# Basic Physics of Nuclear Medicine/Computers in Nuclear Medicine

## Introduction

This is a developing chapter for a Wikibook entitled Basics Physics of
Nuclear Medicine.

Computers are widely used in almost all areas of Nuclear Medicine today.
The main application for our purposes is imaging acquisition and
processing. This chapter outlines the design of a generalised digital
image processor and gives a brief introduction to digital imaging.

Before considering these topics, some general comments are required
about the form in which information is handled by computers as well as
the technology which underpins the development of computers so that a
context can be placed on our discussion.

## Binary Representation

Virtually all computers in use today are based on the manipulation of
information which is coded in the form of binary numbers. A binary
number can have only one of two values, i.e. 0 or 1, and these numbers
are referred to as binary digits -- or
`<b>`{=html}bits`</b>`{=html}, to use computer jargon. When a piece of
information is represented as a sequence of bits, the sequence is
referred to as a `<b>`{=html}word`</b>`{=html}; and when the sequence
contains eight bits, the word is referred to as a
`<b>`{=html}byte`</b>`{=html} -- the byte being commonly used today as
the basic unit for expressing amounts of binary-coded information. In
addition, large volumes of coded information are generally expressed in
terms of `<b>`{=html}kilobytes`</b>`{=html},
`<b>`{=html}megabytes`</b>`{=html} etc. It is important to note that the
meanings of these prefixes can differ slightly from their conventional
meanings because of the binary nature of the information coding. As a
result, `<b>`{=html}kilo`</b>`{=html} in computer jargon can represent
1024 units -- 1024 (or 2^10^) being the nearest power of 2 to one
thousand. Thus, `<b>`{=html}1 kbyte`</b>`{=html} can refer to 1024 bytes
of information and `<b>`{=html}1 Mbyte`</b>`{=html} can represent 1024
times 1024 bytes. To add a bit of confusion here, some hardware
manufacturers refer to a megabyte as 1 million bytes and a gigabyte as a
1,000 million bytes. Its not a simple world, it seems!

Binary coding of image information is needed in order to store images in
a computer. Most imaging devices used in medicine however generate
information which can assume a continuous range of values between preset
limits, i.e. the information is in
analogue form. It is therefore necessary
to convert this analogue information into the discrete form required for
binary coding when images are input to a computer. This is commonly
achieved using an electronic device called Analogue-to-Digital
Converter (**ADC**).
Furthermore, since many display and photographic devices used in
medicine are designed for handling images in analogue format, it is
necessary to reconvert the discrete, binary data when outputting images
from a computer using a Digital-to-Analogue
Converter (**DAC**).

## Development of Modern Computers

The development of modern computers has been almost totally dependent on
major developments in material science and digital electronics which
have occurred over the last thirty years or so. These developments have
allowed highly complex electronic circuitry to be compressed into small
plastic packages called integrated
circuits. These packages contain tiny
pieces of silicon (or other semiconductor material) which have been
specially manufactured to perform complex electronic processes. The
pieces of silicon are generally referred to as **silicon chips**. Within
the circuitry of a chip, a relatively high electronic voltage can be
used to represent the digit \'1\' and a relatively low voltage can be
used to represent the binary digit \'0\'. Thus the circuitry can be used
to manipulate information which is coded in the form of binary numbers.

An important feature of these electronic components is the very high
speed at which the two voltage levels can be changed in different parts
of the circuitry. This results in the ability of the computer to rapidly
manipulate the binary information. Furthermore, the tiny size of modern
integrated circuits has allowed the manufacture of computers which are
very small physically and which do not generate excessive amounts of
heat -- previous generations of computers having occupied whole rooms,
which required cooling because they were built using larger electronic
components such as valves and transistors. Thus modern computers are
capable of being mounted on a desk in an environment which does not
require air-conditioning. In addition, the ability to manufacture
integrated circuits using mass production methods has given rise to
enormous decreases in costs -- which has contributed to the phenomenal
explosion of this technology in recent years.

Before beginning, it is worth noting that the information in this
chapter is likely to change by the time the chapter is read, given the
ongoing, rapid developments in this field. The treatment here is
therefore focused on general concepts -- and you should note that
current technologies and techniques may well differ from those described
here. In addition, note that mention of any hardware or software product
in this chapter does in no way intend support for such a product and its
use in this discussion is purely for illustrative purposes.

## Hardware

The figure below shows a block diagram of the major hardware components
of a general-purpose computer. It shows that a computer consists of a
central communication pathway, called a **bus**, to which dedicated
electronic components are connected. Each of these components is briefly
described below.

!Block diagram of a general-purpose
computer

**Central Processing Unit (CPU)** This component is based, in many
modern computers, on an integrated circuit called a
microprocessor. Its function is to act as
the *brains* of the computer where instructions are interpreted and
executed, and where data is manipulated. The CPU
typically contains two sub-components -- the Control Unit (**CU**) and
the Arithmetic/Logic Unit (**ALU**).

The Control Unit is used for the interpretation of instructions which
are contained in computer programs as well as the execution of such
instructions. These instructions might be used, for example, to send
information to other components of the computer and to control the
operation of these other components. The ALU is primarily used for data
manipulation using mathematical techniques -- for example, the addition
or multiplication of two numbers.

Important features of individual microprocessors include word length,
architecture, programming flexibility and speed. An indicator of speed
is the clock rate and values for common
microprocessors are given in the following table. Note that the clock
rate on its own does not provide a complete indication of computer
performance since the specifications of other components must also be
considered.

  Microprocessor   **Manufacturer**   **Clock Rate (MHz)**   **Example Microcomputer**
  ---------------- ------------------ ---------------------- ---------------------------
  Pentium          Intel              60-200                 IBM-PC Compatible
  PowerPC 604e     Motorola           160-350                Power Macintosh
  Turbo SPARC      Sun                170                    SPARC Station 5
  STP1031LGA       Sun                250                    Ultra SPARC II
  Pentium II       Intel              233-450                IBM-PC Compatible
  PowerPC 750      Motorola           233-500                Power Macintosh
  Alpha 21164      DEC                300-625                DEC Alpha
  Pentium 4        Intel              1,300-1,700            IBM-PC Compatible

\
**Main Memory** This component typically consists of a large number of
integrated circuits which are used for the storage of information which
is currently required by the computer user. The circuitry is generally
of two types -- Random Access
Memory (**RAM**) and Read Only
Memory (**ROM**). RAM is used for the
short-term storage of information. It is a volatile form of memory since
its information contents are lost when the electric power to the
computer is switched off. Its contents can also be rapidly erased -- and
rapidly filled again with new information. ROM, on the other hand, is
non-volatile and is used for the permanent storage of information which
is required for basic aspects of the computer\'s operation.

**Secondary Memory** This component is used for the storage of
information in permanent or erasable form for longer-term purposes, i.e.
for information which is not currently required by the user but which
may be of use at some later stage. There are various types of devices
used for secondary memory and some of their features are summarised in
the following table. RAM is also included in the table for comparison.

The type of technology used is generally based on magnetic materials --
similar to those used for sound recording in hi-fi systems. Here, the
information is stored by controlling the local magnetism at different
points of the storage material and is retrieved by detecting this
magnetism. The local magnetism, as might be expected, can assume one of
two magnetic states because of the binary nature of the coded
information. Materials such as plastic tapes and disks, which have been
coated with a layer of magnetic material, are used. Magnetic
tapes generally consist of open reels or
enclosed cassettes, while magnetic disks generally consist of flexible
(or **floppy**) disks or disks made from a harder plastic. Floppy
disks, like magnetic tapes, can be removed
from the computer system and are used for external storage of
information. Memory sticks can also be used
for this purpose. They also allow the capability of transporting
information between computers. Hard disks, on
the other hand, are usually fixed inside the computer and thus cannot be
easily removed -- although removable versions are also in use, e.g. the
iPod and similar devices.

  Device                **Capacity (Mbyte)**   **Access Time**   **Erasable?**
  --------------------- ---------------------- ----------------- ---------------
  Magnetic Tape         500-16,000             minutes           Yes
  Floppy Disk           0.3-1.5                200-500 ms        Yes
  Hard Disk             1,000-300,000          20-80 ms          Yes
  Removable Hard Disk   100-100,000            100-200 ms        Yes
  Optical Disk          250-4,700              100-500 ms        Yes/No
  RAM                   256-4,000              10-100 ns         Yes

\
Recent developments in our understanding of magnetic and optical
characteristics of materials has led to the production of the so-called
optical disk -- which is similar to the
compact disk (**CD**) used in hi-fi systems. Three general types of
optical disk are available: those containing programs supplied by
software companies (as in the **CD-ROM** disk), those which can be
written to once by the user (as in the **CD-R** disk) and those which
are erasable (as in the **CD-RW** disk and the magneto-optical disk).
The Digital Versatile Disk (**DVD**) is a likely successor to the CD-ROM
-- the first generation having a storage capacity of 4,700 Mbyte (4.7
Gbyte) and the second generation an expected 17,000 Mbyte (17 Gbyte).

A general difference between tape and disk as a secondary storage medium
results from the sequential nature of
the access to information stored on tape, in contrast to the
random nature of the access provided by
disks. As a result, disk-based media are typically faster for
information storage/retrieval than those based on tape. A number of
modern designs of secondary memory are therefore based on hard magnetic
disks for routine information storage, with floppy disks used for
back-up storage of small volumes of information and optical disks for
back-up storage of larger volumes of information.

**Input/Output Devices** These components are used for user-control of
the computer and generally consist of a
keyboard, display
device and
printer. A wide range of technologies
are used here -- the details however are beyond the scope of this
chapter. These components also include devices such as the
mouse, joystick
and trackpad, which are used to enhance
user-interaction with the computer.

**Computer Bus** This consists of a communication pathway for the
components of the computer -- its function being somewhat analogous to
that of the central nervous system. The types of information
communicated along the bus include that
which specifies data and control instructions as well as the memory
addresses where information is to be stored/retrieved. As might be
anticipated, the speed at which a computer operates is dependent on the
speed at which this communication link works. This speed must be
compatible with that of other components, such as the CPU and main
memory.

## Software

There is more to computer technology than just the electronic hardware.
In order for the assembly of electronic components to operate,
information in the form of data and computer instructions is required.
This information is generally referred to as
**software**. Computer instructions are
generally contained within computer
programs.

Categories of computer program include:

-   **Operating systems** -- which are used for operating the computer
    and for managing various resources of the computer. Examples of
    operating systems are Windows,
    MacOS X, Linux and UNIX;
-   **Application packages** -- which are for the use of routine users
    of the computer. These packages include programs which are used for
    word-processing (e.g. MS Word), spreadsheets
    (e.g. MS Excel), databases (e.g. FileMaker
    Pro), graphics (e.g. Adobe
    Illustrator) and digital image
    processing (including software used for operating specific medical
    imaging scanners);
-   **Programming packages** -- which are used for writing programs.
    Examples of common computer languages which are used for writing
    programs are C (and its many
    variants) and Java. A number
    of additional pieces of software are required in order for such
    programs to be written. These include:
    -   An **Editor** for writing the text of the program into the
        computer (which is similar to programs used for
        word-processing);
    -   A **Library** of subroutines -- which are small programs for
        operating specific, common functions;
    -   A **Linker** which is used to link the user-written program to
        the subroutine library;
    -   A **Compiler** or **Interpreter** for translating user-written
        programs into a form which can be directly understood by the
        computer, i.e. it is used to code the instructions in digital
        format.

:   These programming functions and more are combined in packages which
    generate an Integrated Development
    Environment (IDE),
    a good example being Xcode.

## Digital Image Processor

Computers used for digital image processing generally consist of a
number of specialised components in addition to those used in a
general-purpose computer. These specialised components are required
because of the very large amount of information contained in images and
the consequent need for high capacity storage media as well as very high
speed communication and data manipulation capabilities. Digital image
processing involves both the
manipulation of image data and the analysis of such information. An
example of image manipulation is the computer enhancement of images so
that subtle features are displayed with greater clarity. An example of
image analysis is the extraction of indices which express some
functional aspect of an anatomical region under investigation. Most
medical imaging systems provide extensive image manipulation
capabilities with a limited range of image analysis features. Systems
for processing nuclear medicine images (including SPECT and PET) also
provide extensive data analysis capabilities. This situation arises
because of the functional, in contrast to an anatomical, emphasis in
nuclear medicine.

A generalised digital image processor is shown in the following figure.
The shaded components at the bottom of the diagram are those of a
general-purpose computer which have been described above. The digital
image processing components are those which are connected to the image
data bus. Each of these additional components is briefly described
below. The shaded components at the top of the diagram are external
devices which are widely used in medical imaging systems.

!Block diagram of a digital image processing
computer

**Imaging System** This is the device which produces the primary image
information. Examples of these devices include CT
scanners, ultrasound
machines, x-ray fluorography
systems, MRI
systems, gamma
cameras, PET
scanners and computed
radiography
systems "wikilink"). The
device is often physically separate from the other components as in the
CT scanner, but may also be mounted in the same cabinet as the other
components -- as is the case for ultrasound machines. Image information
produced by the imaging system is fed to the image acquisition circuitry
of the digital image processor.

Connections from the digital image processor to the imaging system are
generally also present, for controlling specific aspects of the
operation of the imaging system, e.g. gantry movement of a SPECT camera.
These additional connections are not shown in the figure for reasons of
clarity.

**Image Acquisition** This component is used to convert the analogue
information produced by the imaging system so that it is coded in the
form of binary numbers. The type of device used for this purpose is
called an Analogue-to-Digital Converter (**ADC**). The image acquisition
component may also include circuitry for manipulating the digitised data
so as to correct for any imaging aberrations. The type of device which
can be used for this purpose is called an **Input Look-Up Table**.
Examples of this type of data manipulation include pre-processing
functions on ultrasound machines and logarithmic image transformation in
digital fluorography systems.

**Image Display** This component is sometimes referred to as a **display
controller** and its main use is to convert digital images into a form
which is suitable for a visual display device. It includes a
Digital-to-Analogue Converter (**DAC**) when a CRT
monitor is used, for instance, and a digital
visual interface (DVI) when a digital display such
as an LCD monitor is used. The image display
component can also include circuitry for manipulating the displayed
images so as to enhance their appearance. The type of device which can
be used for this purpose is called an **Output Look-Up Table**. Examples
of this type of data manipulation include post-processing functions on
ultrasound machines and **windowing** functions on Nuclear Medicine
systems. Other forms of image processing provided by the image display
component can include image magnification, image rotation/mirroring and
the capability of displaying a number of images on one screen. This
component can also allow for the annotation of displayed images with the
patient name and details relevant to the patient\'s examination.

**Image Memory** This component typically consists of a volume of RAM
which is sufficient for the storage of a number of images which are of
current interest to the user.

**Image Storage** This component generally consists of magnetic disks of
sufficient capacity to store large numbers of images which are not of
current interest to the user and which may be transferred to image
memory when required.

**Image ALU** This component consists of an ALU designed specifically
for handling image data. It is generally used for relatively
straight-forward calculations, such as image subtraction in
DSA and the reduction of
noise through averaging a sequence of images.

**Array Processor** This component consists of circuitry designed for
more complex manipulation of image data and at higher speeds than the
Image ALU. It typically consists of an additional CPU as well as
specialised high speed data communication and storage circuitry. It may
be viewed as a separate special-purpose computer whose design has traded
a loss of operational flexibility for enhanced computational speed. This
enhanced speed is provided by the capability of manipulating data in a
parallel fashion as opposed to a sequential fashion (which is an
approach widely used in general-purpose computing). This array processor
is used, for example, for calculating Fast Fourier
Transforms and for image
reconstruction calculations in
cross-sectional imaging modalities, such as CT, SPECT and MRI.

**Image Data Bus** This component consists of a very high speed
communication link designed specifically for image data.

## Digital Imaging

The digitisation of images generally consists of two concurrent
processes -- sampling "wikilink") and
quantisation "wikilink"). These two
processes are described briefly below and a consideration of the storage
requirements for digital images follows.

**Image Sampling** This process is used to digitise the spatial
information in an image. It is typically achieved by dividing an image
into a square or rectangular array of sampling points. Each of the
sampling points is referred to as a picture element -- or
pixel to use computer jargon.

!An illustration of a digital image obtained when an original,
consisting of a central dark region with the brightness increasing
towards the periphery, is digitised with N=8 and G=4 (i.e.
m=2)")

The process may be summarised as the digitisation of an analogue image
into an N x N array of pixel data. Examples of values for N are 128 for
a nuclear medicine scan, 512 for CT and MRI scans, 1024 for a DSA image,
and 2048 for a computed radiograph image and digital radiograph. Note
that N has values which are integer powers of 2, because of the binary
nature of modern computing techniques.

Naturally, the larger the number of pixels, the closer the spatial
resolution of the digitised image approximates that of the original
analogue image -- see the images below.

**Image Quantisation** This process refers to the digitisation of the
brightness information in an image. It is typically achieved by
representing the brightness of a pixel by an integer whose value is
proportional to the brightness. This integer is referred to as a \'pixel
value\' and the range of possible pixel values which a system can handle
is referred to as the gray scale. Naturally,
the greater the gray scale, the closer the brightness information in the
digitised image approximates that of the original, analogue image -- see
the images below.

The process may be considered as the digitisation of image brightness
into G shades of gray. The value of G is dependent on the binary nature
of the information coding. Thus G is generally an integer power of 2,
i.e. G=2^m^, where m is an integer which specifies the number of bits
required for storage. Examples of values of G are 256 (m=8) in
ultrasonography, 1024 (m=10) in DSA and 4096 (m=12) in nuclear medicine.

!A bone scan of a patient\'s hand displayed with digital image
resolutions of 256x256x8 bits, 32x32x8 bits and 256x256x2
bits.

**Digital Image Resolution** The number of bits, b, required to
represent an image in digital format is given by\

```{=html}
<div class="center">
```
`<big>`{=html}`<span style="color:red;">`{=html}b = N x N x
m`</span>`{=html}`</big>`{=html}

```{=html}
</div>
```
The following table shows the number of bits required to represent
images which are digitised at different spatial resolutions for a range
of gray scales. It is seen that very large values are required to
achieve the resolution used in medical imaging (you might confirm as an
exercise that the asterisked value in the Table represents 0.25 Mbytes).
The resulting amounts of computer memory needed to store such images are
therefore quite large and processing times can be relatively long when
manipulating such large volumes of data. This feature of digital images
gives rise to the need for dedicated hardware for image data which is
separate from the components of a general-purpose computer as we have
described above -- although this distinction is vanishing with ongoing
technological developments.

  N x N         **m = 8**     **m = 10**   **m = 12**
  ------------- ------------- ------------ ------------
  128 x 128     131,072       163,840      196,608
  256 x 256     524,288       655,360      786,432
  512 x 512     2,097,152\*   2,621,440    3,145,728
  1024 x 1024   8,388,608     10,485,760   12,582,912
  2048 x 2048   33,554,432    41,943,040   50,331,648

  : The number of binary digits (bits) required to store images which
  have been digitised to spatial resolutions of N x N pixels and to 2^m^
  shades of gray.

## Digital Image Processing

**Contrast enhancement** as an example of a very common form of digital
processing of images (also referred to as **windowing**) is described
below. It is just one from the wide range of data manipulation processes
which are available on modern systems. Contrast enhancement is a form of
gray-level transformation where the real pixel values in an image are
replaced by processed pixel values for display purposes. !An
illustration of the gray-level transformation required for contrast
enhancement of images with 256 shades of gray (i.e. m=8). In this
example, the unprocessed data is transformed so that all pixels with a
pixel value less than 50 are displayed as black, all pixels with a pixel
value greater than 150 are displayed as white and all pixels with pixel
values between 50 and 150 are displayed using an intermediate shade of
gray.. In this example, the unprocessed data is transformed so that all pixels with a pixel value less than 50 are displayed as black, all pixels with a pixel value greater than 150 are displayed as white and all pixels with pixel values between 50 and 150 are displayed using an intermediate shade of gray.")
The process is generally performed using the Output Look-Up Table
section of the image display component of the digital image processor.
As a result, the original data in image memory is not affected by the
process, so that from an operational viewpoint, the original image data
can be readily retrieved in cases where an unsatisfactory output image
is obtained. In addition, the process can be implemented at very high
speed using modern electronic techniques so that, once again from an
operational viewpoint, user-interactivity is possible.

An example of a Look-Up Table (**LUT**) which can be used for contrast
enhancement is illustrated in the following figure. This information is
usually presented using a graph of the real pixel values stored in image
memory versus the pixel values used for display purposes. The process is
generally controlled using two controls on the console of the digital
image processor -- the **LEVEL** control and the **WINDOW** control. It
should be noted that variations in the names for these controls, and in
their exact operation, exist between different systems but the general
approach described here is sufficient for our purposes. It is seen the
figure that the LEVEL controls the threshold value below which all
pixels are displayed as black and the WINDOW controls a similar
threshold value for a white output. The simultaneous use of the two
controls allows the application of a gray-level window, of variable
width, which can be placed anywhere along the gray scale. Subtle
gray-level changes within images can therefore be enhanced so that they
are displayed with greater clarity. A common application of this form of
digital image processing in nuclear medicine is the removal of
background counts from images.

Other forms of contrast enhancement used in Nuclear Medicine include
using LUTs with logarithmic, exponential or other nonlinear input/output
relationship -- the logarithmic LUT being used for instance to
accommodate a broad range of counts on one grey scale. Colour LUTs are
also popular, where the digital contrast resolution of an image is
represented by a range of different colours -- a rainbow scheme for
example as shown in the following figure -- or by hues of one or a small
number of colours.

  ---------------------------------------------------- ---------------------------------------- ------------------------------------------------
  ![](twoHandsGreyscale.jpg "twoHandsGreyscale.jpg")   ![](rainbowCLUT.gif "rainbowCLUT.gif")   ![](twoHandsRainbow.jpg "twoHandsRainbow.jpg")
  ---------------------------------------------------- ---------------------------------------- ------------------------------------------------

```{=html}
<div class="center">
```
Graphical representation of a Colour Look-Up Table (CLUT) which converts
a grey scale into a spectrum of rainbow colours.\
The red, green and blue channels are represented by the plots with the
respective colours.\
Click **HERE** to
view a QuickTime movie (\~9 Mbyte) demonstrating the effects of various
CLUTs on a fused SPECT/CT study.

```{=html}
</div>
```
Other examples of digital image processing are illustrated in figure
below:\
!Image processing options based on the same unprocessed bone scan shown
in top
left.

A detailed treatment of the image processing capabilities of a modern
personal computer is provided at this `<b>`{=html}external
link`</b>`{=html}.

Other common forms of digital image processing include applying
geometric transformations to images so as to magnify or zoom in on
specific details or to correct for geometric distortions introduced by
the imaging system. Image zoom can be readily achieved in the display
controller using a **pixel replication** process, where each pixel is
displayed N^2^ times, where N is the zoom factor, as illustrated below:

!Illustration of a pixel replication process used to display an image
zoomed by a factor of
two.

A disadvantage of this approach however is that zoomed images can have a
blocky appearance reflecting the larger size of each effective pixel.
Although the application of a smoothing filter can reduce this
pixelation effect, a more visually pleasing
result can be generated using spatial
interpolation techniques. Here, the pixel
values of unknown pixels are estimated using the pixel values of known
neighbouring pixels. Suppose the image above is zoomed again and suppose
that this time the known pixels are distributed to the corners of the
zoomed image, as shown in the following figure:

!Illustration of image interpolation applied to zooming an image by a
factor of
two.

The task of the interpolation process is to calculate the pixel values
of the unknown pixels based on the known pixel values of the corner
pixels. The simplest approach is linear interpolation where a linear
relationship between the pixel values of the known pixels is assumed.
Let\'s suppose that we wish to estimate the pixel value of the pixel
shaded red in the figure above. In the case of two-dimensional linear
interpolation, also called **bilinear interpolation**, the first step is
to calculate the pixel value of the pixel at position
`<span style="color:red;">`{=html}(x,0)`</span>`{=html}, shaded yellow
on the top line of the matrix, as follows:

```{=html}
<div class="center">
```
`<span style="color:red;">`{=html}`<big>`{=html}P(x,0) = x P(1,0) +
(1-x) P(0,0)`</big>`{=html}`</span>`{=html}

```{=html}
</div>
```
where `<span style="color:red;">`{=html}x`</span>`{=html}: fractional
distance along the horizontal axis.

Similarly, the pixel value of the pixel shaded yellow on the bottom line
of the matrix, at position
`<span style="color:red;">`{=html}(x,1)`</span>`{=html}, is calculated
using the following equation:

```{=html}
<div class="center">
```
`<span style="color:red;">`{=html}`<big>`{=html}P(x,1) = x P(1,1) +
(1-x) P(0,1)`</big>`{=html}`</span>`{=html}

```{=html}
</div>
```
Finally, the pixel value for the unknown red-shaded pixel in the figure
can be obtained by linear interpolation between these two calculated
(yellow-shaded) pixel values, as follows:

```{=html}
<div class="center">
```
`<span style="color:red;">`{=html}`<big>`{=html}P(x,y) = y P(x,1) +
(1-y) P(x,0)`</big>`{=html}`</span>`{=html}

```{=html}
</div>
```
where `<span style="color:red;">`{=html}y`</span>`{=html}: fractional
distance along the vertical axis.

This process is applied to all unknown pixels in an image. Example
images are shown below:

!Images magnified by pixel replication and interpolated
zoom.

Functions other than linear can also be applied, e.g. two-dimensional
**polynomial** and **cubic spline** interpolation, to effect a more
visually pleasing result. Remember however that the interpolated data is
not real and the approach simply generates estimates of pixel values in
an effort to improve the blocky nature of the simple zoom technique
considered earlier. We\'ll be considering further applications of image
interpolation in our chapter on X-ray
CT.

## The Fourier Transform -- A Pictorial Essay

The treatment of the Fourier Transform
(**FT**) in many textbooks uses a level of mathematics which is quite
alien to many students of the medical sciences. The treatment here will
adopt a different approach, based on a pictorial essay, in an attempt to
convey more effectively the concept on which the transform is based. It
does in no way substitute for a rigorous mathematical treatment, and is
solely aimed at supporting your understanding of **image filtering**.

!Illustration of the use of the Fourier Transform. The FT and its
inverse allow us to convert image data from the spatial to spatial
frequency domains and vice versa,
respectively.

This presentation will demonstrate that images can be thought about from
both **spatial** and **spatial frequency** perspectives. The spatial
perspective is the conventional way of presenting image data and relates
to real world parameters such as distance and time. An image may also be
considered as consisting of a large number of spatial frequencies
interacting with each other. This aspect will be examined using a fairly
simple image to begin with, and then by considering a more complicated
one, namely a chest radiograph, i.e. an example of medical image which
consists of a very broad range of spatial frequencies. The FT transforms
the image data from the **spatial** representation to the **spatial
frequency** representation and the **inverse FT** performs the reverse
operation -- see the figure on the right.

An image of a sinusoidal brightness pattern running at an angle of 45
degrees to the horizontal is shown below in panel (a). A plot of pixel
values along the line AB demonstrates this sinusoidal pattern in one
dimension as demonstrated in panel (b). It can also be represented in a
different fashion by plotting the amplitude of the sine waves present in
panel (b) against their spatial frequency. This is shown in panel (c).
This latter plot confirms that there is just one spatial frequency
dominating the image, as might be expected. This type of plot is called
a **1D Fourier spectrum** and utilises a one-dimensional FT of the image
data. Note that panels (b) and (c) illustrate the frequency information
in just one dimension.

When the frequency information is displayed in 2 dimensions, as in panel
(d), it is called a **2D Fourier spectrum** and this is achieved using a
2D FT of the image data. It demonstrates spatial frequencies for the
vertical and horizontal image dimensions along its vertical and
horizontal axes, with the origin at the centre (shown by the tiny white
dot). Two other dots are seen in panel (d), with slight horizontal
streaking beside them -- one to the upper left and the other to the
bottom right of the origin. These correspond to the frequency of the
sine wave in panel (a). Since the FT generates both positive and
negative values for the frequency, the resulting two frequencies are
displayed on either side of the origin as indicated.

```{=html}
<div class="center">
```
!(a) Image of a sinusoidal pattern. (b) Image brightness profile along
the line AB in (a) demonstrating the presence of one strong single
spatial frequency as confirmed by the 1D Fourier spectrum of (b) in (c).
Finally, the 2D FT or Fourier spectrum of (a) is illustrated in
(d). Image of a sinusoidal pattern. (b) Image brightness profile along the line AB in (a) demonstrating the presence of one strong single spatial frequency as confirmed by the 1D Fourier spectrum of (b) in (c). Finally, the 2D FT or Fourier spectrum of (a) is illustrated in (d).")\

```{=html}
</div>
```
A more complicated 2-D Fourier spectrum is obtained when a chest
radiograph is transformed to the spatial frequency domain as illustrated
in the next figure. The transformed data show a broad range of spatial
frequencies, with significant vertical and horizontal features, as might
be expected from the horizontal ribs and vertical vertebral column
displayed in the radiograph.

```{=html}
<div class="center">
```
!A chest radiograph is illustrated in (a) with its 2-D Fourier spectrum
in (b). The spatial frequency data show a broad range of values with
significant vertical and horizontal features associated with the
vertebral column and ribs,
respectively. with its 2-D Fourier spectrum in (b). The spatial frequency data show a broad range of values with significant vertical and horizontal features associated with the vertebral column and ribs, respectively.")\

```{=html}
</div>
```
A potential use of the FT and its inverse is the removal of unwanted or
corrupt data from a digital image and this process is illustrated in the
final figure below. An extreme example of a corrupt image may be
generated by adding together the two images just analysed, as in panel
(a). The Fourier spectrum in panel (b) portrays the frequency
characteristics of the summed image. The undesirable features
attributable to the sinusoidal pattern may be removed by editing of the
data in the frequency domain as in panel (c), before the inverse FT is
performed to recover an image largely free of artifact, as in panel (d).

```{=html}
<div class="center">
```
!The use of the FT and its inverse to remove unwarranted information
from an image. (a) An image obtained by adding the sine wave and chest
radiography images together with its equivalent Fourier spectrum in (b).
The unwanted interference caused by the sinusoidal brightness pattern
can be removed by editing the spatial frequency information as shown by
the blackened areas in (c). The inverse FT then recovers the original
chest image largely undistorted as shown in (d). Further refinement of
the editing process would ideally allow complete restoration of the
image
quality. An image obtained by adding the sine wave and chest radiography images together with its equivalent Fourier spectrum in (b). The unwanted interference caused by the sinusoidal brightness pattern can be removed by editing the spatial frequency information as shown by the blackened areas in (c). The inverse FT then recovers the original chest image largely undistorted as shown in (d). Further refinement of the editing process would ideally allow complete restoration of the image quality.")\

```{=html}
</div>
```
The key finding to be gleaned from these three examples is that the
**spatial** and **spatial frequency** representations of the image data
are entirely equivalent. The frequency representation has numerous
advantages in terms of data manipulation. In general the FT and its
inverse provide us with the tools to transform the data from the real
world to one of spatial frequency and vice versa.

A more formal treatment of the Fourier transform is presented in our
chapter on Fourier
Methods.

Grundlagen der Nuklearmedizin/ Computer in der


# Basic Physics of Nuclear Medicine/Fourier Methods

Fourier methods are described below in
quite general terms. We will see that any waveform can be broken up
mathematically into sine waves of different
amplitudes and frequencies using the Fourier
transform. We will also see that the
resulting Fourier Spectrum can be modified to enhance and/or suppress
frequencies of interest. The aim of this exercise is to treat the
Modulation Transfer Function (**MTF**) and the filtration stage of the
tomographic image reconstruction process in greater detail than
previously.

## Periodic Functions

A periodic waveform, is a function
which repeats itself regularly over a given interval of time or space. A
good example is a sine wave or a square
wave. When the waveform fluctuates with
respect to time, the wave can be characterised by its **frequency** (see
the figure below), which is defined as the number of cycles passing a
given point each second. Frequency is expressed in units of cycles per
second, or hertz (Hz). A good example of such a periodic waveform in
medicine is the electrocardiogram (ECG).

!A sine wave in the temporal
domain.

When the waveform fluctuates with respect to distance (see the following
figure), the wave is characterised by its **spatial frequency**, which
is defined as the number of cycles per unit distance, e.g. cycles per
mm.

!A sine wave in the spatial
domain.

A medical imaging example of this latter type of fluctuation is the Pb
bar pattern which is widely used to determine the spatial resolution of
an imaging system (see the following figure). In this case, the spatial
frequency is expressed in, for example, **line pairs per mm**, since
each section of Pb and its adjacent void is referred to as a **line
pair**.

!(a) An image of a test object used to determine the limiting spatial
resolution of a gamma camera. (b) An image of a test object used to
determine the limiting spatial resolution of a radiographic imaging
system. (c) A plot of the number of counts (or the radiographic density)
against distance for well resolved regions of image (a) or
(b). An image of a test object used to determine the limiting spatial resolution of a gamma camera. (b) An image of a test object used to determine the limiting spatial resolution of a radiographic imaging system. (c) A plot of the number of counts (or the radiographic density) against distance for well resolved regions of image (a) or (b)."){width="380"}

## Fourier Series

Almost all periodic functions of interest in medical imaging can be
represented by a Fourier Series. This
approach considers that any periodic waveform can be represented by the
sum of a series of sine and cosine waves as follows:

```{=html}
<div class="center">
```
$f(x) = \frac{1}{2}a_0 + a_1 \cos (x) + a_2 \cos (2x)... + a_n \cos (nx) + b_1 \sin (x) + b_2 \sin (2x)... + b_n \sin (nx)$

```{=html}
</div>
```
A square wave, for example, can thus be represented by:

```{=html}
<div class="center">
```
$f(x) = \frac{4h}{\pi} \left ( \sin (x) + \frac{1}{3} \sin (3x) + \frac{1}{5} \sin (5x) + \frac{1}{7} \sin (7x) + ... \right )\ \ (1)$

```{=html}
</div>
```
where *h* is the amplitude of the square wave.

The opposite consideration is also true mathematically, i.e. a square
wave can be constructed by adding together a large number of sine waves
of different frequencies and amplitudes. The addition of the first four
terms of equation (1) is demonstrated in the following figure. The first
term (sin *x*) is shown in panel (a), the addition of the second term to
the first term is shown in panel (b) and so on. Notice that the
fundamental, or first harmonic \[panel
(a)\], has the same frequency as the square wave and that the higher
frequencies progressively build up the shape of the square waveform
\[panels (b)-(d)\]. We can conclude that the higher frequencies
contribute to the sharpness of the sides of the square wave.

!Illustration of the addition of sine waves to approximate a square
wave.{width="380"}

## Fourier Spectrum

The Fourier Series can also be represented as a frequency
spectrum. For example, the amplitudes
of the frequency components for the square wave in equation (1) are
plotted against spatial frequency in the following figure. Note that the
Fourier Spectrum can be used to identify the frequencies and amplitudes
of the sine waves which contribute to make up a given waveform. Note
also that plots of amplitude versus distance are generally referred to
as the **spatial domain** representation, and plots of amplitude versus
spatial frequency as the **frequency domain** representation.

!The Fourier Spectrum for a square
wave.{width="280"}

## Fourier Transform

The Fourier transform is an elegant mathematical technique for
converting data from the spatial domain to the frequency domain (see the
next figure). In other words, the frequencies and amplitudes of the sine
waves which make up any waveform can be readily determined by taking the
Fourier transform of that waveform.

!The Fourier Transform can be used to generate the Fourier Spectrum of
any waveform, including the square wave
above.{width="380"}

The Fourier transform is used widely throughout medical imaging, where
its applications include:

-   determining the spatial resolution of imaging systems,
-   spatial localisation in magnetic resonance
    imaging,
-   analysis of Doppler
    ultrasound
    signals,
-   image filtering in emission and transmission computed tomography.

The **Inverse Fourier Transform** is the mathematical technique used for
coverting data in the opposite direction, i.e. from the frequency domain
to the spatial domain -- see the following figure:

!The Inverse Fourier Transform can be used to generate any waveform of
interest, including the square wave
above.{width="380"}

In conclusion, the Fourier Transform (**FT**) allows us to identify the
component sine waves of a waveform, and the Inverse Fourier Transform
(**IFT**) allows us to construct a waveform from its component sine
waves.

Finally, it should be noted that the computation of Fourier Transforms
using a digital computer is generally achieved using a special algorithm
called the Fast Fourier Transform (FFT).

## The Dirac Delta Function

An interesting case to consider from a medical imaging perspective is
the delta function:

!The delta function on the left and its Fourier Spectrum on the
right.{width="380"}

The Fourier Transform of the Dirac delta function tells us that it is
composed of an infinite number of sine waves, each of the same
amplitude. If we begin to broaden this function -- as in the following
figure -- we see that the low frequency sine waves have a high amplitude
and that the amplitudes of the sine waves decrease as the spatial
frequency increases:

!The Fourier Transform of a broadened delta
function.{width="380"}

## Modulation Transfer Function

This broadening phenomenon is similar to what happens in medical
imaging. We can consider the amplitude versus distance plot in the
previous figure to be similar to a:

-   density profile through an image of a small hole in a sheet of Pb,
    or a
-   count rate profile through an image of a point source of
    radioactivity.

This type of plot is called a Point Spread Function
(PSF) in medical imaging, and its
Fourier Spectrum is called the Modulation Transfer Function (**MTF**):

!Illustration of the MTF of an ideal and an actual imaging
system.{width="240"}

A comparison of general and imaging terms used in this field is shown in
the following table:

  Domain                  General Terminology   Imaging Terminology
  ----------------------- --------------------- ------------------------------------
  **Spatial**             Input Function        Point Spread Function (PSF)
  **Spatial Frequency**   Fourier Spectrum      Modulation Transfer Function (MTF)

In addition, the MTF may be derived from:

:\* a Line Spread Function (**LSF**)

:\* a differentiated Edge Response Function (**ERF**).

Symbolically, we can write:

```{=html}
<div class="center">
```
$I_{\text{actual}} = I_{\text{ideal}} * PSF\,\!$

```{=html}
</div>
```
where $*$ indicates a convolution operation. In other words, the actual
image is obtained when the ideal image is convolved with the PSF of the
imaging system.

To restore the ideal image, all that is required theoretically is to
remove the effect of the PSF (e.g. attempts to solve imaging problems
encountered by the Hubble space
telescope). This can be achieved readily
using the Fourier transform because the convolution process in the
spatial domain is equivalent to a multiplication process in the spatial
frequency domain, i.e.

```{=html}
<div class="center">
```
$FT(I_{\text{actual}}) = FT(I_{\text{ideal}}) \cdot FT(PSF)\,\!$

```{=html}
</div>
```
and therefore,

```{=html}
<div class="center">
```
$FT(I_{\text{ideal}}) = \frac{FT(I_{\text{actual}})}{FT(PSF)}$

```{=html}
</div>
```
The complete restoration process is referred to as a
deconvolution operation and is given by:

```{=html}
<div class="center">
```
$I_{\text{ideal}} = IFT \left \lbrace \frac{FT(I_{\text{actual}})}{FT(PSF)} \right \rbrace$

```{=html}
</div>
```
## Filtering the Fourier Spectrum

The image restoration process discussed above is an example of Fourier
spectrum filtering. In other words, once a Fourier Spectrum has been
generated for an image, it can be filtered so that certain spatial
frequencies can be modified, e.g. enhanced or suppressed. This filtered
spectrum can then be inverse transformed to generate a filtered image
with, for example, sharpened or smoothed features -- as illustrated in
the following figure:

!The Fourier filtering
process.{width="280"}

An example is shown in the following figure in the case of a bone scan
of the hand. Its 2D FFT is shown in the top right panel. The result of
applying a filter which suppresses spatial frequencies below 0.05
cycles/pixel and above 0.3 cycles/pixel, i.e. a bandpass of 3-20 pixels,
is shown in the centre row of the figure, while that of a filter which
suppresses frequencies below 0.01 and above 0.1 cycles/pixels in shown
on the bottom row.

!Illustration of bandpass filtration to selectively suppress high and
low spatial frequencies and preserve intermediate
frequencies.{width="380"}

A feature we need to consider in more detail before proceeding is the
spatial frequency nature of the image data itself. Remember that images
are generally sampled digitally using a square matrix composed of
pixels, the size of which dictate how well a digital image approximates
its analogue counterpart. The resultant digital spatial resolution
places a limit on the maximum spatial frequency that can be
accommodated. The criterion usually applied in digital imaging is based
on the Nyquist-Shannon Sampling Theorem
which implies that:

```{=html}
<div class="center">
```
`<big>`{=html}When an image has spatial frequency components with a
maximum spatial frequency, f, then the image data should be sampled with
a sampling frequency of at least twice f for faithful
reproduction.`</big>`{=html}

```{=html}
</div>
```
This sampling frequency is commonly called the Nyquist
frequency. At lower sampling
frequencies, the resultant digital images can contain artefactual
patterns, called Moiré patterns, and the
phenomenon is sometimes referred to as
aliasing.

## Filtered Back Projection

The streaking inherent in the simple backprojection process makes the
actual image appear as if it had been combined mathematically with a 1/r
function, where `<big>`{=html}r`</big>`{=html} is the radial distance in
the Fourier domain. In filtered backprojection, Fourier filtering can be
used to remove the effect of this **
blurring**.

Symbolically, a measured projection can be considered to be the result
of a convolution with a blurring function:

```{=html}
<div class="center">
```
$P_{\text{measured}} = P \cdot \frac{1}{r}$

```{=html}
</div>
```
The first stage of the filtration process is to calculate the Fourier
transform of the measured projection data, i.e.

```{=html}
<div class="center">
```
$FT(P_{\text{measured}}) = FT(P) \cdot FT \left ( \frac{1}{r} \right )$

```{=html}
</div>
```
The corrected projection, *P*, is then obtained by dividing the FT of
the measured projection by **FT()**, and
taking the inverse transform of the result, i.e.

```{=html}
<div class="center">
```
$P = IFT \left \lbrace \frac{FT(P_{\text{measured}})}{FT \left ( \frac{1}{r}\right )} \right \rbrace$

```{=html}
</div>
```
The **FT()** function is simply that of a
ramp. In other words, when the **FT** of the measured projection is
calculated and the result is divided by this ramp function, then the
corrected projection can be obtained by calculating the **IFT** of this
quotient.

In addition, if some variation is introduced into the
**FT()** function, it is possible to
simultaneously correct for this blurring effect and to enhance or
suppress features in the back-projected image. For example, the blurring
artifact can be removed and, at the same time:

:\* the fine detail can be enhanced (as in the so-called **bone
algorithms** in X-ray CT), or

:\* the image noise can be suppressed (as in the so-called **soft tissue
algorithms** in X-ray CT).

In general, variation in the ramp function can be achieved by
multiplying it by a second function, e.g. a Butterworth
function used in SPECT:

```{=html}
<div class="center">
```
$\text{Amplitude} = \frac{1}{1+(\frac{f}{f_c})^{2n}}$

```{=html}
</div>
```
where:

-   *f*~*c*~: cut-off frequency, which defines the frequency at which
    the amplitude drops by 50%, and
-   *n*: the order of the function.

!A ramp function and a Butterworth function (of variable order and
cut-off frequency) are multiplied to form the Fourier filter used in the
FBP
process. are multiplied to form the Fourier filter used in the FBP process."){width="480"}

Characteristics of a number of filters used in SPECT imaging are shown
in the following table:

+----------------------+----------------------+----------------------+
| Filter               | Equation             | Comment              |
+======================+======================+======================+
| Ram-Lak              | $\text{Amp           | star artifact        |
|                      | litude} = 1\ \,\text | removal; noise       |
|                      | {for}\ f < f_c\,\!$\ | sensitivity          |
|                      | \                    |                      |
|                      | $\text{Am            |                      |
|                      | plitude} = 0\ \,\tex |                      |
|                      | t{for}\ f > f_c\,\!$ |                      |
+----------------------+----------------------+----------------------+
| Butterworth          | $\text{Amplit        | noise reduction      |
|                      | ude} = \frac{1}{1+(\ |                      |
|                      | frac{f}{f_c})^{2n}}$ |                      |
+----------------------+----------------------+----------------------+
| Metz                 | $\text{Amplitude} =  | noise reduction;     |
|                      | \frac{1-\big(1-MTF(f | contrast enhancement |
|                      | )^2\big)^x}{MTF(f)}$ |                      |
+----------------------+----------------------+----------------------+
| Wiener | de} = \frac{1}{MTF(f | contrast enhancement |
|                      | )} \left \lbrace \fr |                      |
|                      | ac{MTF(f)^2}{S+MTF(f |                      |
|                      | )^2} \right \rbrace$ |                      |
+----------------------+----------------------+----------------------+
| Scramp               | $\text{Am            | noise reduction;     |
|                      | plitude} = 1 + \frac | contrast enhancement |
|                      | {f}{f_c} \lbrace MTF |                      |
|                      | (f_c)-1 \rbrace\ \,\ |                      |
|                      | text{for}\ f < f_c$\ |                      |
|                      | \                    |                      |
|                      | $\text{Am            |                      |
|                      | plitude} = 0\ \,\tex |                      |
|                      | t{for}\ f > f_c\,\!$ |                      |
+----------------------+----------------------+----------------------+
| Inverse MTF          | $                    | noise reduction;     |
|                      | \text{Amplitude} = \ | contrast suppression |
|                      | frac{1}{MTF(f)}\ \,\ |                      |
|                      | text{for}\ f < f_c$\ |                      |
|                      | \                    |                      |
|                      | $\text{Am            |                      |
|                      | plitude} = 0\ \,\tex |                      |
|                      | t{for}\ f > f_c\,\!$ |                      |
+----------------------+----------------------+----------------------+
| Hamming              | $\text{Amplitude     | noise reduction      |
|                      | } = 0.54 + 0.46 \cos |                      |
|                      |  \left ( \pi \frac{f |                      |
|                      | }{f_c} \right )\ \,\ |                      |
|                      | text{for}\ f < f_c$\ |                      |
|                      | \                    |                      |
|                      | $\text{Am            |                      |
|                      | plitude} = 0\ \,\tex |                      |
|                      | t{for}\ f > f_c\,\!$ |                      |
+----------------------+----------------------+----------------------+
| Parzen               | $\text{Ampl          | noise reduction      |
|                      | itude} = 2 \left \lb |                      |
|                      | race 1 - \left (\fra |                      |
|                      | c{f}{f_c} \right )^3 |                      |
|                      |  \right \rbrace\ \,\ |                      |
|                      | text{for}\ f < f_c$\ |                      |
|                      | \                    |                      |
|                      | $\text{Am            |                      |
|                      | plitude} = 0\ \,\tex |                      |
|                      | t{for}\ f > f_c\,\!$ |                      |
+----------------------+----------------------+----------------------+
| Shepp-Logan          | $\                   | noise reduction      |
|                      | text{Amplitude} = \f |                      |
|                      | rac{2f_c}{\pi f} \si |                      |
|                      | n \left (\pi \frac{f |                      |
|                      | }{f_c} \right )\ \,\ |                      |
|                      | text{for}\ f < f_c$\ |                      |
|                      | \                    |                      |
|                      | $\text{Am            |                      |
|                      | plitude} = 0\ \,\tex |                      |
|                      | t{for}\ f > f_c\,\!$ |                      |
+----------------------+----------------------+----------------------+
| Hann                 | $\text{Amplitude}    | noise reduction      |
|                      |  = \frac{f}{2} \left |                      |
|                      |  (1+ \cos \pi \frac{ |                      |
|                      | f}{f_c} \right )\ \, |                      |
|                      | \text{for} f < f_c$\ |                      |
|                      | \                    |                      |
|                      | $\text{Am            |                      |
|                      | plitude} = 0\ \,\tex |                      |
|                      | t{for}\ f > f_c\,\!$ |                      |
+----------------------+----------------------+----------------------+

The choice of filter for a given image reconstruction task is generally
a compromise between the extent of noise reduction and fine detail
suppression (and of contrast enhancement in some cases) as well as the
spatial frequency pattern of the image data of interest.

## Further reading

-   Engineering Analysis/Multi-Dimensional Fourier
    Series
-   The Feynman lectures Chapter 50
    Harmonics


# Basic Physics of Nuclear Medicine/X-Ray CT in Nuclear Medicine

The need for attenuation compensation in SPECT imaging has been
understood for many years but it has only been in recent times that
effective techniques have been applied in commercial systems. Gamma rays
get attenuated by the medium through which they pass, as we\'ve seen
earlier,
via photoelectric absorption and Compton scattering. The projections
measured in SPECT imaging therefore contain impressions of this
attenuation which reduce their fidelity for tomographic reconstruction,
as illustrated in the following figure:

!Tomographic slices from a cardiac study reconstructed without
attenuation correction on the top row and with attenuation correction on
the bottom row. Note the more uniform distribution of radioactivity in
the cardiac muscle depicted in the attenuation corrected
images.{width="280"}

Patient specific **attenuation maps** can be used to correct the
measured projections. These can be produced using the gamma camera to
generate a transmission scan of the patient using a radioisotope source
-- see the following figure for an example:

!A posterior-anterior transmission image of the author\'s chest
acquired using an uncollimated gamma camera and an 18.5 MBq point source
of ^99m^Tc at a focal distance of two
metres.{width="160"}

The transmission scan contains the attenuation information for the
volume of interest within the patient and a map of this attenuation
pattern is subsequently applied to each SPECT projection prior to
filtered back projection. A more powerful approach is to use an X-ray CT
scanner to generate the attenuation maps -- see the following figure.
Numerous *hybrid* scanners have been produced as a result which combine
nuclear medicine with CT imaging, as in **SPECT/CT** and **PET/CT** --
see the following figure for an example:

!A SPECT/CT system, with relevant components labelled in the photograph
on the
right.{width="280"}

An additional benefit of these developments is the simultaneous
capability, with sufficient computing power, of blending the nuclear
medicine and CT images so as to generate physiological images
colocalized with images of patient anatomy -- as shown in the following
figure -- and hence improve the diagnostic utility of nuclear medicine
procedures.

!A SPECT image blended with a CT scan. Blending, indicated by the +
sign, in this case involved sub-processes which included image
magnification, contrast enhancement, application of a colour look-up
table and layering of one
image on top of the other using an appropriate
transparency.{width="280"}

Hybrid scanners generally have a single patient couch with the gamma
camera gantry moulded to the front of a CT unit so that the patient can
be transported into the X-ray beam before or after the SPECT
acquisition. Some CT units are of a low resolution design and generate
crude, single slice CT images which are sufficiently accurate for SPECT
attenuation correction but are of insufficient quality on their own for
diagnostic purposes -- see the example below. There are also higher
resolution CT units in use for hybrid scanning with capabilities of
acquiring, for instance, 16-slice helical scans which can compete
directly with the diagnostic capabilities of dedicated CT scanners.

!A low resolution CT scan generated for attenuation correction
applications in a SPECT/CT
system.

This chapter provides an overview of the physical aspects of X-ray CT
scanning from a nuclear medicine perspective. We begin with various
scanner designs and will explore a number of computational techniques
relevant to this form of imaging. The intention is to provide a
foundation for understanding the application of X-ray CT in nuclear
medicine.

## Conventional Radiography

The attenuation of an X-ray beam in the body is used in conventional
radiography to project a shadow onto an image receptor (as shown in the
figure below). These shadowgraphs record a two-dimensional
representation of a three-dimensional object. Small lesions are
therefore not readily identified because of overlapping and underlying
anatomy, image distortion occurs because of unequal magnification
effects and low contrast masses are poorly delineated since scatter
contributes substantially to the image data.

!Bird\'s eye view of a conventional radiographic procedure on the left,
with a chest radiograph displayed on the right using a nuclear
medicine-oriented look-up table -- where low counts are assigned a dark
shade on a grey scale and high counts as whiter
shades.{width="380"}

We can consider that there is a **degeneracy** introduced into such
images, i.e. two anatomically quite different objects may produce the
same film density because the effective attenuation they each produce is
identical, as illustrated in the following figure:

!The origin of Image Degeneracy in conventional radiography. The
attenuation of each element may be characterised by
`<span style="color:red;">`{=html}μ`</span>`{=html}, the linear
attenuation coefficient as in panel (a). The total attenuation of each
of the two columns of elements in panel (b) is identical so that the
image density is the same for quite different anatomical
detail.. The total attenuation of each of the two columns of elements in panel (b) is identical so that the image density is the same for quite different anatomical detail.")

Panel (a) shows a pencil beam of X-radiation of incident intensity,
I~o~, attenuated by the body, represented here for simplicity by four
volume elements each of thickness, Δx, and of different linear
attenuation coefficients, μ~1~ thru μ~4~. The transmitted intensity,
when no scattered radiation is detected, is given by I, which is
dependent on the sum of these linear attenuation coefficients, according
to the exponential attenuation equation. Panel (b) illustrates that two
adjacent columns of volume elements can generate identical values of the
transmitted intensity, I, even though the two columns contain volume
elements with different linear attenuation coefficients. In other words,
what is detected in the integrated contribution of the linear
attenuation coefficients without any information about the contribution
from any individual volume element.

One way to improve this situation is to use X-Ray Computed Tomography
(**CT**). Note that the image reconstruction algorithms developed for
X-ray CT can also be applied to gamma camera images (as in **SPECT**) to
overcome limitations of planar nuclear imaging and to images generated
using positron-emitting radioisotopes (as in **PET**).

## Back Projection

We have considered this method of computerized image reconstruction in a
previous chapter in the context of SPECT. Our intention here is to
consider this reconstruction method in the context of X-Ray CT.

The following figure is a representation of how back projection works.
The basic premise is that any attenuation of the X-ray beam is assumed
to have occurred uniformly along the entire ray path.

!Back projection
reconstruction.{width="280"}

Thus, the result of the back projection of the first profile, P~1~ is to
put the values 7 and 9 in both elements of the first and second rows,
respectively. The second profile, P~2~ adds a
`<span style="color:green;">`{=html}4`</span>`{=html} to the top right
element, 1 to the bottom left element and 11 to the other two elements
when back projected. The other profiles are treated in similar fashion.
Following subtraction of an offset (16 in this case) and renormalisation
(division by 3 in this case) of the data set, the final image is
obtained. You should confirm for yourself that this final matrix
satisfies all projections.

The principle is demonstrated in more detail in the following figure:

!A demonstration of simple back projection: (a) An X-ray tube scans a
phantom, consisting of a radio-dense object in an otherwise uniform
container, and generates the profile as shown for the back projection
process. (b) Four profiles generated by scanning at slightly different
angles around the phantom. (c) The image reconstructed from just four
projections. An X-ray tube scans a phantom, consisting of a radio-dense object in an otherwise uniform container, and generates the profile as shown for the back projection process. (b) Four profiles generated by scanning at slightly different angles around the phantom. (c) The image reconstructed from just four projections."){width="380"}

A single profile is back projected to give a dark stripe across the
entire image plane \[panel (a)\]. As we scan the phantom from many
directions and back project the ray profiles onto the image plane
\[panel (b)\], an image of the radio-dense dot, albeit a poor one,
begins to resolve \[panel (c)\]. As the number of projections increases,
the quality improves but some blurring will always remain in the image.

This blurring can be removed using what is known as Filtered Back
Projection.

## Filtered Back Projection

The blurring inherent in the simple back projection process can be
suppressed mathematically using filtering techniques and the overall
reconstruction process is then called **Filtered** Back Projection. In
this technique (see the following figure), the attenuation profile is
first filtered to counteract the effect of sudden density changes which
cause the blurring in simple back projection. Note that the graphical
representation in the figure is very crude as the filtered profile may
be a quite complicated function of distance as illustrated in panel (c).
The filter is also referred to as a **convolution filter**, or
**convolution kernel**. After completion of the filtration process, the
Inverse Fourier Transform is applied to the data for each profile before
the back projection process is undertaken. Notice that the reconstructed
image \[panel (d)\] is free of the **star** artefact generated in
unfiltered back projection.

!A demonstration of filtered back projection: (a) An X-ray tube scans a
phantom and generates the profile which is filtered using the Fourier
transform. (b) Four filtered profiles generated by scanning at slightly
different angles around the phantom. (c) A detailed view of the effect
of the filtration process on one ray profile. (d) The reconstructed
image is free of the **star**
artefact. An X-ray tube scans a phantom and generates the profile which is filtered using the Fourier transform. (b) Four filtered profiles generated by scanning at slightly different angles around the phantom. (c) A detailed view of the effect of the filtration process on one ray profile. (d) The reconstructed image is free of the star artefact."){width="380"}

Most CT scanners offer a choice of filters which may be selected by the
operator to enhance either soft tissue features or bone detail in
images. Indeed, the image can be post-processed using a different filter
after the scan has been completed if so desired. Images can be generated
to enhance bone detail on the one hand and to observe subtle low
contrast masses on the other, without the need to re-scan the patient.
Thus, the filter choice will have a major impact on image quality. Two
common filters used in X-ray CT are those due to Ramachandran &
Lakshminarayanan (Ram-Lak) and to Shepp & Logan. In SPECT, Butterworth,
Hanning and Metz are used, as you will
remember.
The ramp filter on its own compensates for artefacts introduced by the
simple back projection process but does not compensate for the
increasing noise content of the data with increasing spatial frequency.
The Ram-Lak is a ramp filter which leaves all frequencies unchanged
except those above a cut-off, which are eliminated. Most of the other
filters suppress higher frequencies to varying degrees. Sometimes these
filters are referred to as **algorithms**. The term **kernel** is also
used. Images of an axial tomogram reconstructed with a soft tissue and
with a bone algorithm are shown below as examples:

!A tomogram reconstructed using (a) a soft tissue filter, and (b) using
a bone filter. Notice the subtle edge enhancement effect generated using
the bone filter and the more smooth nature of the soft tissue
filtration. a soft tissue filter, and (b) using a bone filter. Notice the subtle edge enhancement effect generated using the bone filter and the more smooth nature of the soft tissue filtration."){width="280"}

There\'s an excellent online CT reconstruction simulator available via
the Biomedical Imaging
Group at the EPFL.
As an exercise, you might like to explore your understanding of filtered
back projection by varying the type of filter used prior to
reconstruction. You might also like to explore the influence of the
number of angles used for profile acquisition on the quality of the
reconstructed image.

## CT Scanning Geometries

!First generation CT scanner
design.

Most CT scanner designs use some form of mechanical gantry which allows
movement of the X-ray tube and the detector around the patient. The
X-ray beam was collimated to a pencil beam in early designs and consists
of a wide fan beam in modern systems. The thickness of the slice,
typically 1 to 10 mm, is generally defined by pre-patient collimation
using motor driven adjustable wedges external to the X-ray tube. Even
with relatively large fan beams the volume of tissue irradiated is still
small quite compared with projection radiography and scatter is a less
serious problem. The detectors have collimators placed in front of them
to minimise the impact of out of slice scatter. To minimise the effects
of beam hardening CT scanners use a heavily filtered beam (the Siemens
Somatom Plus uses 2.7 mm Al plus 0.2 mm Cu, for instance) operating at
120 -- 140 kVp. Note that even with the best radiographic geometry, such
as in the third generation scanners, object magnification is substantial
and the finite size of the focal spot, typically 1 mm, may well limit
the achievable spatial resolution.

The **first generation** of CT scanner used what is referred to as
**Translate-Rotate** geometry. The original EMI Mk 1 scanner (see the
figure on the right), for instance, used a pencil X-ray beam and a
single detector (a NaI:Tl scintillator coupled to a photomultiplier
tube).
During the translational motion of the gantry, the transmitted X-ray
beam was sampled 160 times to produce a single profile. A rotation
through 1 degree with the X-ray beam off then followed before a new
profile was obtained. This procedure was completed until the patient\'s
head was scanned from 180 different angles.

!Second generation CT scanner
design.

The whole procedure took about 5 minutes and depended on the patient
remaining completely still. The patient\'s head was usually clamped to
minimise patient motion. In addition, their head was surrounded by a
water bag which helped to overcome problems associated with afterglow in
the NaI(Tl) scintillator and the very substantial signal dynamic range
that would otherwise be present at the detectors. Subsequent generations
of scanners have sought to decrease the scan time to a few seconds to
minimise the movement artifact.

The **second generation** of scanner addressed this issue by using a
small fan beam (see the next figure) with multiple detectors -- up to 30
in some designs. The detectors collect multiple data readings during a
translation so that fewer translations and rotations are required. Each
of the detectors collects its own limited profile during each
translation. Typically, a rotation of 30 degrees was employed between
translational movements so that only six rotational movements were
required to obtain the data for one slice.

Scan times of between 5 to 90 seconds were achieved with this design.
The detector elements were scintillators but the water bag was dispensed
with and body scans could be performed for the first time. The use of
multiple detectors gave a quantum leap in performance.

The **third generation** of CT scanner decreased scan times even further
by using a **Rotate-Rotate** geometry (see the figure on the right).
Most scanners today are of the third generation type. A typical machine
employs a large fan beam such that the patient is completely encompassed
by the fan. The number of detector elements is typically in the
hundreds, e.g.

:\* the GE Hispeed Advantage has 852 elements 1.0 mm apart;

:\* the Siemens Somatom Plus 4 has 768 elements 1.1 mm apart; and

:\* the Toshiba Xpress/SX has 896 elements 1.03 mm apart.

!Third generation CT scanner
design.

The detector elements are aligned along the arc of a circle centred on
the focus of the X-ray tube. The X-ray tube and detector array rotate as
one through 360 degrees during which time several hundred discrete
profiles are obtained. Both solid state detectors and pressurised xenon
gas detectors are used for data acquisition.

Because the fan beam totally encompasses the patient, translational
motion is not needed and the tube and detector array rotate as one about
the patient. The number of profiles recorded depends on the chosen scan
parameters but is at least a few hundred and can be in excess of a
thousand. Scan times can be as low as a second. Although the dynamic
range of the detectors is very high, some manufacturers control
excessive variations in signal strength by using bow-tie shaped filters
chosen to suit the body or head shape. Such filters generally attenuate
the peripheral part of the fan beam to a greater extent than the central
part. It also helps overcome the effects of beam hardening and to
minimise patient skin dose in the peripheral part of the field of view.

A number of variants on this geometry have been developed, which include
those based on offsetting the centre of rotation and the use of a
**flying focus** X-ray tube.

!Fourth generation CT scanner
design.

The **fourth generation** of CT scanner uses a **Rotate-Fixed Ring**
geometry where a ring of fixed detectors completely surrounds the
patient. The X-ray tube rotates inside the detector ring through a full
360 degrees with a wide fan beam. Only those detectors which see the
beam are activated at any one time which means that only a small
fraction of thousands of detector elements (e.g. the Picker PQ-5000V has
4800 detector elements 1.1 mm apart) are in use at any given time. Scan
time is comparable with third generation scanners but the radiographic
geometry is poor because the X-ray tube must be closer to the patient
than the detectors, i.e. the geometric magnification is large.

The disadvantages of poor geometry noted above has been alleviated very
neatly by the so called **nutating** geometry. The X-ray tube is
external to the detector ring but slightly out of the detector plane.
Scan times as low as 0.6 s can be achieved using machines of this type
with interscan delays of as little as 100 ms. Excellent resolution (0.35
mm) can be obtained by using as many several thousand detector elements.

## Helical Scanning

Innovations in slip-ring technology, beginning with the Siemens Somatom
Plus and the Toshiba TCT 900S, have enabled the X-ray tube to rotate
continuously in the same direction which overcomes problems of interscan
delays. When the continuous motion of the X-ray tube is combined with a
continuous advance of the patient table along the axis of the scanner we
have helical (also called spiral) scanning, as illustrated the following
figure. Typical table velocities are 1 -- 10 mm/s; a complete 360 degree
rotation can be achieved in 0.5 s and the nominal fan beam thickness is
1 mm or greater.

!Illustration of helical
scanning.

In the context of helical scanning a parameter called **pitch** is
defined as the:

```{=html}
<div class="center">
```
`<span style="color:green;">`{=html}ratio of the distance that the
patient couch moves in one rotation to the slice
thickness.`</span>`{=html}

```{=html}
</div>
```
In other words, for a couch advance of 10 mm and a nominal slice width
of 10 mm, the pitch is 1. Pitch values are typically in the range of 1
to 2 depending on the required spatial resolution in the direction of
the couch motion. Its a *coverage* indicator, in other words.

A significant reduction in scan times results from helical scanning
because large volumes can be scanned contiguously without any gaps and
with little influence from patient motion. The Somatom Plus-4A, for
instance, allows up to 80 slices to be scanned contiguously in 60
seconds of non-stop scanning. Single breathhold scanning of the thorax
is therefore possible and reconstruction, with minimal loss of spatial
resolution, of sagittal and coronal planes is feasible.

Axial reconstruction in helical scanning is illustrated in the following
figure. It is seen that no axial slices are actually irradiated during
the scan and that any transaxial image data needs to be estimated from
the acquired helical record. One estimation technique is referred to as
z-interpolation, because it applies data interpolation techniques along
the z-dimension of the acquired data, i.e. the patient\'s head-to-foot
axis.

!Illustration of transaxial reconstruction in helical
scanning.

The filtered back projection reconstruction process is therefore
modified to include an z-interpolation step prior to Fourier filtration,
as illustrated in the following flow diagram:

!The image reconstruction process in helical
scanning.

## Interpolation & Extrapolation

Before proceeding, let\'s briefly review the topic of interpolation and
extrapolation to clarify our basic understanding and to put the
subsequent discussion in context. Let\'s start with a simple
one-dimensional case, as illustrated in the following graph. Suppose
we\'ve taken measurements A and B and plotted them on the graph, and
wish to estimate the value at an in-between point, C. Such an estimation
process is called **interpolation**, and if we assume that there\'s a
linear relationship between variables X and Y as illustrated in our
graph, the process of estimating point C is called **linear
interpolation**. Other functions can also be used as interpolatants for
more complex data than we\'re considering here.

!Illustration of interpolation and
extrapolation.

If we also require knowledge about point D on our graph, i.e. a point
outside the measured range, we refer to the estimation process as
extrapolation, and linear extrapolation is illustrated in our graph
above.

Mathematically we can write that given:

```{=html}
<div class="center">
```
$A \Rightarrow (x_1, y_1)\ \ B \Rightarrow (x_2, y_2)\ \ C \Rightarrow (x_3, y_3)$

```{=html}
</div>
```
then the Y-value of point C is given by:

```{=html}
<div class="center">
```
$y_3 = y_1 + \frac{(x_3 - x_1)(y_2 - y_1)}{(x_2 - x_1)}$

```{=html}
</div>
```
using linear interpolation. You might like to develop a similar equation
as an exercise which can be used to determine the Y-value of point, D.

Since helical scanning is a three-dimensional phenomenon, we need to
develop some two-dimensional perspectives on the situation before
proceeding with our discussion -- as illustrated in the following
figure:

!Three perspectives of the helical scan in the top left
corner.

We\'ll use the side view in the following discussion.

## Interpolation in Helical CT

A large number of computational methods have been developed for helical
scanning. The **360^o^ linear interpolation algorithm** is illustrated
in the following figure as an example:

!A side view of a helical scan and parameters used in the 360^o^ linear
interpolation algorithm (360LI), where Z~ref~ indicates the position of
the axial slice to be
interpolated., where Zref indicates the position of the axial slice to be interpolated.")

The following equation can be developed on the basis of our earlier
discussion for linear interpolation of the transaxial data of interest:

```{=html}
<div class="center">
```
$P_z(i, \alpha) = (1-w)P_j(i, \alpha) + wP_{j+1}(i, \alpha)\,\!$

```{=html}
</div>
```
where:

:   P~z~(i, α) is the interpolated projection for an angle, α, at
    position Z~ref~;
:   P~j~(i, α) is the helical projection measured in rotation j at
    position Z~j~ and projection angle α;
:   P~j+1~(i, α) is the helical projection measured in rotation j+1 at
    position Z~j+1~ and projection angle α; and

```{=html}
<div class="center">
```
$w = \frac{Z_{\text{ref}} - Z_j}{d}$

```{=html}
</div>
```
This interpolation process is applied throughout all data points in the
helix to generate estimates of the axial data, prior to filtered
backprojection. Higher order interpolation is also possible where data
points from, say, four rays are used in the estimation. In addition,
sophisticated z-axis filters can be applied for these estimations in
multi-slice CT.

## Volume Imaging

It is possible to produce a number of slices per rotation when the fan
beam of a CT scanner is broadened along the z-axis and two-dimensional
detectors are used. The fan beam can be formed into the shape of a cone
on this basis and concepts such as **volume imaging** and **cone beam
CT** can be applied. Terms such as **multi-slice** and
**multi-detector** CT can also be used in this context, although as you
will see the second of these two terms is probably the more appropriate.

The 2-D detectors are solid-state devices and the simplest example is
the **Matrix Array**, as illustrated below:

!Illustration of four modes of multi-slice acquisition for a matrix
array CT
detector.{width="280"}

The array can consist of (say) 912 columns by 16 rows of identical
detectors, each 1.25 mm square, curved to fit an arc of the x-ray tube
rotation. Eight of these columns are shown in panel (a) above being
irradiated by an X-ray beam of width 6 mm, with the implication that
data for four 1.25 mm axial slices can be acquired simultaneously. More
rows, up to sixteen in this case, can be irradiated simultaneously with
this arrangement when the x-ray beam is broadened to encompass the width
of the detector array.

Flexibility can be built into this design by coupling the outputs of
adjacent detector rows, as illustrated in panels (b), (c) and (d), where
the outputs of 2, 3 and 4 rows, respectively, are summed to
simultaneously generate four 2.5 mm thick slices, or four 3.75 mm
slices, or (you guessed it!) four 5 mm slices.

A more flexible design is provided by the **Adaptive Array** detector,
as illustrated in the next figure:

!Illustration of an adaptive array CT
detector.{width="280"}

Rather than columns of square detectors, this array uses columns of
detectors of variable width, such that the width in the two central
columns is relatively narrow, e.g. 1 mm as in the figure above, with the
column width increasing towards the periphery, e.g. from 1.5 mm, through
2.5 mm to 5 mm as in our figure. The reason for this pattern is
explained below.

Four modes of operation of this detector array are shown in the
following figure:

!Illustration of four modes of multi-slice acquisition for an adaptive
array CT
detector.{width="280"}

We can see in panel (a) that two 0.5 mm slices can be acquired when a
1 mm thick fan beam is aligned with the central columns of the adaptive
array. In panel (b) we can see that four 1 mm slices can be acquired
using a fan beam of thickness 4 mm. Detector coupling is illustrated in
panels (c) and (d), where the outputs of detectors in the four central
columns are coupled so as to simulate two 2.5 mm wide detection columns
and hence acquire data for four 2.5 mm slices. How do you think this
coupling process is applied in panel (d) to generate four 5 mm slices?

A second advantage of the adaptive array is that the number of
individual detector elements on each row can be considerably reduced,
from 16 to 8 in the case we\'ve just discussed. This adds greatly to the
speed with which data can be generated by the array and reduces the
number of computations necessary for uniformity and other corrections
that must also be applied to the measured data.

The interpolation process in multi-slice helical CT is illustrated in
our final figure where the situation in a 4-slice system using a
**360^o^ multi-slice filtered interpolation algorithm** is shown:

!Illustration of the 360MFI
algorithm.{width="280"}

The number of detector rows in Multi-Detector CT (MDCT) in 2010 is
typically 8, 16 or 64, or greater -- even a 320-slice scanner has been
developed.

## CT Image Display

After the computer calculates the linear attenuation coefficient for
each pixel via filtered back projection, the values are normalised to
the value for water as a reference, scaled and presented as a Hounsfield
or CT-number which is defined as:

```{=html}
<div class="center">
```
$H = 1000 \frac{\mu_m - \mu_{\text{water}}}{\mu_{\text{water}}}$

```{=html}
</div>
```
where μ~m~ and μ~water~ are the linear attenuation coefficients for the
tissue material and for water, respectively. The CT-number of water is
therefore zero. CT-numbers for a number of tissues are shown in the
following table:

  Tissue                CT-Number (H)
  --------------------- ---------------
  **Lung**              -300
  **Fat**               -90
  **White Matter**      30
  **Gray Matter**       40
  **Muscle**            50
  **Trabecular Bone**   300-500
  **Cortical Bone**     600-3,000

The image is usually presented on a computer monitor using a grey scale.
The grey scale is chosen to encompass all or some part of the entire
range of CT-numbers by selecting a suitable window level and window
width. The window width is the range of CT-numbers we select for display
and the window level is, usually but not always, the central CT-number
about which the window is chosen. Typically, the highest number is
assigned to white and the lowest number to black with all intervening
numbers assigned intensities on a linear scale. Thus, air will be
displayed as black and cortical bone will appear relatively bright.

Examples of image display manipulation are shown in the following two
figures. In the first, the same image of a slice through a patient\'s
liver is displayed using a relatively narrow window (high contrast) and
also with a wide window. The image with the narrower window appears
noisier, but this is merely a reflection of the fact that the gray scale
spreads over a narrow range of CT-numbers.

!Effect of window width and level on CT image display. (a) Level = 50;
Width = 200. (b) Level = 50; Width = 400. The image in (a) is displayed
with greater contrast and appears noisier than that in
(b). Level = 50; Width = 200. (b) Level = 50; Width = 400. The image in (a) is displayed with greater contrast and appears noisier than that in (b)."){width="380"}

In the second figure, a relatively narrow window has been used to
highlight pathology:

!Effect of window width and level on CT image display. (a) Level =
-600; Width = 1700. (b) Level = -60; Width = 400. The image in (a)
displays the lung tissue more clearly, whereas the image in (b)
highlights pulmonary
metastases. Level = -600; Width = 1700. (b) Level = -60; Width = 400. The image in (a) displays the lung tissue more clearly, whereas the image in (b) highlights pulmonary metastases."){width="380"}

## Scanned Projection Radiography

Most CT systems use scanned projection radiography (**SPR**), also known
as a *scout view* and *scanogram*, to establish a consistent set of
anatomical markers for all subsequent procedures. In SPR, the fan beam
is not rotated about the patient but is fixed, and the patient couch is
moved slowly through the beam. Each detector element measures a varying
amount of radiation intensity as the anatomy changes between the focus
and the detector. These discrete detector signals are digitised but not
processed to any great extent. The computer stores the signals from all
the detectors as a series of line images. The width of each line is
small, typically 1.5 -- 2 mm, corresponding to the finite width of the
fan beam. Subsequently, a projection radiograph is synthesized by making
a composite of these line scan images with the grey scale suitably
chosen to display the anatomy with an adequate contrast -- see an
example image below:

!Lateral view SPR image of a patient\'s head acquired prior to CT
scanning.{width="280"}

The volume of tissue to be examined in CT mode can be defined once the
SPR image has been generated. The CT procedure can then be automatically
controlled by the computer once the appropriate scanning factors have
been selected.