COMPUTED TOMOGRAPHY
IMAGE RECONSTRUCTION
Presented By: Gunjan Patel(MS-Medical Software )(B.E.-Biomedical Engg.)
(PGQ-Quality Management)
History of Image Reconstruction
1917 Radon has developed mathematical solution to the problems of image reconstruction from of a set of projection.Utilization in solving problems in astronomy and optics.1961 finally these techniques were used in medical field.
CT Image Reconstruction
For an N×N image, we have N unknowns to estimate the digital image reconstruction.
2
pixelpixel
IMAGE RECONSTRUCTION
BACK PROJECTION METHOD
The oldest methodNot used in commercial ct scannersMethod is analogous to a graphic reconstructionProcessing part is simple and directEach projection can not contribute originally formal of profileSome produces images are ‘Starred’ and ‘blurring’ that makes unsuitable for medical diagnosis
A sinogram is a special x-ray procedure that is done with contrast media (x-ray dye) to visualize any abnormal
opening (sinus) in the body
BACK PROJECTION METHOD
• Start from a projection value and back-project a ray of equal pixel values that would sum to the same value
• Back-projected ray is added to the estimated image and the process is repeated for all projection points at all angles
• With sufficient projection angles, structures can be somewhat restored
Example:
Problem:
Problems with back-projection include mainly severe blurring in the computed images
Iterative reconstruction
Successive approximation method Iterative least squares techniquesAlgebraic reconstruction
Hounsfield used this technique in his First EMI BRAIN SCANNER
Iterative methods are not use in today commercial scanners
Example:
Successive approximation method to obtain an image of attenuation coefficients from the measured intensity form Object sliceThe attenuation coefficient of the object are unknown before hand
Calculation of Method: Click
Analytical methodsCurrent Commercial scanner uses this methodA mathematical technique known as convolution or filtering Technique employs a spatial filter for remove blurring artifacts.2 types of method1) Filtered back projection2) Fourier filtering
1. Filtered back projection
Spatial Filter
(-)
(-)
(-)
(+) (+)
(+)
1. Filtered back projection
This technique elimination the unwanted cusp like tails of the projection.The projection data are convoluted with suitable processing function before back projectionThe filter function has negative side lobes surrounding a positive core, so that in summing the filtered back projection - positive and negative contribution that cancel outside the central core The constructed image resemble Original object
2. Fourier filtering
A property of the Fourier transformRelates the projection data in the spatial domain to the Frequency domain
The 1D Fourier transform of the projection of an
image at an angle θ
The 1D Fourier transform of the projection of an
image at an angle θ
The slice of the 2D Fourier
transform at the same angle
The slice of the 2D Fourier
transform at the same angle
Fourier Transform to Projection
Fourier Slice Theorem
P(t)
f(x,y)
t
y
x
X-rays
Ky
Kx
F(Kx,Ky)
F[P(t)]
Mathematical Illustration• 2D Fourier transformation:
• The slice of the 2D Fourier transform at kx=0 is given by:
and at ky=0 is given by
From Projections to Image
y
x
Ky
Kx
F-1[F(Kx,ky)]
f(x,y) P(t) F(Kx,Ky)
Reconstruction of Object• Interpolation can be used in the frequency domain to re-grid the
radial sampling to uniform sampling• Inverse DFT can then be efficiently used to compute the object
Freq. domain Interpolation IDFT Computed Object
References
• http://www.slideshare.net/NYCCT1199/ct-reconstruction-methods
• http://en.wikipedia.org/wiki/Iterative_reconstruction• Handbook of Biomedical Instrumentation-
R.S.Khandpur
Queries !!!