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
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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
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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
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Geometry of 2D Radon transform
• Input space coordinates x, y
• Input function f(x, y)
• Output space coordinates , s
• Output function F(, s)
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Formulas of pure RT and IRT
π2
0
dsincos,F),f(
dsincos,cossinf),F(
yxyx
tststs
Radon transform
Inverse Radon transform
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Example of Example of Radon Radon
TransformTransform
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Full circle in RT
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Shifted full circle in RT
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Empty circle in RT
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Shifted empty circle in RT
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Thin stick in RT
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Shifted thin stick in RT
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Full triangle in RT
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Shifted full triangle in RT
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Full square in RT
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Shifted full square in RT
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Empty square in RT
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Shifted empty square in RT
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| x |2/3 + | y |2/3 ≤ 1 in RT
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| x | + | y | ≤ 1 in RT
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| x |3/2 + | y |3/2 ≤ 1 in RT
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| x |2 + | y |2 ≤ 1 in RT
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| x |6 + | y |6 ≤ 1 in RT
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| x |n + | y |n ≤ 1 for n in RT
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2D Gaussian in RT
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Shifted 2D Gaussian in RT
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Six 2D Gaussians in RT
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Smooth elliptic object in RT
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Radon Radon Transform Transform PropertiesProperties
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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
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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
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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
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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
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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);
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Radon Radon Transform - Transform -
ReconstructionReconstruction
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Reconstruction from 32 angles
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Reconstruction from 64 angles
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Reconstruction from 96 angles
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Reconstruction from 128 angles
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Reconstruction from 180 angles
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Reconstruction from 256 angles
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Reconstruction from 360 angles
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Reconstruction from 512 angles
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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