Mathematical Preliminaries for 2-D Image Reconstructions · • Sinogram • Reconstruction with...
Transcript of Mathematical Preliminaries for 2-D Image Reconstructions · • Sinogram • Reconstruction with...
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Chapter 2: Mathematical PreliminariesMathematical Preliminaries for 2-D Image
Reconstructions
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Contents
• Review of Fourier transform and filtering operations
• Projection data and imaging systems
• Radon transform
• Central slice theorem
• Sinogram
• Reconstruction with simple backprojection
• Filtered-backprojection
• Other analytical reconstruction methods.
• Matlab examples
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The Inverse Problem
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The inverse problem:
?
General Inverse Problem and Image Reconstruction
• Given a set of measured output signal
• Given a known system response function
• Given the statistical description of the data
what should be the input signal that gave rise to the output data?
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Convolution Theorem Revisited
),(),(),( yxhyxfyxg
),(),(),(),( yxgyxfyxgyxf FFF -1
The convolution theorem enables one to perform convolution operationas multiplication process in spatial frequency domain.
By using the Fast Fourier Transform (FFT) algorithms, the convolutionoperation can be performed very efficiently !! This provide a practical wayfor modeling linear shift-invariant systems …
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PET image reconstruction 2D Reconstruction
Each parallel slice is reconstructed independently (a 2D sinogramoriginates a 2D slice)
Slices are stacked to form a 3D volume f(x,y,z)
2D reconstruction
Plane 5
Slice 5etc
etc
2D reconstruction
Plane 4
Slice 4
2D reconstruction
Plane 3
Slice 3
2D reconstruction
Plane 2
Slice 2
2D reconstruction
Plane 1
Slice 1
2D Reconstruction
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Matlab Examples
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Sampling of a Signal
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Review of Key ConceptsTwo Dimensional Sampling
-n -m
-n -m
)-,-δ(),(
)-,-δ()(
),,,(),(),(
ymyxnxymxnf
ymyxnxx,yf
yxyxyxfyxf ss
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Fourier Transform of Sampled Image
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ymv
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),(F1
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),()(F
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The result: Replicated F(u,v), or “islands” every 1/x in u, and 1/y in v.
NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019
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An sufficiently small sampling interval is important …
Review of Key Concepts (5)Two Dimensional Sampling
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The Nyquist Theorem
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Two Dimensional Sampling
Nyquist Theorem:In order to restore the original function, the sampling rate must be greater than twice the highest frequency component of the function.
Nyquist Sampling Interval:The maximum sampling interval allowed without introduce aliasing is
cux
21
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Example – Fourier Transform of a Continuous Function
f(x,y)
Aliasing due to insufficient sampling
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The Discrete Fourier Transform
and
the Equivalency of Spatial Domain and Frequency
Domain Representations of a Signal
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Review of Key ConceptsDiscrete Fourier Transform
Nyquist sampling theorem indicates that all information containedin a continuous but band-width limited signal can be carried by justN samples.
Fourier transform is a no-loss transform.
So it make sense that we would only need a finite number (N) ofFourier coefficients to carry the same amount of information …
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Review of Key ConceptsDiscrete Fourier Transform
0uu- esfrenquenci negative the toingcorrespond are 1-N..., N/2,nuu0 esfrenquenci positive the toingcorrespond are 1-N/21,...,n
zero) is frenquency (spatialcomponent DC the toingcorrespond 0n
1-1,2,...N,0n ,
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kn efF
NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Fourier Transform
The discrete Fourier transform (DFT) is defined as
And the inverse DFT is defined as
1-1,2,...N,0k ,1 1
0
2
N
n
Nnkj
nk eFN
f
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Typical Filtering Operation in Fourier Domain
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Review of Fourier Transform and Filtering Spectral Filtering
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Review of Fourier Transform and Filtering Spectral Filtering
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Projections
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Projection Data from Early X-ray CT Systems
• Uses a collimator to keep exposure to a slice• Builds image from multiple projections• We will assume parallel rays for now.
• Actually how first generation scanners worked.• Translate source and single detector across body for one angle• Then rotate source and detector to get next angle
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Projection Data from Early X-ray CT Systems
From Computed Tomography, Kalender, 2000.
b
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dttIIeII
b
a )()/ln( 0
)(
0
:object the through passing
after beam the of intensity The
Data measured bytranslating the detector is atypical projection data.
NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019
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A Typical Gamma Camera
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Ring of PhotonDetectors
• Radionuclide decays,emitting +.
• + annihilates with e– fromtissue, forming back-to-back511 keV photon pair.
• 511 keV photon pairs detectedvia time coincidence.
• Positron lies on line definedby detector pair (known as achord or a line of response ora LOR).
Detect Pair of Back-to Back 511 keV Photons
Detect Radioactive Decays
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Example -- MRI of the Brain
T2 ContrastTE = 102 ms, TR = 6000 ms
• Long TR minimizing T1 effect• GM, WM decayed, CSF partially
decayed large contrast betweenGM/WM and CSF
• Small contrast between GM and WM(WM is darker due to shorter T2)
Proton DensityTE = 17 ms, TR = 6000 ms
T1 ContrastTE = 17 ms, TR = 600 ms
• Short TR building up T1 effect• TR falls between T1’s for GM, WM
but much shorter than that for CSFGM/WM should built up more(brighter in the image) that CSF.
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How Do We Describe Projections in Math?
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The Line Integral Projection
• Measures the integral propertya physical object along astraight line.
• Projection data is natruallyacquired by successive samplingthe object at given angularintervals.
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NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Signals and Systems
Line Impulse Signal (1)
otherwiselyx
xwhere
lyxyxL
,0sincos,0
)(
)sincos(),(
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NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Signals and Systems
Line Impulse Signal (1)
)sincos(),( lyxyxL
• It can be used to measure the spatial resolution of a givenimaging system.
• It is used to calculate the line-integral projection data for agiven 2-D object.
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NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Signals and Systems
Line Impulse Signal (2)
×
x’
2-D integral
The value of the projection function p(x’)at this point is the integral of the functionof f(x,y) along the straight line:x’=xcos+ysin
dxdyxyxyxfxp )sincos(),(),(
The integral of a line impulse function and a given 2-D signal gives theprojection data from a given view …
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Radon Transform (1)
ydyxyxf
dxdyxyxyxf
yxfxp
)cossin,sincos(
)sincos(),(
)],([),(
R
The Radon transform of a 2-D function is defined as
where)(xp
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NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Signals and Systems
Radon Transform and Sinogram• Radon transform maps a 2-D function f(x,y) into a sinogram.
),(),( yxfx RP
Any point represented by polarcoordinates (r,) will simplyfollow the equation x’=rcos(-)in the sinogrsm space.
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NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Signals and Systems
Radon Transform and Sinogram
degree0
degree180
Sinogram is the basic dataformat for reconstruction
x
Sinogram is a 2-D functionrepresenting the originalfunction f(x,y) into theprojection data space.
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NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Signals and Systems
Radon Transform and Sinogram
http://tech.snmjournals.org/cgi/content-nw/full/29/1/4/F3
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NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Signals and Systems
Radon Transform and Sinogram
http://tech.snmjournals.org/cgi/content-nw/full/29/1/4/F3
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Radon Transform and Sinogram
• Is Radon transform a no-loss transform?
• Can we restore the original function f(x,y) from the sinogram?
• Under what condition that the f(x,y) can be reconstructed exactly?
• How to perform the reconstruction?
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What Exactly Does Projection Data Tell Us?
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Central Slice Theorem
NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019
http://engineering.dartmouth.edu/courses/engs167/12%20Image%20reconstruction.pdf
F{p ( , x’)} = F(r,)
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Central Slice Theorem
NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019
http://engineering.dartmouth.edu/courses/engs167/12%20Image%20reconstruction.pdf
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Central Slice Theorem
NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019
http://engineering.dartmouth.edu/courses/engs167/12%20Image%20reconstruction.pdf
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Let’s consider the 2D FFT of an arbitrarily given function
F(u,v) = ∫ ∫ f(x,y) e -i 2π (ux + vy) dx dy
In polar coordinates within the spatial-frequency domain, letu = cos and v = sin ,
Then
F(,) = ∫ ∫ f(x,y) e -i 2π (x cos + y sin ) dx dy
If we treat (x cos + y sin ) as a constant, then exp [ -i 2π (x cos + y sin )] could be written as a linear phase shift, which is the Fourier transform of a shifted delta function. Let’s write the complex exponential as the FT of a function, F(,) = ∫y ∫ xf(x,y) F{[x’ – (x cos + y sin I]} dx dy ,
and
F(,) = ∫y ∫x f(x,y) {∫ x’ [x’ – (x cos + y sin I] e-i 2πx’ dx’ } dx dy ,
Central Slice Theorem
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Examples
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Square signal 2-D sinc function
Delta function 2-D DC plane
2-D line impulse 2-D line impulse
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Properties of Fourier Transform
NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019 Fourier Transform
Spatial Domain
Spatial Frequency Domain
Linear shifting in spatial domain simply adds some linear phase to
the pulse Magnitude is unchanged
ajexfaxf 2)]([)]([ FF
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F(,) = ∫ x ∫ y f(x,y) {∫ x’ [( x cos + y sin - x’)] e-i 2πρx’ dx’}dx dy .
At this point, we can change the order of the integration operators as
F{p ( , x’)}= ∫ x’ [ ∫y ∫x f(x,y) ( x cos + y sin - x’) dx dy] e-i 2πρx’ dx’ .
Recall how we wrote the projection as a double integral of f(x,y)where a delta function performs the line integral,
p ( , x’) = ∫ x ∫ y [((x,y) (x cos + y sin - x’)]dx dy
We take the Fourier Transform of p ( , x’) :
F{p ( , x’)}= ∫ x’ [ ∫y ∫x f(x,y) ( x cos + y sin - x’) dx dy] e-i 2πρx’ dx’
which is exactly what we wrote for F(ρ, ) above! Therefore, we have
F{p (x’)} = F(ρ, )
Central Slice Theorem
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Central Slice Theorem
NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019
http://engineering.dartmouth.edu/courses/engs167/12%20Image%20reconstruction.pdf
F{p ( , x’)} = F(r,)
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NPRE 435, Principles of Imaging with Ionizing Radiation, Fall 2019
Projection Theorem or Central Slice TheoremAn alternative proof:
See Page 77, Foundations of Medical Imaging, Z. H. Cho.