ho kyung kim pusan national universitybml.pusan.ac.kr/lectureframe/lecture/graduates/... · linear...
TRANSCRIPT
Linear Systems
Ho Kyung [email protected]
Pusan National University
Medical Imaging Detectors
2
0.16 μR (45 photons/mm2)
16 μR (4500 photons/mm2) 24 μR (6720 photons/mm2)
1.6 μR (450 photons/mm2)
Dirac function
Quantum image as the sample distribution
• 𝑞 𝐫 ∑ 𝛿 𝐫 𝐫
Dirac 𝜹 function
• Generalized function, not a "well‐behaved" function
• 𝛿 𝑥 𝑥0 for 𝑥 𝑥undefined for 𝑥 𝑥
• 𝛿 𝑥 𝑥 d𝑥 1 w/ a dimension of the inverse of the argument [𝑥 ]
• Shifting property 𝑓 𝑥 𝛿 𝑥 𝑥 d𝑥 𝑓 𝑥 if 𝑎 𝑥 𝑏0 otherwise
‒ 𝑓 𝑥 𝛿 𝑥 𝑥 d𝑥 𝑓 𝑥 𝛿 𝑥 𝑥 d𝑥 𝑓 𝑥 𝑓 𝑥 |
‒ Sampling or evaluating a function at a specified position 𝑥 𝑥
• 𝑓 𝑥 𝛿 𝑥 𝑥 𝑓 𝑥 𝛿 𝑥 𝑥 𝑓 𝑥‒ Multiplication w/ the 𝛿 function does not result in the sample value alone
‒ But, it results in a 𝛿 function scaled by the sample value 𝑓 𝑥
3
4
0.16 μR (45 photons/mm2)
16 μR (4500 photons/mm2) 24 μR (6720 photons/mm2)
1.6 μR (450 photons/mm2)
Distribution theory
Quantum image (image consisting of a distribution of quanta) must be interpreted using distribution theory
• Measured only thru the use of a sampling function 𝜙 𝑥‒ Describe the measurement process
• Sometimes called an aperture function
‒ Describe the sensitivity profile of a detector
• e.g. Consider a signal 𝑑 (proportional to the number of interacting quanta) when measuring 1‐D quantum image 𝑞 𝑥) w/ a detector of width 𝑎
• 𝑑 𝑘 𝑞 𝑥 d𝑥/
/ 𝑘 𝑞 𝑥 ∏ d𝑥
‒ 𝑘 = a constant relating the number of interacting quanta to the detector output signal
‒ Sampling function 𝜙 𝑥 ∏
5
Signal transfer
Contrast
• Relative brightness difference b/w two locations
• 𝐶𝐫 𝐫
𝐫 𝐫where 𝑗 in, out
• Large‐area contrast‐transfer factor T
• Contrast transfer is related to the spatial resolution of a system
‒ A blurring system transfers the small‐area contrast poorly!
6
Noise transfer
Contrast transfer says nothing about the transfer of image noise
Noise transfer affects the output image quality severely
Define a noise‐transfer factor?
• T ???
‒ 𝜎 E ∆𝑑 E 𝑑 E 𝑑
• Ergodic when ensemble average = spatial average
7
8
The same unity noise varianceCorrelation over only a very short distance Correlation over only a greater distance
Linear systems
Linearity
• System transfer characteristic 𝑆• Superposition 𝑆 ℎ 𝑥 ℎ 𝑥 𝑆 ℎ 𝑥 𝑆 ℎ 𝑥• Proportionality 𝑆 𝑎ℎ 𝑥 𝑎𝑆 ℎ 𝑥
Impulse‐response function, IRF
• irf 𝑥, 𝑥 𝑆 𝛿 𝑥 𝑥• 𝑆 𝛿 𝑥 𝑥 𝛿 𝑥 𝑥 irf 𝑥, 𝑥 irf 𝑥, 𝑥• IRF = PSF (point‐spread function) for a 2‐D imaging system
9
Linear & shift‐invariant (LSI) systems
• irf 𝑥, 𝑥 irf 𝑥 𝑥• Application of the Fourier transform to describe the spatial correlations b/w image locations
Limitations of LSI systems
• Non‐linear response
‒ H&D (Hurter‐Driffield) curve of x‐ray film
‒ Linearization
• Shift‐variant response
‒ Distortion of the periphery of images from x‐ray image intensifier
‒ Restricted analysis to central regions
• Spectral effects
‒ Usually negligible
• Noise non‐stationarity
‒ Non‐uniform noise through an image
10
Convolution
• If ∆𝑥 is small (relative to the width of the IRF), each rectangle can be represented as a 𝛿 fnt. positioned at 𝑥 𝑗∆𝑥 & scaled by ℎ 𝑗∆𝑥 ∆𝑥
• 𝑆 ℎ 𝑥 ∑ ℎ 𝑗∆𝑥 irf 𝑥, 𝑗∆𝑥 ∆𝑥
• If ∆𝑥 → 0, 𝑆 ℎ 𝑥 ℎ 𝑥 irf 𝑥, 𝑥 d𝑥′‒ Called a "superposition integral"
11
Area of a rectangle = ℎ 𝑗∆𝑥 ∆𝑥
Convolution integral for a shift‐invariant IRF
• 𝑆 ℎ 𝑥 ℎ 𝑥 irf 𝑥 𝑥 d𝑥′
• 𝑆 ℎ 𝑥 ℎ 𝑥 ∗ irf 𝑥• Describe the output signal when the input ℎ 𝑥 is passed thru an LSI system
• Describe a deterministic system only
• Describe only the expectation value of a stochastic system response
System characteristic function
Response to a sinusoidal input ℎ 𝑥 𝑒
• 𝑑 𝑥 𝑆 𝑒 irf 𝑥 𝑒 d𝑥′ 𝑒 irf 𝑥 𝑒 d𝑥′ 𝑒 T 𝑢
‒ Sinusoidal input produces a sinusoidal output at the same frequency, scaled by T 𝑢‒ Characteristic function T 𝑢 ℱ irf 𝑥 (Fourier pairs)
12
Eigenfunctions
Eigenvalues
Express the output for ℎ 𝑥 having the FT 𝐻 𝑢
• 𝑑 𝑥 𝑆 ℎ 𝑥 𝑆 𝐻 𝑢 𝑒 d𝑥
• 𝑑 𝑥 𝐻 𝑢 T 𝑢 𝑒 d𝑢
Express 𝑑 𝑥 as the IFT of 𝐷 𝑢
• 𝑑 𝑥 𝐷 𝑢 𝑒 d𝑢
Then we have
• 𝐷 𝑢 𝐻 𝑢 T 𝑢
13
MTF
Consider the transfer of sinusoidal signals
• ℎ 𝑥 𝑎 𝑏𝑒
• Input modulation 𝑀
• Output signal 𝑑 𝑥 𝑆 𝑎 𝑏𝑒 𝑎𝑆 𝑒 𝑏𝑆 𝑒 𝑎T 0 𝑏T 𝑢 𝑒
• Output modulation 𝑀 𝑀
14
Modulation transfer function
• MTF 𝑢
‒ T 0 = area under the IRF (Fourier DC theorem)
Optical transfer function (OTF)
• OTF 𝑢
• Retain phase‐transfer information
LSF
Line‐spread function
• Response to a "line" delta function normalized to unity area
• One‐directional system response
• lsf 𝑥 𝑥,
,
,
,
For shift‐invariant systems
• lsf 𝑥,
,
Fourier pairs
• OTF 𝑢 ℱ lsf 𝑥
1‐D MTF evaluation
• MTF 𝑢 MTF 𝑢, 𝑣 |
15
Correlation integral
• 𝑑 𝑥 , 𝑥 𝑥 𝑓 𝑥 ℎ 𝑥 𝑥 d𝑥′
• Stationary 𝑓 𝑥 & ℎ 𝑥 in 𝑥: 𝑑 𝑥 𝑓 𝑥 ℎ 𝑥 𝑥 d𝑥′
• 𝑑 𝑥 𝑓 𝑥 ⋆ ℎ 𝑥
• 𝑓 𝑥 ⋆ ℎ 𝑥 ℎ 𝑥 ⋆ 𝑓 𝑥
• 𝑓 𝑥 ⋆ ℎ 𝑥 𝑓 𝑥 ∗ ℎ∗ 𝑥 ℎ 𝑥 ∗ 𝑓∗ 𝑥
16
Complex conjugate
Fourier Transform
Forward
• 1‐D 𝐷 𝑢 𝑑 𝑥 𝑒 d𝑥
• 2‐D 𝐷 𝑢, 𝑣 𝑑 𝑥, 𝑦 𝑒 d𝑥d𝑦
• Vector 𝐷 𝐤 𝑑 𝐫 𝑒 𝐤·𝐫d 𝐫
Inverse
• 1‐D 𝑑 𝑥 𝐷 𝑢 𝑒 d𝑢
• 2‐D 𝑑 𝑥, 𝑦 𝐷 𝑥, 𝑦 𝑒 d𝑢d𝑣
• Vector 𝑑 𝐫 𝐷 𝐤 𝑒 𝐤·𝐫d 𝐤
17
Discrete Fourier Transform
Forward for a sequence of 𝑁 values 𝑑 for 0 𝑛 𝑁 1• 𝐷 DFT 𝑑 ∑ 𝑑 𝑒 /
• Result in a sequence of 𝑁 complex values 𝐷 for 0 𝑚 𝑁 1
• Central position 𝑥 0 corresponds to 𝑛 1
Inverse
• 𝑑 DFT 𝐷 ∑ 𝐷 𝑒 /
• Central frequency 𝑢 0 corresponds to 𝑚 0
Problems
• Aliasing
• Spectral leakage & side lobes
• Truncation & windowing
• Zero‐position & phase errors
• Frequency wrap‐around
• Scaling factors & units
18
Sampling
Sampling
• Evaluation of 𝑑 𝑥 at 𝑥 𝑛𝑥 : 𝑑 𝑑 𝑛𝑥 𝑑 𝑥 | for 0 𝑛 𝑁 1
Use of the shifting property of 𝛿 functions
• 𝑑 𝑥 | 𝑑 𝑥 𝛿 𝑥 𝑥 d𝑥 𝑑 𝑥 𝛿 𝑥 𝑥 d𝑥 𝑑 𝑥
• 𝑑 𝑥 𝑑 𝑥 ∑ 𝛿 𝑥 𝑥 ∑ 𝑑 𝑛𝑥 𝛿 𝑥 𝑥 ∑ 𝑑 𝛿 𝑥 𝑥‒ 𝑑 𝑥 = a sequence of scaled 𝛿 functions; units [𝑑 𝑥 𝑥 ]
‒ 𝑑 𝑥 = "presampling" signal
• ℱ 𝑑 𝑥 𝐷 𝑢 ∗ ∑ 𝛿 𝑢
‒ Sampling frequency = 1/𝑥‒ Cutoff frequency = 1/2𝑥
19
20
AliasesAliasing
Sampling theorem
Any band‐limited function having infinite extent and no component frequencies at frequencies greater than 𝑢 𝑢 can be fully determined from an infinite set of discrete samples if sampled at a frequency greater than 𝑢 2𝑢 where 𝑢 is called the
Nyquist sampling frequency
A rule of thumb
• 𝑢 𝑓 𝑢 𝑓 2𝑢
• 𝑓 1.2, the "Kell" factor
21
How to recover 𝑑 𝑥 from 𝑑 ; or recover 𝐷 𝑢 from ℱ 𝑑 𝑥 ?
• (No aliasing) Isolate the primary spectral component from its aliases by multiplying the rectangular function 𝑥 ∏ 𝑥 𝑢 having a value 𝑥 over 1/2𝑥 𝑢 1/2𝑥
• Recovered signal 𝑑 𝑥 𝑑 𝑥 ∗ sinc ∑ 𝑑 𝛿 𝑥′ 𝑛𝑥 sinc 𝜋 d𝑥′
• 𝑑 𝑥 ∑ 𝑑 𝛿 𝑥′ 𝑛𝑥 sinc 𝜋 d𝑥′ ∑ 𝑑 sinc 𝜋
‒ Called the "sinc interpolation" (convolution of the sample values w/ the sinc function)
22
2AL
A 2AL
k
k = 1/2L
k = 1/L
23
Stochastic Process
A random (stochastic) process gives rise to random fluctuations in a signal, represented as a random variable
Not possible to predict the future values of a RV, but possible to determine its statistical properties
Expected value
• E 𝑎 𝛼𝜆 𝛼 d𝛼‒ 𝜆 𝛼 = the probability of the RV 𝑎 having the value 𝛼
Variance
• 𝜎 E Δ𝑎 E 𝑎 E 𝑎 E 𝑎 E 𝑎
Autocorrelation
• R 𝑥 , 𝑥 𝑥 E 𝑎 𝑥 𝑎∗ 𝑥 𝑥• Correlation of 𝑎 𝑥 w/ itself at a location displaced by 𝑥
Autocovariance
• K 𝑥 , 𝑥 𝑥 E Δ𝑎 𝑥 Δ𝑎∗ 𝑥 𝑥 R 𝑥 , 𝑥 𝑥 E 𝑎 𝑥 E 𝑎∗ 𝑥 𝑥• Correlation of 𝑎 𝑥 w/ itself at a location displaced by 𝑥 about the expected values
24
WSS random process
Stationary expected value & variance (i.e. fixed values)
• E 𝑎 lim→
∑ 𝑎
• 𝜎 lim→
∑ 𝑎 E 𝑎 lim→
∑ 𝑎 ∑ 𝑎
Random processes
• Strict‐sense stationary (SSS): all statistical properties are stationary w/ 𝑥• Wide‐sense stationary (WSS): (at least) E 𝑎 & R 𝑥 , 𝑥 𝑥 are stationary
‒ R 𝑥 , 𝑥 𝑥 R 𝑥‒ K 𝑥 , 𝑥 𝑥 K 𝑥
• Fourier pairs: NPS 𝑢 ℱ K 𝑥 for WSS random process
25
ErgodicWSS random process
• Being "ergodic" means that expected values can be determined equivalently from ensembleaverages or spatial averages
• An estimate of the autocovariance is then given by the sample autocovariance:
‒ K , 𝑥 ∆𝑎 𝑥 ∆𝑎∗ 𝑥 𝑥 d𝑥′
‒ K 𝑥 lim→
K , 𝑥 ∆𝑎 𝑥 ⋆ ∆𝑎∗ 𝑥
NPS of an ergodic WSS random process
• NPS 𝑢 lim→
E ∆𝑎 𝑥 𝑒 d𝑥 lim→
E ℱ ∆𝑎 𝑥
‒ ℱ ∆𝑎 𝑥 = FT of the zero‐mean function ∆𝑎 𝑥 truncated to the region 𝑋/2 𝑥 𝑋/2
• 𝜎 E 𝑎 𝑥 𝑎∗ 𝑥 E 𝑎 𝑥 E 𝑎∗ 𝑥 E ∆𝑎 𝑥 ∆𝑎∗ 𝑥 K 𝑥 |
• 𝜎 NPS 𝑢 d𝑢
26
Ergodic WSCS random process
Strict‐sense cyclostationary (SSCS)
• Periodic stochastic processes
• Invariant statistical properties to a shift of any multiple of a period
Wide‐sense cyclostationary (WSCS)
• Only the mean & correlation are invariant in a given period
• E 𝑎 𝑥 𝑛𝑥 E 𝑎 𝑥• R 𝑥 𝑛𝑥 , 𝑥 𝑥 𝑛𝑥 R 𝑥 , 𝑥 𝑥
Sensing function
• 𝑎 𝑥 ∑ 𝑎 𝑠 𝑥 𝑛𝑥• E 𝑎 𝑥 E 𝑎 ∑ 𝑠 𝑥 𝑛𝑥
• R 𝑥 ∑ R 𝑛𝑥 𝜏 𝑥
‒ 𝜏 𝑥 𝑠 𝑥 𝑠 𝑥 𝑥 d𝑥′ 𝑠 𝑥 ⋆ 𝑠 𝑥
27
In digital imaging systems w/ a sensing function 𝑠 𝑥 ∑ 𝛿 𝑥 𝑛𝑥• Digital sampling 𝑎 𝑥 ∑ 𝑎 𝛿 𝑥 𝑛𝑥
• E 𝑎 𝑥 E 𝑎 ∑ 𝛿 𝑥 𝑛𝑥
• R 𝑥 ∑ R 𝑛𝑥 𝛿 𝑥 𝑛𝑥 R 𝑥 ∑ 𝛿 𝑥 𝑛𝑥
• K 𝑥 ∑ K 𝑛𝑥 𝛿 𝑥 𝑛𝑥 K 𝑥 ∑ 𝛿 𝑥 𝑛𝑥
‒ Units [𝑎 𝑥 𝑥 ]
NPS of an ergodic WSCS random process
• NPS 𝑢 ℱ K 𝑥 NPS 𝑢 ∗ ∑ 𝛿 𝑢
‒ Units [𝑎 𝑥 𝑥 ]
‒ Units of NPS 𝑢 [𝑎 𝑥 𝑥]
28
Rose Model
SNR for the detection of a uniform object (𝑞 , 𝐴) in a uniform background (𝑞 ) for uncorrelated Poisson‐distributed noise
• Contrast 𝐶
• Rose signal ∆𝑆 𝑞 𝑞 𝐴
• Rose SNR ∆SNR∆
𝐶 𝐴𝑞
‒ Lesion detectability ∝ object contrast 𝐶 and the square root of object area & radiation dose 𝐴𝑞
• ∆SNR ≳ 5 for reliable detection of a uniform object
Limitations
• Restricted to "signal known exactly" (SKE) & "background known exactly" (BKE) detection task
• Valid only for uncorrelated Poisson‐distributed noise
• Image noise is generally not uncorrelated noise Poisson distributed
‒ Additive system noise & statistical correlations due to x‐ray or secondary quanta scatter
29
30
NPS
2‐D NPS for ergodic WSS random process 𝑑 𝑥, 𝑦
• NPS 𝑢, 𝑣 lim, →
E ∆𝑑 𝑥, 𝑦 𝑒 d𝑥d𝑦
• NPS 𝑢, 𝑣 lim, →
E ℱ , ∆𝑑 𝑥, 𝑦 lim, →
E 𝑆∆ , , 𝑢, 𝑣
• Units [𝑑 𝑥, 𝑦 𝑥 ]
1‐D NPS
• 𝑑 𝑥 𝑑 𝑥, 𝑦 d𝑦
• NPS 𝑢 lim, →
E ∆𝑑 𝑥 𝑒 d𝑥
• NPS 𝑢 lim, →
E ∆𝑑 𝑥, 𝑦 𝑒 d𝑦 d𝑥
• NPS 𝑢 NPS 𝑢, 𝑣 |
• Units [𝑑 𝑥, 𝑦 𝑥 ]
• Autocovariance K 𝑥 lim→
K 𝑥, 𝑦 d𝑦 (central‐slice theorem)
31
Sample spectrum of ∆𝑑 𝑥, 𝑦
Zero‐frequency NPS
Fourier DC theorem
• NPS 𝑢 | K 𝑥 d𝑥 E ∆𝑎 𝑥 ∆𝑎 𝑥 𝑥 d𝑥
• NPS 0 depends on the extent to which ∆𝑎 𝑥 may be correlated
Average correlation length 𝑋
• K 𝑥 d𝑥 𝑋 K 0 𝑋 𝜎
• NPS 𝑢 | 𝑋 𝜎
• NPS 𝑢, 𝑣 | , ∬ 𝜎 d𝑥d𝑦 𝐴 𝜎
32
2‐D avg. correlation area
NPS measurements w/ an aperture (the sampling function) that is large relative to any correlation distance
• 𝑑 𝑥, 𝑦 𝑎 𝑥, 𝑦 d𝑥′d𝑦′/
//
/ 𝑎 𝑥, 𝑦 ∗ ∏ ,
• NPS 𝑢, 𝑣 NPS 𝑢, 𝑣 sinc 𝜋𝑋𝑢 sinc 𝜋𝑌𝑣• Zero‐frequency value NPS 0,0 NPS 0,0
• NPS 𝑢, 𝑣 constant NPS 0,0 for all frequencies if 𝐴 is large w.r.t. any correlation distance
• 𝜎 NPS 𝑢, 𝑣 d𝑢d𝑣
• 𝜎 NPS 0,0 sinc 𝜋𝑋𝑢 sinc 𝜋𝑌𝑢 d𝑢d𝑣 NPS 0,0
• NPS 0,0 𝐴𝜎
33
NPS of uncorrelated quanta
NPS
• NPS 𝑢, 𝑣 E 𝑞 𝑥, 𝑦 E 𝑞
Autocovariance
• K 𝑥, 𝑦 E ∆𝑞 𝑥 , 𝑦′ ∆𝑞 𝑥 𝑥, 𝑦 𝑦E ∆𝑞 𝑥, 𝑦 for 𝑥 0, 𝑦 00 for 𝑥 0, 𝑦 0
• K 𝑥, 𝑦 E 𝑞 𝛿 𝑥, 𝑦‒ No statistical correlation at any position 𝑥, 𝑦 w/ any other position 𝑥′, 𝑦′
Zero‐frequency values
• NPS 𝑢, 𝑣 | , K 𝑥, 𝑦 d𝑥d𝑦 E 𝑞 𝛿 𝑥, 𝑦 d𝑥d𝑦 E 𝑞
• 𝜎 NPS 𝑢, 𝑣 d𝑢d𝑣 E 𝑞 d𝑢d𝑣 undefined!
34
NEQ
NPS is arbitrary or specific to a particular system
Noise‐equivalent number of quanta
• To obtain an absolute scale of noise, express image noise in terms of the number of Poisson‐distributed input photons per unit area 𝑞 at each spatial frequency
• NEQ 𝑞, 𝑢
‒ Units [𝑞]‒ T 𝑢 = system characteristic function describing signal transfer from input to output
‒ T 𝑢 𝐺 MTF 𝑢
‒ 𝐺 �̅�/𝑞 = system large‐area gain factor
• NEQ 𝑞, 𝑢̅
/
Systems having a non‐linear response & exhibiting only small‐signal linearity
• NEQ 𝑞, 𝑢
35
(Squared) expected output signal
Output noise power
Physical meaning of NEQ
• Image quality on an absolute scale, independent of specific system parameters
• Number of Poisson‐distributed quanta that would produce the same SNR given an ideal detector
• Greater NEQ lower image noise
• Ideal system: NEQ 𝑞, 𝑢 𝑞
‒ No frequency dependence & the best possible NEQ for an input 𝑞
System aperture
• Fundamental measure of resolution in a noise‐limited imaging system
• 𝑎𝐤
𝟎d 𝐤
Ideal observer SNR
• Performance of the observer in detection tasks
• SNR ∆𝑆 𝑢 NEQ 𝑢 d𝑢
36
Frequency contents of an object ∆𝑠 𝑥 to be detected
DQE
Detective quantum efficiency
• Measure of the effective fraction of incident Poisson‐distributed quanta contributing to image SNR
• Measure of system performance (while NEQ is a measure of image quality)
• DQE 𝑞, 𝑢,
Descriptive (practical) DQE
• Expressed in terms of parameters determined from measured image data
• Only applicable to a system for the Poisson‐distributed & uncorrelated input quanta
‒ e.g. x‐ray quanta in a "flat‐field" image
‒ SNR 𝑢 𝑞
• DQE 𝑞, 𝑢̅
=̅
• DQE 𝑞, 𝑢 DQE 𝑢‒ In the absence of additive system noise (e.g. electronic noise) or multiplicative noise (e.g. fixed‐pattern
noise)
37
𝑞 𝑋 [R (fluence per R)]
• Φ 𝐸 𝐸 d𝐸
‒ Φ 𝐸 = normalized incident x‐ray spectrum
‒ 𝐸 = fluence per unit exposure for mono‐energetic beam w/ 𝐸
‒ Mass energy absorption coefficient for air [cm2/g]
‒ X‐ray energy 𝐸 [keV]
‒ Electronic charge 𝑒 = 1.602210‐19 C‒ W‐value of air 𝑊 = 33.97 eV
‒ Charge liberated in air by 1 R 𝑄 = 2.58010‐4 C/kg/R
38
39
Mono‐energetic beam
Spectral beam
Conceptual DQE
• SNR ‐transfer form of DQE
• DQE 𝑢
Stochastic DQE
• DQE 𝑢̅ ̅
‒ Measure of the NPS produced by a deterministic system relative to the actual NPS
‒ Stochastic nature of the imaging system causes degradation of the DQE
Predictive DQE
• DQE formalism in terms of the design parameters
• e.g. DQE 𝑢⋯
⋯ ⋯
40
Expected NPS when passed thru a deterministic system
Expected NPS when passed thru a real (stochastic) system