filtered backprojection - pusan national...
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Filtered Backprojection
Ho Kyung [email protected]
Pusan National University
Introduction to Medical Engineering
Outline
• Sinogram
• Radon transform
• Backprojection
• Central slice theorem
• Filtered backprojection
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How can we determine ?
c + d = 1, a + b = 0,a + c = 1, b + d = 0 .........
C = 1 !A = B = D = 0 !
Light
Box / holes
Eyes
Brain3Taken from WA Kalender's Text Material (2000)
Image reconstruction
• Inverse problem
– Usually inconsistent, ill‐posed problem• Noise corruption• Huge size of 𝐴• 𝐴 does not exist• Solution is not unique• Solution is unstable
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[System characteristic]{Image} = {Measurement}
Solving methods
• Iterative solvers (discretization)‒ Series expansion methods (e.g., ART)‒ Statistical methods‒ Optimization (regularization)
• [MLEM, LS, ART] + [priori/penalty]
• Analytic (kernel) solver‒ 𝜇 𝐱 d𝐱 𝑘 𝐱 𝑝 𝐱 𝐱
Sinogram
• Collection of projection data as a function of 𝜃
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0 180
Detector num
ber
𝜃 𝜃
𝜃
𝑢
𝑢
𝑠
𝐼
𝐼 𝑢; 𝜃 𝐼 𝑒 𝐱
𝑝 𝑢; 𝜃 ln𝐼𝐼 𝜇 𝐱 d𝑠
Projection= ray sum= line integral= Radon transform¶
= x‐ray transform
¶ Radon transform is an integral over a plane in 3D while the x-ray transform is described as a line integral for any dimension
𝑢 𝑢
𝑢 𝑢
𝜃 𝜃
Projection and Radon transform
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• Consider the 2D parallel‐beam geometry.– 𝜇 𝑥, 𝑦 = the distribution of the linear attenuation coefficient in the 𝑥𝑦‐plane with a diameter
FOV– 𝐼 = the unattenuated intensity of the x‐ray beams– 𝑟, 𝑠 = a new coordinate system defined by rotating 𝑥, 𝑦 over the angle 𝜃
• Transformations
– 𝑟𝑠
cos 𝜃 sin 𝜃sin 𝜃 cos 𝜃
𝑥𝑦
–𝑥𝑦
cos 𝜃 sin 𝜃sin 𝜃 cos 𝜃
𝑟𝑠
– with the Jacobian
• 𝐽 cos 𝜃 sin 𝜃sin 𝜃 cos 𝜃 1
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• For a fixed angle 𝜃, the measured intensity profile as a function of 𝑟 is given by:
– 𝐼 𝑟 𝐼 𝑒,
, 𝐼 𝑒,
,
• 𝐿 , = the line that makes an angle 𝜃 with the 𝑦‐axis at distance 𝑟 from the origin
• Considering the polyenergetic x‐ray spectrum:
– 𝐼 𝑟 𝐼 𝐸 𝑒, ,
, d𝐸
– Practically, it is assumed that x rays are monochromatic
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• Transforming each intensity profile into an attenuation profile (log transform):
– 𝑝 𝑟 ln 𝜇 𝑟 cos 𝜃 𝑠 sin 𝜃 , 𝑟 sin 𝜃 𝑠 cos 𝜃 d𝑠,
• 𝑝 𝑟 = the projection of the function 𝜇 𝑥, 𝑦 along the angle 𝜃• 𝑝 𝑟 = zero for |r| FOV/2• Sufficient to measure 𝑝 𝑟 for 𝜃 ranging from 0 to 𝜋 as far as parallel‐
beam geometry because of concurrent beams
• Sinogram: 2D dataset 𝑝 𝑟, 𝜃 by stacking all these projections 𝑝 𝑟
• Radon transform: the transformation of any function 𝑓 𝑥, 𝑦 into its sinogram 𝑝 𝑟, 𝜃– 𝑝 𝑟, 𝜃 ℛ 𝑓 𝑥, 𝑦 𝑓 𝑟 cos 𝜃 𝑠 sin 𝜃 , 𝑟 sin 𝜃
𝑠 cos 𝜃 d𝑠– Periodic in 𝜃 with period 2𝜋: 𝑝 𝑟, 𝜃 2𝜋– Symmetric in 𝜃 with period 𝜋 : 𝑝 𝑟, 𝜃 𝜋
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• Projection and back‐projection of a single dot:
Sampling
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• In practice, available are a limited number 𝑀 of projections or views and a limited number 𝑁 of detector samples– Discrete sinogram 𝑝 𝑛∆𝑟, 𝑚∆𝜃
• A matrix with 𝑀 rows and 𝑁 columns• ∆𝑟 = detector sampling distance• ∆𝜃= the rotation interval b/w subsequent views
• To limit aliasing: ∆ ∆
or ∆𝑟 ∆ (two samples/beam width ∆𝑠)
• What is the min number of views?– maxdist 𝐿 , , 𝐿 , maxdist 𝐿 , , 𝐿 ,
– maxdist 𝐿 , , 𝐿 ,·
°
– maxdist 𝐿 , , 𝐿 ,
– number of views per 360° 𝜋 · number of detector samp– e.g., if FOV = 50 cm & ∆𝑠 = 1 mm 𝑁 = 1000 & 𝑀 = ~3000
𝑝 𝑟
∆𝑠
sampledprojection
smoothedprojection
samplingat ∆𝑟
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• Typically the number of views per 360 is on the order of the number of detector channels– GE scanners with 888 detector channels acquire 984 views per rotation– Siemens scanners with 768 detector cells use 1056 views per 360
• Means to improve sampling include quarter detector offset and in‐plane focal spot wobbleor deflection
Backprojection
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• How to reconstruct the distribution 𝜇 𝑥, 𝑦 (or, 𝑓 𝑥, 𝑦 ) for the given sinogram 𝑝 𝑟, 𝜃 ?– Assign the value 𝑝 𝑟, 𝜃 to all points 𝑥, 𝑦 along a line 𝑟, 𝜃 and repeat
this for 𝜃 ranging from 0 to 𝜋 backprojection:– 𝑏 𝑥, 𝑦 ℬ 𝑝 𝑟, 𝜃 𝑝 𝑥 cos 𝜃 𝑦 sin 𝜃 , θ d𝜃
– Note that the resultant image is blurred a simple backprojection is unsatisfactory
– Discrete version of BP:– 𝑏 𝑥 , 𝑦 ℬ 𝑝 𝑟 , 𝜃 ∑ 𝑝 𝑥 cos 𝜃 𝑦 sin 𝜃 , 𝜃 ∆𝜃
– Note that the discrete positions rn generally do not coincide with the discrete values 𝑥 cos 𝜃 𝑦 sin 𝜃
– Interpolation is required; the corresponding projection value is calculated by interpolation b/w its neighboring measured values (called “pixel‐driven” or “voxel‐driven” backprojection).
Projection theorem
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• More mathematically instead of the previous intuitive answer we need a mathematical expression for the inverse Radon transform: 𝑓 𝑥, 𝑦 ℛ 𝑝 𝑟, 𝜃
• Projection theorem, called central (or Fourier) slice theorem:
– 𝐹 𝑘 , 𝑘 𝑓 𝑥, 𝑦 𝑒 d𝑥d𝑦 or 𝑃 𝑘 𝑝 𝑟 𝑒 · d𝑟
– If 𝜃 is variable, 𝑃 𝑘 𝑃 𝑘, 𝜃
– The projection theorem states that 𝑃 𝑘 𝐹 𝑘 , 𝑘 with
𝑘 𝑘 cos 𝜃𝑘 𝑘 sin 𝜃
𝑘 𝑘 𝑘
– “The 1D FT w.r.t. variable 𝑟 of the Radon transform of a 2D function is the 2D FT of that function”
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𝜃 135°
𝜃 135°
𝜃 135°
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• Proof:
𝐹 𝑘 , 𝑘 𝑓 𝑥, 𝑦 𝑒 d𝑥d𝑦
𝑓 𝑥, 𝑦 𝑒 · · · · d𝑥d𝑦
𝑓 𝑟 · cos 𝜃 𝑠 · sin 𝜃 , 𝑠 · cos 𝜃 𝑟 · sin 𝜃 𝑒 · · · · · · d𝑠d𝑟
𝑓 𝑟 · cos 𝜃 𝑠 · sin 𝜃 , 𝑠 · cos 𝜃 𝑟 · sin 𝜃 𝑒 · d𝑠d𝑟
𝑓 𝑟 · cos 𝜃 𝑠 · sin 𝜃 , 𝑠 · cos 𝜃 𝑟 · sin 𝜃 d𝑠 𝑒 · d𝑟
𝑝 𝑟, 𝜃 𝑒 · d𝑟
𝑃 𝑘, 𝜃
Direct Fourier reconstruction
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1. Calculate the 1D FT of all the projections 𝑝 𝑟, 𝜃 : ℱ 𝑝 𝑟 𝑃 𝑘2. Put all the values of the 1D function P(k) on a polar grid to obtain the 2D function 𝑃 𝑘, 𝜃 ,
which is equal to 𝐹 𝑘 , 𝑘3. Calculate the 2D IFT of 𝐹 𝑘 , 𝑘 : ℱ 𝐹 𝑘 , 𝑘 𝑓 𝑥, 𝑦
• This method requires the interpolation in step 2, and which causes artifacts, making this method less popular than the filtered backprojection method.
Filtered backprojection (FBP)
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• To avoid interpolation:
• FBP scheme:① Filter the sinogram 𝑝 𝑟, 𝜃 : ∀𝜃: 𝑝∗ 𝑟 𝑝 𝑟 ∗ 𝑞 𝑟 or 𝑃∗ 𝑘 𝑃 𝑘 · 𝑘② Backproject the filtered sinogram 𝑝∗ 𝑟, 𝜃 : ∀𝜃: 𝑓 𝑥, 𝑦 𝑝∗ 𝑥 cos 𝜃 𝑦 sin 𝜃 , 𝜃 d𝜃
ramp filter
convolution kernel
𝑓 𝑥, 𝑦 𝑃 𝑘, 𝜃 𝑘 𝑒 d𝑘d𝜃 with 𝑟 𝑥 cos 𝜃 𝑦 sin 𝜃
𝑃∗ 𝑘, 𝜃 𝑒 d𝑘d𝜃
𝑝∗ 𝑟, 𝜃 d𝜃
𝑝∗ 𝑟, 𝜃 𝑝 𝑟 , 𝜃 𝑞 𝑟 𝑟 d𝑟′
𝑞 𝑟 ℱ 𝑘 𝑘 𝑒 d𝑘
with 𝑟 𝑥 cos 𝜃 𝑦 sin 𝜃
with 𝑟 𝑥 cos 𝜃 𝑦 sin 𝜃
𝑃∗ 𝑘, 𝜃 𝑃 𝑘, 𝜃 𝑘
𝑝∗ 𝑟, 𝜃 𝑃∗ 𝑘, 𝜃 𝑒 d𝑘
Filter functions
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• In practice, the continuous filter 𝑘 is not useful because of its divergent nature– The useful Fourier content is limited to frequencies smaller than 𝑘 1 ∆𝑠⁄ 1 2∆𝑟⁄– Ram‐Lak filter (Ramachandran & Lakshiminarayanan): difference of a block & a triangle
– In space domain: 𝑞 𝑟
Usually, frequencies slightly below 𝑘 are unreliable because of aliasing & noise• Application of a smoothing window (Hanning (𝛼 = 0.5), Hamming (𝛼 = 0.54), Shepp‐Logan,
Butterworth) suppresses the highest spatial frequencies & reduces these artifacts
• 𝐻 𝑘𝛼 1 𝛼 cos for 𝑘 𝑘
0 for 𝑘 𝑘
Hamming window
Hanning window
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Fan‐beam FBP
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• Changed coordinates from 𝑟, 𝜃 to 𝛾, 𝛽 :– 𝛾 = the angle b/w a particular ray & the center line of the corresponding fan– 𝛽 = the angle b/w the source and the 𝑦‐axis– Fan‐angle = the angle formed by the fan– Required a range from 0 to (𝜋+ fan‐angle) to include all line measurements
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• Rebinning: reordering the data into parallel data with interpolation
• Adaptive FBP algorithm:
– Recall the FBP method: 𝑓 𝑥, 𝑦 𝑝 𝑟 , 𝜃 𝑞 𝑥 cos 𝜃 𝑦 sin 𝜃 𝑟′ d𝑟′d𝜃//
• 1/2 for compensating the modification of the integration limits from 0 to 2• Coordinate transforms: 𝜃 𝛾 𝛽 and 𝑟 𝑅 sin 𝛾
– 𝑓 𝑥, 𝑦 𝑝 𝑟 , 𝛽 𝑞 𝑥 cos 𝛾 𝛽 𝑦 sin 𝛾 𝛽 𝑅 sin 𝛾 ′ 𝑅 sin 𝛾 ′d𝛾′d𝛽
– After a few calculations
• 𝑓 𝑥, 𝑦 𝑅 cos 𝛾′ · 𝑝 𝑟 , 𝛽 𝑞 𝛾 𝛾′ d𝛾′d𝛽
– Modified FBP weighted with 1/𝐿– Inner integral is a convolution of 𝑝 𝑟, 𝛽 , weighted with 𝑅 cos 𝛾, with a modified filter kernel
𝑞 𝛾
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• For the detectors lying on a straight line perpendicular to the center line of the fan;– Changed coordinates from 𝑟, 𝜃 to 𝑡, 𝛽 :
• 𝑡 = the distance from the origin to the ray through 𝑥, 𝑦 measured parallel to the detector array
– The weighted FBP:
• 𝑓 𝑥, 𝑦/
· 𝑝 𝑡 , 𝛽 · 𝑞 𝑡 𝑡′ d𝑡′d𝛽
– 𝑈 = the projection of the source‐to‐point distance onto the central ray of the fan
• The strategy to remove or reduce the intrinsic blur artifacts is the backprojection after taking the proper filter function on the projection data
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2
0)()(
21),( dhpyxf
Taken from WA Kalender's Text Material (2000)
Wrap‐up
• Sinogram
• Radon transform
• Backprojection
• Central slice theorem
• Filtered backprojection
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