filtered backprojection_good
TRANSCRIPT
IMAGE RECONSTRUCTION
Principles behing Tomography
MRIX-ray CT
SPECT PET
Tomographic images
History
• Tomography: τωµϖσ/tomos = slice, to cut a slice with a sharp knife
• graphy=description/make an image of something
• A few years– 1917: Joseph Radon– 1968: Hounsfield – Cormarck Nobel price (79)– 1971: EMI first CT-scanner– 1970th X-ray Computed Tomography (CT)– 1970-80 Single Photon Emission Computed Tomography (SPECT), Position Emission
Tomography (PET) – 1980-90: Magnetic Resonance Imaging (MRI)
The Scintillation Camera• Measure photons
• Determine where on the detector the photon impinge
• Images reflect where the radio-pharmaceutical are located and the amount.
Basics of SPECT Imaging
PET/CT
Basics of PET Imaging
Projection & Sinogram
Sinogramt
θ
Sinogram:All projections
P(θ,t)
f(x,y)
t
θ
y
x
X-rays
Projection:All ray-sums in a direction
π
Computed Tomography
P(θ,t) f(x,y)P(θ,t)
f(x,y)
t
θ
y
x
rays
Computed tomography (CT):Image reconstruction fromprojections
BACKPROJECTION: STEP 1
• 0 0 100 0 0Mathematically align center of
emission profile with point in
slice which will represent the location of the
center
Emission Profile 1
Matrix for slice
BACKPROJECTION: STEP 2
• 0 0 100 0 0
Following a pathparallel to the line
to the COR, add thevalue in emissionprofile divided by
the dimension of the matrix to voxels
along path.
Emission Profile 1
Matrix for slice
0
0
0
0
0
0
0
0
0
0
20
20
20
20
20
0 0
0 0
0 0
0 0
0 0
BACKPROJECTION: STEP 3
•
Repeat steps1 and 2
for anotheracquisition
angle.
Emission Profile 2
Matrix for slice
00
1000
0
0
0
20
0
0
0
0
20
0
0
20
20
20
20
40
0 0
0 0
20 20
0 0
0 0
BACKPROJECTION:STEP 4, Etc.
•Repeat steps
1 and 2for rest ofacquisition
angles.
20
0
20
0
20
0
20
20
20
0
20
20
20
20
80
0 20
20 0
20 20
20 0
0 20
Reconstruction By Backprojection
•
1 Projection 2 Projections 4 Projections
30 Projections 60 Projections 120 Projections
Fourier Transformation
[ ]
[ ] dudvevuFvuFFyxf
dxdyeyxfyxfFvuF
vyuxj
vyuxj
∫ ∫
∫ ∫∞
∞−
∞
∞−
+−
∞
∞−
∞
∞−
+−
==
==
)(21
)(2
),(),(),(
),(),(),(
π
π
FourierTransform
f(x,y) F(u,v)
ImageSpace
FourierSpace
Fourier Slice Theorem
v
u
F(u,v)
P(θ,t)
f(x,y)
t
θ
y
x
X-rays
θ
F[P(θ,t)]
From Projections to Image
y
x
v
u
F-1
[F(u,v)]
f(x,y) P(θ,t) F(u,v)
Filtered Backprojection
f(x,y) f(x,y)
P(θ,t) P’(θ,t)
1) Convolve projections with a filter2) Backproject filtered projections
FILTERED BACKPROJECTION
•-20 -30 100 -30 -20
Note RAMP filtering has put negativevalues in to bebackprojected.
Emission Profile 1
Matrix for slice
-4
-4
-4
-4
-4
-6
-6
-6
-6
-6
20
20
20
20
20
-6 -4
-6 -4
-6 -4
-6 -4
-6 -4
FILTERED BACKPROJECTION 2
•
Repeatfor anotheracquisition
angle.
Emission Profile 2
Matrix for slice
-20-30
100-30
-20
-8
-10
16
-10
-8
-10
-12
14
-12
-10
16
14
14
16
40
-10 -8
-12 -10
14 16
-12 -10
-10 -8
Example: Projection
SinogramIdeal Image
Projection
Projection
Example: Backprojection
Projection
Example: Backprojection
Sinogram Backprojected Image
Example: Filtering
Filtered SinogramSinogram
Example: Filtered Backprojection
Filtered Sinogram Reconstructed Image
Some examples
Filtered Backprojection
• In the Spatial Domain, the blurring of backprojection varies as one over the distance from the actual location:– Blur = 1 / |d|
• In the Frequency Domain, this becomes one over the absolute value of the spatial frequency:– Blur = 1 / |f|
• Thus to correct for the blur we apply the inverse filter which is just the absolute value of the spatial frequency, or RAMP FILTER:
– RAMP = |f|Amp
f
Reconstruction By Backprojection
•
1 Projection 2 Projections 4 Projections
30 Projections 60 Projections 120 Projections
Reconstruction By Filtered Backprojection
•
1 Projection 2 Projections 4 Projections
30 Projections 60 Projections 120 Projections
Reconstruction By Backprojection
•
1 Projection 2 Projections 4 Projections
30 Projections 60 Projections 120 Projections
Reconstruction By Filtered Backprojection
•
1 Projection 2 Projections 4 Projections
30 Projections 60 Projections 120 Projections
Filtered Backprojection (Fourier)
2
0
( , ) ( ) i rf x y P e d dπ
πρθ ρ ρ ρ θ
∞
−∞
= ∫ ∫
Reconstruction filter
•The ramp filter is multiplied by a low-pass filter
•An commonly used filter is the Butterworth filter
Frequency
Amplitude Amplitude
Frequency
Amplitude
* =
Ramp Low-pass filter Resultant filter
Frequency
1.01.01.0
Filtered Backprojection - cont
• Noise reduced but at the expense of spatial resolution
2
0
( , ) ( ) ( ) i rf x y P LP e d dπ
πρθ ρ ρ ρ ρ θ
∞
−∞
= ∫ ∫
Principle of Filtered Backprojection
1. Collect projection data,2. Fouriertransform the projection,3. Multiply with the ramp filter,4. Multiply with low-pass filter if necessary,5. Make inverse of the Fouriertransformed
projection,6. Backproject result onto an image plane,7. Repeat (2-6) for all projection angles.