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ECE 6258 Russell M. Mersereau
ECE6258 Lecture 41
Reconstructing MultidimensionalSignals from Projections
ECE 6258 Russell M. Mersereau
Motivation
A
projection
is like an “x-ray photograph” of a
multidimensional function.
The reconstruction of a signal from its projections isequivalent to estimating a 3-D structure from aseries of x-ray photographs take from differentdirections.
It is the only signal processing problem to beawarded a Nobel Prize (Medicine, 1971)
ECE 6258 Russell M. Mersereau
A Computed Tomography System
u
object
u
) ˆ
( 1
u
p
) ˆ
( 2
u
p
u
) ˆ
( 3
u
p
ECE 6258 Russell M. Mersereau
Formal Definition
Formally, a projection is a mapping from
R
N
to
R
M
for
M
N
In the special case
v d v u v u x u p
v
u
x
cos
sin
sin
cos
v d v u x u p
0
ECE 6258 Russell M. Mersereau
Applications
X-ray photos (computed tomography)
Positron emission tomography (PET)
Electron microscopy
Marginal probability distributions
Fan-beam radio telescopes
Cross-Borehole measurements in seismicsignal processing
NMR imaging
ECE 6258 Russell M. Mersereau
The Problem
Given
for several
I
= 1,2,…,
P
, estimate
x
(
u
,
v
).
Approaches:
Multidimensional Fourier transform
Radon Transform (spatial domain)
)
ˆ
(
u
p
i
ECE 6258 Russell M. Mersereau
Slices and Projections
ECE 6258 Russell M. Mersereau
Continuation
Therefore,
)
0 ,
(
)
(
μ
X
u
p
D
↔
μ
ν
u
)
(
0
u
p
ECE 6258 Russell M. Mersereau
Playing with the sampling intervals
By varying the sampling interval with the projectionangle, we can place samples on the same horizontaland vertical lines as the 2-D DFT.
ν
μ
ECE 6258 Russell M. Mersereau
Results (1-D Interpolation)
16 projections
32 projections 128 projections
64 projections
ECE 6258 Russell M. Mersereau
Radon Transform
We begin with the inverse Fourier transform and convert the frequency
variables to polar coordinates.
Recall that
ECE 6258 Russell M. Mersereau
Radon Transform (2)
This gives
ECE 6258 Russell M. Mersereau
Convolution Backprojection Algorithm
This suggests the following algorithm
Filter each projection by |
ω
|.
Back project at the angle of the projection.
Add all the backprojections together.
Correct the DC value.
Iterative approach
ECE 6258 Russell M. Mersereau
Results
ECE 6258 Russell M. Mersereau
Results
