11 12 13 14 CT Radon Central Slice Theorem FBP etcwork2007/11 12 13 14 CT Radon Central Slice... ·...

Medical ImagingMedical Imaging

Image Reconstruction from Projections

Prof Ed X. Wu

Transmission MeasurementTransmission Measurement

µ(x,y)

unknownabsorption cross-section

X-r

ay s

ou

rce

( measurable attenuation,

shadowgram )

I = Ioexp − A(x, y)dl

L∫

⎛ ⎝ ⎜

⎞ ⎠ ⎟

I(ϕ,

ξ)

XX--Ray TomographyRay Tomography

I(ϕ,

ξ)

X-r

ay s

ou

rce

Uses X-rays to generate several shadowgrams I(ϕ,ξ).


I(ϕ,ξ)

X-ray source


I(ϕ,ξ)

X-ray so

urce


I(ϕ,

ξ)

X-r

ay s

ou

rce

To obtain image from projection datause inverse radon transform.

µ(x,y)

unknownabsorption cross-section

Image Reconstruction from Projections:

• Radon Transform

• Projection Slice Theorem

• Image Reconstruction from Projection Data- Method I: Fourier Reconstruction- Method II: Backprojection Filtering- Method III: Fourier Filtered Backprojection- Method IV: Convolution Filtered Backprojection

OverviewOverview

Radon TransformRadon Transform

I = Ioexp − µ(x, y)dlL∫

⎛

⎝ ⎜

⎞

⎠ ⎟

g ≡ lnIoI

⎛ ⎝ ⎜

⎞ ⎠ ⎟ (Signal)

X-ra

y so

urce

The Radon transform g(s,θ) of a function µ(x,y) is defined as its line integral along a line inclined at an angle θ from

the y-axis and at a distance s from the origin.

x’

x

y

θs

g(s,θ)

µ(x,y)unknown

absorption

y’

g(s,θ ) = µ(x, y)dlL∫

µ(x,y) g(s,θ)

θ


The Radon transform g(s,θ) of a function µ(x,y) is the one-dimensional projection of µ(x,y) at an angle θ.

g(s,θ ) = µ(x, y)dlL∫

The Radon transform maps the spatial domain (x,y) to the domain (s,θ).

Each point in the (s,θ) space corresponds to a line in the spatial domain (x,y).

x’

x

y

θ

sg(s,θ)

µ(x,y)unknownabsorption

y’s

µ(x,y) g(s,θ)


θ

s

cos sin 0

( , ) ( , ) ( , )

( , ) ( cos sin )

L x y s

g s x y dl x y dl

x y x y s dxdy

θ θ

θ µ µ

µ δ θ θ

+ − =

∞ ∞

−∞ −∞

= =

= + −

∫ ∫

∫ ∫

Equation of line (red) given s and θ;

y(x) = m x + c

m = - cosθ / sinθc = y(0) = s/cos(90- θ)

= s/sinθ=> y = - cosθ /sinθ x + s/sinθ

0 = - sin θ y - cosθ x + s

0 = x cosθ + y sinθ - s

s cosθ

s si

nθs

/ sin

θ

y

x

L

More Radon TransformMore Radon Transform

'

( , ) ( , ) ( cos sin )

( 'cos 'sin , 'sin 'cos ) ( ' ) ' '

( 'cos 'sin , 'sin 'cos ) '

( cos 'sin , sin 'cos ) '

x s

g s x y x y s dxdy

x x y y x y x s dx dy

x x y y x y dy

s y s y dy

θ µ δ θ θ

µ θ θ θ θ δ

µ θ θ θ θ

µ θ θ θ θ

=

= + −

= = − = + −

= = − = +

= − +

∫∫∫∫∫

∫

'cos 'sin

'sin 'cos

' cos sin

' sin cos

x x y

y x y

x x y

y x y

θ θθ θ

θ θθ θ

= −= +

= += − +

Achieved by rotating coordinate system so that

the integration line is along y’-axes: x’

x

y

θsy’

Radon Transform: Example 1Radon Transform: Example 1

A(x,y) = δ(x, y) (point at center)

[ ] 0, 0

( , ) ( , ) ( cos sin )

( , ) ( cos sin ) ( ) ( )x y

g s x y x y s dxdy

g s x y s s s

θ δ δ θ θ

θ δ θ θ δ δ= =

= + −

= + − = − =∫∫

SinogramSinogram

Line in Radon spaceLine in Radon space

S=0S=0

x’

x

y

θ

s

y’

δ(x,y)

(s,θ)


A(x,y) = δ(x-1,y) (point on x-axis)

[ ] 1, 0

( , ) ( 1, ) ( cos sin )

( , ) ( cos sin ) (cos )

cos

x y

g s x y x y s dxdy

g s x y s s

Non zero s

θ δ δ θ θ

θ δ θ θ δ θ

θ= =

= − + −

= + − = −

− => =

∫∫

SinogramSinogram

-1

-0.5

0

0.5

1

-6 -4 -2 0 2 4 6

s

θ

x’

x

y

θ

s

y’ δ(x-1, y)

Example II: A(x,y) = δ(r,φ) (arbitrary point)


-1.5

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0 2 4 6

s

θ

φ

r

SinogramSinogram

x’

x

y

θ

s

y’φr

[ ] [ ]cos , sin

( , ) ( , ) ( cos sin )

( , ) ( cos sin ) cos( )

cos( )

x r y r

g s A x y x y s dxdy

g s x y s r s

Non zero r s

φ φ

θ δ θ θ

θ δ θ θ δ θ φ

θ φ= =

= + −

= + − = − −

− => − =

∫∫δ(r,φ)

x’

x

yExample III: A(x,y) = δ(r,φ) +

y’


φ

r

-1.5

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0 2 4 6

s

θ

φ

δ(x-1,y)

δ(x-1,y)

δ(r,φ)


2D Real Space 2D Radon Space

θ(0 to 180 degree)

s


2D Real Space 2D Radon Space

θ(0 to 180 degree)

s

Where these structures are in Radon space?

Properties of Radon TransformProperties of Radon Transform


• Radon Transform



OverviewOverview

Projection Theorem( also “Central Slice Theorem” or Projection Slice Theorem)

If g(s,θ) is the Radon transform of a function f(x,y), then the one-dimensional Fourier transform G(ωs,θ) with respect to s of the projection g(s,θ) is equal to the central slice, at angle θ, of the two dimensional Fourier transform F(ωx, ωy) of the function f(x,y).

x

y

θ

g(s,θ)

µ(x,y)

2D-space domain of µ(x,y)

ωx

ωy

θ

2D-frequency domain of µ(x,y)

s1D-Fourier

transform F(g(s))


If g(s,θ) is the Radon transform of a function f(x,y), then the one-dimensional Fourier transform G(ωs,θ) with respect to s of the projection g(s,θ) is equal to the central slice, at angle θ, of the two dimensional Fourier transform F(ωx, ωy) of the function f(x,y).

x

y

θ

g(s,θ)

µ(x,y)

Proof: Assignment

ωx

ωy

θ

s1D-Fourier

transform F(g(s))


Clues ?

2D-space domain of µ(x,y) 2D-frequency domain of µ(x,y)

1D-FTF(g(s))

-8 -6 -4 -2 0 2 4 6 8ξ [cm]

0

1

2D-FTF(µ(x,y))

s

under viewing angle θ = 00

g(s,

00 )

ωxωy

ωxωy

Projection Theorem


2D-space domain of µ(x,y) 2D-frequency domain of µ(x,y)

1D-FTF(g(s))

-8 -6 -4 -2 0 2 4 6 8ξ [cm]

0

1

2D-FTF(µ(x,y))

s

under viewing angles θ = 00 &θ = 900

g(s,

0 o

r 90

0)

ωxωy

ωxωy


µ(x,y)

Preparing for Class Project

• Computer skills (MatLab or C from ELEC2201) ?

• Skills in numerical computation ?


• Radon Transform



OverviewOverview

Projection Theorem

How can we use Projection slice theorem to reconstruction

spatial distribution of absorption profile µ(x,y) ?

I. Fourier Reconstruction

Object Spaceµ(x,y)

2 DimensionalFourier-Object SpaceF(ωx,ωy) = FT(µ(x,y))

Radon Spaceg(s,θ)

1 DimensionalFourier-Radon Space

G(ωs, θ) = FT(g(s))

X-ray absorption measurements yield

Radon Transform

1-D Fourier Transform

FT(g(s))

Many slices

at different θ fill 2D Fourier-object space

2-D InverseFourier

Transform

Fourier Image Reconstruction with Projection Theorem

Fourier Reconstruction

x

y

θ

g(s,θ)

µ(x,y)

ωx

ωy

θ

s1D-Fourier

transform F(g(s))


1D-Fourier transform F(g(s,θ1))of one projection

2D-Inverse Fourier transform


x

y

g(s,θ1,2)

µ(x,y)

ωx

ωys 1D-Fourier transform F(g(s))


1D-Fourier transform F(g(s,θ1,2))

of two projection



x

y

g(s,θ1,2,3,4)

µ(x,y)

ωx

ωy

s

1D-Fourier transform F(g(s))


1D-Fourier transform F(g(s,θ1,2,3,4))

of 4 projection



x

y

θ

g(s,θ1,2, ...8)

µ(x,y)

ωx

ωy

θ

s1D-Fourier

transform F(g(s))


1D-Fourier transform F(g(s,θ1,2,... ,8))

of 8 projection



1D-Fourier transform F(g(s,θ1,2, ... , 16))

of 16 projection



1D-Fourier transform F(g(s,θ1,2, ... , 64))

of 64 projection


Influence of # of Projections

64 projections16 projections8 projections

1 projections 2 projections 4 projections


ωx

ωy

θ

Problem:Points in 2D Fourier Space are not on rectangular grid.

=> Inverse Fourier transform not trivial.


A practical algorithm:


How is the Fourier reconstruction method connected with

the backprojection method?

II. Backprojection FilteringII. Backprojection Filtering

0

ˆ( , ) ( , ) ( cos sin , ) b x y f x y g g s x yB dπ

θ θ θ θ≡ ≡ = = +∫

1 1 2 2( , ) ( ) ( ) ( ) ( )g s g s g sθ δ θ φ δ θ φ= − + −

Example: Backproject 2 projections (g1 and g2) only

2 projections represented in Randon space

Backproject 2 Projections

b(x,y) = g1(s1) + g2 (s2 ) = b(r,φ)

1 1 2 2cos( ), cos( )s r s rφ φ φ φ= − = −

The value of the backprojection Bg is evaluated by integrating g(s,θ) over θ for

all lines that pass through that point.

Backprojection operatorBackprojection operator

Backprojection & Radon TransformBackprojection & Radon Transform

Real Space Radon Space, g(s,θ)

x

y

θ

s

The backprojection at (r, φ ) is the integration of g(s,θ) along the sinusoid s = r cos(θ−φ)

WHY? (Optional)

(r, φ)s = r cos(θ−φ)

•-1.5

•-1

•-0.5

•0

•0.5

•1

•1.5

•-4 •-2 •0 •2 •4

•s

This is why we sawThis is why we saw……

Backprojection & Radon TransformBackprojection & Radon Transform

Object Simple

Backprojection (BP)

Sinogram

Backprojection

Measurement

Difference between

Object & BP image

θ(0 to π)

s

Backprojection Operator: MathematicsBackprojection Operator: Mathematics

It can be shown from

that the backprojected Radon transform data g or Rf (i.e., the simple backprojection image)

2 2

ˆ ( , )

1 ( , )

f x y Bg BRf

f x yx y

≡ =

⎛ ⎞⎜ ⎟= ⊗⎜ ⎟+⎝ ⎠

Therefore backprojection of radon transform gives the original image convolved with 1/sqrt(x2+y2). This results in blurred image.

What could you do to get back f(x,y)?

∫∫

∫−+=

=

dxdysyxyxf

dlyxfsgL

)sincos(),(

),(),(

θθδ

θ

),sincos(),(ˆ0

θθθθπ

dyxsgBgyxf +==≡ ∫

Proof?

Assignment

Image Reconstruction by Backprojection FilteringImage Reconstruction by Backprojection Filtering

( ) 1/ 22 2

ˆ ( , )

( , )

f x y Bg BRf

f x y x y−

≡ =

= ⊗ +

Use a filter! - But what filter?

( ) ( )( ) ( ) ( )( )( ) ( )

1/ 2 1/ 22 2 2 2

1/ 22 2

ˆ ( , ) ( , ) ( , )

( , ) x y

F f x y F f x y x y F f x y F x y

F f x y ω ω−

− −= ⊗ + = +

= +

Use convolution theorem:

( )( ) ( ) ( ) ( )( )),(

),( ),(ˆ 2/1222/1222/122

yxfF

yxfFyxfF yxyxyx

=

++=+−

ωωωωωω

( ) ( )( ) ( )( )1/ 22 2ˆ ( , ) ( , ) ) ( ,x yIF F f x y IF F f x y f x yω ω = =+

SQRT SQRT -- FiltersFilters

Filter

(HP)ω = ω x

2 + ω y2( )1/ 2

ω −1 = ω x2 + ω y

2( )−1/ 2Filter

(LP)

ωx

ωx

ωyωy

Backprojection Filtering Algorithm:

(1) Get Radon transform g(s,θ) of f(x,y)by performing tomographic X-tray imaging.

(2) Backproject the Radon transform data.

(3) Take Fourier transform of backprojected data.

(4) Multiply with filter sqrt(ωx2+ ωy

2)

(5) Perform inverse Fourier Transform to obtain f(x,y)

f (x, y) = IF2 ω • F2 B Rf( )( )( )

Image Reconstruction by Backprojection FilteringImage Reconstruction by Backprojection Filtering

Backprojection Filtering Reconstruction Backprojection Filtering Reconstruction & Fourier Reconstruction& Fourier Reconstruction

f (x, y) = IF2 ω • F2 B Rf( )( )( )

f (x, y) = IF2 PST Rf( )( )( )

= IF2 F1,s g(s,θn )( )n=1

N∑

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Backprojection Filtering Method:

Fourier Reconstruction Method:

There are also other techniques !!!!

PST := Projection Slice Theorem or Central Slice Theorem

Note that re-griding is required

Radon TransformRadon Transform2D Real Space 2D Radon Space

θ

s

Why such 2D Radon Transform is important ?- Formulation of x-ray attenuation measurement in CT

- Mathematics for later use in image reconstruction- Inverse Radon Transform possible?

),sincos(ˆ),(0

θθθθπ

dyxsgyxf +== ∫ssss dsiGsgwith ωωθωωθ )exp(),(),(ˆ ∫

∞

∞−

=

with G(ω s ,θ ) = F1,s g(s,θ)( )

III. Fourier Filtered BackprojectionIII. Fourier Filtered Backprojection

The inverse Radon transform is obtained in two steps:

(1) Each projection is filtered by a one dimensional filter whose frequency response is |ωs|.

(2) The result of step (1) is backprojected to yield f(x,y).

),sincos(ˆ),(0

θθθθπ

dyxsgyxf +== ∫

( )

),(),(

)exp(),(),(ˆ

,11

1

θθω

ωωθωωθ

sgFFwith

dsiFsgwith

ss

ssss

=

= ∫∞

∞−

Inverse Radon Transform Theorem:

Proof ?

ProofProof

)](exp[),(),( yxyxyx ddyxiFyxf ωωωωωω += ∫ ∫∞

∞−

∞

∞−

The inverse Fourier transform is given by:

Rewriting in polar coordinates results in:

)]sincos(exp[),(),(2

0 0

θωωθθωθωπ

ddyxiFyxf ssssp += ∫ ∫∞

Changing the limits of integration we get:

)]sincos(exp[),(),(0∫ ∫ +=

∞

∞−

π

θωθθωθωω ddyxiFyxf sssps

Since the Projection Slice Theorem(1D Fourier transform with respect to s of Radon transform equals slice through 2D Fourier transform at angle θ of the object function f)

∫∫

∫ ∫

=+=

⎭⎬⎫

⎩⎨⎧

=∞

∞−

ππ

π

θθθθθθ

θωωθωω

00

0

),(ˆ),sincos(ˆ),(

]exp[),(),(

dsgdyxgyxf

ddsiGyxf ssss

x

y

θ

g(s,θ)

µ(x,y)

ωx

ωy

θ

s1D-Fourier

transform F(g(s))

=


Fourier Filtered Backprojection Reconstruction Fourier Filtered Backprojection Reconstruction && Backprojection Filtering ReconstructionBackprojection Filtering Reconstruction

),sincos(ˆ),(0

θθθθπ

dyxsgyxf +== ∫

( )

),(),(

)exp(),(),(ˆ

,11

1

θθω

ωωθωωθ

sgFFwith

dsiFsgwith

ss

ssss

=

= ∫∞

∞−

f (x, y) = IF2 ω • F2 B Rf( )( )( )

f (x, y) = B IF1,s ωs • F1,s Rf( )( )( )Fourier Filtered Backprojection Method

Backprojection Filtering Method

Fourier Filtered Backprojection ReconstructionFourier Filtered Backprojection Reconstruction

discrete implementation:

basic concept:

||ωω| | -- FiltersFilters

RAM - LAK Shepp-Logan

Hamming Lowpass Cosine

IV. Convolution Filtered BackprojectionIV. Convolution Filtered Backprojection

page 446

),sincos(ˆ),(0

θθθθπ

dyxsgyxf +== ∫

ssss dsiGsgwith ωωθωωθ )exp(),(),(ˆ ∫∞

∞−

= with G(ω s ,θ ) = F1,s g(s,θ)( )

= ω sG(ω s ,θ) sgn(ω s ) exp(iω ss)dωs−∞

∞∫

= IF1 ωsG(ωs ,θ ){ }[ ]⊗ IF1 sgn(ω s ){ }[ ]=

1

i2π⎛ ⎝ ⎜

⎞ ⎠ ⎟

∂g(s,θ )

∂s

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ⊗

−1

iπs

⎡ ⎣ ⎢

⎤ ⎦ ⎥

=1

2π 2

⎛

⎝ ⎜

⎞

⎠ ⎟

∂g(t ,θ )

∂t

⎡ ⎣ ⎢

⎤ ⎦ ⎥

−∞

−∞∫

1

s − tdt Hilbert Transform

convolution theorem

Convolution Filtered BackprojectionConvolution Filtered Backprojection

The inverse Radon transform is obtained in three steps:

(1) Each projection is differentiated with respect to s.

(2) A Hilbert transformation is performed with respect to s.

(3) The result of step (2) is backprojected to yield f(x,y).

f (x, y) = 1 / 2π( )B Hs Ds Rf( )( )( )

convolution

SummarySummary

f (x, y) = 1 / 2π( )B Hs Ds Rf( )( )( )Convolution Filtered Backprojection – Method IV:

f (x, y) = IF2 ω • F2 B Rf( )( )( )

f (x, y) = B IF1,s ωs • F1,s Rf( )( )( )Fourier Filtered Backprojection - Method III

Backprojection Filtering - Method II:

f (x, y) = IF2 F1,s g(s,θn )( )n=1

N∑

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Fourier Reconstruction - Method I:

ProjectsProjects

Groups of 3Groups of 3--4 work on the same problem but with 4 work on the same problem but with different approaches. Consult each other and different approaches. Consult each other and

divide work whenever possible. divide work whenever possible.

Presentation in classPresentation in class

Will be graded as ~6 Will be graded as ~6 homeworkshomeworks ( or ~10% Grade).( or ~10% Grade).

Group ProjectsGroup Projects

f (x, y) = 1 / 2π( )B Hs Ds Rf( )( )( )D. Convolution Filtered Backprojection - Method IV:

f (x, y) = IF2 ω • F2 B Rf( )( )( )

f (x, y) = B IF1,s ωs • F1,s Rf( )( )( )C. Fourier Filtered Backprojection - Method III:

B. Backprojection Filtering - Method II

f (x, y) = IF2 F1,s g(s,θn )( )n=1

N∑

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

A. Fourier Reconstruction - Method I:

25 students => 25 students => 55 groups of groups of 55Each group will write reconstruction program in Each group will write reconstruction program in MatlabMatlab::

E. Iterative Reconstruction

11 12 13 14 CT Radon Central Slice Theorem FBP etcwork2007/11 12 13 14 CT Radon Central Slice... ·...

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