Cascaded Systems Analysis
Ho Kyung [email protected]
Pusan National University
Medical Imaging Detectors
Noise Transfer
Image quality basically depends on # quanta interacting w/ an imaging system
Information from the quanta detected by a detector should be expressed as a final image without any loss; but
• Complex transfer mechanism: the imaging system consists of multiple processes from detecting x‐ray quanta to displaying the final image
• Inefficient transfer: un‐optimally designed system can include additional factors degrading image quality
Cascaded‐systems analysis
• Represent the complex system as a cascade of elementary stages
‒ Quantum amplification
‒ Deterministic blurring
‒ Quantum scattering
• Use transfer theory to describe the transfer of signal/noise thru the system
• Predict system performance based on design parameters
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Quantum amplification
Quantum gain or selection
Conversion of quanta from one form to another
• e.g. X‐ray quanta to optical quanta in a scintillator
• 𝑞 𝐫 𝑔𝑞 𝐫
‒ 𝑞 𝐫 = quantum image
‒ 𝑔 = gain; RV characterized by a mean 𝑔 & variance 𝜎
Mean number of quanta
• 𝑞 �̅�𝑞
NPS
• NPS 𝐤 �̅� NPS 𝐤 𝑞 𝜎
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Binomial selection
Special case of the quantum amplification process (𝑔 1) Describe the quantum efficiency of a detector
• Each quantum is either detected (or transferred thru this selection stage w/ prob. �̅�) or not (w/ prob. 1 �̅�)
• 𝑔 = RV having a value of 1 or 0 only
• Variance 𝜎 �̅� 1 �̅�
Noise transfer
• NPS 𝐤 �̅� NPS 𝐤 𝑞 �̅� 1 �̅� �̅� NPS 𝐤 𝑞 𝑞 �̅�
‒ If NPS 𝐤 ≪ 𝑞 (significant correlated noise) ⇒ NPS 𝐤 𝑞 𝑔 1 𝑔‒ If NPS 𝐤 𝑞 (uncorrelated input quanta), ⇒ NPS 𝐤 𝑞 𝑔
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Uncorrelated noiseCorrelated noise
Deterministic blur
Blur resulting from a convolution (linear filtering) of the input w/ a PSF
Input signal redistribution w/ a weighting given by the PSF
• 𝑑 𝐫 𝑞 𝐫 ∗ psf 𝐫
Signal
• �̅� 𝑞‒ Analog output
Noise transfer
• NPS 𝐤 NPS 𝐤 MTF 𝐤
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Quantum scatter
Random relocation of an input quantum to a new location w/ a prob. of PSF
Redistribution w/ probabilities
• Note that the deterministic blur is the redistribution of signal by weights
• 𝑞 𝐫 𝑞 𝐫 ∗ psf 𝐫
Signal
• 𝑞 𝑞
NPS
• NPS 𝐤 NPS 𝐤 𝑞 MTF 𝐤 𝑞
‒ NPS 𝐤 NPS 𝐤 MTF 𝐤 𝑞 1 MTF 𝐤
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Uncorrelated noiseCorrelated noise
Deterministic blur Additional noise
7Cunningham et al. IEEE EMBC & CMBEC (1995)
Cascade of elementary stages
Amplification + scattering
• e.g. Conversion of x‐ray photons to light quanta & their scatter w/i a scintillator
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Hypothetical detector
Incident x‐ray quanta
• 𝑞• NPS 𝑢 𝑞
1) Selection of interacting x rays
• 𝑞 𝑞 𝛼• NPS 𝑢 𝑞 𝛼
2) Conversion from x rays to light
• 𝑞 𝑞 𝛼𝑚
• NPS 𝑢 𝑞 𝛼𝑚 1
3) Spatial spreading of light
• 𝑞 𝑞 𝛼𝑚
• NPS 𝑢 𝑞 𝛼𝑚 1 MTF 𝑢 𝑞 𝛼𝑚
4) Selection of light quanta
• 𝑞 𝑞 𝛼𝑚𝛽
• NPS 𝑢 𝑞 𝛼𝑚 𝛽 1 MTF 𝑢 𝑞 𝛼𝑚𝛽
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Quantum‐Sink Analysis
DQE of a detector system w/ 𝑀 Poisson gain stages
• e.g. Photo‐multiplier tube
• DQE 𝑢⋯
⋯⋯
‒ P ∏ 𝑔
‒ P = # quanta at the 𝑗th stage normalized to # input quanta
‒ Called the "particle‐based analysis"
• DQE degrades if 1 or P 1
‒ System w/ a "quantum sink" at the 𝑗th stage‒ P = quantum efficiency of the detector
‒ P 1 always; called the "primary quantum sink"
• To avoid a secondary quantum sink, P 10‒ Not predict a secondary quantum sink at non‐zero spatial frequencies
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Quantum‐accounting diagram
Plot P as a function of the stage number 𝑗
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Fourier‐based analysis
• 2nd‐order statistics
• DQE 𝑢⋯
⋯ ⋯
• Poisson excess: 𝜀 1
‒ Poisson amplification (𝜎 𝑔 ): 𝜀 0
‒ Deterministic gain (a gain w/ no random variability, 𝜎 0): 𝜀 1
• Each stage either be an amplification or a scattering (not both)
‒ Amplification stage: MTF 𝑢 1 (gain w/ no blur)
‒ Scattering stage: 𝑔 1 & 𝜀 1 (blur w/ no gain)
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If 𝜀 ≪ 1;
• DQE 𝑢⋯
⋯ ⋯⋯
• P 𝑢 ∏ �̅� MTF 𝑢‒ P 𝑢 ≲ 1 degrades the DQE
‒ P 𝑢 1 has no effect on the DQE
To avoid a secondary quantum sink, P 𝑢 10 & P 0 100• Assumption: MTF 𝑢 0.33, where 𝑢 = max. freq. of interest
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𝛼 0.5
�̅� 10
𝛽 0.02
Minor 2ndary q sink
Dominant 2ndary q sink
1.0 1.1 1.25 1.4 1.7
2.0 2.5 3.3 5.0 10.0
Digital Metrics
Digital detector = an array of (discrete) detector elements
Detector element ('del' or 'pixel')
• Produce a signal proportional to # quanta interacting in the element
• Regarded as a spatial integrator of image quanta
‒ Integration of quanta in a detector element is represented as a convolution integral
• (1‐D) 𝑑 𝑘 𝑞 𝑥 d𝑥/
/
• 𝑑 𝑘 𝑞 𝑥 ∏ d𝑥
• 𝑑 𝑘𝑞 𝑥 ∗ ∏ 𝑑 𝑥 |
𝑑 𝑥 = presampling detector signal
• Analog (continuous) sample function
• Detector signal for all possible detector‐element positions
• Evaluation at 𝑥 𝑛𝑥 , or 𝑑 𝑥 | gives the detector signal
• 𝑑 𝑥 𝑘𝑞 𝑥 ∗ ∏ 𝑥/𝑎 ⇔ 𝐷 𝑢 𝑄 𝑥 T 𝑢
• Aperture MTF MTF 𝑢 sinc 𝜋𝑎 𝑢
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Digital MTF
Sampling
• The process of evaluating a function
‒ 𝑑 𝑘𝑞 𝑥 ∗ ∏ 𝑑 𝑥 |
Evaluating 𝑑 𝑥 at 𝑥 𝑛𝑥 for all 𝑛:• 𝑑 𝑥 𝑑 𝑥 ∑ 𝛿 𝑥 𝑛𝑥 ∑ 𝑑 𝛿 𝑥 𝑛𝑥
‒ Infinite train of 𝛿 functions scaled by the detector values 𝑑
• ℱ 𝑑 𝑥 𝐷 𝑢 ∗ ∑ 𝛿 𝑢
‒ Aliases of 𝐷 𝑢 at spacings of 𝑢 1/𝑥‒ Aliasing (overlap of aliases): distortion of the image signal at 𝑢 𝑢 1/ 2𝑥
Effect of the digital detector
• Attenuate spatial frequencies by the presampling MTF, MTF 𝑢‒ If 𝛾 1 (𝑎 𝑥 ), MTF 𝑢 sinc 𝜋𝑥 𝑢 ; the 1st zero at 𝑢 1/2𝑥
‒ If 𝛾 1 (𝑎 𝑥 ), MTF 𝑢 sinc 𝜋𝑎 𝑢 ; increasing bandwidth & more aliasing
• Introduce aliasing
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AliasesAliasing
Digital NPS
Discrete values 𝑑 : neither WSS nor WSCS random processes
Presampling detector signal 𝑑 𝑥 : WSS random process
Sampled signal 𝑑 𝑥 (an array of 𝛿‐functions scaled by 𝑑 ): WSCS random process
• NPS 𝑢 NPS 𝑢 ∗ ∑ 𝛿 𝑢 NPS 𝑢 ∑ NPS 𝑢
• Note again that the sampling theorem states that frequencies above the cut‐off frequency 𝑢1/ 2𝑥 cannot be represented w/ sample obtained w/ a uniform sampling frequency of 𝑢1/𝑥
Truncation to the cut‐off freq. range (= convolution w/ a sinc fnt = sinc interpolation)
• Estimate of 𝑑 𝑥 : 𝑑 𝑥 ∑ 𝑑 sinc 𝜋 𝑑 𝑥 ∗ sinc 𝜋𝑥 𝑢
• NPS 𝑢 NPS 𝑢 𝑥 ∏ 𝑥 𝑢
• 𝑑 𝑥 𝑑 𝑥 only if there is no aliasing of the presampling NPS
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Aliases centered at 𝑢 𝑛/𝑥
Fundamental presampling NPS
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Digital NPS
• NPS 𝑢/
E DFT ∆𝑑
‒ ∆𝑑 𝑑 E 𝑑‒ Numerical estimate of the NPS of 𝑑 𝑥
• NPS 𝑢 NPS 𝑢 𝑥 NPS 𝑢 NPS 𝑢 ∑ NPS 𝑢
‒ For 𝑢 𝑚/𝑁𝑥 & 𝑚 1
2‐D digital NPS
• NPS 𝑢, 𝑣 E DFT ∆𝑑 ,
1‐D NPS of 2‐D noise process represented by a digital image
• NPS 𝑢 E DFT ∑ ∆𝑑 ,
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Undesirable aliases
Noise variance from NPS
• 𝜎 NPS 𝑢 d𝑢 𝑥 NPS 𝑢 d𝑢/
/
• 𝜎 NPS 𝑢 ∑ NPS 𝑢 d𝑢/
/
Pixel variance
• 𝜎 ∑ ∆𝑑
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Digital NEQ & DQE
Digital NEQ is defined only for 𝑢 𝑢 1/ 2𝑥
• NEQ 𝑞, 𝑢
‒ For 𝑢 𝑚/𝑁𝑥 & 𝑚 1
‒ Note that MTF 𝑢 includes the "aperture" MTF
For linear digital systems
• NEQ 𝑞, 𝑢/
• NEQ 𝑞, 𝑢∑
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Digital DQE
• DQE 𝑞, 𝑢,
∑
Signal aliasing
• Viewed as a form of image noise, hence resulting in additional artifacts & image degradation
• Not included in NEQ & DQE because being neither WSS nor WSCS
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Cascaded Model for A Hypothetical FPD
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E 𝑞 𝑞
𝑊 𝐤 𝑞
Stage 1 [mm‐2]: selection of x‐ray quanta that interact in phosphor
• Quantum selection stage
‒ 𝛼 = RV having 0 & 1 only w/ an expected value 𝛼 (known as quantum efficiency)
• Quantum image: 𝑞 𝑥, 𝑦 𝑞 𝑥, 𝑦 𝛼• Expected value: 𝑞 𝛼𝑞• NPS: 𝑊 𝑢, 𝑣 𝛼𝑞
Stage 2 [mm‐2]: conversion to optical quanta in screen [mm‐2]
• 𝑚 optical quanta per interaction w/ a variance 𝜎‒ 𝜎 accounts for all variations in the conversion gain, incl. Swank noise & a polychromatic x‐ray beam
• 𝑞 𝑥, 𝑦 𝑞 𝑥, 𝑦 𝛼𝑚• 𝑞 𝛼𝑚𝑞
• 𝑊 𝑢, 𝑣 𝛼𝑚 𝑞 1 𝛼𝑚𝑞
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Stage 3 [mm‐2]: scattering of optical quanta in phosphor
• Optical quanta scatter w/ the same (normalized) psf 𝑥, 𝑦‒ Neglecting variable interacting depths
• 𝑞 𝑥, 𝑦 𝑞 𝑥, 𝑦 𝛼𝑚 ∗ psf 𝑥, 𝑦• 𝑞 𝛼𝑚𝑞
• 𝑊 𝑢, 𝑣 𝛼𝑚 𝑞 1 T 𝑢, 𝑣 𝛼𝑚𝑞
Stage 4 [mm‐2]: selection of optical quanta that interact
• A fraction 𝛽 of all optical quanta interacts somewhere in the photodiode array
‒ Including the coupling efficiency & quantum efficiency
• 𝑞 𝑥, 𝑦 𝑞 𝑥, 𝑦 𝛼𝑚 ∗ psf 𝑥, 𝑦 𝛽• 𝑞 𝛼𝑚𝛽𝑞
• 𝑊 𝑢, 𝑣 𝛼𝑚 𝛽 𝑞 1 T 𝑢, 𝑣 𝛼𝑚𝛽𝑞
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Stage 5 [mm2]: spatial integration of interacting optical quanta in elements
• Deterministic blur stage
‒ Integral of 𝑞 𝑥, 𝑦 over 𝑎 & 𝑎 ⇒ detector presampling signal
‒ 𝑘 = scaling factor relating 𝑞 & 𝑑
• 𝑑 𝑥, 𝑦 𝑘 𝑞 𝑥, 𝑦 𝛼𝑚 ∗ psf 𝑥, 𝑦 𝛽 ∗ ∏ ,
• �̅� 𝑘𝑎 𝑎 𝛼𝑚𝛽𝑞
• 𝑊 𝑢, 𝑣 𝑘 𝑎 𝑎 𝛼𝑚 𝛽 𝑞 1 T 𝑢, 𝑣 𝛼𝑚𝛽𝑞 sinc 𝜋𝑎 𝑢 sinc 𝜋𝑎 𝑣
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Stage 6 [mm2]: output from discrete detector elements
• Sampling stage w/ pitches of 𝑥 & 𝑦
• 𝑑 𝑥, 𝑦 𝑘 𝑞 𝑥, 𝑦 𝛼𝑚 ∗ psf 𝑥, 𝑦 𝛽 ∗ ∏ , ∑ ∑ 𝛿 𝑥 𝑛 𝑥 , 𝑦 𝑛 𝑦
• E 𝑑 𝑥, 𝑦 E 𝑑 , 𝑘𝑎 𝑎 𝛼𝑚𝛽𝑞
• 𝑊 𝑢, 𝑣 𝑊 𝑢, 𝑣 ∑ ∑ 𝑊 𝑢 , 𝑣
• 𝑊 𝑢, 𝑣 𝑊 𝑢, 𝑣 ∑ ∑ 𝑊 𝑢 , 𝑣
1D NPS [mm2]
• 𝑊 𝑢 𝑊 𝑢, 𝑣 𝑊 𝑢, 𝑣 ∑ 𝑊 𝑢
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DQE
• DQE 𝑢∑
‒ MTF 𝑢 𝑇 𝑢 sinc 𝜋𝑎 𝑢
‒ 𝐹 𝑢 1 T 𝑢, 𝑣 sinc 𝜋𝑎 𝑢
• Assumptions
‒ 𝑚𝛽 100 ⇒ 𝑚𝛽 ≪ MTF 𝑢
‒ Poisson conversion gain from x rays to light quanta ⇒ 𝜀 /𝑚 ≪ 1
• DQE 𝑢∑
‒ No noise aliasing if MTF 𝑢 ≪ 1 for 𝑢 𝑢 1/2𝑥
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Low‐resolution scintillator
• Correlated quantum noise on detector array
‒ System MTF is limited by psf 𝑥 or T 𝑢 not by the detector element size 𝑎‒ T 𝑢 causes the quantum noise in the optical image incident on the photodiode array to be correlated,
which reduces the noise bandwidth
• DQE 𝑢∑
• If T 𝑢 ≪ 1; DQE 𝑢 𝛼
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High‐resolution photoconductor
• Uncorrelated quantum noise on detector array
‒ System MTF is limited by the detector‐array aperture function not by the photoconductor
‒ T 𝑢 is approximately constant over frequencies passed by sinc 𝜋𝑎 𝑢
• DQE 𝑢∑
‒ Note that ∑ sinc 𝜋𝑎 𝑢
• 𝛾 = detector fill factor in the 𝑥 direction
• DQE 𝑢 𝛼𝛾 𝛾 sinc 𝜋𝑎 𝑢
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𝛼 1 𝛾 1.0
0.75
0.5
0.25
Video‐based portal imaging system
35Bissonnette et al. MP (1997)
a‐Si FPD
36Siewerdsen et al. MP (1997)
Non‐linear response near detector saturation
Image blur
Lag
Zero‐freq. counting statistics
a‐Se FPD
37Zhao et al. MP (1997); Zhao & Rowlands, MP (1997)