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Image reconstruction: Part 1 – understanding filtered back projection, noise and image acquisition.
Schofield R1, King L2, Tayal U1, Castellano I2, Stirrup J3, Pontana F4, Earls J5, Nicol
E1
1Department of Cardiovascular CT, Royal Brompton Hospital, London, UK
2Joint department of Physics, The Royal Marsden, London, UK
3 Department of Cardiology, Royal Berkshire Hospital, Reading, UK
4 Department of Cardiovascular Imaging, Lille University Hospital, France
5 George Washington University Hospital, Washington DC, USA
Keywords: Cardiovascular CT, Filtered Back Projection, Sinogram, Iterative
Reconstruction
Authors
Rebecca Schofield MBChB [email protected]
Laurence King MSc [email protected]
Upasana Tayal PhD [email protected]
Elly Castellano PhD [email protected]
Jim Stirrup MD(Res) [email protected]
Francois Pontana PhD [email protected]
James Earls MD [email protected]
Ed Nicol* MD [email protected]
*corresponding author: Dr Ed Nicol, Department of Cardiovascular CT, Royal
Brompton Hospital, London, SW3 6NP, UK. Tel: 02073528121, email:
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Word count: 3697
Conflict of Interest: The authors have no conflicts of interest to declare.
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Abstract
Image reconstruction is an increasingly complex field in CT. Iterative Reconstruction
(IR) is at present an adjunct to standard Filtered Back Projection (FBP)
reconstruction, but could become a replacement for it. Due to its potential for
scanning at lower radiation doses, IR has received a lot of attention in the medical
literature and all vendors offer commercial solutions. Its use in cardiovascular CT
has been driven in part due to concerns about radiation dose and image quality.
This paper is the first manuscript of a pair. It aims to review the basic principles of
CT scanning, to describe image reconstruction using Filtered Back Projection, and to
identify the physical processes that contribute to image noise which IR may be able
to compensate for. The aim is to enable cardiovascular imagers to understand what
happens to the raw data prior to the reconstruction process so they may have a
better appreciation of the strengths and weaknesses of the various reconstruction
techniques available.
The second manuscript of this pair will discuss the various vendor permutations of IR
in more detail, including the most recent machine learning based offerings, and
critically appraise the current clinical research available on the various IR techniques
used in cardiovascular CT.
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Introduction
The future of CT will, almost inevitably, involve shorter scanning times and reduced
radiation dose (following the “As low as reasonably achievable (ALARA) consistent
with the imaging task” principle). The radiation dose required to image the patient is
determined by the degree to which image noise can be tolerated by the imager: the
lower the image noise required, the higher the radiation dose that must be used. The
standard method of image reconstruction in CT is known as Filtered Back Projection
(FBP). FBP is computationally efficient, but the filtering step enhances the noise in
the CT image. Iterative reconstruction (IR) techniques have existed since the origins
of CT scanning1 but a lack of computing power initially prevented their use in
standard clinical practice. By virtue of the iterative nature of the reconstruction, IR
images are less noisy than FBP ones. Thus IR potentially enables reduced radiation
dose imaging compared to FBP, although it is not in itself a dose reduction tool.
Improvements in processing power have enabled IR to be re-introduced into clinical
imaging over recent years. The implementation of IR has involved intense
commercial competition and vendors have developed specific proprietary
techniques. This hinders detailed understanding of the modelling processes used
and, as a result, IR techniques are often difficult to understand with confusing
nomenclature adding to the challenge for clinicians.
Undoubtedly cardiovascular CT has benefited greatly from advances in CT
technology such as prospective ECG gating and ECG based tube modulation to
reduce doses, and weight/BMI based reductions in tube voltage. The cardiovascular
CT community is primed to accept the next technological advancements. Although
IR techniques are deemed acceptable in thoracic imaging2 -9, we would advocate
caution before transferring these findings into routine cardiovascular CT clinical
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practice, particularly when the imaging tasks are different, such as the assessment
of non-calcified plaque (NCP) burden in coronary arteries. The ability of IR
techniques to detect minor irregularities in small structures (such as coronary
arteries) is an example of an area that requires robust validation and has been
identified as a potential pitfall of several IR techniques10.
To understand IR techniques, it is essential to have a robust understanding of the
basic imaging physics of CT. To highlight what the different IR techniques offer and
‘do’ to the images we first present a basic introduction to the physics of CT imaging,
an explanation of FBP reconstruction techniques, and a review of sources of image
noise.
Physics of CT data acquisition
Within the CT x-ray tube, electrons are emitted from a cathode and accelerated
across a vacuum to a target anode by a high voltage (the tube voltage or kV)
between the cathode and anode. Typically, in CT the tube potential ranges between
70-140kV. The number of electrons accelerated towards the anode determines the
tube current (mA). The electrons strike the anode at the focus and release their
energy as heat and x-ray photons. Due to the various physical processes involved,
and the stochastic nature of the electrons interacting in the target anode, the x-ray
photons are emitted with a range of x-ray energies described by the x-ray spectrum
(Figure 1). The peak energy of the spectrum is determined by the set tube voltage.
Lower energy x-rays are removed from the x-ray beam by physical filters before
reaching the patient; this is desirable as lower energy x-rays have a high probability
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of attenuation in the patient before reaching the detector and so contribute to patient
dose but not the image.
Pre-patient collimators restrict the shape and direction of the ‘primary’ x-ray beam
emanating from the x-ray focus so that it just covers the active detector array, thus
reducing unnecessary irradiation of areas not being imaged. Post-patient anti-scatter
grids absorb x-rays that have undergone scatter in the patient and have diverged
from the path of the primary x-ray beam, as these x-rays would provide erroneous
positional and attenuation information.
X-ray photons that are transmitted through the patient are absorbed in the CT
detector. All modern CT detectors are made from scintillators (i.e. materials that
release light photons via fluorescence following excitation by x-ray photons). The
current generation of commercial detectors are made from rare earth ceramic
materials. An ideal detector has high atomic density to give a high x-ray photon
interaction probability, is predictable (releases a number of light photons linearly
proportional to the x-ray intensity over a wide range) and is rapidly responsive (has
an instantaneous fluorescent response with no after-glow). The light photons are
then converted to electrical signal in photodiodes. This electrical signal is digitised
and collected to form the raw projection data.
A CT detector array consists of a series of rows of detector elements aligned axially
across the patient. There are typically 16-320 detector rows, and 800-1000 detector
elements in each detector row. During a CT acquisition some or all of the detector
rows may be active. The individual detector elements do not respond in a perfect
way to x-ray photons and individual ones will have slightly different responses to x-
ray signal. This may be corrected by calibration under controlled conditions. In the
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future, photon counting detectors, currently in development, may overcome some of
the shortfalls of use of scintillators.
On modern CT scanners, the x-ray source and detector array are mounted opposite
one other in a rotating gantry. During a CT acquisition, the gantry rotates around the
patient whilst the active detector elements are sampled about 1000 times per
second. Each time the detector is sampled a projection is acquired. The projection is
akin to an x-ray of a thin slab of the patient (Figure 2). In order to reconstruct the CT
images, a set of projections acquired over a little more than half a gantry rotation, or
a full gantry rotation, is needed.
In order to scan the required anatomy, the patient is moved through the gantry.
During axial or ‘step and shoot’ scanning the gantry makes one revolution around the
stationary patient while projection data is acquired, followed by an incremental table
movement, and this repeats until the entire desired anatomy is imaged. During
helical scanning, the scanner continuously acquires projection data as the patient
table moves through the gantry. Helical scanning allows a volume scan to be
acquired much faster. In addition, helical scanning allows axial images to be
reconstructed at overlapping intervals. The scan pitch is defined as the table
movement for a 360-degree rotation of the x-ray source divided by the nominal x-ray
beam width. Typical ranges of clinical pitches are 0.7 - 1.4, although 0.1 – 3.2 are
possible using specialist CT scanners. During dynamic scanning, the scanner
continuously acquires projection data at one patient table position. This allows the
selected patient anatomy, e.g. the heart, to be scanned over a period of time.
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CT Image Reconstruction
Line integrals, attenuation profiles and the sinogram
When an X-ray beam passes through a patient, the intensity of the beam decreases
exponentially with distance travelled due to the x-ray photons interacting with the
tissues in the body (Beer’s Law). On exiting the patient, the remaining x-ray photons
are absorbed by the CT detector array and are converted into an electronic signal.
The attenuation of the x-ray beam along a given ray path is determined by the sum
over all the tissues along that ray path of the product of the length of the ray path
through each tissue (x) and the effective linear attenuation coefficient () of that
tissue averaged over the x-ray spectrum. This is called a line integral. The x-ray
projection (or attenuation profile) is made up of the set of line integrals along all the
ray paths in the x-ray beam. Figure 2 illustrates two x-ray projections of an
anthropomorphic chest phantom at right angles to each other. The resultant
attenuation profiles appear quite different.
The CT raw data for an axial slab through the patient is a set of x-ray projections
acquired at multiple gantry angles for just over half a gantry rotation or for a full
gantry rotation. The CT raw data for an axial slab is thus the full set of line integrals
through the patient for all ray paths in the x-ray beam at all gantry angles. This is
called the Radon Transform.
In modern CT, attenuation profiles are typically acquired a thousand times per 360°
rotation of the gantry. The sinogram is a way of showing the raw acquisition data as
a function of gantry angle in a two-dimensional matrix. Figure 3 illustrates the
sinogram for a single detector row. The x-axis of the sinogram represents the gantry
angle. The y-axis shows the attenuation profile measured at that gantry angle. Each
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point in the attenuation profile is provided by a detector element along the detector
row. For a given coordinate (x,y) in the patient, the contribution of that coordinate to
each attenuation profile will manifest itself as a sinusoidal signal. This behaviour
gives the name to this particular representation of the raw acquisition data. Note that
the sinogram can be thought of as the graphical representation of the Radon
transform. The sinogram for a modern CT scanner with a multi-row detector will be a
three-dimensional matrix, with the z-axis showing the sinogram from each active
detector row.
Filtered Back Projection
The mathematical problem of recreating a 2D image from a series of line integrals is
complex and beyond the scope of this manuscript. For the interested reader, the
mathematical challenges are discussed with relation to tomographic imaging by
Cormack11 and other authors. Instead, we use words and pictures to explain how the
standard CT reconstruction algorithm, Filtered Back Projection, works.
In simple Back Projection (BP), the attenuation profile at each gantry angle is ‘back-
projected’ across image space. The attenuation value in the sinogram is divided by
the number of image pixels along the direction of the projection from x-ray source to
detector, and the average attenuation value thus obtained is assigned to these
pixels; if the pixel values were to be added up along the direction of the projection,
the original attenuation value in the sinogram would be recovered. This is repeated
for each gantry angle. The final back-projected image is then the sum of all the back-
projected attenuation profiles. The simple BP process is graphically represented in
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Figure 4. Whilst simple BP is conceptually straight forward to understand, it does not
give clear images.
The BP ‘blur’ can be counteracted by applying a spatial frequency filter to the
attenuation profiles prior to back projection. The same steps of image reconstruction
are shown in Figure 5, but this time a simple ramp filter has been applied to the
attenuation profiles making up the sinogram before back-projection. This is Filtered
Back Projection. Applying the mathematical filter to data before back-projection is
computationally easy to do in a step-wise fashion as each new attenuation profile is
acquired. This is the reason that filtered back projection can be used to reconstruct
axial CT images in very short times: large volume scans take tenths of a second to
reconstruct on modern scanners.
The ramp filter is a mathematical function that suppresses low spatial frequency
components of the attenuation profiles. Rapid spatial changes in the attenuation
profile, for example where a boundary between high and low density anatomical
structures exists, are enhanced. This has the effect of suppressing blurring and
enhancing edges in the image, as is observable in Figures 4 and 5. However, this
filter also enhances image noise, which exists in the raw signal primarily at high
spatial frequencies. This gives FBP CT images their characteristically mottled or
speckled appearance.
The ramp filter is mathematically mandated to remove the blur from back-projection,
but it is possible to combine it with different strength filters (kernels) to enhance the
spatial resolution of the final image to varying degrees depending on the application
(e.g. high-resolution vascular kernels, or smoother soft tissue kernels); greater image
noise is the penalty of achieving higher spatial resolution.
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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). It is also assumed that 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 centre of the detector element, rather than continuously
throughout the patient slab being imaged and the whole detector element.
The CT image
The CT image is a two-dimensional attenuation map of the imaged slab through the
patient. Each point in the map is assigned a linear attenuation coefficient relative to
water, which is defined as the Hounsfield Unit (HU). The Hounsfield Unit for air and
water are -1000 and 0 respectively by definition. The image typically consists of a
matrix of 512×512 pixels.
The CT image is not a perfect rendition of the linear attenuation coefficients of the
patient’s anatomy. There is an uncertainty associated with each pixel value arising
from the x-ray detection process, the assumptions made about the interactions of x-
rays with the body, and the image reconstruction process. The uncertainty can
manifest itself as local fluctuations in CT number, referred to as image noise, or
artefacts such as streaks or banding. However manifested, such image noise is
unwelcome because it reduces the quality of the CT image, potentially rendering it
non-diagnostic.
Image noise is usually measured as the standard deviation of CT numbers (in
Hounsfield units) in a region of interest. Noise can also be evaluated by a more
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subjective “quality” or “texture” which is more difficult to quantify but which can be an
issue with higher order iterative reconstruction.
Contributions to image noise from the x-ray detection process
There are several mechanisms that introduce noise within the data acquisition
process. In basic terms image noise can be divided into ‘statistical noise’, ‘structural
noise’ and ‘electronic noise’ which are discussed in detail below.
Statistical Noise
Statistical noise is caused by the fluctuation in the raw x-ray signal due to the
probabilistic nature of x-ray production at the x-ray tube, attenuation of x-rays by the
patient, and absorption in the detector. These variations arise from the “quantum”
nature of x-rays and so statistical noise is also known as ‘quantum noise’. The
statistical variation is inversely proportional to the square root of the number of x-ray
photons. The noise component therefore becomes a smaller proportion of the signal
when the number of x-rays increases. Increasing the tube potential or tube current-
time product (mAs) increases the number of x-ray photons produced and as a result
the statistical noise is reduced.
Structural Noise
Structural noise is typically created by variations in the structure of the detector and
the differing response of individual elements. Structural noise is not usually a major
component of image noise and is minimised by regular calibration of the detector
under controlled conditions.
Scattered X-rays
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Interactions of the x-ray photons within the body alter both the energy and direction
of travel of photons within the patient. This creates scattered x-rays. Scattered
photons create spurious signal if they are absorbed by the detector, adding to the
image noise. More scatter is created by low-density objects (i.e. those with a low
atomic number, such as fat). The amount of scatter produced increases with the tube
potential. Focused anti-scatter grids at the detector reduce scatter and are routinely
used in CT.
Electronic noise
Electronic circuits in the detector system are the main cause of electronic noise. This
noise carries no diagnostic information and is a constant low-amplitude signal,
independent of the x-ray signal, so that is only a significant contributor to image
noise when the amplitude of the measured x-ray signal in the detector is also low.
This occurs when the number of x-rays reaching the detector is low, for example in
very low dose protocols or when scanning obese patients who provide significant
attenuation of the x-ray beam.
The electronic noise can be minimised by reducing the amount of electronic coupling
between the photodiode and the digital read-out. Detectors featuring a fully
integrated electronic system (integrated detectors) in which the scintillator,
photodiode, pre-amplifier and analogue-to digital converter are all in the same chip
have been shown to have significantly reduced noise and superior image quality12.
The main constraints to universal use of single chip systems are manufacturing
ones.
To demonstrate these types of noise visually, Figures 6a – 6d were created from
simulated raw projection data from an axial CT image of an anthropomorphic chest
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phantom. The data was manipulated before filtered back projection to demonstrate
the presence of different types of noise.
Contributions to image noise from the image reconstruction process
In FBP, the projection data is filtered using convolution techniques. These are
computationally efficient, but convolution enhances the differences between
neighbouring data points in the attenuation profile, and hence magnifies noise in the
data. This noise is propagated through the reconstruction process into the final
image. Image noise and image texture are both highly dependent on the
reconstruction kernel selected: smooth kernels suppress noise at the expense of
spatial resolution, and sharp kernels enhance small and linear features such as
coronary arteries or stents but also increase image noise.
Contributions to image noise from beam hardening and photon starvation
In order to reconstruct axial images using FBP, it is assumed that the x-ray beam
has a constant effective photon energy along the ray path through the patient. In
reality the x-ray beam contains x-rays of many different energies (i.e. a spectrum of
x-ray energies). The attenuation in the patient is preferentially greater for lower
energy x-rays. The x-ray beam spectrum further along the ray path therefore
preferentially contains higher energy x-rays with greater penetration (a ‘harder’ x-ray
beam), resulting in lower attenuation and lower measured CT numbers along the
remaining beam path. Attempts to overcome this problem include physical pre-
filtering of the x-ray beam to obtain a pre-hardened beam, or additional post-
processing of the FBP image. In cardiovascular CT, beam hardening causes
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artefacts adjacent to densely calcified atherosclerotic plaques, prosthetic heart
valves, intravascular wires, implanted devices and intravascular stents. The size and
shape of the dense object becomes distorted as regions of artificially low attenuation
that can be misinterpreted as low attenuation plaques or intravascular thrombus can
be created. Beam hardening accounts for much of the false positive findings in CT
coronary angiography.
When x-rays pass through an object of high attenuation, i.e. large bones or metallic
devices such as permanent pacemakers or implantable cardiac defibrillators, the
resultant x-ray signal at the detector can become so low that there is essentially no
information to contribute to the image from that projection; this photon starvation can
result in a streak artefact with complete loss of signal downstream of the increased
attenuation structure making images difficult to interpret. This can account for false
positive perfusion defects in the myocardium adjacent to the dense thoracic vertebral
bodies, decreased attenuation within a coronary stent, or even HU changes adjacent
to high concentration of iodine enhanced blood pool in the great vessels or cardiac
chambers.
Challenges of FBP and how we can address them
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 that
we are used to. Statistical noise aside, any other aberrations in the acquisition data
can produce dramatic image artefacts, such as streak or banding artefacts.
However, although these FBP images can be quite grainy and noisy, radiologists
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who have ‘grown up’ looking at them often feel more comfortable reviewing these
images rather than IR ones even as the concern for patient radiation dose has driven
the acceptable signal-to-noise ratio progressively lower and lower. In part two of this
paper, we will examine Iterative Reconstruction techniques, which provide the
opportunity to reduce image noise for a given radiation dose by manipulating the raw
acquisition data and / or processing the reconstructed image and thus increase the
acceptability of low dose CT images.
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sinogram-affirmed iterative reconstruction versus standard dose filtered back
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McCollough CH.Electronic Noise in CT Detectors: Impact on Image and Noise
Artefacts Duan X et al AJR 2013);201(4)626-632.
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Figure 1: Filtered X-ray Bremsstrahlung spectra produced by X-ray tube potentials of 50 and 100 kV
keV= kilo-electron volts, energy of photons.
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Figure 2: the charts show the integrated attenuation profile (arbitrary units) of an
anthropomorphic chest phantom against detector position, for an AP and lateral
projection respectively. The attenuation profiles appear to be quite different in these
directions, although some rotational symmetry exists. Ray paths (a) and (b) for each
projection correspond to a low and high attenuation signal respectively.
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Figure 3: Two example CT slices and their simulated sinograms over a 180°
acquisition are given. The first image is a cylindrical water phantom with simulated
high and low-attenuation objects: a uniform cylindrical phantom looks identical from
all projection angles but the simulated objects in this example move position relative
to one another, giving a sinogram with sinusoidal traces. The second image is an
anthropomorphic phantom with clinical detail, with more complex attenuation profiles
and thus a more complex sinogram.
Attenuation
Projection
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Figure 4: Simple Back projection (a) a slice of an anthropomorphic chest phantom
is shown, with a single attenuation profile in the AP direction. When this single profile
is projected back across reconstructed image space, the projection image (b) is
formed. Image (c) is a back-projection of two perpendicular projection profiles; image
(d) is formed from back-projection of four profiles; (e) from eight profiles; and (f) from
720 profiles. At this point the image is starting to resemble the subject that was
imaged, but with considerable blurring. The simple back-projection process
inherently produces blurred images.
fed
cba
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Figure 5: Filtered back projection - taking the same attenuation profiles from an
imaged slice of an anthropomorphic chest phantom (a), the corresponding filtered
back projected images to the simple back projected images in Figure 4 (b-f) are
shown here, created from (b) one, (c) two, (d) four, (e) eight and (f) 720 filtered back-
projected attenuation profiles. The filtering process recovers spatial resolution in the
final image.
fed
cba
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Figure 6: Filtered Back Projection (FBP) reconstruction (a) was recreated from raw
data generated from a CT image of an anthropomorphic phantom with no added
noise. Reconstruction (b) was carried out after addition of Gaussian noise to the raw
data, simulating increased statistical noise in the acquisition. Figure (c) was created
after altering a single data point in the raw data to a value of zero, representing the
case where a single detector element has failed for a single projection during
acquisition, resulting in a line artefact (high-lighted by white arrow). Figure (d) was
created after changing just 5% of the raw data points to minimum or maximum
values (“salt & pepper noise”), indicating a large degree of structural noise.
a b
c d
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