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

2

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?

Noise in MRI

3

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

Noise in MRI

4

Noise in MRI

5

Noise in MRI

6

1- Noise in MRI

Noise in MRI

7

Scanner K-spaceX-space

(complex) magnitude

F-1 | . |

Signal acquisition (Single coil)

Noise in MRI

8

K-spaceX-space

(complex)magnitude

F-1 | . |

Acquisition Noise (single coil)

Complex Gaussianσ2Κ

Complex Gaussianσ2n=σ2Κ /|Ω|

Ricianσ2n

Noise in MRI

9

F-1

Signal acquisition (Multiple coil)

Noise in MRI

10

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

Noise in MRI

11

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

Noise in MRI

12

2- Basic Noise Models

Noise in MRI

13

K-spaceX-space

(complex)magnitude

F-1 | . |

Rician Model

Complex Gaussianσ2Κ

Complex Gaussianσ2n=σ2Κ /|Ω|

Ricianσ2n

Noise in MRI

1414

Complex

Magnitude

Rician

Rayleigh

Rician Model

Noise in MRI

15

RicianGaussian

σn Meaning of σn?

SNR=A / σn

Noise in MRI

16

Non-central chi model

Complex Gaussianσ2i=σ2Κi /|Ω|

σ21

σ22

σ23

σ24

ρ12

ρ14

ρ34

ρ23

Particular case:

σ2i= σ2j = σ2nρij=0

Noise in MRI

17

Non-central chi model

Complex Gaussianσ2n|

σ2n

σ2n

σ2n

σ2n

Sum of Squares

Non central chiL, σ2n|

Noise in MRI

18

Non-central chi model

Non Central Chi

Central Chi

Noise in MRI

19

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).

Noise in MRI

20

Example: Conventional approach

Rician

Noncentral-chi

Noise in MRI

21

Example: LMMSE filter

Rician

Noncentral-chi

Noise in MRI

22

3- More Complex Models

Noise in MRI

23

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

Noise in MRI

24

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!!!

Noise in MRI

25

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.

Noise in MRI

26

B- Sampling of the k-space:

Noise in MRI

27

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)

Noise in MRI

28

Survey of statistical models for MRI

Noise in MRI

29

4- Example: the non stationary nc-chi

approximation

Noise in MRI

30

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

Noise in MRI

31

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

Noise in MRI

32

σeff(x) Leff(x)

Noise in MRI

33

• 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

Noise in MRI

34

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!!!

Noise in MRI

35

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)

Noise in MRI

36

• 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:

Noise in MRI

37

σ2B

σ2S

Where is noise estimated?

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

Noise in MRI

38

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

Noise in MRI

39

5- Other models

Noise in MRI

40

SENSE: (non stationary Rician)

σ2R(x) x-dependant noise

σ2n = σ2K (r / |Ω|)

Noise in MRI

41

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

Noise in MRI

42

6- Conclusions

Noise in MRI

43

• 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:

Noise in MRI

44

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