5.3 Image Reconstruction from Projections
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Agenda
Image reconstruction from projections:
• The reconstruction problem
• Principles of Computed Tomography (CT) • The Radon transform
• The Fourier-slice theorem
• Reconstruction by filtered back-projections
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The Reconstruction Problem
• Consider a single object on a uniform background (suppose that this is a cross section of 3D region of a human body).
• Background represents soft, uniform tissue and the object is also uniform but with higher absorption characteristics.
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The Reconstruction Problem
• A beam of X-rays is emitted and part of it is absorbed by the object.
• The energy of absorption is detected by a set of detectors.
• The collected information is the absorption signal.
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The Reconstruction Problem
• A simple way to recover the object is to back-project the 1D signal across the direction in which the beam came.
• This simply means duplicating the signal across the 1D beam.
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The Reconstruction Problem
• Werotatethe position of the source-detector pair and obtain another 1D signal.
• Werepeatthe procedure and add the signals from the previous back- projections.
• Wecannowtellthat the object of interest is located at the central square.
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The Reconstruction Problem
• After a large number of views have been back- projected in this manner, the original point is reconstructed as a diffuse “blob”.
• We only consider projections from 0 to 180 degrees as projections differing 180 degrees are mirror images of each other.
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The Reconstruction Problem
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Principles of X-ray CT
• ThegoalofCTistoobtaina3D representation of the internal structure of an object by X-raying it from many different directions.
• Back-projecting the image would result in slices through the body.
• A 3D representation is then obtained by stacking the slices.
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First-generation CT scanner
• Single source, single detector
• Veryslow
– e.g., need 785,000 measurements
– 1 ms per measurement – 785 s (13 min!)
• Modern CT scanners collect data in < 1s.
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First-generation CT scanner
https://www.youtube.com/watch?v=fNaCxhhhZTE
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2nd generation CT scanner
• Similar to first generation, but multiple detectors
• Translation still required
• faster scan than 1st generation scanner
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2nd generation CT scanner
www.youtube.com/watch?v=Ni4Hsi3GhXo
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3rd generation CT scanner
• Multiple detectors (500- 1000) cover entire fan beam through patient.
• Both detectors and x-ray source rotate in synchrony
• much faster (<1s per scan)
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3rd generation CT scanner
http://www.youtube.com/watch?v=bdf0kXn5Eeg
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Ring artifact in 3rd generation scanner
• error in detector results in a bad measurement at each projection view.
• artifact is a ring of erroneous μ
• error in 0.1% can result in observable rings
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Ring artifact in 3rd generation scanner
Fig. 1 Sinogram illustrating the effect of one detector with sensitivity reduced to 90%
Fig. 2 Sinogram illustrating the effect of one detector with sensitivity reduced to 95%
Reconstruction
of the data in Fig. 1 (10% detector error)
Reconstruction
of the data in Fig. 2 (5% detector error)
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4th generation CT scanner
• Onlyx-raysource rotates, detectors remain fixed.
• Avoidsringartifact
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4th generation CT scanner
http://www.youtube.com/watch?v=AWVz3yke_bY
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3D image reconstructed from slices .....
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Spiral (Helical) CT
• Componentsrotatecontinuouslytobuildupa3D image.
• Could be 3rd or 4th generation designs
• Rotationcanbeasfastastwopersecond.
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Modern CT scanner instrumentation
Main components:
• Gantry: Contains x- ray source, detector arrays and electronic controlling the collection of data.
• Sliding Table: Slides into the gantry.
• Computer
– Controls the sliding table, which moves to the table so
that the desired body part is scanned – Reconstruct the image
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Radon Transform
Think of every point , on the line, projected onto the unit vector (, ) have a length of . That is:
, ⋅ , =
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Radon Transform
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Radon Transform
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Radon Transform
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Radon Transform and Sinogram
• The representation of the Radon transform (, ) as an image with and as coordinates is called a sinogram.
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Why is it called “sinogram”?
=cos =cos
www.youtube.com/watch?v=XA7GXhPbRT0
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Backprojection
Mathematical Description of Backprojection:
For a fixed rotation angle , and a fixed distance
, backprojecting the value of the projection (,) is equivalent to copying the value (,) to the image pixels belonging to the line + = .
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Backprojection
• Projecting the entire profile for a fixed angle (i.e., (, )) yield:
, =, =(cos+sin,)
= cos + sin
(, )
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Backprojection
• This equation holds for every angle : , =(cos+sin,)
• The final image is formed by integrating over all the backprojected images:
, = ,
Backprojection results in blurred images.
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The Fourier-Slice Theorem
• Let the 1D FT of a projection with respect to (at a
given angle) be: , = ,
• Substituting the projection , :
,
= , cos+sin−
= , cos+sin−
= ,
(by sifting property)
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The Fourier-Slice Theorem
, = ,
Let = cos and v = sin
, = , ,
which is the 2D FT of the image (, ) evaluated at the indicated frequencies , :
, = (cos,sin)
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The Fourier-Slice Theorem
• The resulting equation , =
( cos , sin ) is known as the Fourier- slice theorem.
• It states that the 1D FT of a projection (at a given angle θ) is a slice of the 2D FT of the image.
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The Fourier-Slice Theorem
• We could obtain (, ) by evaluating the FT of every projection and inverting them.
• However, this procedure needs irregular interpolation which introduces inaccuracies.
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Filtered Backprojection
• The 2D inverse Fourier transform of (, ) is , = ,
• Express the integral in polar coordinate with
= cos, = sin
• The differential is equal to in
polar coordinate
• The integral in polar coordinate becomes
, = cos,sin
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Filtered Backprojection
• Using the Fourier-slice theorem,
, = ,
• With some manipulation:
, = ,
• Thetermcos +sin=andis independent of :
, = ,
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Filtered Backprojection
(,)= ,
• For a given angle , the inner expression is the 1-D Fourier transform of the projection multiplied by a ramp filter ||.
• This is equivalent in filtering the projection with a high-pass filter with Fourier Transform || before backprojection.
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Filtered Backprojection
(,)= ,
Problem: the filter () = || is not integrable in the inverse Fourier transform as it extends to infinity in both directions.
• It should be truncated in the frequency domain.
• The simplest approach is to multiply it by a box filter in the frequency domain, but ringing will be noticeable.
• Windows with smoother transitions are used.
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Filtered Backprojection
• Ringing decreases with these windows.
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Filtered Backprojection
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Filtered Backprojection
Filtered backprojection is obtained by the following steps:
1. Compute the 1-D Fourier transform of each projection (with padding to mitigate the effect of wraparound error)
2. Multiply each Fourier transform by the filter function || (multiplied by a suitable window, e.g., Hamming).
3. Obtain the inverse 1-D Fourier transform of each resulting filtered transform.
4. Backproject and integrate all the 1-D inverse transforms from step 3.
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Filtered Backprojection
Backprojection Box-windowed FBP Hamming FBP
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Filtered Backprojection
• Ringing is more
Box-Windowed FBP
Hamming FBP
pronounced in box-windowed FBP
• Ringing is reduced in Hamming FBP, at the expense of slight blurring
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Filtered Backprojection
Box-Windowed FBP Hamming FBP
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