RADON TRANSFORM A small introduction to RT, its inversion and applications Jaromír Brum Kukal, 2009

RADON TRANSFORMA small introduction to RT,

its inversion and applications

Jaromír Brum Kukal, 2009

Johann Karl August Radon• Born in Děčín (Austrian

monarchy, now North Bohemia, CZ) in 1887

• Austrian mathematician living in Vienna

• Discover the transform and its inversion in 1917 as pure theoretical result

• No practical applications during his life

• Died in 1956 in Vienna

Actual applications of inverse Radon transform

1. CT – Computer Tomography

2. MRI – Magnetic Resonance Imaging

3. PET – Positron Emission Tomography

4. SPECT – Single Photon Emission Computer Tomography

Geometry of 2D Radon transform

• Input space coordinates x, y

• Input function f(x, y)

• Output space coordinates , s

• Output function F(, s)

Formulas of pure RT and IRT

π2

0

dsincos,F),f(

dsincos,cossinf),F(

yxyx

tststs

Radon transform

Inverse Radon transform

Example of Example of Radon Radon

TransformTransform

Full circle in RT

Shifted full circle in RT

Empty circle in RT

Shifted empty circle in RT

Thin stick in RT

Shifted thin stick in RT

Full triangle in RT

Shifted full triangle in RT

Full square in RT

Shifted full square in RT

Empty square in RT

Shifted empty square in RT

| x |2/3 + | y |2/3 ≤ 1 in RT

| x | + | y | ≤ 1 in RT

| x |3/2 + | y |3/2 ≤ 1 in RT

| x |2 + | y |2 ≤ 1 in RT

| x |6 + | y |6 ≤ 1 in RT

| x |n + | y |n ≤ 1 for n in RT

2D Gaussian in RT

Shifted 2D Gaussian in RT

Six 2D Gaussians in RT

Smooth elliptic object in RT

Radon Radon Transform Transform PropertiesProperties

Radon transform properties

1. Image of any f + g is F + G

2.Image of cf is cF for any real c

3. Rotation of f causes translation of F in

4.Scaling of f in (x,y) causes scaling of F in s

5.Image of a point (2D Dirac function) is sine wave line

Radon transform properties

6. Image of n points is a set of n sine wave lines

7. Image of a line is a point (2D Dirac function)

8. Image of polygon contour is a point set

Radon transform realization Space domain:• Pixel splitting into four subpixels• 2D interpolation in space domain• 1D numeric integration along lines

Frequency domain:• 2D FFT of original• Resampling to polar coordinates• 2D interpolation in frequency domain• Inverse 2D FFT brings result

Inverse transform realization

Filtered back projection in space domain:• 1D HF filtering of 2D original along s• Additional 1D LF filtering along s• 2D interpolation in space domain• 1D integration along lines brings result Frequency domain:• 2D FFT of original• Resampling to rectangular coordinates• 2D interpolation in frequency domain• 2D LF filtering in frequency domain• Inverse 2D FFT brings result

RT and IRT in Matlab • Original as a square matrix D (2n2n) of nonnegative numbers• Vector of angles alpha• Basic range alpha = 0:179• Digital range is better alpha = (0:2^N -1)*180/2^N• Extended range alpha = 0:359 • Output matrix R of nonnegative numbers• Angles alpha generates columns of RR = radon(D,alpha); D = iradon(R,alpha); D = iradon(R,alpha,metint,metfil);

Radon Radon Transform - Transform -

ReconstructionReconstruction

Reconstruction from 32 angles








Radon transform applicationsNatural transform as result of measurement: 1. Gamma ray decay from local density map2. Extinction from local concentration map3. Total radioactivity from local concentration map4. Total echo from local nuclei concentration map5. 3D reality is investigated via 2D slices

Artificial realization:1. Noise – RT – noise – IRT simulations2. Image decryption as a fun3. TSR invariant recognition of objects

Documents

RADON TRANSFORM A small introduction to RT, its inversion and applications Jaromír Brum Kukal, 2009