Digital Image Processing, Image reconstruction of CT images, principles of computed tomography, CT scanners generations, radon transform, Fourier Slice theorem, back projection, filtered back-projection

1. Image
Reconstruction
GROUP 1

2. Contents
• Introduction
• Back projection
• Principles of Computed Tomography (CT)
• Generations of CT
• Radon Transform
• Analytical Reconstruction Methods: Fourier-Slice Transform,
Parallel-Beam Filtered Back projections, Fan- Beam Filtered
Back projections
• Iterative Reconstruction

3. What is Image Reconstruction?
• Image Reconstruction in CT is a mathematical process that
generates tomographic images from X-ray projection data
acquired at many different angles around patient.
• In this presentation, we examine the problem of reconstructing
an image from a series of projections, with a focus on X-ray
computed tomography (CT).

4. CT Scanner Components

5. Back Projection

6. Back Projection

7. Back Projection

8. Back Projection

9. Back Projection
• Simplest reconstruction procedure.
• Assumptions:
• Rays: ideal straight lines.
• Image: dimensionless points.
• Procedure:
• Estimate of the density at a point by simply summing (integrating) all the rays
that pass through it at various angles.
• Problems
• Finite number of rays per projection,
• Finite number of projections,
• Interpolation is required.

10. Back Projection
Image is Blurred

11. Principles of Computed tomography (CT)
• The goal of X-ray computed
tomography is to obtain a 3-D
representation of the internal
structure of an object by X-
raying the object from many
different directions.
• Computed tomography
attempts to get information by
generating slices through the
body.
• A 3-D representation then can
be obtained by stacking the
slices.

12. First Generation (G1) CT scanners
• Pencil X-ray beam and a
single detector.
• For a given angle of rotation,
the source and detector pair
is translated incrementally
along the linear direction
shown.
• A projection, is generated by
measuring the output of the
detector at each increment
of translation.

13. Second Generation (G2) CT scanners
• The beam used is in the
shape of a fan.
• This allows the use of
multiple detectors, thus
requiring fewer translations
of the source/detector pair.

14. Third Generation (G3) scanners
• G3 scanners employ a bank of
detectors long enough (on the
order of1000 individual
detectors) to cover the entire
field of view of a wider beam.
• Consequently, each increment
of angle produces an entire
projection, eliminating the need
to translate the source/detector
pair, as in the geometry of
G1and G2 scanners

15. Fourth Generation (G4) scanners
• A circular ring of detectors
(on the order of 5000
individual detectors), only
the source has to rotate.
• The key advantage of G3
and G4 scanners is speed.

16. CT Image scan formation

17. Phase of CT Scan
• Scanning the patient or data acquisition
• X-ray generator
• X-rat tube
• X-ray filtration System
• Detector System
• Reconstruction
• Simple back projection
• Iterative method
• Analytical method- 2 D Fourier Analysis and Filtered Back Projection
• Display

18. Projections and Radon Transform
• A straight line in Cartesian
coordinates can be
described either by its slope-
intercept form, y = ax
+ b or as the equation
shown in the image.

19. Geometry of a parallel beam
Radon Transform
Line integral

20. Radon Transform
• For a fixed rotation angle θk, and a fixed distance ρk, back-
projecting the value of is g(θk, ρk) equivalent to copying the
value g(θk, ρk) to the image pixels belonging to the line xcosθk
+ ysinθk = ρj
• Repeating the process for all the values ρj, having a fixed line θk
in the following expression for the image values:
This equation holds for every angle θ

21. Radon Transform (Cont.)
• The final image is formed by integrating over all the back-
projected images:
Back projection provides
blurred images. We will
reformulate the process to
eliminate blurring. (Filter)

22. Sinogram
• The representation of the
random transform g(p,Θ) as an
image with p and Θ as
coordinates is called sinogram.
• A sinogram is a special x-ray
procedure that is done to
visualize any abnormal opening
(sinus) in the body, following
the injection of contrast media
(x-ray dye) into the opening.

23. Iterative Reconstruction
• An iterative reconstruction starts with an assumption (all points
in the matrix have same value) and compares this assumption
with a measured value, make the corrections to bring two into
agreement and repeats the process over and over again until
assumed and measured value are same or within acceptable
limits.
• Limitations:
• Difficult to obtain accurate ray sum because of quantum noise and patient
motion.
• Takes too long to generate the reconstructed image because the iteration
can done only after all projection data sets have been obtained.

24. Analytical Method Reconstruction
• Current commercial scanners uses this method.
• A mathematical technique known as Convolution or filtering is
used.
• Technique employs spatial filter to remove blurring artifacts.
• Two major types:
• Fourier Reconstruction Algorithm
• Filtered back projection

25. The Fourier Slice Theorem
• It relates to the 1D Fourier transform of a projection with the 2D
Fourier transform of the region of the image from which
projection was obtained.
• It is the basis of image reconstruction method.

26. The Fourier Slice Theorem
2D F.T

27. The Fourier Slice Theorem

28. The Fourier Slice Theorem

29. The Fourier Slice Theorem

30. The Fourier Slice Theorem
H
F
LF

31. The Fourier Slice Theorem
• Let the 1D F.T of a projection with respect to p at a given angle
be:
• Substituting the projection g(p, Θ) by the ray-sum:

32. The Fourier Slice Theorem
Equation (5.11-11) is known as Fourier Slice Transform

33. Illustration of Fourier-Slice Transform
Theorem

34. Filtered Back Projection
• Same as back projection except that the image is filtered, or
modified to exactly counterbalance the affect of sudden density
changes, which causes blurring (star like pattern) in simple back
projection.

35. The complete Back Projection
• Compute the 1-D Fourier transform of each projection.
• Multiply each Fourier transform by the filter function |w| which,
as explained above, has been multiplied by a suitable (e.g.,
Hamming) window.
• Obtain the inverse 1-D Fourier transform of each resulting
filtered transform.
• Integrate (sum) all the 1-D inverse transforms from step 3.
• Because the filter function is used, this image reconstruction
approach is called filtered back-projection.

36. Summary
• Image Reconstruction Basics.
• Image reconstruction methods:
• Back Projection method
• Iterative method
• Analytical method
• CT Scanners and its components

37. References
• Digital Image Processing Image Restoration and Reconstruction
Image (slidetodoc.com)
• Digital Image Processing Book

38. THANK YOU