Example%20of%20Aliasing

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Example of Aliasing
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Sampling and Aliasing in Digital Images

• Array of detector elements
• 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 gt FN ?
  sampled lt 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.
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Sampling and Aliasing in Digital Images

• Example sampling pitch of 100 mm ? FN 5
  cycles/mm When input f gt 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

c.f. Bushberg, et al. The Essential Physics of
Medical Imaging, 2nd ed., pp. 285-286.
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Aliasing due to Reciprocating Grid Failure
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• Noise is anything in the image that is not the
  signal we are interested in seeing.
• Noise can be structured or Random.

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Structure Noise

• Noise which comes from some non-random source
  breast parenchyma, hum bars in CRTs.
• The design goal in making an imaging system is to
  reduce structure or system noise to below the
  level of the random noise.

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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.

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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.

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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 k5.
• (remember, good resolution means D is small)

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Contrast Resolution

• Ability to detect a low-contrast object Related
  to how much noise there is in the image ? SNR
• As SNR ? the CR ?

• Rose criterion SNR gt 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.
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Statistics as image models
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Gaussian Probability Distribution Function

• Gaussian (normal) distribution
• ltXgt the mean
• and s 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.
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FOR GAUSSIAN PROBABILITY DISTRIBUTION
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FOR GAUSSIAN PROBABILITY DISTRIBUTION
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GAUSSIAN (NORMAL) PROBABILITY DISTRIBUTION
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ASSUMPTIONS FOR A NORMAL PROBABILITY DISTRIBUTION

• SAMPLE SELECTED FROM A LARGE POPULATION
• SAMPLE HOMOGENEOUS
• STOCHASTIC RANDOM MEASUREMENT PROCESS
• NO SYSTEMATIC ERRORS AFFECTING THE RESULTS

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GAUSSIAN (NORMAL) STATISTICAL DISTRIBUTIONS

• MEAN - 1 STD lt X lt MEAN 1 STD
• CONTAINS 68.3 OF MEASUREMENTS
• MEAN - 2 STD lt X lt MEAN 2 STD
• CONTAINS 95.5 OF MEASURMENTS
• MEAN - 3 STD lt X lt MEAN 3 STD
• CONTAINS 99.7 OF MEASUREMENTS

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GAUSSIAN (NORMAL) PROBABILITY DISTRIBUTION
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Poisson Probability Distribution Function

• Poisson distribution
• 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
• Radioactive decay
• Quantum mottle

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Probability Distribution Functions

• Probability of observing an observation in a
  range integrate area (for G)
• 1 s 68.25
• 1.96 s 95
• 2.58 s 99
• Error bars and confidence intervals
• P(x) very similar to G(x) when s vx ? use G(x)
  as approx.
• Can adjust the noise (s) 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.
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GAUSSIAN (NORMAL) DISTRIBUTION
EXP - ( X - X ) 2 / 2 ? 2
?
(2 ?)0.5
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COMPARISON OF VARIOUS STATISTICAL DISTRIBUTIONS
OF PROBABILITY FOR COIN FLIPPING
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Quantum Noise

• N mean photons/unit area
• s vN, from P(x) ? s2 (variance) N
• Relative noise coefficient of variation s/N
  1/vN (? with ? N)
• SNR signal/noise N/s N/vN vN (? 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.
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Noise Frequency the Wiener Spectrum W(f)

• 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 (s2) 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.
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Detective Quantum Efficiency

• DQE metric describing overall system SNR
  performance and dose efficiency
• DQE
• SNR2in N (? SNR vN)
• SNR2out
• DQE(f)

DQE(f0) QDE
c.f. Bushberg, et al. The Essential Physics of
Medical Imaging, 2nd ed., p. 282.
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Contrast Detail (C-D) Curves

• 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 high-contrast, small
  detail
• Better contrast resolution low-contrast

c.f. Bushberg, et al. The Essential Physics of
Medical Imaging, 2nd ed., p. 287.
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Receiver Operating Characteristic Curves

• 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.
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Receiver Operating Characteristic Curves

• 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/(TPFN) Sensitivity
• FPF FP/(FPTN)
• Specificity (1-FPF) TNF
• ROC curve sensitivity vs. 1-specificity
  usu. _at_ five threshold levels

c.f. Bushberg, et al. The Essential Physics of
Medical Imaging, 2nd ed., pp. 288-289.
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ROC Questionnaire 5 Point Confidence Scale
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The ROC Cookbook
Rank Signal (Lesion) Detection On A Scale of 1
to 5.   1.                 Almost certainly NOT
present. 2.                 Probably NOT
present 3.                 Equally likely to be
Present or Not Present. 4.                
Probably PRESENT 5.                 Almost
certainly PRESENT   Make a table of the number of
cases receiving each rank for both the positive
and negative images.
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The survey
  Categories  
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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.    
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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.
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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.