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What to measure when measuring noise in

MRI

Santiago Aja-Fernández

sanaja@tel.uva.es

UNIVERSIDAD DE VALLADOLID E. T. S. INGENIEROS DE TELECOMUNICACIÓN

Antwerpen 2013

Noise in MRI

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Motivation:

• Many papers and methods to estimate noise out of MRI data.

• Noise estimation vs. SNR estimation• In single coil systems, variance of noise is a “good”

measure.• Complex systems: what are we really measuring? Is the

variance of noise still valid?

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Outline1. Noise in MR acquistions2. Basic Models

– Rician

– Non-central chi

3. More Complex Models– Correlated multiple coils

– Parallel MRI: SENSE and GRAPPA

4. The nc-chi example– Non-stationary noise

– Effective values

5. Other models

10. Conclusions

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1- Noise in MRI

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Scanner K-spaceX-space

(complex) magnitude

F-1 | . |

Signal acquisition (Single coil)

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K-spaceX-space

(complex)magnitude

F-1 | . |

Acquisition Noise (single coil)

Complex Gaussianσ2Κ

Complex Gaussianσ2n=σ2Κ /|Ω|

Ricianσ2n

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F-1

Signal acquisition (Multiple coil)

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F-1

Acquisition noise (Multiple coil)

Complex Gaussianσ2Κi

Complex Gaussianσ2i=σ2Κi /|Ω|

ρ12

σ2Κ2

σ2Κ3

σ2Κ4

σ21

σ22

σ23

σ24ρ2

3ρ34

ρ14

ρ12

ρ14

ρ34

ρ23

σ2Κ1

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Acquisition noise:

• Noise in receiving coils is complex Gaussian (assuming no post-processing)

• Noise in x-space related to noise in k-space• Final distribution will depend on the

reconstruction

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2- Basic Noise Models

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K-spaceX-space

(complex)magnitude

F-1 | . |

Rician Model

Complex Gaussianσ2Κ

Complex Gaussianσ2n=σ2Κ /|Ω|

Ricianσ2n

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Complex

Magnitude

Rician

Rayleigh

Rician Model

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RicianGaussian

σn Meaning of σn?

SNR=A / σn

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Non-central chi model

Complex Gaussianσ2i=σ2Κi /|Ω|

σ21

σ22

σ23

σ24

ρ12

ρ14

ρ34

ρ23

Particular case:

σ2i= σ2j = σ2nρij=0

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Non-central chi model

Complex Gaussianσ2n|

σ2n

σ2n

σ2n

σ2n

Sum of Squares

Non central chiL, σ2n|

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Non-central chi model

Non Central Chi

Central Chi

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Basic models:

• Rician: relation between σ2n and variance of noise in Gaussian complex data.

• Nc-chi: also relation between σ2n and Gaussian Variance.

• Nc-chi: many times, interesting parameter σ2n·L

EM(x)2=A(x)2+2L·σ2n

SNR=A(x)/L1/2·σn• Usually: equivalence between L·σ2n (nc-chi)

and σ2n (Rician).

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Example: Conventional approach

Rician

Noncentral-chi

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Example: LMMSE filter

Rician

Noncentral-chi

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3- More Complex Models

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Limitation of the nc-model:

Only valid if:• Same variance of noise in each coil• No correlation between coils• No acceleration• Reconstruction done with sum of squares

Real acquisitions.• Correlated, different variances• Accelerated• Reconstructed with different methods

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A- Effect of Correlations

σ21

σ22

σ23

σ24

ρ12

ρ14

ρ34

ρ23

=

221

22221

11221

LLL

L

L

σσσ

σσσσσσ

Σ

Sum of Squares

Nc-chi no longer valid!!!

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Nc-chi approximation if effective parameters are used

Relative errors in the PDF for central-chi approximation as a function of the correlation coefficient. A) Using effective parameters. B) Using the original parameters.

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B- Sampling of the k-space:

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In real acquisitions:

• Different variances and correlations → effective values and approximated PDF.

• Subsampling → modification of σ2i in x-space

• Reconstruction method → output may be Rician or an approximation of nc-chi

• The variance of noise (σ2i) becomes x-dependant → σ2i(x)

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Survey of statistical models for MRI

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4- Example: the non stationary nc-chi

approximation

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Nc-chi approximation

σ21

σ22

σ23

σ24

ρ12

ρ14

ρ34

ρ23

=

221

22221

11221

LLL

L

L

σσσ

σσσσσσ

Σ

Sum of Squares

Nc-chi approximation if effective parameters

are used

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Effective number of coils as a function of the coefficient of correlation. A) Absolute value. B) Relative Value.

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σeff(x) Leff(x)

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• Params. σeff(x) and Leff(x) both depend on x.

• Good news:

Leff(x)· σ2eff(x)= tr(Σ)=σ21+…+ σ2L=L·< σ2i >

• The product is a constant:

Leff(x)· σ2eff(x)=L·σ2n

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Implications:

• Equivalence between effective and real parameters

• Product: easy to estimate

Leff(x)· σ2eff(x)= L·σ2n=mode EML2(x)x / 2

• In some problems: only product needed:I2(x)= EML2(x)x - 2L·σ2n

• NOTE: If effective values are used for σ2eff(x), also for Leff(x)·

σ2eff(x)·L → Wrong!!!

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Implications:

• My point of view: σ2eff(x) and Leff(x) should be seen as a single parameter:

σ2L = Leff(x) · σ2eff(x)and therefore

σ2eff(x) = σ2L / Leff(x) • Some applications do need σ2eff(x)

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• Param. σ2eff(x) now depends on position.• The dependence can be bounded. SNR → 0

σ2B= σ2n(1+<ρ2>(L-1)) SNR → ∞

σ2S= σ2n(1+<ρ>(L-1)) Total:

σ2eff(x)=(1-φ(x)) σ2S + φ(x) σ2Bwith

φ(x)=(SNR2(x)+1)-1

X-dependant noise:

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σ2B

σ2S

Where is noise estimated?

What to estimate: σ2eff(x), σ2n,σ2B, σ2S…

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Implications:

• Estimation over the background → underestimation of noise.

• Use of a single value of σn → error is most areas.

• Noise is higher in the high SNR.• Main source of non-stationarity: correlation

between coils.• High correlation: lower effective coils and

higher noise.• Possible source of error: σ2eff(x)·L

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5- Other models

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SENSE: (non stationary Rician)

σ2R(x) x-dependant noise

σ2n = σ2K (r / |Ω|)

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σeff(x)

GRAPPA: nc-chi approx.

Leff(x) (σ2eff(x) Leff(x) )1/2

σ2n = σ2K / (r |Ω|)Product σ2eff(x)·Leff(x) is not a

constant

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6- Conclusions

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• Be sure what you need in your application.• Follow the whole reconstruction pipeline to

be sure which is your “original” noise.• Useful: simplified models to make process

easier.• Be sure how noise affects the different slides.

Conclusions:

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Questions?

What to measure when measuring noise in

MRI

Santiago Aja-Fernández

sanaja@tel.uva.es

UNIVERSIDAD DE VALLADOLID E. T. S. INGENIEROS DE TELECOMUNICACIÓN

Antwerpen 2013