filtered backprojection_good

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


• 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


•

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 Filtered Backprojection

•




•



Reconstruction By Filtered Backprojection

•



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.

filtered backprojection_good

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