Rician Noise Removal in Diffusion

Tensor - MRIThesis Defense

Saurav BasuSchool of Computing

University of Utah

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Organization• Brief overview of DT MRI

• Goals for this thesis

• Motivation :

- why noise removal ?

- why Rician noise ?

- previous DT-MRI filtering methods

• Rician Bias Correction Filter

• Results and discussion

• Conclusion: summary, future work

• Questions ?

March-April 2006

DT-MRI is themost recent in aseries of astonishingbreakthroughs in brain imaging

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HyperStreamLines used to Visualize White Matter Fibres in the brain

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Diffusion Tensor

Brief overview of DT-MRI

Imaging technique to compute a 3x3 matrix (D)Characterizes diffusion of water across brain tissue

•Symmetric

•Positive Definite

•All eigenvalues are positive

Used to study structure of brain fibresKey: More diffusion along fibres than across

fibres

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‣ Visualize the eigen values of D over the volume to infer connectivity and structure

Tensor Orientation: Principle Eigen VectorTensor Anisotropy: directional characteristics

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How is the tensor computed ?

A0

Ai

gi

Stejskal Tanner equation

Known: b, gi

Measured: A0, Ai

Find DMost Common:Linear Least Squares on

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Goals for this Thesis:

• What is the best way to filter DT-MRI data?

• How do current filtering methods compare ?

• Is there a better way of doing filtering?

Answer these questions.

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Motivation

• Why is DT - MRI filtering important?

• Why is it important to account for Rician noise in the filtering process?

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• DT MRI plagued by low SNR

‣ Multiple Scans needed to increase SNR

‣ Issues: long acquisition time, patient comfort system throughput

‣ Mis-registration issues (motion artifacts)

‣ Partial voluming (volume averaging) : voxel covers a non homogeneous tissue region

Why DT - MRI filtering?

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• what is Rician noise? how does it arise in DT MRI?

• how does it effect tensors?

• previous filtering methods

Why Rician noise removal?

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Rician noise in DT MRI ?

•DWI 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

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Rician Noise

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

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p(x|A)

ARice Distribution

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p(x)

ANormal Distribution

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10000 samples , sigma=20

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How does Rician noise affect estimated tensors?

previous studies show

noise trace and FA

“However, when we performed Monte Carlo simulations with Rician noise with diffusion tensors characteristic of those in the human brain we found FA and trace can be incorrectly estimated when tensors are aligned with gradient directions.”

aligned tensors: noise FA trace

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Tensor Splitting Gradient direction

Tensor aligned with gradient direction

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This tells us FA can be overestimated or underestimated depending on how a person sits inside the scanner !

Bottom Line!

It is important to consider Rician noise in filtering process

Can Seriously affect the validity of clinical studies using these FA estimates.

Previous filtering approaches2

categories

DWI spaceTensor Space

1) Non linear smoothing for reduction of systematic errors. Parker(2000)

2) Constrained Variational approach Wang, Vemuri (2004)

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

1) Riemannian Space filtering Pennec (2004)

Very effective techniques, but do not explicitly handling Rician noise as part of

the filtering process.

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Others: Median filtering, K- Space(Fourier Domain) methods

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• DWI Space filter

• 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

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MAP Image Reconstruction

•A Prior Model

•A Likelihood or Noise Model

3.Optimization Scheme (maximize posterior)

3 Key components

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

From Baye’s Rule:

constant for a given noisy image u0

MAP Formulation

Given: Noisy Image u0

estimate

Output: Clean/Filtered Image u

Known: p(u0|u) has a Rician distribution

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posterior likelihood prior

maximizewith

gradient ascent

Capture some prior knowledge about the filtered image.Example: enforce smoothing criteria on the image

Captures the noise model on the data

Essentially says : What is the probabilityof the clean image given that i have a particularnoisy image.

For gradient ascent we need to take derivatives!

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Likelihood Term

Taking derivative w.r.t u ,

Rician attachment term or

Bias correction term

After Substituting for Rice pdf

The Likelihood Term:

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We use a Gibb’s prior with an energy functional which enforces a smoothness without blurring edges

The Prior Term:

Gibb’s prior Energy functional

conductance

weighing factor

edge preserving smoothing prior

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Combining the Rician correction term with the variational of the energy functional we get the

update equation for the filtered image

Derivative of likelihood term

Variational of energy functional

Implementation:

Modify PDE Diffusion Filtering to use this modified update term

1. u = vector image of 7 or more DWIs2 Bias correction term term computed independently for each component of the vector

Note:

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ResultsWe compared 4 different filtering methods

on both synthetic and real data sets

DWI SpaceTensor Space

1. Anisotropic Diffusion without Rician attachment

2. Rician Bias Correction filter

1. Anisotropic Diffusion in euclidean space. 2. Anisotropic Diffusion on the Riemannian manifold

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Error Metrics

•Tensor Components

•Fractional Anisotropy (FA)

3.Trace

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Synthetic Data -1•10x10x4 volume of tensors

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

• Synthetic rician noise, baseline image intensity=250

Clean Noisy (SNR=15)

Gradient Directions:

( 1 0 1 )(-1 0 1 )( 0 1 1 )( 0 1 -1)( 1 1 0 )(-1 1 0 )

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Aniso DWI Rician DWI

DWI Space Filters

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Euclidean Riemannian

Tensor Space Filters

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• To check whether variability in directions affects results we generated a torus with tensors oriented in all possible directions.

• Ran the filtering on the torus data set

Synthetic Data Set -2 : Hollow Torus

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Clean Tensor

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Noisy Tensor (sigma=10)

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Euclidean Filtering

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Riemannian Filtering

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Aniso DWI Filtering

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Rician DWI Filtering

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Real Data Results

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

How do we evaluate filtering performance quantitatively?

➡ use repeated scans of the same subject.

Solution:

p(x/A) is the Rician pdf

Maximize (Brent’s| Golden search method)

ML Estimate:

LIKELIHOOD FUNCTION

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ML Estimator versus Averaging for generating Ground truth

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•5 scans of healthy volunteer•Resolution: 2 mm x 2 mm x 2 mm 3.3T scanner , scan time 12 mins.

Real Data Filtering Results

About the Real Data:

Added Rician noise SNR levels of 10,15 and 20 with respect to white matter signal level.and ran our filtering methods.

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Clean

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Sigma=10

Euclidean

Riemannian

Aniso

Rician

Noisy Image

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Noisy Image

Euclidean

Riemannian

Aniso

Rician

Noisy Image

Euclidean

Riemannian

Aniso

Rician

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Noisy Image

Euclidean

Riemannian

Aniso

Rician

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Noisy Image

Euclidean

Riemannian

Aniso

Rician

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Discussion

• Rician Filter : best RMS error performance.

• Real Data: Filtering DWIs better.

• Riemannian filtering: overall performs poorly

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Conclusions

• Bias effects of Rician noise

• New Rician-bias correction filter

• Systematic comparison

Summary & Contributions

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• ML method: low noise DWIs

• Filtering tools:

✦ Rician Bias Correction Filter

✦ Riemannian Space Tensor Filter

✦ Anisotropic Diffusion Filter on tensors

✦ Anisotropic Diffusion Filter on DWIs

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• DT-MRI Ground Truth: investigate ML methods

• Noise effects:

•fiber-tractography,

•diagnostic decisions.

• Rician Noise model : Tensor estimation

Future Work:

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Acknowledgments

Advisor:Dr. P. T. Fletcher

Committee: Dr. Ross T. Whitaker, Dr. Tolga Tasdizen

Gordon for help with Deft and teem

Josh for help with ITK

Dr. Guido Gerig, Dr. Wei Lin from UNC for providing us the real DT-MRI data

VIPER, NAMIC : Funding.

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Thanks, Question?

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Euclidean Space

Gradient

neighbors

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Riemannian Space:

Gradient neighbors

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