceTensor Space

Anisotropic DiffusionParker(2000)

Constrained Variational approach Wang, Vemuri (2004)

Bayesian regularization using Gaussian markov random fields. Martin (2004)

Riemannian Space filtering Pennec (2004)

Very effective techniques, but do not explicitly handling Rician noise as part of the filtering process.

• Based on maximum a posteriori (MAP) approach to image reconstruction

• In statistics MAP estimation is used to obtain a point estimate of an unobserved quantity based on empirical data

Rician Bias Correction Filter

• Given an initial noisy image u0 we want to estimate the clean image u.

• We know that p(u0|u) has a Rician distribution.

• To estimate the clean value we want to maximize p(u|u0)

From Baye’s Rule

constant for a given noisy image u0

posterior

noise model(likelihood )

prior

prior: what is pdf of the unobserved data (clean image) whichwe are trying to estimate?noise model (likelihood) : what is the conditional probability of the observed data( noisy image) , given a particular value of the unobserved data (clean image)? posterior: The probability of the unobserved data (clean image)given the observed data (noisy image)

maximizewith

gradient ascent

Rician likelihood term

Using the Rician pdf for the noise model we get

rician likelihood term

Taking derivative w.r.t. u,

Rician attachment Term

The Rician attachment term can be combined with any image prior.

We use a Gibbs prior with Perona Malik Energy functional.

Combining with the prior:

Gibb’s prior

Perona Malik energy

conductance

weighing factor

edge preserving smoothing prior

Combining the Rician correction term with prior we get the update equation for the

filtered image

Preliminary ResultsWe compared 4 different filtering

methods on both synthetic and real datasets

DWI SpaceTensor Space1.Anisotropic Diffusion

without Rician attachment

2. Rician Bias Correction filter

1.Anisotropic Diffusion in Euclidean space

2.Anisotropic Diffusion on the Riemannian manifold

Error Metrics:1.RMS error in tensor components2.Fractional Anisotropy3.Trace

•Parameters optimized for RMS error in tensor components.

•For both synthetic and real data we used 7 images for each slice (6 gradient directions + 1 baseline)

Synthetic Data Results

• 10x10x4 volume of tensors

• 2 tensor orientations (along gradient and splitting the gradient directions)

• Synthetic rician noise

CleanNoisy

(SNR=15)

Aniso DWIRician DWI

DWI Space Filters

Euclidean Riemannian

Tensor Space Filters

Real Data Results

Issue: No ground truth data available for DT-MRI !

How do we evaluate filtering performance quantitatively?

• we developed a method to estimate a ground truth data from repeated scans of the same object

• if {xi} is a set of intensities for the same voxel in N repeated scans we find the ML estimate of the true value A by maximizing the log likelihood function:

Solution:

p(x|A) is the Rician pdf

• Generated ground truth from 5 scans• added known Rician noise (SNR=10,15,20)• compared errors as before

Clean Coronal Slice Noisy Coronal Slice(SNR=15)

Aniso DWI Rician DWI

Both Aniso-DWI and Rician DWI gave very good results with Rician being marginally better

DWI Space

Euclidean Tensor Riemannian Tensor

Tensor Space