Image Reconstruction from Projections
Antti Tuomas Jalava
Jaime Garrido Ceca
Overview
Reconstruction methods Fourier slice theorem & Fourier
method Backprojection Filtered backprojection Algebraic reconstruction
Diffractive tomography Display of CT images Tissue characterization with CT
Projection Geometry
Problem: Reconstructing 2D Image. Given parallel-ray projections.
1D projection (Radon transform). Density distribution Ray AB
Integral evaluated for different values of the ray offset t1. 1D projection or Radon transform.
dxdytyxyxfdsyxftpAB
11 sincos,,
1sincos tyx yxf ,
tp
The Fourier Slice Theorem
1D Fourier Transform of 1D projection of 2D image
is equal to the radial section (slice or profile) of the 2D Fourier Transform of the 2D image at the angle of the projection.
dxdyvyuxjyxfvuF 2exp,,
dtwtjtpwP 2exp
,, wFvuFwP
sin
cos
wv
wu
The Fourier Slice Theorem
How to obtain f(x,y) applying Fourier Slice Theorem: Assumption: we have projections available at all angles from 0º to 180º.
1. From projections, we take their 1D Fourier transform.
2. Fill the 2D Fourier Space with the corresponding radial sections.
3. Take an inverse 2D Fourier transform to obtain
Problem: finite number of projections available
Solution: Interpolation is needed in 2D Fourier space.
yxf ,
Backprojection
Simplest reconstruction procedure Assumptions:
Rays: Ideal straight lines. Image: dimensionless points.
Procedure Estimate of the density at a point by simply summing (integrating )
all the rays that pass through it at various angles.
sincos
,0
yxt
where
dtpyxf
Problem:
•Finite number of rays per projection
•Finite number of projections
Interpolation is required.
Backprojection
BP produces a spoke-line pattern blurring details. Finite number of projections produces streaking artifacts.
Reconstructed image modeled by convolution between PSF (impulse response) and the original image.
Solution: Applying deconvolution filters to the reconstructed image.
Filtered BP technique.
Point density function
Filtered Backprojection
After some manipulations, we get:
where
In practice, smoothing window should be applied to reduce the amplification of high-frequency noise.
0
, dtqyxf
dwwtjwwPtq 2exp
Filter is represented by this function:
Ramp filter
Discrete Filtered Backprojection
Projection in frequency domain is manipulated:
2/
2/
2exp
22
12 N
Nm
mkN
jW
mp
WN
WkP
Frequency axis discretized
Finite number of samples
Samples at the sampling rate 2W
tp
Discrete Filtered Backprojection
The filtered projection may then be obtained as:
2/
2/
22exp
2222exp
N
N
tN
Wkj
N
Wk
N
WkP
N
WdwwtjwwPtq
2/
2/
2exp
222
2
N
N
mkN
jN
Wk
N
WkP
N
W
W
mq
• Problem: control noise enhancement
• Solution we apply hamming window:
2/
2/
2exp
2222
2
N
N
mkN
jN
WkG
N
Wk
N
WkP
N
W
W
mq
WN
Wk
N
WkG
2cos04654,0
2
tq
Discrete Filtered Backprojection
Finally, we get this expression :
Algorithmic:1. Measure projection.
2. Compute filtered projection.
3. Backproject the filtered.
4. Repeat 1-3 all projection angles
L
lll yxq
Lyxf
l1
sincos,~
Original
Back
projection
1°
Filtered
back
projection
1°
Filtered
back
projection
10°
Algebraic Reconstruction Techniques
Projections seen as set of simultaneous equations. Kaczmarz method
Iterative method. Implemented easily.
Assumptions: Discrete pixels. Image density is constant within each cell.
Equations
MMMNMM
NN
NN
pfwfwfw
pfwfwfw
pfwfwfw
2211
22222121
11212111
Contribution factor of the nth image element to the mth ray sum.
Algebraic Reconstruction Techniques
Karzmarz method take the approach of successively and iteratively projecting an initial guess and its successors from one hyperplane to the next.
In general, the mth estimate is obtained from the (m-1)th estimate as:
Because the image is updated by altering the pixels along each individual ray sum, the index of the updated estimate or of the iteration is equal to the index of the latest ray sum used.
mmm
mmm
mm www
pwfff
11
Algebraic Reconstruction Techniques
Characteristics worth:
ART proceed ray by ray and it is iterative Small angles between hyperplanes
Large number or iterations It should be reduced by using optimized ray-access schemes.
M>N noisy measurements oscillate in the neighborhood of the intersections of the hyperplanes.
M<N under-determined. Any a priori information about image is easily
introduced into the iterative procedure.
Approximations to the Kaczmarz method
We could rewrite reconstruction step at the nth pixel level as:
Corrections could also be multiplicative:
m
mmmn
mn N
qpff 1,0max
Number of pixels crossed by the mth ray.
True ray sum
Computed ray sum
m
mmn
mn q
pff 1
Approximations to the Kaczmarz method
Generic ART procedure:1. Prepare an initial estimate
2. Compute ray sum
3. Obtain difference between true ray sum and the computed ray sum and apply the correction.
4. Perform Steps 2 and 3 for all rays available.
5. Repeat Steps 2-4 as many times as required.
0f
mq
Original
2.
1.
178 angles
dt =
1 voxel width
3.
Original again
5.
6.
4.
Imaging with Diffraction Sources
Non ionizing radiationUltrasonicElectromagnetic (optical or thermal)
Refraction and diffraction
Fourier diffraction theorem
Imaging with Diffraction Sources
When an object, f(x,y), is illuminated with a plane wave the Fourier transform of the forward scattered fields measured on line TT’ gives the values of the 2-D transform, F(w1,w2), of the object along a circular arc in the frequency domain, as shown in the right half of the figure.
= measured attenuation coefficient. = attenuation coefficient of water When K = 1000 units are called Hounsfield Units
Air: -1000 HU Water: 0 HU Bone 1000 HU
Study 86 healthy infants aged 0-5 years White matter: 15 HU to 22 HU Gray matter: 23 HU to 30 HU Difference between grey and white matter exactly 8 HU (In all measurements) Boris P, Bundgaard F, Olsen A. Childs Nerv Syst. 1987;3(3):175-7
Display of CT Images
1
µ
µKHU
µµ
Microtomography
µ-scale CT Volume: few
Nanotomography already introduced.
Biomedical use: Both dead and alive (in-vivo) rat and mouse scanning. Human skin samples, small tumors, mice bone for
osteoporosis research.
3cm
Estimation of Tissue Components with CT
Manual segmentation of tumor by radiologist
Parametric model for the tissue composition Gaussian mixture model
Method to estimate the parameters of the model EM algorithm
Gaussian Mixture Model (i)
Fit M gaussian kernels to intensity histogram
Gaussian Mixture Model (ii)
Intensity value for voxel is a Gaussian random variable. Parameters for ith tissue: Probability that voxel belonging to that tissue gets value x
M number of different tissues in tumor = the fraction of belonging to ith tissue (probability).
Tumor as whole: PDF is a mixture of M Gaussians
2
2
2exp
2
1|
i
i
i
ii
µxxp
iii µ ,
i
M
i i11
MMi ,...,,,..., 1
M
i iii xpxp1
||
Gaussian Mixture Model (iii)
Tumor as whole: PDF is a mixture of M Gaussians
Probability of parameter set
If nothing is known about
Find that maximizes likelihood
MMi ,...,,,..., 1
M
i iii xpxp1
||
xpxpp
xp
||
|| xpcxp
N
jjxpxpxL
1
||ˆ|
Gaussian Mixture Model (iv)
Probability that jth voxel with value belongs to the ith tissue type
EM algorithm (iterative, chapter 8) ->
,
|
,
,||,|
j
iji
j
jj xp
xp
xp
ixpipxip
jx
newi
newi
newi ,,
Ending Remarks
Some image manipulation tasks can be performed in 1D in radon domain (edge detection etc.).
Reconstruction heavily dependent on reconstruction algorithm (method).
MRI images are usually reconstructed with Fourier method (according to book).
CT allows fast 3D imaging So does MRI. MRI has better sensitivity especially with
soft tissues.