ASM Metals HandBook Vol. 17 - Nondestructive Evaluation and Quality Control
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Fig. 31
The process of filtering according to the convolution theorem. Filtering operations can be performed as
a convolution (*) of the spatial functions, (top) or as a multiplication (×) of
the Fourier transform of these
functions (bottom). Either technique can be used in the filtered-backprojection reconstruction process.
Because the blurring or convolution process is represented in the spatial frequency domain as a simple functional
multiplication, the blurred image conceptually can be easily restored to a closer representation of the original object
distribution. If the Fourier transform of the image, G( ), is filtered by multiplying by the function H'( ) where H'( ) =
1/H( ), the result is the original object frequency distribution, F( ). In practice, this restoration is limited by the
frequency limit of H( ) leading to division by zero, and by the excessive enhancement of noise along with the signal at
frequencies with small H( ) values. This restoration process or "deconvolution" of the blurring function can likewise be
performed as a convolution in the spatial domain as illustrated in Fig. 30.
Backprojection is the mathematical operation of mapping the one-dimensional projection data back into a two-
dimensional grid. This is done intuitively by radiographers in interpreting x-ray films. If a high-density inclusion or
structure is visible in two or more x-ray films taken at different angles, the radiographer can mentally backproject along
the corresponding ray paths to determine the intersection of the rays and the location of the structure in space.
Mathematically, this is done by taking each point on the two-dimensional image grid and summing the corresponding
projection value from each angle from which projection data were acquired. This backprojection process yields a
maximum density at the location of the structure where the lines from the rays passing through this structure cross (Fig.
3a). The resultant image is not an accurate representation of the structure, however, in that these lines form a star artifact
(Fig. 3a) extending in all directions from the location of the high-density structure. For a very large number of projection
angles, the density of this structure is smeared over the entire image and decreases in amplitude with 1/r, where r is the
distance from the structure. This simple backprojected image, f
b
, can be represented by the convolution of the true image,
f, with the blurring function, 1/r, or:
f
b
(r, ) = f(r, ) * (1/r)
(Eq 19)
With ideal data, this blurring function can be removed by a two-dimensional deconvolution or filtering of the blurred
image. The appropriate filtering function can be determined by using the convolution theorem to transform Eq 19 into its
frequency domain equivalent of:
F
b
( , ) = F( , ) (1/ )
(Eq 20)
where the function (1/ ) is the Fourier transform of 1/r in polar coordinates. Dividing both sides by (1/ ) yields:
F( , ) = F
b
( , )
(Eq 21)
Therefore, the corrected image, f, can be obtained by determining a simple backprojection, f
b
, taking its Fourier
transform, filtering, and then taking the inverse Fourier transform to get back into spatial coordinates. Alternatively, the
filter function, , can be transformed into spatial coordinates, and the backprojected image can be corrected by a two-
dimensional convolution operation. Equations 19, 20, and 21 become somewhat more complex when evaluated in
rectangular coordinates rather than as represented in polar coordinates.
This approach tends to produce poor results with actual data. Filtering out this blurring function from the projection data
prior to backprojection, however, is quite effective and is the basis for the most commonly used reconstruction method,
the filtered-backprojection reconstruction technique.
Filtered-Backprojection Reconstruction. According to the central slice theorem discussed in the section "Direct
Fourier Reconstruction Technique" in this Appendix, the Fourier transform of the one-dimensional projection data
through a two-dimensional distribution is equivalent to the radial values of the two-dimensional Fourier transform of the
distribution. Consequently, the filtering operation performed in Eq 21 can be performed on the projection data prior to
backprojection. This is the conceptual basis for the filtered-backprojection reconstruction technique (Ref 52, 53), and it is
illustrated schematically in Fig. 3(b).
As with other filtering operations, this correction can be implemented as a convolution in the spatial domain or as a
functional multiplication in frequency domain. Fourier filtered backprojection is performed by taking the measured
projection data, Fourier transforming it into the frequency domain, multiplying by the ramp-shaped filter (which
enhances the high spatial frequencies relative to the low frequencies), taking the inverse Fourier transform of this
corrected frequency data, and then backprojecting the filtered projection data onto the two-dimensional grid.
If the filtering is performed in the spatial domain by convoluting the measured projection data with a spatial filter
equivalent to the inverse Fourier transform of , the process is often referred to as the convolution-backprojection
reconstruction method. The frequency filter has the shape of a ramp, enhancing the high frequency (or sharp detail
information) of the projection data. Therefore, the convolution function, or kernel, has the expected shape for a
sharpening filter. The positive central value is surrounded by negative values that diminish in magnitude with distance
from the center. Because of the similar results obtained by Fourier filtering and convolution filtering and the uncertainty
with which the filtering approach is often implemented on specific systems, the terms filtered-backprojection
reconstruction and convolution-backprojection reconstruction are sometimes used interchangeably.
The filtered-backprojection reconstruction technique is the method that is commonly used in both medical and industrial
tomography systems. The method is more tolerant of measured data imperfections than some of the other techniques.
Filtered-backprojection reconstruction provides relatively fast reconstruction times and permits the processing of data
after each view is acquired. With appropriate computational hardware, the displayed image can be available almost
immediately after all the data have been acquired.
Variations in the filter functions used result from the physical limitations encountered with actual data (particularly its
finite quantity) and the boundary assumptions made in the calculations. Additional windowing filters are sometimes
combined with the theoretically derived filter to smooth or sharpen the image with no additional processing time.
Another variation used in special situations consists of combinations of the reconstruction techniques, such as performing
a filtered-backprojection reconstruction followed by iterative processing. This may be beneficial in cases where the data
set is limited and additional known information on the object's shape and content can be applied through the iterative
steps to provide an improved image.
Iterative Reconstruction Techniques
Another broad category of methods is the iterative reconstruction algorithms. With this approach, an initial guess is made
of the density distribution of the object. This initial guess may result from knowledge of the nominal design of the object
or may assume a homogenous cylinder of some density. The computer then calculates the projection data values that
would be measured for this assumed object. Each calculated value is compared to the corresponding measured projection
data value, and the difference between these values is used to adjust the assumed density values along this ray path. This
correction to the assumed distribution is applied successively for each measured ray. An iteration is completed when the
image has been corrected along all measured rays, and the process begins again with the first ray. With each iteration, the
approximated distribution (or reconstructed image) improves its correspondence to the object distribution.
There are numerous variations of iterative processing that may be additive or multiplicative, weighted or unweighted,
restricted or unrestricted, and may specify the order in which the projection data are processed. Some of the more
common techniques are the iterative least squares technique, the simultaneous iterative reconstruction technique, and the
algebraic reconstruction technique (Ref 54, 55, 56).
Iterative reconstruction techniques are rarely used. They require all data to be collected before even the first iteration can
be completed, and they are very process intensive. Iterative techniques may be beneficial for selected situations where the
data set is limited or distorted. Iterative techniques can be effective with incomplete sets of projection data or with
irregular data collection configurations. Known information about the object, such as the design, material densities, or
external contours, can be used, along with optimization criteria, to aid in determining the solution. This can allow the
reconstruction process to be less sensitive to missing or inaccurate projection data. In addition, reconstruction-dependent
corrections can be incorporated into these techniques, such as spectral correction for multiple materials or attenuation
corrections in emission computed tomography (SPECT) of radionuclide distributions.
References cited in this section
48.
R. Brooks and G. DiChiro, Principles of Computer Assisted Tomography (CAT) in Radiographic and
Radioisotopic Imaging, Phys. Med. Biol., Vol 21, 1976, p 689-732
49.
G.T. Herman, Image Reconstruction From Projections: Implementation and Applications, Springer-
Verlag,
1979
50.
G.T. Herman, Image Reconstruction From Projections: The Fundamentals of Computerized Tomography,
Academic Press, 1980
51.
B.P. Flannery, H.W. Deckman, W.G. Roberge, and K.L. D'Amico, Three-Dimensional X-
Ray
Microtomography, Science, Vol 237, 18 Sept 1987, p 1439-1444
52.
G.N. Ramachandran and A.V. Laksh-minarayanan, Three-
Dimensional Reconstruction From Radiographs
and Electron Micrographs: III. Description and Application of the Convolution Method,
Indian J. Pure
Appl. Phys., Vol 9, 1971, p 997
53.
L.A. Shepp and B.F. Logan, The Fourier Reconstruction of a Head Section, Trans. IEEE, Vol NS-
21, 1974,
p 21-43
54.
R. Gordon, R. Bender, and G.T. Herman, Algebraic Reconstruction Techniques (ART) for Three-
Dimensional Electron Microscopy and X-Ray Photography, J. Theor. Biol., Vol 29, 1970, p 471-481
55.
P. Gilbert, Iterative Methods for the Three-Dimensional Reconstruction of an Object From Projections,
J.
Theor. Biol., Vol 36, 1972, p 105-117
56.
G.T. Herman and A. Lent, Iterative Reconstruction Algorithms, Comput. Biol. Med., Vol 6, 1976, p 273-
294
Industrial Computed Tomography
Michael J. Dennis, General Electric Company, NDE Systems and Services
Appendix 2: Computed Tomography Glossary
• Afterglow
• A detrimental property of a scintillator in which the emission of light continues for a period of time after
the exciting radiation has been discontinued.
• Algorithm
• A procedure for solving a mathematical problem by a series of operations. In computed tomography, the
mathematical process used to convert the transmission measurements into a cross-sectional image. Also
known as reconstruction algorithm.
• Aliasing
• An artifact in discretely measured data caused by insufficient sampling of high-frequency data, with this
high-frequency information falsely recorded at a lower frequency. See Nyquist sampling frequency.
• Aperture, collimator
• The opening of the collimator enabling variations in the shape of the measured x-ray beam.
• Archival storage
• Long-term storage of information. In computed tomography, this can be accomplished by storing the
information on digital magnetic tape, on optical disk, or as images on x-ray film.
• Array, data
• An ordered set or matrix of numbers.
• Array, detector
• An ordered arrangement of individual detector elements that operates as a unit, such as a linear detector
array.
• Array processor
• A special-purpose computer processor used to perform special functions at high speeds.
• Artifact
• An error or false structure in an image that has no counterpart in the original object.
• Attenuation of x-rays
• Absorption and scattering of x-rays as they pass through an object.
• Backprojection
• The process of mapping one-dimensional projection data back into a two-dimensional image array. A
process used in certain CT reconstruction algorithms.
• Beam hardening
• The increase in effective energy of a polyenergetic (for example, x-ray) beam with increasing attenuation
of the beam. Beam hardening is due to the preferential attenuation of the lower-energy, or soft, radiation.
• Capping
• Artifact characterized by dark shading or lowering of pixel values toward edges; opposite of cupping.
Sometimes caused by an overcompensating beam hardening correction to the data.
• CAT
• See computed tomography.
• Collimator
• The x-ray system component that confines the x-ray beam to the required shape. An additional collimator
can be located in front of the x-ray detector to further define the portion of the x-ray beam to be
measured.
• Computed tomography (CT)
• The collection of transmission data through an object and the subsequent reconstruction of an image
corresponding to a cross section through this object. Also known as computerized axial tomography,
computer-assisted tomography, or CAT scanning.
• Contrast-detail-dose diagram
• A plot of the minimum percent contrast needed to resolve a feature versus the feature size. The ability to
visualize low-contrast structures tends to be limited by image noise, while small high-contrast structures
are resolution limited.
• Contrast scale
• The change of CT number that occurs per given change in linear attenuation coefficient. Contrast scale is
dependent on the x-ray beam energy and the scaling parameters in the reconstruction process.
• Contrast sensitivity
• The ability to differentiate material density (or linear attenuation coefficient) differences with respect to
the surrounding material.
• Convolution
• A mathematical process used in certain reconstruction algorithms. An operation between two functions in
which one function is blurred or smeared by another function.
• Coronal plane
• A medical term for a plane that divides the body into a front and back section. A y-z planar presentation,
which may be mathematically constructed from a series of cross-sectional slices. See sagittal plane and
reformation, planar.
• Crosstalk, detector
• The unwanted pickup of the signal associated with neighboring detector elements. Crosstalk may be
caused by radiation scatter or electronic interference.
• CT numbers
• The numbers used to designate the x-ray attenuation in each picture element of the CT image.
• Cupping
• Artifact characterized by dark shading in the center of the field of view. May be caused by beam
hardening.
• Data acquisition system (DAS)
• The components of a CT system used to collect and digitize the detected x-ray signal.
• Density resolution
• The measure of the smallest density difference of an image structure that can be distinguished.
• Detectability, low contrast
• The minimum detectable object size for a specified percent contrast between the object and its
surroundings as measured with a test phantom.
• Detective quantum efficiency (DQE)
• The fraction of the beam intensity needed to produce the same signal-to-noise ratio as a particular
detector if it were replaced with an ideal detector. Also known as the quantum detection efficiency.
• Detector aperture
• The physical dimensions of the active area of the detector, including any restriction by the detector
collimation. Used in particular for the detector dimension in the plane of the CT scan.
• Detector, x-ray
• Sensor array used to measure the x-ray intensity. Typical detectors are high-pressure gas ionization,
scintillator-photodiode detector arrays, and scintillator-photomultiplier tubes.
• Digital radiography (DR)
• Radiographic imaging technology in which a two-dimensional set of x-ray transmission measurements is
acquired and converted to an array of numbers for display with a computer. Computed tomography
systems normally have DR imaging capabilities. Also known as digital radioscopy.
• Display resolution
• Number of picture elements (pixels) per unit distance in the object.
• Dual energy imaging
• The process of taking two identical scans (DR or CT) at two different x-ray energies and processing the
data to produce alternate images that may be insensitive to beam hardening artifacts or may be
particularly sensitive to physical density, or the effective atomic number of the materials contained in the
object.
• Dynamic range
• The range of operation of a device between its upper and lower limit. This range can be given as a ratio
(example 100:1) of the upper to lower limits, the number of measurable steps between the upper and
lower limits, the number of bits (needed to record this number of measurable steps), or the minimum and
maximum measurable values.
• Edge enhancement
• A mathematical manipulation in which rapid density changes are enhanced or sharpened by means of
differentiation or high-pass filters. This operation will also increase image noise.
• Effective atomic number
• For an element, the number of protons in the nucleus of an atom. For mixtures, an effective atomic
number can be calculated to represent a single element that would have attenuation properties identical to
those of the mixture.
• Effective x-ray energy
• The monoenergetic beam energy that is attenuated to the same extent by a thin absorber as is the given
polyenergetic x-ray beam.
• Field-of-view (FOV)
• The maximum diameter of an object that can be imaged. Also known as scan FOV. The object dimension
corresponding to the full width of a displayed image (display FOV).
• Focal spot
• The area of the x-ray tube from which the x-rays originate. The effective focal-spot size is the apparent
size of this area when viewed from the detector.
• Fourier transformation
• A mathematical process for changing the description of a function by giving its value in terms of its
sinusoidal spatial (or temporal) frequency components instead of its spatial coordinates (or vice versa).
• Full width at half maximum (FWHM)
• A parameter that can be used to describe a distribution such as the point spread function.
• Geometries, CT
• The geometrical configuration and mechanical motion to acquire the x-ray transmission data. Typically,
parallel beam data are acquired from translate-rotate data acquisition, and fan beam data are obtained
from rotate-only movement of the object (or of the source and detector about the object).
• Histogram
• A plot of frequency of occurrence versus the measured parameter. In a CT image, the plot of the number
of pixels with a particular CT number value versus CT number.
• Histogram equalization
• An image display process in which an equal number of image pixels is displayed with each displayable
gray shade or color.
• Hysteresis
• Dependence of a measured signal on the previous measurement history.
• Intensity
• The energy per unit area per unit time incident on a surface.
• Ionization
• The process in which neutral atoms become charged by gaining or losing an electron.
• Iterative reconstruction
• A method of reconstruction that produces an improving series of estimated images that is repeated until a
suitable quality level is obtained.
• Kernel
• A function that is convoluted with a data function. The kernel in the convolution-backprojection
reconstruction algorithm can be modified to enhance sharpness or reduce noise. Also known as
convolution kernel.
• Kiloelectron volt (keV)
• A unit of energy usually associated with individual particles. The energy gained by an electron when
accelerated across 1000 V.
• Linear attenuation coefficient
• The fraction of an x-ray beam per unit thickness that a thin object will absorb or scatter (attenuate). A
property proportional to the physical density and dependent on the atomic number of the material and the
energy of the x-ray beam.
• Linearity, detector
• A measure of the consistency in detector sensitivity versus the radiation intensity level.
• Matrix
• An array of numbers arranged in two dimensions (rows and columns).
• Megaelectron volt (MeV)
• A unit of energy usually associated with a particle. The energy gained by an electron accelerated across
1,000,000 V.
• Modulation transfer function (MTF)
• A measure of the spatial resolution of an imaging system that involves the plot of image contrast (system
response) versus the spatial frequency (line pairs per millimeter) of the contrast variations on the object
being imaged. A plot of these two variables gives a curve representing the frequency response of a
system. The MTF can also be determined from the Fourier transform of the point spread function.
• Noise
• Salt-and-pepper appearance of an image caused by variations in the measured data. Image noise can be
affected by choice of reconstruction algorithm and by image processing, such as sharpening and
smoothing operations.
• Noise, quantum (or photon)
• Noise due to statistical variations in the number of x-ray photons detected. An increase in the number of
photons measured decreases the relative quantum noise.
• Noise, structural
• Unwanted detail of an object from overlying structures or due to granularity or material structure.
• Noise, structured
• Nonrandom noise, such as due to reconstruction algorithm filtering or systemic errors in the
measurements.
• Nyquist sampling frequency
• The ability to characterize a signal from a series of discrete samples requires that the signal be sampled at
a minimum of twice the frequency of the highest frequency contained in the signal. The undersampling of
a signal causes the high-frequency signals to mimic or appear as lower-frequency signals, an effect
termed aliasing.
• Partial volume artifact
• Streaking or shadowing artifacts caused by high-density structures partially intercepting the finite-sized
measured x-ray beam. The effect is due to the summation of multiple ray paths in an inhomogenous
mixture.
• Partial volume effect
• The effect of measuring a density lower than the true structure density due to the fact that only a part of
the structure is within the measured voxel, that is, within the full slice width or resolution element.
• Phantom
• A test object used to measure the response of the CT system. Phantoms can be used to calibrate the
system or to measure performance parameters.
• Photodiode
• A semiconductor device that permits current flow proportional to the intensity of light striking the diode
junction.
• Photomultiplier tube (PMT)
• A light-sensitive vacuum tube with multiple electrodes providing a highly amplified electrical output.
• Photon
• A quantum, or particle, of electromagnetic radiation (such as light or x-rays).
• Pixel
• Shortened term for picture element. A pixel is the smallest displayable element of an image; a single
value of the image matrix. A pixel corresponds to the measurement of a volume element (voxel) in the
object.
• Point spread function (PSF)
• The image response of a system to a very small, high-amplitude object; the image blurring function.
• Polychromatic
• Photons with a mixture of colors, or energies. Same as polyenergetic.
• Polychromatic artifacts
• Image distortion such as image shading or low-density bands between high-density structures caused by
the polychromatic x-ray beam. When an x-ray beam travels a longer path or through a high-absorption
region, the low-energy components of the polychromatic beam are preferentially absorbed, resulting in a
higher effective beam energy and creating artifacts in the image.
• Polychromatic x-ray spectra
• An x-ray beam that consists of a range of x-ray energies. The maximum amplitude and effective energy
of an x-ray beam are always less than the corresponding voltage applied to the tube. The maximum
energy of an x-ray beam corresponds to the peak x-ray tube voltage.
• Projection
• A set of contiguous measured line integrals (projection data points) through an object. May be parallel ray
or divergent (fan) ray projection data set. Also called a view.
• Projection data
• The logarithm of the normalized transmitted intensity data. The line integral of the linear attenuation
coefficient through an object.
• Quantum detection efficiency
• See detective quantum efficiency.
• Radionuclide
• A specific radioactive material.
• Reconstruction
• The process by which raw digitized detector measurements are transformed into a cross-sectional CT
image.
• Reformation, planar
• A displayed image comparable to a measured CT image, but along a planar section that cuts across a
series of measured CT slices. The computer selects the data corresponding to the selected plane from a
stack of CT slices through a volume of the object.
• Reformation, 3-D surface
• A perspective display of a structure; an image that appears as a photograph of the structure.
• Resolution, high contrast
• A measure of the ability of an imaging system to present multiple high-contrast structures and fine detail;
measurements made with line pair gage or bar resolution phantom. See spatial resolution.
• Resolution, low contrast
• The minimum detectable spacing between specified objects for a specified percent contrast between the
objects and their background as measured with a test phantom.
• Sagittal plane
• A medical term for a plane that divides the body into left and right sides. A y-z planar presentation, which
can be mathematically constructed from a series of cross-sectional slices. See coronal plane and
reformation, planar.
• Scan
• The operation of acquiring the data for a CT or DR image. Sometimes used in reference to the CT or DR
image.
• Scattering of x-rays
• One of the two ways in which the x-ray beam interacts with matter and the transmitted intensity is
diminished in passing through object. The x-rays are scattered in a direction different from the original
beam direction. Such scattered x-rays falling on the detector do not add to the radiological image.
Because they reduce the contrast in that image, steps are usually taken to eliminate the scattered
radiation. Also known as Compton scatter.
• Scintillator
• A material that produces a rapid flash of visible light when an x-ray photon is absorbed. The scintillator is
coupled with a light-sensitive detector to measure the x-ray beam intensity.
• Sharpening
• Image-processing technique that enhances edge contrast and increases image noise. The opposite of
smoothing.
• Slice
• The cross-sectional plane through an object that is scanned to produce the CT image. See also
tomographic plane.
• Slice thickness
• The height or z-axis dimension of the measured x-ray beam normally measured at the center of the object.
The slice thickness along with the image resolution defines the size of the measured volume
corresponding to a pixel CT value.
• Slit-scan radiography
• Method of producing an x-ray image in which a thin x-ray beam produces the image one line at a time. A
method used to reduce measured scatter radiation. The DR mode of most CT systems uses a slit-scan
technique with a linear detector array.
• Smoothing
• Image-processing technique that averages the image data, reducing edge sharpness and decreasing image
noise. Opposite of sharpening.
• Spatial resolution
• A measure of the ability of an imaging system to represent fine detail; the measure of the smallest
separation between individually distinguishable structures. See resolution, high contrast, point spread
function, and modulation transfer function.
• Spectrum, x-ray
• The plot of the intensity or number of x-ray photons versus energy (or wavelength).
• Test phantom
• See phantom.
• Tomographic plane
• A section of the part imaged by the tomographic process. Although in CT the tomographic plane or slice
is displayed as a two-dimensional image, the measurements are of the materials within a defined slice
thickness associated with the plane.
• Tomography
• From the Greek "to write a slice or section." The process of imaging a particular plane or slice through an
object.
• View
• A set of measured parallel or fan beam data for a particular angle through the center of the object. Also
known as a profile or projection.
• Voxel
• Shortened term for volume element. The volume within the object that corresponds to a single pixel
element in the image. The box-shaped volume defined by the area of the pixel and the height of the slice
thickness.
• Window, display
• The range of CT values in the image that are displayed. The display window can normally be adjusted
interactively by the operator to view different density ranges in the reconstructed image data.
Industrial Computed Tomography
Michael J. Dennis, General Electric Company, NDE Systems and Services
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