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UCSD BENG 280A - Noise and SNR

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1Thomas Liu, BE280A, UCSD, Fall 2008Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2008MRI Lecture 6: Noise and SNRThomas Liu, BE280A, UCSD, Fall 2008What is Noise?Fluctuations in either the imaging system or the objectbeing imaged.Quantization Noise: Due to conversion from analogwaveform to digital number.Quantum Noise: Random fluctuation in the number ofphotons emittted and recorded.Thermal Noise: Random fluctuations present in allelectronic systems. Also, sample noise in MRIOther types: flicker, burst, avalanche - observed insemiconductor devices.Structured Noise: physiological sources, interferenceThomas Liu, BE280A, UCSD, Fall 2008Quantization NoiseSignal s(t)r(t) = s(t) + q(t)Although the noise is deterministic, it is useful to model the noise as a random process. Quantization noiseq(t)Quantized Signal r(t)Thomas Liu, BE280A, UCSD, Fall 2008Physiological NoiseSignal Intensity Pulse Ox (finger)Cardiac component estimated in brain Image Number (4 Hz)Perfusion time series: Before Correction (cc = 0.15)Perfusion time series: After Correction (cc = 0.71)Perfusion Time Series2Thomas Liu, BE280A, UCSD, Fall 2008Noise and Image QualityPrince and Links 2005Thomas Liu, BE280A, UCSD, Fall 2008Thermal NoiseFluctuations in voltage across a resistor due to randomthermal motion of electrons.Described by J.B. Johnson in 1927 (therefore sometimescalled Johnson noise). Explained by H. Nyquist in 1928.! V2= 4kT " R " BWVariance in VoltageResistanceBandwidthTemperatureThomas Liu, BE280A, UCSD, Fall 2008Thermal Noise! V2= 4kT " R " BWAt room temperature, noise in a 1 k# resistor isV2/ BW =16 $10%18 V2/ HzIn root mean squared form, this corresponds to V/BW = 4 nV/ Hz .Example : For BW = 250 kHz and 2 k# resistor, total noise voltage is 2 "16 $10-18" 250 $103= 4 µVThomas Liu, BE280A, UCSD, Fall 2008Thermal Noise! Noise spectral density is independent of frequency upto 1013 Hz. Therefore it is a source of white noise.Amplitude distribution of the noise is Gaussian.3Thomas Liu, BE280A, UCSD, Fall 2008Signal in MRI! Recall the signal equation has the formsr(t) = M(x, y,z)e" t /T2( r )e" j#0texp " j$G%( )0t&' r(%)d%( ) * + , - &&&dxdydz! Faraday's LawEMF = "#$#t$= Magnetic Flux = B1(x, y,z) % M(x, y,z)dV&B0Thomas Liu, BE280A, UCSD, Fall 2008Signal in MRI! Signal in the receiver coilsr(t ) = j"0B1xyM( x, y,z)e# t /T2( r )e# j"0texp # j$G%( )0t&' r(%)d%( ) * + , - &dVRecall, total magnetization is proportional to B0Also "0=$B0.Therefore, total signal is proportional to B02Thomas Liu, BE280A, UCSD, Fall 2008Noise in MRI! Primary sources of noise are :1) Thermal noise of the receiver coil2) Thermal noise of the sample. Coil Resistance : At higher frequencies, the EM wavestend to travel along the surface of the conductor (skineffect). As a result, Rcoil " #01/2 $ Ncoil2 " #01/2" B01/ 2Sample Noise : Noise is white, but differentiationprocess due to Faraday's law introduces a multiplicationby #0. As a result, the noise variance from the sampleis proportional to #02.Nsample2 "#02" B02Thomas Liu, BE280A, UCSD, Fall 2008SNR in MRI! SNR =signal amplitudestandard deviation of noise"B02#B01/ 2+$B02If coil noise dominatesSNR " B07 / 4If sample noise dominatesSNR " B04Thomas Liu, BE280A, UCSD, Fall 2008Random Variables! A random variable X is characterized by its cumulativedistribution function (CDF)Pr( X " x) = FX(x)The derivative of the CDF is the probability densityfunction(pdf)fX(x) = dFX(x) / dxThe probability that X will take on values between twolimits x1 and x2 is Pr(x1" X " x2) = FX(x2) # FX(x1) = fX(x)dxx1x2$Thomas Liu, BE280A, UCSD, Fall 2008Mean and Variance! µX= E[X]= xfX(x)dx"##$%X2= Var[X]= E[ X "µX( )2]= (x "µX)2fX(x)dx"##$= E[X2] "µX2Thomas Liu, BE280A, UCSD, Fall 2008Gaussian Random Variable! fX(x) =12"#2exp $(x $µ)2/ 2#2( )( )µX=µ#X2=#2Thomas Liu, BE280A, UCSD, Fall 2008Independent Random Variables! fX1,X2(x1, x2) = fX1(x1) fX2(x2)E[X1X2] = E[X1]E[X2]Let Y = X1+ X2 then µY= E[Y]= E[X1] + E[X2]=µ1+µ2E[Y2] = E[X12] + 2 E[ X1]E[X2] + E[X22] = E[X12] + 2µ1µ2+ E[X22]"Y2= E[Y2] #µY2= E[X12] + 2µ1µ2+ E[X22] #µ12#µ22# 2µ1µ2="X12+"X225Thomas Liu, BE280A, UCSD, Fall 2008Signal Averaging! ! We can improve SNR by averaging. Let y1= y0+ n1y2= y0+ n2The sum of the two measurements is 2y0+ n1+ n2( ).If the noise in the measurements is independent, then the variances sum and the total variance is 2"n2SNRTot=2y02"n= 2SNRoriginalIn general, SNR # Nave# TimeThomas Liu, BE280A, UCSD, Fall 2008Noise in k-space! ! Recall that in MRI we acquire samples in k - space.The noise in these samples is typically well describedby an iid random process. For Cartesian sampling, the noise in the image domainis then also described by an iid random process.For each point in k -space, SNR =S(k)"n whereS(k) is the signal and "n is the standard deviation ofeach noise sample.Thomas Liu, BE280A, UCSD, Fall 2008Noise in image space! ! Noise variance per sample in k - space is "n2.Each voxel in image space is obtained from the Fourier transform of k - space data. Say there are N points in k - space. The overall noise variancecontribution of these N points is N"n2.If we assume a point object, then all points in k - space contribute equally to the signal, so overall signal is NS0. Then overall SNR in image space is SNR #NS0N"n= NS0"nTherefore, SNR increases as we increase the matrix size. Thomas Liu, BE280A, UCSD, Fall 2008Signal Averaging! ! We can improve SNR by averaging in k - spaceIn general, SNR " Nave" Time6Thomas Liu, BE280A, UCSD, Fall 2008Effect of Readout Window! ! ADC samples acquired with sampling period "t.Thermal noise per sample is #n2$ "f =1"tIf we double length of the readout window, thenoise variance per sample decreases by two.The noise standard deviation decreases by 2, andthe SNR increases by 2.In general, SNR $ TRe ad= Nkx"tThomas Liu, BE280A, UCSD, Fall 2008SNR and Phase Encodes! ! Assume that spatial resolution is held constant.What happens if we increase the number of phaseencodes? Recall that "y=1Wky. Thus, increasingthe number of phase encodes NPE, decreases #ky andincreases FOVy.If we double the number of phase encodes, each pointin image space has double the number of k - space linescontributing to its signal. The noise variances sum. The SNR therefore goes up by 2.In general SNR$ NPE Thomas Liu, BE280A, UCSD, Fall 2008Overall SNR! ! SNR"Signal#n"$x$y$z#nPutting everything together, we find thatSNR" NaveNxNPE$t$x$y$z = Measurement Time


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UCSD BENG 280A - Noise and SNR

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