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 !!!