Transcript NUCLEAR MEDICINE – COUNTING STATISTICS

Example of Aliasing
Sampling and Aliasing in Digital Images
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Array of detector elements
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Sampling (pixel) pitch
Detector aperture width
The spacing between samples
determines the highest frequency
that can be imaged
Nyquist frequency: FN = 1/2D
If a frequency component in an
image > FN → sampled <
2x/cycle: aliasing
Wraps back into the image as a
lower frequency
Moiré pattern, spoke wheels
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., p. 284.
Sampling and Aliasing in Digital Images
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Example: sampling pitch of 100
mm → FN = 5 cycles/mm When
input f > FN then the spatial
frequency domain signal at f is
aliased down to:
fa = 2FN – f
Not noticeable with patient
Antiscatter grids
Aperture blurring - signal averaging
across the detector aperture
MTF(f)=FT{rect(a)}=sinc(af)=
sin(a f)
a f
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., pp. 285-286.
Aliasing due to Reciprocating Grid Failure
• Noise is anything in the image that is not
the signal we are interested in seeing.
• Noise can be structured or Random.
Structure Noise
Noise which comes from some non-random
source: breast parenchyma, hum bars in
CRT’s.
The design goal in making an imaging system
is to reduce structure or system noise to
below the level of the random noise.
Random or Quantum Noise
Noise resulting from the statistical nature of the
signal source is random or quantum noise.
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In imaging, the signal is light in the form of
photons being emitted randomly in time and
space.
Because we are working with a random
source, we can use statistics to describe the
behavior of the image noise.
Rose Model
• The information content of a finite amount of light
is limited by the finite number of photons, by the
random character of their distribution, and by the
need to avoid false alarms (false positives).
• The measure of how well an object (signal) can be
seen against a background of varying signal
strength (noise) is the signal to noise ratio: S/N.
Rose Model
• To see an object of a given diameter (resolution) you must have
sufficient contrast and S/N.
• In an ideal system, where the only noise is quantum noise, the
diameter, D, which can be resolved is given by:
• D2 x n2 = k2/C2
where C is the contrast of the detail, n is the number
of photons/sq cm in the image, and k is the threshold
S/N ratio.
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Most people use k=5.
• (remember, good resolution means D is small)
Contrast Resolution
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Ability to detect a lowcontrast object Related to
how much noise there is in
the image → SNR
As SNR ↑ the CR ↑
Rose criterion: SNR > 5 to
reliably identify an object
Quantum noise and
structure noise both affect
the conspicuity of a target
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., p. 281.
Statistics as image models
Gaussian Probability Distribution
Function
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Gaussian (normal) distribution:
<X> the mean
G( x )  ke
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1 x x 
 
2   
2
x and σ describe the shape
Many commonly encountered
measurements of people and things
make for this kind of distribution
(Gaussian) hence the term “normal”
e.g., the height of 1000 third grade
school children approximates a
Gaussian
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., p. 275.
FOR GAUSSIAN PROBABILITY
DISTRIBUTION
MEAN
 Xi
i
X=
N
VARIANCE

2=

i
( Xi - X )2
(N - 1)
FOR GAUSSIAN PROBABILITY
DISTRIBUTION
STANDARD DEVIATION
 =
2

~
=
X
GAUSSIAN (NORMAL) PROBABILITY DISTRIBUTION
ASSUMPTIONS FOR A NORMAL
PROBABILITY DISTRIBUTION
• SAMPLE SELECTED FROM A LARGE
POPULATION
• SAMPLE = HOMOGENEOUS
• STOCHASTIC = RANDOM
MEASUREMENT PROCESS
• NO SYSTEMATIC ERRORS AFFECTING
THE RESULTS
GAUSSIAN (NORMAL) STATISTICAL
DISTRIBUTIONS
• MEAN - 1 STD < X < MEAN + 1 STD
– CONTAINS 68.3 % OF MEASUREMENTS
• MEAN - 2 STD < X < MEAN + 2 STD
– CONTAINS 95.5 % OF MEASURMENTS
• MEAN - 3 STD < X < MEAN + 3 STD
– CONTAINS 99.7 % OF MEASUREMENTS
GAUSSIAN (NORMAL) PROBABILITY DISTRIBUTION
Poisson Probability
Distribution Function
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Poisson distribution:
m x m
P( x ) 
e
x!
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m = mean, shape governed by one
variable
P(x) difficult to calculate for large
values of x due to x!
X-ray and g-ray counting statistics
obey P(x)
Used to describe
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Radioactive decay
Quantum mottle
Poisson Distribution
0.4
0.35
0.3
0.25
m=1
m=2
0.2
m=4
m=6
0.15
m=8
m=10
m=20
0.1
0.05
0
0
10
20
30
40
Probability Distribution
Functions
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Probability of observing an
observation in a range: integrate
area (for G):
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1 σ = 68.25%
1.96 σ = 95%
2.58 σ = 99%
Error bars and confidence
intervals
P(x) very similar to G(x) when σ
≈ √x → use G(x) as approx.
Can adjust the noise (σ) in an
image by adjusting the mean
number of photons used to
produce the image
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., pp. 276 - 277.
GAUSSIAN
(NORMAL)
DISTRIBUTION
EXP[ - ( X - X ) 2 / 2  2 ]
 (2 )0.5
COMPARISON OF VARIOUS STATISTICAL
DISTRIBUTIONS OF PROBABILITY FOR
COIN FLIPPING
PROBABILITY OF RESULT
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
BINOMIAL
15
20
25
30
NUMBER OF HEADS
POISSON
35
GAUSSIAN
40
45
Quantum Noise
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N = mean photons/unit area
σ = √N, from P(x) → σ2 (variance) = N
Relative noise = coefficient of variation = σ/N = 1/√N (↓ with ↑ N)
SNR = signal/noise = N/σ = N/√N = √N (↑ with ↑ N)
Trade-off between SNR and radiation dose: SNR ↑ 2x → Dose ↑ 4x
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., p. 278.
Noise Frequency: the Wiener Spectrum W(f)
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Although noise appears random,
the noise has a frequency
distribution
Example: ocean waves
Take a flat-field x-ray image (still
has noise variations) Fourier
Transform (FT) the flat image →
Noise Power Spectrum: NPS(f)
NPS(f) is the noise variance (σ2) of
the image expressed as a function
of spatial freq. (f)
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., p. 282.
Detective Quantum Efficiency
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DQE: metric describing
overall system SNR
performance and dose
efficiency
DQE =
2
SNRout
SNRin2
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SNR2in = N (→ SNR = √N)
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MTF (f )
2
SNR2out =
DQE(f=0) = QDE
NPS (f )
DQE(f) = k MTF (f )
N  NPS(f )
2
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., p. 282.
Contrast Detail (C-D) Curves
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Spatial resolution: MTF(f)
Contrast resolution: SNR
Combined quantitative: DQE(f)
Qualitative: C-D curve
C-D phantom: holes in plastic of ↓
depth and diameter
What depth hole at which
diameter can just be visualized
Connect the dots → C-D line
Better spatial resolution: highcontrast, small detail
Better contrast resolution: lowcontrast
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., p. 287.
Receiver Operating Characteristic
Curves
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The ROC curve is essentially a
way of analyzing the SNR
associated with a specific
diagnostic task Az: area under the
curve – concise description of the
diagnostic performance of the
systems (including observers)
being tested
Measure of detectability
Az = 0.5 guessing
Az = 1.0 perfect
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., p. 291.
Receiver Operating Characteristic
Curves
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Diagnostic task: separate
abnormal from normal
Usually significant overlap in
histograms
Decision criterion or threshold
Based on threshold: either normal
(L) or abnormal (R)
N cases: 2 x 2 decision matrix
TPF= TP/(TP+FN)= Sensitivity
FPF = FP/(FP+TN)
Specificity = (1-FPF) = TNF
ROC curve: sensitivity vs. 1specificity usu. @ five threshold
levels
c.f. Bushberg, et al. The Essential Physics of Medical
Imaging, 2nd ed., pp. 288-289.
ROC Questionnaire: 5 Point Confidence Scale
The ROC Cookbook
Rank Signal (Lesion) Detection On A Scale of 1 to 5.
1.
2.
3.
4.
5.
Almost certainly NOT present.
Probably NOT present
Equally likely to be Present or Not Present.
Probably PRESENT
Almost certainly PRESENT
Make a table of the number of cases receiving each rank for both the positive and negative
images.
The survey
Categories
Image
Rank
(Certainty
that a
lesions is
present)
1
2
3
4
5
Total
Number
of Images
Certainly
Not
Probably
Not
Unsure
Probably
Present
Certainly
Present
Positive
Images
2
14
34
34
16
100
Negative
Images
24
51
51
21
3
150
Make the Cumulative Table
Make a second table with a cumulative ranking: Add the cells so that the lowest rank has
the total of all possibilities, the next has all but the lowest rank, the next all but the two
lowest rank, etc.
Cumulative
Rank
Positive
Images
Negative
Images
1+2+3+4+5
2+3+4+5
3+4+5
4+5
5
100
98
84
50
16
150
126
75
24
3
Normalize the Data to One.
Divide the positive image values by 100
Divide the negative image values by 150.
Put them in a new table.
Cumulative
Rank
Positive
Images
Negative
Images
1+2+3+4+5
2+3+4+5
3+4+5
4+5
5
1
.98
.84
.50
.16
1
.84
.5
.16
.02
Probability of
calling a signal
when a signal is
present.
Probability of
calling a signal
when a signal is
absent.
Plot the Curve
Plot the results. The straight line is a pure guess line. The area under the curve is Az, a measure of overall image
performance. Az = 0.5 is equivalent to pure guessing. The greater the area under the curve, the better the system
under test performs the task.
ROC Curve
1
0.9
0.8
0.7
TPF
0.6
0.5
evaluation
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
FPF
0.6
0.7
0.8
0.9
1