Answering Physics questions
Diagnostic Imaging Physics
MPHY-501
Bushburg Chapter 4
Image Quality
During the next two years you will learn the basics of Diagnostic Imaging. Ultimately after this period you will enroll in a Medical Physics Residency Program and finally take the Board examination that, once you pass, will certify you as a medical physicist.
1
Image Quality: Dose versus information content
Do Not
maximize image quality
Do
optimize image quality
Why?
Unnecessary Dose
Unnecessary Heat
Unnecessary Scan Times
Unnecessary Anesthesia Duration
Spatial Resolution
In digital imaging the pixel has a size. Bushberg says the pixel size sets the limit on resolution in many cases it is not the pixel size that sets the limit on resolution
what about magnification?
Open discussion on factors that limit resolution
Availability of photons (signal) to fill up a del (detector element)
Del size
Magnification factor
Motion (blurring)
Other factors related to blurring (not motion, like scatter)
Depth of field / partial volume
A Few Imaging Terms:
Resolution
Field of View (FOV)
Depth of Field
Exposure Time
Intensity
Sampling Matrix
Display
Reconstruction
Sampling Time
Point Spread Function
Source to Object Distance (SOD)
Source to Image Distance (SID)
Magnification
In Plane
Projection Imaging
Volumetric Imaging
Imaging Field of View (FOV)
Line imaging (1D)
Spectroscopy
Projection Imaging (2D)
Mammography: 5cm X 5cm
Chest x-ray: 35cm X 45cm
Volumetric Imaging (3D)
MRI: 45cm X 45CM X 45CM
CT: 45cm X 45cm X ??cm
Point-Spread /Impulse-Response Function (PSF)
Think of two point light sources separated by a distance X.
The intensity from these point sources falls off radially from the source
Resolved if at full width ½ Maximum (FWHM)
intensity are separated.
Otherwise, if overlapping points, cannot be resolved
Point
Intensity
Intensity
Distance
Distance
Point Spread
100%
100%
50%
Point Spread Function (PSF) Describes the Extent of Blurring
Basic measurements of resolution in imaging systems
Can be multi dimensional PSF(x,y,z)
Phantoms of small specks in its material or, in the case of a two dimensional PSF(x,y), a wire that is normal to the imaging plane
When PSF is the same throughout the imaging plane it is called Shift Invariant
When PSF is not the same throughout the imaging plane it is called nonstationary
Because most detector elements are rectangular in shape the point spread function is inherently rectangular
Line Spread Function (LSF)
Point Spread Function
Edge Spread Function (ESF)
Used to determine effects of glare or scatter
Similar to Point spread function or Line spread function just single-sided
Line Spread Function
Edge Spread Function
Convolution
Convolution can be used for:
Smoothing (Using a Rectangle Function)
Fourier Analysis (Using a Comb Function)
Convolution With Rectangle (RECT) Function
H Kernel Sum G H Kernel Sum G H Kernel Sum G H Kernel Sum G H Kernel Sum G
42.4 x 0.2 42.4 42.4 42.4 42.4
87.8 x 0.2 87.8 x 0.2 87.8 87.8 87.8
36.2 x 0.2 62.8 36.2 x 0.2 62.7 36.2 x 0.2 62.7 36.2 62.7 36.2 62.7
71.5 x 0.2 71.5 x 0.2 58.4 71.5 x 0.2 58.4 71.5 x 0.2 58.4 71.5 58.4
76 x 0.2 76 x 0.2 76 x 0.2 57.6 76 x 0.2 57.6 76 x 0.2 57.6
20.7 20.7 x 0.2 20.7 x 0.2 20.7 x 0.2 65.4 20.7 x 0.2 65.4
83.4 83.4 83.4 x 0.2 83.4 x 0.2 83.4 x 0.2 63.6
75.5 75.5 75.5 75.5 x 0.2 75.5 x 0.2
62.2 62.2 62.2 62.2 62.2 x 0.2
S
S
S
S
S
sH = 23.3
sG = 3.4
20%
RECT =
Shift, Multiply, Add
This is the function of Convolution
In this case represented in discrete form
Gray Scale Value
Data (H) Smoothed as a Result of Box-Car Average (G)
H 42.4 87.8 36.200000000000003 71.5 76 20.7 83.4 75.5 62.2 G 62.7 58.440000000000005 57.56 65.420000000000016 63.56
More Kernels
Negative going part of function provides Edge Enhancement
Various Kernels
Rectangular Smoothing Kernel -5 -4 -3 -2 -2 -1 0 0 0 1 2 2 3 4 5 0 0 0 0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0 0 0 0 Sinc Function Edge Enhancement Kernel -5 -4 -3 -2 -2 -1 0 0 0 1 2 2 3 4 5 0 0 0 -0.15 -0.15 0.35 0.6 0.6 0.6 0.35 -0.15 -0.15 0 0 0 Gaussian Smoothing Kernel -5 -4 -3 -2 -2 -1 0 0 0 1 2 2 3 4 5 0 0 0 0.05 0.05 0.2 0.5 0.5 0.5 0.2 0.05 0.05 0 0 0 Delta Function Kernel -5 -4 -3 -2 -2 -1 0 0 0 1 2 2 3 4 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
Pixel Numer
Kernel Value
Relationships Between: PSF, LSF, ESF
Mechanisms of Blurring
Thin Scintillator: low blurring
Thick Scintillator: hi blurring
Thin Pixel: low blurring
Thick Pixel: hi blurring
X-ray
X-ray
Visible Photons
Frequency Domains: Time Domain, Space Domain
Time Domain
Time Oscillations: Frequency is represented ad Cycles/Unit time
Cycles per second
Space Domain
Spatial Oscillations: Frequency is represented ad Cycles/Unit distance
Cycles per millimeter
Time (seconds)
Or
Distance (millimeters)
Sampled Frequency = Cycles/16 seconds
or
Cycles/16 millimeters
Actual Frequency = Cycles/3 seconds
or
Cycles/3 millimeters
Nyquist Sampling Theorem: To accurately reproduce a signal:
Sampling must occur at or greater than twice the signals frequency
Fourier Series
Method for decomposing a function into a sum of Sign Waves
Representation of a Sign Wave:
g(x) = a sin (2p fx + y)
Where:
a = Amplitude
f = Frequency
y = Phase
Fourier Analysis
Adapted from: Bushburg and http://mriquestions.com/fourier-transform-ft.html
s(t) = sin (wt) + 1/3 sin (3wt) + 1/5 sin (5wt) + 1/7 sin (7wt)
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B$48:$B$256 58.201950959802474 79.383236227059825 103.28538427428327 129.61361031780251 158.04320879545011 188.22355795471628 219.78244406483279 252.33065192235964 285.46676503564146 318.78211628754298 351.86582802002738 384.30987938129397 415.71413843997726 445.69129700542106 473.87164729298172 499.90764152382019 523.47817822572495 544.29256237208392 562.09409051868352 576.66321672290258 587.82026020007243 595.42762132352721 599.39147863844778 599.66294596026353 596.23867528713026 589.16089809078107 578.51690447650537 564.43796663577234 547.09771986853502 526.71002114204043 503.52631159650582 477.83251552582954 449.94551407815879 420.20923716609923 388.99042178497751 356.67408905174091 323.65879574609647 290.35171891655574 257.16363417284589 224.50384959733211 192.77515775542702 162.36886806176207 133.65998076785667 107.00256209042237 82.725377518758364 61.127837155586121 42.476303097367236 27.000804395114471 14.892200110035503 6.2998254519898751 1.3296500308969712 4.2970935345294947E-2 2.455656756646647 8.5379518817703115 18.214843468810386 31.366986579083459 47.832176056220163 67.407346999587901 89.851079160243017 114.88657437273665 142.20507030216424 171.46964840476753 202.31938913858124 234.373823177981 267.23762373537653 300.50548211973489 333.76710640174304 366.61228153716638 398.63592854198151 429.44310032457139 458.65385256145964 485.90792954416366 510.86920720669536 533.22983853792516 552.71405025351555 569.08154390312529 582.1304594670944 591.69986489261873 597.67174086600949 599.97243634287349 598.57357688518528 593.49241460274709 584.79161538321318 572.57848603476793 557.00365087309228 538.25919407425124 516.57629070383177 492.2223556388052 465.49774554441694 436.73205458056418 406.28004952264934 374.51729442899978 341.83551881574766 308.63778646339892 275.33352443811077 242.33347363467914 210.04462311609461 178.86519072438074 149.17971186674899 121.35429704698836 95.732116630874486 72.62916853190859 52.3303810144495 35.086098678290966 21.108994962990664 10.571449249971522 3.6034209105542345 0.29084651925035132 0.67457999945634128 4.7498887727142005 12.466512125520865 23.729281073848597 38.399292080600787 56.295620150562655 77.197550175276831 100.84729900870727 126.95319470237652 155.19327369052539 185.21925156127102 216.66081844231172 249.13020602625386 282.22696991053084 315.54292827137732 348.66719596329165 381.19125195851854 412.71397762990767 442.84660374009218 471.21750512585345 497.47678394491402 521.30058496004699 542.39508964019115 560.50013981937389 575.39244622292927 586.88834229031011 594.84604933164155 599.16742508188304 599.79917408770632 596.73350499941125 590.0082266615093 579.70628181688858 565.95472417538372 548.92315146254111 528.82161377376679 505.89802303008673 480.43509548465852 452.74686498804965 423.17481001412011 392.08364221184013 359.8568084227461 326.89176163759595 293.59505921542103 260.37734881856011 227.64830390205282 195.81157121791489 165.25979264663894 136.36976275158696 109.49778177807877 84.975261408545805 63.104637467915239 44.155639987745019 28.361966630351276 15.918400499529753 6.9784078838633832 1.6522455596058307 5.6009957432365809E-3 2.0587822315607127 7.7864674179406279 17.118017111295444 29.938345468619275 46.089339599224132 65.371809568320288 87.547945003080827 112.34424800390809 139.45490618935966 168.5455642750226 199.25744767153125 231.21178724544734 264.01449067251508 297.26100277156598 330.54129487665915 363.44492171361611 395.56608341457769 426.50863024084521 455.89094829083842 483.35066593783262 508.54912295303859 531.17554719642158 550.95088736419723 567.63125452379518 581.01092999189996 590.92490245945851 597.25090307333744 599.91091337599926 598.87212750580863 594.14735679116257 585.79487174859605 573.91768343349713 558.66227300651985 540.21678518393173 518.80870785204047 494.70206646297697 468.19416781330619 439.61193336437753 409.30786732552843 377.65570922568975 345.04582459015808 311.88039056923878 278.56843589431986 245.52079633351264 B$48:$B$256 138.45301300100016 145.68947868913244 144.22931528113313 133.01218749063986 118.37861681792397 112.01013056128048 125.42696387739058 163.82552171907093 222.58871438763993 288.50477438338754 345.26373204940597 380.5154553128549 390.93340732745776 382.80029003478217 368.0204495142429 357.84881570484379 357.70500005675115 365.65854285906937 375.01901850492624 379.18712760494219 375.79995749702732 367.82424728504265 361.19253253459283 360.66656439659329 366.62043876098028 374.79000237590964 379.14699523306882 376.07240857229766 367.12666040518064 358.49290847282333 357.16884765317229 366.02870747987856 380.74151876630879 390.65100990987173 383.54377320443575 351.97389913225214 297.77397823547994 232.19843999695235 171.4089846825475 129.49146254398519 112.51970544700855 116.61861460246237 130.83901006047722 143.12519801647031 146.12195802695868 139.84478958289418 130.11671138310572 124.05896663298066 125.46099940745475 132.71345726621374 140.31387453279231 142.77132402425087 138.33636871247367 130.17388824493571 124.23673465144739 125.18764663504381 132.99017069028662 142.46342947532571 146.33575693280898 140.16425687431769 126.20377228046746 113.71193257608698 115.09823963779155 139.70259180194984 188.46424448625663 252.43952610090724 316.16251203030544 364.27682002594975 388.11213022976284 388.90877584337181 376.19127718829708 362.3604676952815 356.49943939208651 360.63347600334419 370.16412369919112 377.81349220722279 378.5274444604558 372.43716284758125 364.30492088909364 360.05507668034727 362.71946245027925 370.40099772157902 377.54569207935265 378.71815432415974 372.46782993619388 362.73329476863972 356.67161940216272 359.98734295037445 372.47497124477036 386.57643672165011 390.17476119597472 372.45581116741562 329.82631100117891 268.65646228936413 203.19653260349048 149.55790105338502 118.7553601491765 112.31249099238048 122.55519907815119 137.22393349441109 145.79989802554076 144.18008753502704 135.42546063850784 126.58332221459403 123.73978090494181 128.28739915660037 136.47989865045514 142.28106997675175 141.53807989667766 134.77940352432111 126.86080405209123 123.68322539667003 128.07145916448547 137.49927801569521 145.29278060277113 144.83889631945553 134.41009848777605 119.64741880131717 111.91534072516976 123.03660311357605 159.04171864088246 216.29627631345869 282.23736086227808 340.54410355646297 378.20122983725776 390.8967915787548 384.09100787855971 369.40498245257953 358.4271910274141 357.28216664335713 364.69920549945249 374.25270602221605 379.11845804536028 376.41363061264445 368.64162857181464 361.61938893111278 360.39896944099604 365.84855547800032 374.07079742020937 379.02632742232703 376.69766841383819 368.10523896250618 359.10806314595112 356.84602811993949 364.781508567439 379.31739925413996 390.25441054442393 385.25266027398578 356.14986049989818 303.78816330402367 238.63814016808027 176.67631908611304 132.49560404905401 113.11372097919781 115.56928021762013 129.36668616610655 142.27363323298363 146.29351605356854 140.72769281040081 131.00365370084467 124.34153878236515 125.00187044974606 131.88970058146145 139.73113530957545 142.84594295692867 139.01537303082941 130.97041689861288 124.55852389199525 124.74978978019134 132.04362079424629 141.67587950314672 146.37174075739185 141.2027895850984 127.68893095714074 114.51510449558207 114.04188610204606 136.1830698991015 182.83835166190414 245.95209888138203 310.43227537605401 360.59361381422013 386.90012253973941 389.62094934012657 377.66995720565399 363.45452070268868 356.60465012349312 359.88335517665121 369.20467057185311 377.32760309908144 378.80182691424187 373.21906399927747 365.01043631759444 360.1776447077844 362.17205842521707 369.57433847894504 377.04568663874988 378.94899285207271 373.32946706787095 363.64018277588639 356.90740145304449 359.20029585494319 371.01696720847195 385.45920421085094 390.6264386836454 375.29160792423414 334.96703437653071 275.06542234014876 209.28926597957391 153.88760936600551
Position in millimeters
Grey Scale
Fourier Transform: Transforms from Spatial Domain to Frequency Domain
Some Fourier Transformations
http://mriquestions.com/fourier-transform-ft.html
http://mriquestions.com/fourier-transform-ft.html
Recapture of Convolution
Fun Facts: Fourier Transform Versus Convolution
Fourier Transform computation is faster than the convolution
Convolution procedures in CT are often run in the frequency domain
Kernels used in CT are offend described in the frequency domain
Ramp
Sheep-Logan
Bone kernel
B41
Fourier Transform is used to perform filtering procedure in the filtered back-projection for CT reconstruction
Inverse Fourier Transforms are used for image reconstruction in MRI (Time Domain ? Space Domain)
Modulation Transfer Function (MTF)
The limiting resolution can be defined at
MTF > 10%
http://sciencewise.info/resource/line_spread_function/Line_spread_function_by_Wikipedia
The optical transfer function of a well-focused (a), and an out-of-focus optical imaging system without aberrations (d). As the optical transfer function of these systems is real and non-negative, the optical transfer function is by definition equal to the modulation transfer function (MTF). Images of a point source and spoke target are shown in (b,e) and (c,f), respectively. Note that the scale of the point source images (b,e) is four times smaller than the spoke target images.
How to Define (Limiting) Resolution
10%
10%
Nyquist Frequency:
D
Sampling Pitch
a
Aperture Width
+
0
–
Dexel = Detector Element
Pixel = Picture Element (2D)
Voxel = Volume Element (3D)
Resolution
resolution is defined by the frequency measured in line pairs per millimeter (lp/mm).
At a given resolution, the ability to see the two squares as separate entities will be dependent on grey scale level.
The bigger the separation in the grey scale between the squares and space between them, the more robust is the ability to resolve the squares.
This grey scale separation is known as contrast (at a specified frequency).
Question: Whats Nu? Answer: ½ of everything.
Aliasing Effect:
Example: for a sampling system that Has a
Nyquist frequency of 5 cycles/mm
Where Fin = spatial frequency that is sampled
1 2 3 4 5
Cycles/mm
1 2 3 4 5
Cycles/mm
1 2 3 4 5
Cycles/mm
1 2 3 4 5
Cycles/mm
1 2 3 4 5
Cycles/mm
1 2 3 4 5
Cycles/mm
Aliased from higher frequencies
Fin= 2
Fin= 3
Fin= 4
Fin= 6
Fin= 7
Fin= 8
Aliased Signal is: FN + (Fin FN)
Signal Aliases to: FN – (Fin FN)
Presampled MTF
D
Sampling Pitch
a
Aperture Width
+
0
–
D
Sampling Pitch
Aperture Width
+
0
–
a
Angled Pitch (Higher Frequency, Better Resolution?)
Square Dexel MTF
FT
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 1 2 3 4 5 6 7 8 9 10 11
D
a
Cycles/mm
MTF
Resolution Limited Blurring Test (Template Use):
Test Object
Test Result
Two Dimensional
Why?
Contrast Contrast Contrast Contrast Contrast Contrast Contrast Contrast
CONTRAST
Background Noise, difference between grey levels, glare, blurring, spatial separation
Contrast Resolution
Detect subtle changes in grey scale & distinguish them from noise
Pertains to Signal to noise ratio
Not focused on small objects
Ability to distinguish small changes in signal of anatomical structures and background noise
Accuracy and Precision
Sensitivity and specificity
Hitting the target
Sensitive
Accurate
Hitting the same place anywhere target or not
Grouping
Specificity
Precision
Noise in Image
Accurate NOT Precise
NEITHER Accurate NOR Precise
Precise NOT Accurate
Precise AND Accurate
Sources of Noise
Film Grain size (under magnification)
Electronic Noise
Can add to the noise
shielded against or corrected through biasing the circuit or similar
Can be random thermal or similar
Use of averaging mitigates this
Structure Noise
Gain differences in array detector
Take one image with radiation source on and no object in the FOV (channel gain correct)
Take second image with no radiation on and no object in FOV (offset image correct)
Contamination of detector
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