# 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.

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!(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).")\

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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.

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```
!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.")\

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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).

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!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.")\

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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
