radon transform a small introduction to rt, its inversion and applications jaromír brum kukal, 2009
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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
Reconstruction from 64 angles
Reconstruction from 96 angles
Reconstruction from 128 angles
Reconstruction from 180 angles
Reconstruction from 256 angles
Reconstruction from 360 angles
Reconstruction from 512 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