Image reconstruction in CT is mostly a mathematical process however, this presentation tries to explain the complicated process of image reconstruction in a visual way, mainly focusing om Filtered back projection, Iterative Reconstruction and AI based image reconstruction.

2. Introduction
 The objective of computed tomography (CT) is to reconstruct two-dimensional (2D) or three-
dimensional (3D) images of internal structures from collected signals through an object.
 Steps for CT Image Formation: All CT systems use a three step process
 Scan or Data Acquisition GET DATA
 Image Reconstruction USE DATA
 Image Display DISPLAY DATA
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3. History
 History of reconstruction algorithms that can be traced back to as early as 1917, when J. Radon, an
Austrian mathematician, first presented a mathematics solution for reconstruction as Radon
Transform
 William H. Oldendorf (1963) developed a direct back projection method.
 Later, the idea of filtered back projection was first proposed by Bracewell and Riddle (1967),
probably the most influential development in this area.
 Gordon et al. (1970) proposed the algebraic reconstruction technique (ART), which could produce a
good reconstruction when projections are not uniformly distributed or limited
 Special algorithms (convolution back-projection algorithms) were soon introduced. These algorithms
were developed by Ramachandran and Lakshminarayanan (1971) and later used by Shepp and Logan
(1974) to improve image quality and processing time.
 Originally, CT images were reconstructed with an iterative method called algebraic reconstruction
technique (ART). Due to lack of computational power, this technique was quickly replaced by simple
analytic methods such as filtered back projection (FBP). FBP was the method of choice for decades,
until the first iterative reconstruction (IR) technique was clinically introduced in 2009
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4. Prior to CT reconstruction
 Various preprocessing procedures are applied to the actual acquired projection data prior to CT
image reconstruction
 air calibration scans: influence of the bow tie filter is characterized, characterize differences in
individual detector response
 A dead pixel correction algorithm: replaces dead pixel data with interpolated data from
surrounding pixels
 Scatter correction algorithms: Adaptive noise filtration methods algorithms; to reduce the impact
of noise
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5. Sequence of image construction

6. Problem of CT
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7. Total linear Attenuation


 





n
i
n
t
t
t e
N
e
N
N 1
2
1
(
0
)
..
..........
0

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
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n
i
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 ......
..........
2
1 =
Where,

8. Image Reconstruction Algorithm
 In CT, reconstruction algorithms are used by the computer to solve the many mathematical equations
necessary for information from the detector array to be converted to information suitable for image
display
 The process of using the raw data to create an image is called image reconstruction. It is a
mathematical process that generates tomographic images from X-ray projection data acquired at many
different angles around the patient. This process is a computer-intensive task and one of the most
crucial steps in the CT imaging process.
 Out of many algorithms used in image reconstruction, these algorithms can be divided into three
mathematical methods of image reconstruction:
 Back-projection
 Analytical Methods: Fourier analysis and filtered back projection
 Iterative Methods: Simultaneous Reconstruction, Ray by Ray Correction, Point by Point correction,
Algebraic reconstruction technique
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9. Few Terminologies
 Raw Data
 Image Data
 Sinogram
 Convolution
 Fourier Transform
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10. RAW DATA VERSUS IMAGE DATA
 All of the thousands of bits of data acquired by the system with each scan are called raw data.
 Once raw data have been processed, i.e. reconstructed, so that each pixel is assigned a Hounsfield
unit value, an image can be created; the data included in the image are now referred to as image
data
 The reconstruction that is automatically produced during scanning is often called prospective
reconstruction.
 The same raw data may be used later to generate new images. This process is referred to as
retrospective reconstruction
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11. SINOGRAM
 “Sinogram”, which is simply the 2-D array of
data containing the projections in differenty
angles
 Fig. shows the cross section of a body phantom
(b) and its sinogram (a). The two air pockets
located near the center of the phantom (b) are
clearly visible in the sinogram (a) as two dark
sinusoidal curves near the center, and the five
high-density ribs near the periphery of the
phantom (b) are depicted as bright sinusoidal
curves (a).
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SINOGRAM

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SINOGRAM

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SINOGRAM

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SINOGRAM

16. SINOGRAM
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SINOGRAM

18. CONVOLUTION
 Convolution, a general-purpose algorithm, is a technique of filtering in the space domain. This will become
clear during the discussion of the filtered back-projection algorithm.
 Spatial Frequency Filtering: High-Pass Filtering: aka; edge enhancement or sharpness; intended to sharpen
an input image in the spatial domain that appears blurred. First the spatial location image is converted into
spatial frequencies by using the FT, followed by the use of a high-pass filter, which suppresses the low
spatial frequencies to produce a sharper output image.
 Spatial Frequency Filtering: Low-Pass Filtering: uses a low-pass filter to operate on the input image with
the goal of smoothing. The output image will appear blurred. Smoothing is intended to reduce noise and the
displayed brightness levels of pixels; however, image detail is compromised
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19. Fourier Transform
 The Fourier transform, which was developed by the mathematician Baron Jean-Baptiste-Joseph Fourier
in 1807, is widely used in science and engineering.
 In radiology, the Fourier transform is used to reconstruct images of a patient’s anatomy in CT and also in
magnetic resonance imaging (MRI) and various other modalities.
 In the analysis of time-based signals, the Fourier transform converts from the time domain (x-axis
labeled in seconds) to the temporal frequency domain (x-axis labeled in sec−1). While time-series
analysis has applications in radiological imaging in ultrasound Doppler and magnetic resonance imaging,
here we focus on input signals that are in the spatial domain (x-axis labeled typically in mm), which is
relevant to any anatomical medical image. The Fourier transform of a spatial domain signal converts it
to the spatial frequency domain (x-axis labeled in mm−1).
 The Fourier transform is an algorithm that decomposes a spatial or time domain signal into a series of
sine waves that, when summed, replicate that signal. Once a spatial domain signal is Fourier
transformed, the resulting data are considered to be in the frequency domain.
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20. Fourier Transform
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21. Fourier Slice Theorem
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One of the most fundamental concepts in CT image reconstruction is the
“Central-slice” theorem. This theorem states that the 1-D FT of the
projection of an object is the same as the values of the 2-D FT of the
object along a line drawn through the 2-D FT plane at same angle.

22. Fourier Slice Theorem
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The 1-D projection of the object, measured at certain angle , is the same
as the profile through the 2D FT of the object, at the same angle

23. Fourier Slice Theorem
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The 1-D projection of the object, measured at certain angle , is the same
as the profile through the 2D FT of the object, at the same angle

24. Simple Back Projection
 Back-projection, sometimes called the "summation method,“ is
the oldest means of image reconstruction. None of the
commercial CT scanners uses simple back-projection, but it is
the easiest method to describe
 Using simple backprojection, the reconstructed image has a
characteristic 1/r blurring that results from the geometry of
backprojection. To correct for this, a mathematical filtering
operation is required, and that leads to the discussion of
filtered backprojection in the next section.
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25. Simple Back Projection
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26. Simple Back Projection
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27. Analytical Method
 A mathematical technique known as convolution or filtering
 Technique employs a spatial filter for remove blurring artifacts.
 2 types of method
 Filtered back projection
 Fourier filtering
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28. Filtered back projection
 Similar to back projection except the image is filtered or modified to exactly counter balance the
density which causes blurring(Star-pattern) in simple back projection
 The Mathematical filtering step involve convolving the projection data with a convolution “KERNEL”
 With FBP, CT slices are reconstructed from projection data (sinograms) by applying a high pass filter
followed by a backward projection step
 Convolution filter refers to a mathematical filtering of the data designed to change the appearance of
the image.
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29. Filtered back projection
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30. Filtered back projection
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31. Filtered back projection
CONVOLUTION/FILTER:
 It is a mathematical filtering process of projected data by the mathematical filter (Kernel) to
reduce the blurring effect of the projections.
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Ram Lak
/Ramp
Filter
Shepp-
Logan
Filter
Hamming
Filter
Ideal
Reconstruction
Filter in the
absence of Noise.
It has even more
Pronounced High
frequency roll-off, with
Better higher frequency
noise suppression
It incorporates some roll
off at HF, reduction in
amplification at HF has a
profound influence in
terms of reducing HF noise
in the final CT image

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33. Filtered back projection
CONVOLUTION/FILTER:
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34. Filtered back projection
CONVOLUTION/FILTER:
 Siemens; Kernel names have 4 positions.
 Example: B31s.
 Pos.1: kernel type (B=body, C=child head, H=head, U=ultra high resolution, S=special kernel,
T=topo.)
 Pos. 2: resolution (1,...,9. Higher number -> higher resolution)
 Pos. 3: version (0,...,9)
 Pos. 4: scan mode (f=fast, s=standard, h=highres, u=ultrahighres)
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35. Filtered back projection
FILTER vs NOISE DETAIL
NOISE
STANDARD
SMOOTHING
EDGE-ENHANCEMENT

36. Filtered back projection
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37. Filtered back projection
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38. Filtered back projection
CONVOLUTION/FILTER

39. Filtered back projection
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40. Limitations of filtered back projection
 In FBP reconstruction, it is assumed that
 the x-rays travel in straight lines,
 the x-ray photons all have the same energy, and
 the x-ray intensity attenuates exponentially in the body (Beer's Law)
 the x-ray source is an infinitely small focal spot, and that the x-ray interactions occur along a line
between the focal spot and the geometric center of the detector element, rather than
continuously throughout the patient slab being imaged and the whole detector element.
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41. Disadvantage of FBP
 The disadvantages of FBP are that it assumes that the sinogram represents a perfect representation
of the object being imaged, and the filtering step amplifies noise in the acquired signal, producing
the characteristic mottled axial images.
 Any deviation in the acquisition data can produce dramatic image artefacts, such as streak or
banding artefacts.
 Degradation of IQ by high noise, poor low-contrast detectability, and possibly by artifacts resulting
from photon starvation (e.g. streaking) if photon statistics at the detector are very low, e.g. when
imaging obese patients or at low radiation dose.
 Therefore, FBP offers negligible dose reduction potential.
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42. Iterative Reconstruction
 Filtered back projection was the primary method for reconstructing CT images for many decades;
however, advancements in algorithm design coupled with fast computer hardware have led to the use
of clinically useful iterative reconstruction algorithms for CT images.
 Unlike the FBP algorithm, which is based on the theoretical inversion of the Radon transform, IR starts
with an initial guess of the object and iteratively improves on it by comparing the synthesized
projection from the object estimate with the acquired projection data and making an incremental
change to the previous guess.
 It was first described in 1970 as promising method to generate clinical image with low noise.
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43. Iterative Reconstruction
 Hounsfield used this technique in his first EMI brain Scanner.
 IR for MDCT can achieve superior image quality to its analytical counterparts, such as FBP, and at a
similar or even lower radiation dose level.
 IR techniques have been available since the advent of the first CT scanners, but these initial algebraic
reconstruction techniques (ART), and simultaneous iterative reconstruction techniques (SIRT) were
quickly dropped in favor of FBP due to the excessively large computation time required to reconstruct
images.
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44. Iterative Reconstruction
Generic Steps:
 Assumption : (for example all points in the matrix have same value)
 Comparison: (with the measured values)
 Correction(to bring the two into agreement)
 Repetition(of the process until the assumed and measured values are the same or within acceptable limit
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45. Iterative Reconstruction
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46. Iterative Reconstruction
 This method is also called successive approximation method or correction method and image is formed
by 3 ways
 Simultaneous reconstruction
 Ray by Ray correction
 Point-point correction
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47. Iterative Reconstruction
 Simultaneous reconstruction:
 all projections of the entire matrix are calculated at the beginning of the iteration and all
corrections are made simultaneously for each iteration.
 Ray –by-Ray Correction:
 one ray sum is calculated and corrected and these corrections are incorporated into future ray
sums , with the process being repeated for every ray in each iteration.
 Point-by-point correction:
 The calculations and corrections are made for all rays passing through one point , and these
corrections are used in ensuring calculations , again with the process being repeated for every
point.
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48. Iterative Reconstruction
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TYPES
• Iterative algorithm without statistical modelling (Hybrid)
• used originally by Godfrey Hounsfield, however not commercially used due to the inherent limitations of
microprocessors at that time
• Characterized by iterative filtration of data separately performed in projection space and/or in image space.
• The transition from projection to image space, i.e. the actual image reconstruction, usually relies on FBP
.
• only implements modeling of photon statistics
• Iterative algorithm with statistical modelling (Model Based)
• takes into account
• geometry of the CT system, (distances between X-ray tube, iso-center, and detector, and of its components,
(shape and size of the focal spot and the detector elements, as well as to beam and detector geometry).
• noise (photon statistics, photon spectrum emitted by the X-ray tube, the statistical distribution of photons
data acquisition geometry
• object (radiation attenuation)

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50. Iterative Reconstruction
 While clinical scanners operated with FBP, the CT research community spent a significant effort into
the development of advanced IR algorithms, with the goal to enable low-dose CT with high
diagnostic quality.
 Studies showed improved image quality and diagnostic value with IR compared to FBP. Radiation
dose can be reduced with IR by 23 to 76% without compromising on image quality.
 Recent review of 1616 articles (2015) dealing with clinical use of iterative reconstruction concluded
that both subjective and objective measures of image quality were the same or improved without
reported diagnostic compromise compared to older techniques. However, radiologists needed time
working with iterative reconstruction images to become accustomed to the different look and to
gain confidence in the diagnostic capability. Typically, after about 90 days, many radiologists hardly
notice the difference in image appearance.
 Radiation dose can be reduced further with model based IR compared to hybrid IR and FBP
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51. Iterative Reconstruction
 In 2009, the first IR algorithm ;IRIS (iterative reconstruction in image space, Siemens
Healthineers) received FDA clearance.
 Within 2 years, four more advanced IR algorithms received FDA clearance: ASIR (adaptive
statistical iterative reconstruction, GE Healthcare), SAFIRE (sinogram-affirmed iterative
reconstruction, Siemens Healthineers), iDose4 (Philips Healthcare), and Veo (GE Healthcare). The
first three methods are so-called hybrid IR algorithms
 Veo was the first clinical fully iterative IR algorithm, which is one of the most advanced
algorithms so far.
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52. Iterative Reconstruction
Advantage :
 It has many advantage compare with FBP , important physical factors including focal spot and
detector geometry, photon statistic, beam spectrum and scattering can be more accurately
incorporated into IR.
 artifact correction :higher order beam hardening correction
:cone beam artifact correction
:scatter correction
 It is better than FBP in metal artifact reduction
 Better image with reduced dose is possible
 Low image noise
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53. Iterative Reconstruction
 Disadvantage
 It requires immense computing power . longer recon time
 Larger number of iteration is required so time consuming
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54. Iterative Reconstruction
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The process of backward and forward projection is repeated until the difference between
true and artificial raw data is minimized.

55. Iterative Reconstruction
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An example of the clearer images enabled by iterative reconstruction using Philips iDose4
software. The software can enhance the sharpness of the stent and reduce metal and calcium
blooming artifacts to give a clearer view of the coronary artery lumen.

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Standard FBP (a) and iterative (b) reconstructions at the same level of the ascending aorta.
Image noise expressed as the standard deviation of the attenuation (HU) in the region of
interest was significantly lower in images reconstructed using IR (SAFIRE; Siemens)
(circle in b) than in those reconstructed using FBP (circle in a)

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60. Parallel vs Fan-beam
 Every ray-sum in fan-beam “sinogram” has equivalent
point in parallel-beam sinogram
 The red ray in both the parallel and diverging
configurations are the same, and therefore occupy the
same point in sinogram space.
 This behaviour allows us to interpolate the diverging-ray
data into a parallel ray sinogram.
 For fan beam reconstruction, the previous parallel beam
algorithms may be applied after we reformat fan beam
projections into parallel beam projections. This
reformatting process is called rebinning
 Interpolate div ray projections into parallel-beam
sinogram
 Perform reconstruction as if data were collected in
parallel-beam geometry
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61. IMAGE RECONSTRUCTION IN SPIRAL CT:
 Reconstruction Of spiral CT image is the same as that for conventional CT except for interpolation.
 Interpolation is the computation of an unknown value using known values on either side.
 A transverse planar image can be reconstruction at any position along the axis of the pt. i.e. z- axis.
 Data interpolation is performed by a special computer program called an interpolation algorithm.
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Z-AXIS
KNOWN
DATA
KNOWN
DATA
INTERPOLATED
DATA

62. IMAGE RECONSTRUCTION IN SPIRAL CT:
 During Spiral or Helical CT image data are received continuously but when an image is reconstructed,
the plane of image does not contain enough data for image reconstruction so the data needed for
image reconstruction estimated by a special computation method known as ‘Interpolation Algorithm’
which meant the estimation of value b/w two known value.
 The ability to reconstruct an image at any Z- Axis position is due to interpolation.
 Most modern scanners operate in a helical or spiral mode where the x-ray tube and detector system
rotate continuously during data acquisition as the patient table moves through the scanner. Under
these conditions, the projections are not collected on a slice-by-slice basis, and so the reconstruction
techniques described earlier cannot be used directly. However, virtual projections, (or a virtual
sinogram) can be constructed for each required reconstructed slice by suitable interpolation from the
adjacent projections.
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63. INTERPOLATION ALGORITHM
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64. INTERPOLATION ALGORITHM
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180ºlinear interpolation
360º linear interpolation
Results in improved Z-axis
resolution & greatly improved
reformatted image
Blurring in the reformatted
image
Interpolation of the value
separated by180º,half a
revolution of the x-ray tube
The plane of reconstructed
Image interpolated from
Data acquired in one
revolution apart
Interpolation Algorithm

65. INTERPOLATION ALGORITHM
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1800 LINEAR INTERPOLATION:
• Interpolation of the value separated
by180º,half a revolution of the x-ray tube
Results in improved Z-axis resolution
and greatly improved reformatted
image.
• It results thinner slice than 3600
interpolation
It results more noiser image than 3600
interpolation.
• 20% higher noise than conventional
CT.
Allows scanning at higher pitch.
• Less Broadens sensitivity profile.
3600 LINEAR INTERPOLATION:
• The plane of reconstructed image
interpolated from data acquired in
one revolution apart
Blurring in the reformatted image
• Thicker slice than 1800
interpolation.
Less noiser image than 1800
interpolation..
• 20% less noise than conventional
CT.
Allows scanning at lower pitch than
1800 interpolation.
• Broadens sensitivity profile.

66. INTERPOLATION ALGORITHM
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Quality 3600 1800
Z- Axis Resolution Low High
SSP Broaden Narrow
Spatial Resolution Low High
Noise Low High
Pitch Allow Scanning
Pitch> 1
Allow Scanning
Pitch ≤ 1

67. INTERPOLATION ALGORITHM
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68. Cone-Beam Algorithms
 Fan beam approximation algorithms require that the data be consistent, that is, the x-ray beam from
the tube to the detector and the section being imaged must be in the same plane.
 This is no longer the case for large cone angles characteristic of MSCT systems with larger than four
detector rows
 The fan beam approximation algorithms are not very
accurate used with the new generation of MSCT
scanners, so other image reconstruction algorithms
are needed. These algorithms are called cone-beam
algorithms, and they have been developed to eliminate
the cone-beam artifacts
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69. Cone-Beam Algorithms
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70. Cone-Beam Algorithms
 Several cone-beam algorithms have become available for use with the new generation of MSCT
scanners, and they basically fall into two classes:
 exact cone-beam algorithms and
 approximate cone beam algorithms
 Exact algorithms for cone-beam data have not been successful in recent times, they are also
“computationally complex and difficult to implement”
 However, approximate cone-beam algorithms are successful and fall into two categories: 3D and 2D
algorithms. Two such algorithms that have become common place in MSCT scanners are as follows:
 1. Feldkamp-Davis-Kress (FDK), also simply referred to as the Feldkamp-type 3D algorithm
 2. Advanced single-slice rebinning (ASSR) 2D approiximate algorithm
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71. Cone-Beam Algorithms
Feldkamp-Davis-Kress (FDK) Algorithm:
 FDK algorithm simply is an extension of the 2D FBP for fan-beam geometry into a 3D FBP for cone-
beam geometry
 This algorithm is more extensive than the 2D FBP algorithm, and it is more commonplace for use with
cone-beam CT scanners
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72. Cone-Beam Algorithms
Two-Dimensional Approximate Algorithm:
 The main goal of the 2D approximate algorithms “is to provide an image quality close to that of a
three dimensional reconstruction algorithm using two dimensional and back projection methods
 It is based on the notion of rebinning, a term used to describe the resorting of the 3D data collected
from the cone-beam acquisition (geometry) to a set of 2D fan-beam projection data and,
subsequently, use the conventional 2D FBP algorithm to reconstruct transaxial images
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73. Cone-Beam Algorithms
Two-Dimensional Approximate Algorithm:
 Some of the algorithms in this category are:
 Adaptive multiplane reconstruction (AMPR) algorithm
 weighted hyperplane reconstruction algorithm
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Validation of deep
learning–based
reconstruction (DLR)
applied to human
upper abdominal (a,
b) and pelvic
images (c, d). (a) 120
kVp, 35 mAs. (b) 120
kVp, 52 mAs. (c) 120
kVp, 157 mAs. (d) 120
kVp, 175 mAs. Window
level/window width
(WL/WW) = 40/400
HU.

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Comparison of filtered back
projection (FBP) and deep
learning–based
reconstruction (DLR) images
of metal and truncation
artifacts. The window
level/window width (WL/WW)
= 60/350 HU for metal and
40/400 HU for truncation
artifacts.

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80.  Bushberg., 2012. Essential Physics of Medical Imaging, The. Lippincott, Williams & Wilkins.
 Seeram, E. Computed tomography: Physical Principles, Clinical Applications, and Quality.
 Dowsett DJ, Kenny PA, Johnston RE. The physics of diagnostic imaging. 2nd ed. London, England:
Hodder Arnold; 2006.
 Deep Learning–based CT Image Reconstruction: Initial Evaluation Targeting Hypovascular Hepatic
Metastases ;Yuko Nakamura et al
 https://www.dspguide.com/ch25/5.htm
 https://www.12000.org/my_notes/EE518_CT_project/REPORT/index.htm
 https://www.imaios.com/en/e-Courses/e-MRI/The-Physics-behind-it-all/spatial-frequency
 IMAGE RECONSTRUCTION. ALGORITHM-A SET OF RULES OR DIRECTIONS FOR GETTING SPECIFIC OUTPUT
FROM SPECIFIC INPUT. - [PPT Powerpoint] (cupdf.com)
 Filtered BackProjection (FBP) Illustrated Guide For Radiologic Technologists • How Radiology Works
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Reference

81. Reference
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