Example of Aliasing

Example of Aliasing

Sampling and Aliasing in Digital Images

Array of detector elements Array of detector elements Sampling (pixel) pitch Sampling (pixel) pitch Detector aperture width Detector aperture width

The spacing between samples The spacing between samples determines the highest frequency determines the highest frequency that can be imaged that can be imaged

Nyquist frequency: FNyquist frequency: FN N = 1/2= 1/2 If a frequency component in an If a frequency component in an

image > Fimage > FNN → sampled < → sampled <

2x/cycle: 2x/cycle: aliasing aliasing Wraps back into the image as a Wraps back into the image as a

lower frequency lower frequency Moiré pattern, spoke wheelsMoiré pattern, spoke wheels

c.f. Bushberg, et al. The Essential Physics of Medical c.f. Bushberg, et al. The Essential Physics of Medical Imaging, 2Imaging, 2ndnd ed., p. 284. ed., p. 284.

Sampling and Aliasing in Digital Images

Example: sampling pitch of 100 Example: sampling pitch of 100 m → Fm → FN N = 5 cycles/mm When = 5 cycles/mm When

input f > Finput f > FN N then the spatial then the spatial

frequency domain signal at f is frequency domain signal at f is aliased down to: aliased down to:

ffaa = 2F = 2FNN – f – f Not noticeable with patient Not noticeable with patient Antiscatter grids Antiscatter grids

c.f. Bushberg, et al. The Essential Physics of Medical c.f. Bushberg, et al. The Essential Physics of Medical Imaging, 2Imaging, 2ndnd ed., pp. 285-286. ed., pp. 285-286.

Aperture blurring - signal Aperture blurring - signal averaging across the detector averaging across the detector aperture aperture

sin(a f)MTF(f)=FT{rect(a)}=sinc(af)=

a f

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.

• 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.

• Most people use k=5.• (remember, good resolution means D is small)

Contrast Resolution

Ability to detect a low-Ability to detect a low-contrast object Related to contrast object Related to how much noise there is in how much noise there is in the image → SNR the image → SNR

As SNR ↑ the CR ↑ As SNR ↑ the CR ↑

Rose criterion: SNR > 5 to Rose criterion: SNR > 5 to reliably identify an object reliably identify an object

Quantum noise and Quantum noise and structure noise both affect structure noise both affect the the conspicuityconspicuity of a target of a target


Statistics as image models

Gaussian Probability Distribution Function

Gaussian (normal) distribution:Gaussian (normal) distribution:

<X> the mean<X> the mean

and and σσ describe the shape describe the shape

Many commonly encountered Many commonly encountered measurements of people and things measurements of people and things make for this kind of distribution make for this kind of distribution (Gaussian) hence the term “normal” (Gaussian) hence the term “normal” e.g., the height of 1000 third grade e.g., the height of 1000 third grade school children approximates a school children approximates a GaussianGaussian

x

21

2( )x x

G x ke


FOR GAUSSIAN PROBABILITY DISTRIBUTION

MEAN

X =

Xii

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

Poisson distribution:Poisson distribution:

m = mean, shape governed by one m = mean, shape governed by one variable variable

P(x) difficult to calculate for large P(x) difficult to calculate for large values of x due to x! values of x due to x!

X-ray and X-ray and -ray counting statistics -ray counting statistics obey P(x) obey P(x)

Used to describe Used to describe Radioactive decay Radioactive decay Quantum mottleQuantum mottle

( )!

xmm

P x ex

Poisson Distribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 10 20 30 40

m=1

m=2

m=4

m=6

m=8

m=10

m=20

Probability Distribution Functions

Probability of observing an Probability of observing an observation in a range: integrate observation in a range: integrate area (for G): area (for G): 1 1 σσ = 68.25% = 68.25% 1.96 1.96 σσ = 95% = 95% 2.58 2.58 σσ = 99% = 99%

Error bars and confidence Error bars and confidence intervals intervals

P(x) very similar to G(x) when P(x) very similar to G(x) when σσ ≈ √x → use G(x) as approx. ≈ √x → use G(x) as approx.

Can adjust the noise (Can adjust the noise (σσ) in an ) in an image by adjusting the mean image by adjusting the mean number of photons used to number of photons used to produce the imageproduce the image

c.f. Bushberg, et al. The Essential Physics of Medical c.f. Bushberg, et al. The Essential Physics of Medical Imaging, 2Imaging, 2ndnd ed., pp. 276 - 277. 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

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25 30 35 40 45NUMBER OF HEADS

PR

OB

AB

ILIT

Y O

F R

ES

ULT

BINOMIAL POISSON GAUSSIAN

Quantum Noise

N = mean photons/unit area N = mean photons/unit area σσ = √N, from P(x) → = √N, from P(x) → σσ22 (variance) = N (variance) = N Relative noise = coefficient of variation = Relative noise = coefficient of variation = σσ/N = 1/√N (↓ with ↑ N) /N = 1/√N (↓ with ↑ N) SNR = signal/noise = N/SNR = signal/noise = N/σσ = N/√N = √N (↑ with ↑ N) = N/√N = √N (↑ with ↑ N) Trade-off between SNR and radiation dose: SNR ↑ 2x → Dose ↑ 4x Trade-off between SNR and radiation dose: SNR ↑ 2x → Dose ↑ 4x


Noise Frequency: the Wiener Spectrum W(f)

Although noise appears random, Although noise appears random, the noise has a frequency the noise has a frequency distribution distribution

Example: ocean waves Example: ocean waves

Take a flat-field x-ray image (still Take a flat-field x-ray image (still has noise variations) Fourier has noise variations) Fourier Transform (FT) the flat image → Transform (FT) the flat image → Noise Power Spectrum: NPS(f) Noise Power Spectrum: NPS(f) NPS(f) is the noise variance (NPS(f) is the noise variance (σσ22) of ) of the image expressed as a function the image expressed as a function of spatial freq. (f)of spatial freq. (f)


Detective Quantum Efficiency

DQE: metric describing DQE: metric describing overall system SNR overall system SNR performance and dose performance and dose efficiency efficiency

DQE = DQE = SNRSNR22

inin = N = N (→ SNR = (→ SNR =

√N) √N)

SNRSNR22outout = =

DQE(f) = DQE(f) = c.f. Bushberg, et al. The Essential Physics of Medical c.f. Bushberg, et al. The Essential Physics of Medical

Imaging, 2Imaging, 2ndnd ed., p. 282. ed., p. 282.

2

2out

in

SNR

SNR

2( )

( )

MTF f

NPS f

2( )

( )

k MTF f

N NPS f

DQE(f=0) = QDEDQE(f=0) = QDE

Contrast Detail (C-D) Curves

Spatial resolution: MTF(f) Spatial resolution: MTF(f) Contrast resolution: SNR Contrast resolution: SNR Combined quantitative: DQE(f) Combined quantitative: DQE(f) Qualitative: C-D curve Qualitative: C-D curve C-D phantom: holes in plastic of C-D phantom: holes in plastic of

↓ depth and diameter ↓ depth and diameter What depth hole at which What depth hole at which

diameter can just be visualized diameter can just be visualized Connect the dots → C-D line Connect the dots → C-D line Better spatial resolution: high-Better spatial resolution: high-

contrast, small detail contrast, small detail Better contrast resolution: low-Better contrast resolution: low-

contrastcontrast


Receiver Operating Characteristic Curves

The ROC curve is essentially a The ROC curve is essentially a way of analyzing the SNR way of analyzing the SNR associated with a specific associated with a specific diagnostic task Adiagnostic task Azz: area under the : area under the

curve – concise description of the curve – concise description of the diagnostic performance of the diagnostic performance of the systems (including observers) systems (including observers) being tested being tested

Measure of detectability Measure of detectability AAzz = 0.5 guessing = 0.5 guessing

AAzz = 1.0 perfect = 1.0 perfect


Receiver Operating Characteristic Curves

Diagnostic task: separate abnormal Diagnostic task: separate abnormal from normal from normal

Usually significant overlap in Usually significant overlap in histograms histograms

Decision criterion or threshold Decision criterion or threshold Based on threshold: either normal Based on threshold: either normal

(L) or abnormal (R) (L) or abnormal (R) N cases: 2 x 2 decision matrix N cases: 2 x 2 decision matrix TPF= TP/(TP+FN)= Sensitivity TPF= TP/(TP+FN)= Sensitivity FPF = FP/(FP+TN) FPF = FP/(FP+TN) Specificity = (1-FPF) = TNF Specificity = (1-FPF) = TNF ROC curve: sensitivity vs. 1-ROC curve: sensitivity vs. 1-

specificity usu. @ five threshold specificity usu. @ five threshold levelslevels

c.f. Bushberg, et al. The Essential Physics of Medical c.f. Bushberg, et al. The Essential Physics of Medical Imaging, 2Imaging, 2ndnd ed., pp. 288-289. ed., pp. 288-289.

ROC Questionnaire: 5 Point Confidence Scale

Rank Signal (Lesion) Detection On A Scale of 1 to 5. 1. Almost certainly NOT present.2. Probably NOT present3. Equally likely to be Present or Not Present.4. Probably PRESENT5. Almost certainly PRESENT Make a table of the number of cases receiving each rank for both the positive and negative images.

The ROC Cookbook

Categories

Image Rank(Certainty that a lesions is present)

1 Certainly Not

2 Probably Not

3 Unsure

4 Probably Present

5 Certainly Present

Total Number of Images

Positive Images

2 14 34 34 16 100

Negative Images

24 51 51 21 3 150

The survey

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

1+2+3+4+5 2+3+4+5 3+4+5 4+5 5

Positive Images

100

98

84

50

16

Negative Images

150 126 75 24 3

Make the Cumulative Table

Divide the positive image values by 100 Divide the negative image values by 150.

Put them in a new table.

Cumulative Rank

1+2+3+4+5 2+3+4+5 3+4+5 4+5 5

Positive Images

1

.98

.84

.50

.16

Probability of calling a signal when a signal is present.

Negative Images

1 .84 .5 .16 .02 Probability of calling a signal when a signal is absent.

Normalize the Data to One.

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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FPF

TP

F

evaluation

Plot the Curve

Example of Aliasing

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