These slides based almost entirely on a set provided by Prof. Eric Miller Imaging from Projections Eric Miller With minor modifications by Dana Brooks.

These slides based almost entirely on a set provided by Prof. Eric Miller

Imaging from Projections

Eric MillerWith minor modifications by

Dana Brooks


Outline• Problem formulation

– What’s a projection?– Application examples– Why is this interesting?

• The forward problem– The Radon transform – The Fourier Slice Theorem

• The Inverse Problem– Undoing the Radon transform with the help of Fourier– Filtered Backprojection Algorithm

• Complications and Extensions


A ProjectionThe total amount of f(x,y) along the line defined by t and

x

y

t

P t

0t

( , )f x y


Application Examples• CAT scans:

– X ray source moves around the body– f(x,y) is the density of the tissue

• MRI– Not as clear cut what the “projection” is, but in a peculiar way, the math is

the same (remind me to talk about this when we get to the MRI Imaging equation …)

– f(x,y) is the spin density of molecules in the tissue

• Synthetic Aperture Radar– Satellite moves down a linear track collecting radar echoes of the ground– Used for remote sensing, surveillance, …– Again: math is the same (after much pain and anguish)– f(x,y) is the reflectivity of the earth surface


Motivation• In all cases, one observes a bunch of sum or

integrals of a quantity over a region of space: these are “projections”

• The goal is to use a collection of these projections to recover f(x,y).

• Here we will talk about the full data case– Assume we see for all and t

• Limited view tomography a topic for advanced course

P t


The Radon Transform

x

y

t

P t

( , )f x y

ds

, line

,

, cos sin

t

P t f x y ds

f x y x y t dxdy

Polar equation for line:cos sinx y t

Function oft and

So the line exists only where this equation is true


What does it do?Simplest case: f(x,y) a function: only exists at a single point

0 0

0 0

, cos sin

cos sin

P t x x y y x y t dxdy

x y t

• Proof only by limiting argument as products of ’s not well defined• Interpretation:

• A “function” in (t,) space which “is” 1 along a sinusoidal curve and zero elsewhere: note that a point in 2D a curve• Say y0 = 0 and x0 = 1 then this is an “image” which “is” 1 when t = cos


In Pictures

x

y

t

The ImageCalled the Radon Transform(a.k.a.the sinogram)

Kind of 2D impulse response (PSF)

Note that we draw as a rectangular “image” in t and P t

P t


More Examples

t

t


Fourier Slice Theorem

• Key idea here and for a large number of other problems

• Analytically relate the 1D Fourier transform of P to the 2D Fourier transform of f.

• Why?– If we can do this, then a simple inverse 2D

Fourier gives us back f from the “data” P.


Recall 2D Fourier Transform

, , exp 2F u v f x y j ux vy dxdy

Analysis

Synthesis

, , exp 2f x y F u v j ux vy dudv

• “Space” variable x goes with “frequency” variable u• “Space” variable y goes with “frequency” variable v• (u,v) called “spatial frequency domain”


Fourier – Slice Theorem (FST)

• Let F(u,v) be defined as on last slide• Define S(w) as the 1D Fourier transform of P along t for some frequency variable w

• FST says that S is equal to F(u,v) along a line tilted at an angle with respect to the (u,v) coordinate system•To make this more precise …

exp 2S w P t j wt dt


Fourier-Slice

x

y

t

( , )f x y

P t

1D Fourier Transform

u

v

F(u,v) along line

w

Variables w and are the polar form of u and vcos

sin

u w

v w

So FST is: cos , sinS w F w w


Reconstruction Implications• Collect data from lots and lots of projections.•Take 1D FT of each to get one line in 2D frequency space• Fill up 2D spatial frequency space on a polar grid• Interpolate onto rectangular grid• Inverse 2D FT and we are done!!

u

v


An Alternate ApproachFiltered Backprojection

The derivation of this algorithm is perhaps one of the most illustrative examples of how we can obtain a radically differentcomputer implementation by simply re-writing the fundamentalexpressions for the underlying theory

- Kak and Slaley, CTI

• This requires lots of Fourier Transforms• This means we can’t begin processing until we have all slices• Turns out there’s a more efficient way to organize things• This requires “ugly” interpolation, worse at high frequencies


FBP Motivation in Pictures

u

v

By linearity, could in theory break up reconstruction intocontribution from independent“wedges” in 2D Fourier space

u

v

• In practice, we measure over lines. • Idea: build a 2D filter which covers the line, but has the same “weight” as the wedge at that frequency, w• In other words “mush” triangle to a rectangle• Then “sum up” filtered projections

w

• For K projections, the width of the wedge at w is just

2width

w

K

w


FBP Theory

, , exp 2f x y F u v j ux vy dudv Now, change right side from polar to rectangular

cos

sin

u w

v w

dudv wdwd

To get rectangular coordinates in space, polar in frequency:

2

0 0

, , exp 2 cos sinf x y F w w j w x y dwd


FBP Theory IIMake use of two facts:

, ,

cos sin

F w F w

t x y

To arrive at

0

0

, exp 2

cos sin

f x y S w w j wt dw d

Q x y d

exp 2Q t S w w j wt dw

Filter (in space)

Backproject


FBP Interpretation• Recall from linear systems

• So |w| filter is more or less a differentiator. Accentuated high frequency information leads to problems with noise amplification• In practice, roll off response.

Fourier

transformdf t j F

dt

w w


FBP Interpretation

0

cos sinQ x y d

Backprojection: Note that Q(t) needs only one (filtered) projection

Sum up over allangles

Think of this as Q(t) evaluated at the point t = xcos + y sin

x

y

t

0Q t

Region we arereconstructing

• Along this line in “image space” set the value to Q(t0)• All points get a value• Do for all angles• Add up


FBP Example

Orig. Recon

Zoom


Limited data I:Angle decimation


Limited data II:Limited Angle


Artifact Mitigation• Take a more matrix-based “inverse problems” perspective• Discretized Radon transform, data, and object to arrive at a

forward model

• Where C has many fewer rows than columns• Use SVD, TSVD, Tikhonov, or other favorite

regularization scheme to improve reconstruction results• Note: significant move from analytical to numerical

inversion means a basic shift in how we are approaching the problem. No more FBP (at least not easily)

y Cx


Other Fourier Imaging Applications

u

v

u

v

• Standard SAR• Collects data on wedge shaped regions of Fourier space• Very limited view• Similar math to X-ray

• Diffraction tomography• Collects data on petal shaped regions of Fourier space• Very limited view• More sophisticated math than X ray• Arises in geophysical and medical imaging problems


Generalized Radon Transforms

• Radon transform = integral of object over straight lines

• Many extensions– Integration over planes in 3D– Over circles in 2D (different type of SAR)– Over much more arbitrary mathematical structures

(asymptotic case of some acoustics problems with space varying background).

– Of weighted object function (attenuated Radon transform)

These slides based almost entirely on a set provided by Prof. Eric Miller Imaging from Projections Eric Miller With minor modifications by Dana Brooks.

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