Name ………………………………………………………
Advancing Physics AS
John Mascall
The King’s School, Ely
Chapter 1 Imaging
Student Notes
August 2008
The entry below is taken from the 2008 OCR specification which combines the topics to be
taught in Chapters 1 and 3. References to signalling should be ignored at this stage.
PA 1.1: Imaging and signalling
In the context of the digital revolution in communication, this section introduces elementary ideas
about image formation and digital imaging, and about the storage and transmission of digital
information.
The material can be taught using up-to-date contexts such as mobile telephones, use of internet,
email, and medical scanning and scientific imaging including remote sensing. There are
opportunities to address human and social concerns, for example, the consequences of the growth
of worldwide digital communications.
Assessable learning outcomes
Candidates should demonstrate evidence of:
1. knowledge and understanding of phenomena, concepts and relationships by describing and
explaining:
(i) the formation of a real image by a thin converging lens, understood as the lens changing the
curvature of the incident wave-front;
(ii) the storage of images in a computer as an array of numbers that may be manipulated to enhance
the image (vary brightness and contrast, reduce noise, detect edges and use of false colour);
(candidates are not expected to carry out numerical manipulations in the examination; an
understanding of the nature of the processes will be sufficient);
(iii) digitising a signal (which may contain noise); advantages and disadvantages of digital signals;
(iv) the presence of a range of frequencies in a signal (its spectrum);
(v) evidence of the polarisation of electromagnetic waves;
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2. scientific communication and comprehension of the language and representations of physics, by
making appropriate use of the terms:
(i) pixel, bit, byte, focal length and power, magnification, resolution, sampling, spectrum, signal,
bandwidth, noise, polarisation, refractive index (understood as the ratio of speed of light in
vacuum to the speed of light in material of lens);
and by sketching and interpreting:
(ii) diagrams of the passage of light through a converging lens;
(iii) diagrams of wave-forms, and their spectra;
3. quantitative and mathematical skills, knowledge and understanding by making calculations and
estimates involving:
(i) the amount of information in an image = no. of pixels × bits per pixel;
(ii) power of a converging lens P = 1/f, as change of curvature of wave-fronts produced by the lens;
(iii) use of
1 = 1 + 1
v
u
f
(Cartesian convention; linear magnification
m = image height = v
object height u
restricted to thin converging lenses and real images.
(iv) ν = fλ
I
(v) amount of information, I, provides N = 2 alternatives; I = log N;
2
(vi) minimum rate of sampling ≥ 2 × maximum frequency of signal;
(vii) rate of transmission of digital information = samples per second × bits per sample;
(viii) maximum bits per sample, b, limited by the ratio of total voltage variation to noise voltage
variation: b = log2(Vtotal/Vnoise);
and by showing graphically:
(xi) digitisation of an analogue signal for a given number of levels of resolution.
A Revision Checklist for Chapter 1 can be found on the Advancing Physics
CD-ROM.
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Section 1.1: Seeing invisible things
Learning outcomes
●
Images can be formed with many kinds of signals, including ultrasound and all
regions of the electromagnetic spectrum.
●
Images can be recorded electronically by microsensors; an example is the chargecoupled device (CCD); such images are composed of discrete picture elements –
pixels.
●
Images on the atomic scale can be recorded by scanning methods; an example is
the scanning tunnelling microscope (STM).
●
The resolution of an image is the smallest distance over which a change can be
seen.
What you should know about waves
The amplitude of a wave (symbol A) is the maximum displacement of the medium. This is
measured from the undisturbed position. This is measured in metres (m) etc.
The wavelength (symbol ) of a transverse wave is the distance between two neighbouring
crests or troughs. The wavelength of a longitudinal wave is the distance between two
neighbouring compressions or expansions (rarefactions). This is measured in metres (m) etc.
The wave speed (symbol v) is the speed of the wave profile such as a crest or compression
depending on the type of wave. This is measured in metres per second (ms-1) etc.
The time for one complete cycle is known as the period (symbol T). This is measured in
seconds (s). If the period is 0.5 s, one complete wave is made every 0.5 s, so two complete
waves are made every 1.0 s. The number of waves made every second is called the
frequency (symbol f) of the wave. Frequency used to be measured in cycles per second but is
now measured in hertz (Hz).
f = 1/T and T = 1/f
wave speed (m/s) = frequency (Hz) x wavelength (m)
or v = f 
Ultrasound imaging etc
The image shows an ultrasound scan of a 20 week
old foetus. The image is shown on p1 of student text,
and is the first image in
Activity 10S Software based 'Looking at images'
Further information about this image can be found on
File 10I Image 'The variety of uses of scientific
images'
An excellent description of the wave aspects involved
can be seen in the lesson on ‘Ultrasound’ on
www.teachingmedicalphysics.org.uk . Slides 1-15 are
the most relevant here. See also ‘Making an image
with ultrasound’ on pages 3 and 4 of the student text.
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Frequency, speed and wavelength
Time picture
displacement
time
frequency f
time T
time T for
1 oscillation
T=
1
f
Position picture: two different wave speeds compared (same frequency)
higher speed
source
wavelength
longer
lower
speed
speed v
frequency f
wavelength 
 = vT =
v = f
v
T
source
wavelength
shorter
distance
Ultrasound pulses have a high frequency
An ultrasound pulse lasts 1 μs, has a frequency of 10 MHz and travels at approximately
2000 ms-1. You should be able to show that each pulse contains 10 waves and is 2 mm
long.
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The essential physics involved in the distance measurement can be illustrated with:
Activity 30D Demonstration 'Distance measurement with ultrasound'
together with a demonstration of an ultrasonic tape measure. A model of ‘sonar’ can be set
up using the Unilab ultrasound apparatus.
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Activity 10S focuses on the following points
1.
How was the image made?
 The delay times of the reflections tell the scanner where the denser tissue of the
baby is.
 The strengths of the reflections tell the scanner how dense the various tissues
are.
● The information gleaned from the reflections is assembled to form the image as
an array of 256 x 256 pixels each having one of 256 shades of grey (p 2). This
process is known as computer tomography – CT for short.
To study the nature of the image in more detail image we use the magnification tool
to show pixels of the foetus from File 10I Image. Note that we can undo with ‘File:
Revert to saved’.
2.
What is the scale of the image?
If a map has a scale of 1:25000 we mean that a particular distance drawn on the
map represents a distance on the ground that is 25000 times larger. We are saying
that the scale is 1: [size on ground/size on map].
3.
What does the image tell us, and how?
4.
In what way is what you see 'invisible'?
5.
What wavelength of radiation, if any, is involved?
[Why must the frequency of the ultrasound pulses be so high? See p 3.]
6.
What is the resolution of the image (the smallest size of thing that can be
distinguished)? This is the size of one pixel in a digital image. The actual
size of the object being imaged must be known for this. See p 2.
Notes:
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Now work through other examples from Activity 10S Software based 'Looking at images' in
the ICT suite. You should address each of the points 1-6 above. Further information about
each image can be found on
File 10I
Image 'The variety of uses of scientific images'
File 20I
Image 'Astronomical images: Problems of noise and resolution'
File 30I
Image 'X-ray images in medical physics'
File 40I
Image 'Ultrasound images in medical physics'
File 50I
Image 'Magnetic resonance imaging
File 60I
Image 'Gamma rays detecting abnormalities'
File 70I
Image 'Satellite images of towns in Europe'
To get an idea of what you can do to images electronically you should see digital image
capture at work using a Web Cam intended for video conferencing. The image is displayed
directly on the computer screen.
Activity 20D Demonstration 'Electronic image capture'
Page | 7
Images from the Universe
Electronic images are captured using a charge-coupled device (ccd) which is an example
of a microsensor that converts light into digitally stored images. A digital image with 1
million pixels would have been made with a ccd with a 1000  1000 array of elements.
Tom Ang, Digital Photographer’s Handbook, Dorling Kindersley, 2002
Note that images from the Universe such as those shown on page 5 of the student text are
taken with many wavelengths from the electromagnetic spectrum not just with visible light.
Page | 8
The electromagnetic spectrum
See page 6 of the student text for a diagrammatic representation.
Type of
radiation
Approximate
wavelength
in metres
104 to 10-3
Approximate
frequency in
hertz
104 to 1011
Sources of
radiation
Detectors
Notes (uses,
dangers etc.)
Accelerating
charges
Aerial, tuning
circuit
Microwaves
(sub-set of
radio waves)
Infra-red
10-1 to 10-3
109 to 1011
Horn and dish
10-4 to 10-6
1012 to 1014
Accelerating
charges
(magnetron)
Hot objects
Broadcasting
(television and
radio), mobile
telephones
Microwave
oven, radar
Visible light
(710-7) to
(4 10-7)
6 1014
Hot objects,
excited atoms
Ultra-violet
10-7 to 10-9
1015 to 1017
X-rays
10-9
downwards
1017 upwards
Very hot
objects,
fluorescent
lights
(converted to
visible light)
Collision of
fast electrons
with target as
in X-ray tubes
Gamma rays
10-11
downwards
1019 upwards
Radio
Radioactive
atoms
Phototransistor Radiant heaters,
etc.
night vision,
burglar alarms,
remote controls,
thermography,
i.r. lasers used
with optical
fibres and CD
players
Eyes,
Vision
photographic
film
Photographic
Fluorescence,
film,
sun lamps,
fluorescent
killing germs,
screen
astronomy
Photographic
film, ionisation
effects
Medical
diagnosis,
analysis of
matter
Photographic
film, ionisation
effects
Studying
nuclear
structure,
cancer
treatment, food
sterilisation,
measuring
thickness
‘Spy in the sky’ (page 8 of the student text) suggests some uses for satellite imaging. Note
the references to resolution and to the idea of false colour.
Page | 9
‘Seeing’ atoms
The idea of ‘Seeing atoms’ is discussed on pages 8-9 of the student text. This includes
brief detail of the scanning tunnelling microscope [See A-Z for further details].
Courtesy of Matthias Bohringer and colleagues, University of Lausanne
Page | 10
Section 1.2 Information in images
Learning outcomes
●
Images can be stored as an array of pixels, each defined by a number; the amount
of information stored = number of pixels × bits per pixel.
●
Images can be smoothed by suitable averaging; images can be sharpened by
identifying edges; images can be enhanced by increasing contrast or by false
colouring.
●
Quantities that cover a large range of values can usefully be displayed on a
logarithmic (‘times’) scale. Prefixes for scientific units are chosen at multiples of
1000.
●
1 bit of information contains 2 choices (0 or 1); 1 byte of information contains 8
choices (256 = 28 alternatives); information I provides N = 2 I alternatives.
Image processing
We start by considering the possibilities of some image processing software with a
demonstration of the key elements of Activity 50S Software based 'Image processing:
The surface of Mercury'.
The following exercise helps to explain how to smooth sharp edges, remove noise and find
edges (see student text pp 14-15 and Activities 50S, 60S, 70S and 80S).
Various operations:
(a) Replacing each pixel by the mean of its value and those of its neighbours.
(b) Replacing each pixel by the median of its value and those of its neighbours. When a
series of numbers is place in ascending order, the number in the middle is the median.
(c) Subtracting the N, S, E and W neighbours from four times the value of each pixel.
Exercises
Calculate only the values for the squares inside the darkened box to avoid edge effects.
1
Smoothing edges by calculating mean values [operation (a)].
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
becomes
Using the mean rounds off sharp corners.
2
(i) Removing noise by calculating mean values [operation (a)].
1
1
1
1
1
1
1
2
1
1
0
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
becomes
Page | 11
(ii) Removing noise by calculating median values [operation (b)].
1
1
1
1
1
1
1
2
1
1
0
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
becomes
Taking mean and median values both reduce noise. Which process is the most
effective?
…………………………………………………………………………………………………
3
Finding edges by subtracting the values of the neighbours [operation (c)].
(i) An image with an edge.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
becomes
(ii) An image with a uniform brightness gradient.
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
becomes
Areas of uniform brightness are removed, as are areas where there is a uniform
brightness gradient. Regions where the gradient of the brightness changes abruptly
are enhanced.
Now go on to try enhancing a variety of images. The essential points to establish are
smoothing, noise reduction, contrast and edge enhancement.
Activity 50S Software based 'Image processing: The surface of Mercury'
(All should start with this)
Now try at least one example of image enhancement:
Activity 60S 'Image enhancing: Volcanoes on Io’ or Activity 70S 'Medical uses of x-ray
images' or Activity 80S 'Medical uses of ultrasound images
Page | 12
The key points here is that images are made of pixels, that a pixel is defined by a number,
and that modifying images means doing arithmetic on pixels. The most important general
modification to emphasise is averaging which is a good way of removing random or rapid
fluctuations in all sorts of experiments.
[Did you know that you can get some striking and sometimes artistic effects with image
processing software such as PhotoShop or Paint Shop Pro?]
Amount of information and log scales
The ‘Powers of Ten’ video (http://powersof10.com 9 minutes) makes a useful introduction
to this part.
Decimal and binary number systems
Decimal: 102 101 100
2 4 7
The number 247 (two hundred and forty seven) is made up as follows:
(2100) + (410) + (71)
2 2 21 20
1 0 1
The number 101 in binary represents the number (14) + (02) + (11) = 5 in decimal.
Binary:
The two values used in the binary system are the digits 1 and 0. A binary digit is called a
‘bit’. Electronically the idea of bits may be represented by a lamp that is ‘ON’ or ‘OFF’ or by
a voltage that is ‘HIGH’ or ‘LOW’.
Exercise
(a) Convert the following 4-bit binary numbers into decimal numbers:
1111
1010
0010
0111
………..
………..
………..
………..
(b) Convert the following 8-bit binary numbers into decimal numbers:
00010000
11111111
10101010
01010101
………..
………..
………..
………..
(c) Write the following decimal numbers as 8-bit binary numbers:
25
100
255
180
………………….
…………………..
………………….
……………………
Page | 13
The next section uses logarithmic thinking. We use logarithms every time we check the
size of a computer file in bytes (1byte = 8 bits). It will be shown that the space needed to
store a file is a logarithmic measure of the amount of information in it.
A message which can only be 'yes' or 'no' (or 'with sugar' or 'without sugar') has only two
alternatives. We need one bit (zero or one) to store it.
One pixel of a grey scale image can have 256 different alternative values. But 256 bits are
not needed to store it. Eight are enough, because there are 256 different numbers, which
can be represented by eight binary digits, or bits.
The decimal value of the binary number 11111111 is
(127) + (126) + (125) + (124) + (123) + (122) + (121) + (120) which is
(128 + 64 + 32 + 16 + 8 + 4 + 2 + 1) which is 255.
From 0 to 255 there are 256 numbers.
That is, the number N of alternatives which can be represented by information I bits is just
N = 2 I.
When I = 7, N = 27 = 128 and when I = 8, N = 28 = 256.
Thus if you add one bit of information you double the number of alternatives.
Display Material 50O OHT 'Bits and bytes'
Bits and byte s
8 bit = 1 byte
1
Decimal
2 bit 1 bit value
4 bit
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
1
0
2
0
0
0
0
0
0
1
1
3
0
0
0
0
0
1
0
0
4
0
0
0
0
0
1
0
1
5
0
0
0
0
0
1
1
0
6
0
0
0
0
0
1
1
1
7
0
0
0
0
1
0
0
0
8
0
0
0
0
1
1
1
1
15
0
0
0
1
0
0
0
0
16
0
0
0
1
1
1
1
1
31
0
0
1
0
0
0
0
0
32
0
0
1
1
1
1
1
1
63
0
1
0
0
0
0
0
0
64
0
1
1
1
1
1
1
1
127
1
0
0
0
0
0
0
0
128
1
1
1
1
1
1
1
1
255
0
0
0
0
0
0
0
0
256
One 8-bit byte stores 2 8 =256 alternatives
Number of
alternatives
NB There is an alignment error in
Display 50O and in the diagram on
Page 12. See if you can spot it.
21 =2
22 =4
23 =8
24 =16
25 =32
26 =64
27 =128
28 =256
Page | 14
number = baseindex
We now need to introduce the idea of logarithms.
You know that 1000 = 103. In general, we can say that
number = baseindex so 10 is the base in our example, 3 is the
index and 1000 is the number we get when we raise 10 to the
power of 3.
logbase(number) = index
If we know the base and the number we can find the index using the idea of logarithms
where logbase(number) = index. In our case, log10 1000 = 3 because 103 = 1000.
If the base is 2 instead of 10 we have log2 8 = 3 because 23 = 8.
Exercise
(a) Write down log10 1000000 ……………………………………………………………………..
(b) What is the number whose logarithm to the base 10 is 5? …………………………………
(c) Write down log2 32 ………………………………………………………………………………
(d) What is the number whose logarithm to the base 2 is 10? …………………………………
Display Material 60O
OHT ''Plus' and 'times' scales of information'
‘Plus’ and ‘times’ scales of information
The OHT ‘Bits and bytes’ shows that the number of alternative values
which can be represented grows rapidly as the amount of memory used
increases.
N = 2I
log2N = I
If 8 bits of information are used:
number of alternatives N = 28 = 256
In general, if the amount of information is I bits:
number of alternatives N = 2I
If the number of bits increases by one, the number of alternatives doubles.
Information is measured on a ‘plus’ scale; the number of alternatives on a ‘times’
scale.
‘PLUS’ SCALE
(LINEAR)
‘TIMES’ SCALE
(LOGARITHMIC)
amount of information
number of alternatives
increase by equal additions
increase by equal multiples
amount = I
number of alternatives = 2I
amount = log2 N
number of alternatives= N
log 2 (num ber of alternatives) = information I = num ber of bits
Page | 15
We often use logarithmic scales in the form of ladders of quantities where adding one rung
of the ladder multiplies the quantity by some amount. The link between adding one kind of
quantity and multiplying another is the heart of the matter.
Display Material 70O
OHT
'Comparing logarithms of base 2 and 10'
Comparing logarithms base 2 and base 10
Display Material 80O
OHT
'Logarithmic ('times') ladder of distance'
Logarithmic ladder of distance
Logarithms and bases
logarithms of
numbers to
base 2
numbers
10
1024 1000
9
512
8
256
7
128 100
6
64
5
32
4
16
3
logarithms of
numbers to
base 10
A ladder of distances in multiples of metres
Examples
log2 (128) = 7
2
log10 (100) = 2
1018
109
log10 (16) = 1.2
10
3
4
1
2
0
1
210 = 1024
10 = log2 (1024)
1
1
0
103 = 1000
Then index = logarithm base (number)
3 = log10 (1000)
distance to the Sun
1 Tm (tera)
1 Gm (giga)
1 Mm (mega)
3
small town
1 km (kilo)
0
10
human body
1m
10–3
width of a hair
1 mm (milli)
10–6
microchip element
1 m (micro)
10–9
molecule
1 nm (nano)
10–12 atomic nucleus
1 pm (pico)
10–18
A logarithmic scale is one on
which equal spaces correspond
to equal multiples.
In this distance scale each
upward step multiplies the
distance by 1000.
the Earth
10
10–15
If base index = number
1 Em (exa)
1 Pm (peta)
6
10
8
2
nearest stars
1015
1012
log2 (10) = 3.4
(approx)
Equal multiple scales
1021 galaxy
quark
Each downward step divides
the distance by 1000.
Each upward step adds 3 to
the logarithm (base 10) of the
distance.
Each downward step subtracts
3 from the logarithm of the
distance.
× 1000
1 fm (femto)
1 am (atto)
1
× 1000
Page | 16
Display Material 90O OHT 'Logarithmic
('times') ladder of time'
Logarithmic ladder of time
Display Material 100O
OHT
'Logarithmic ('times') ladder of mass'
Logarithmic ladder of mass
A ladder of times in multiples of seconds
A ladder of masses in multiples of grams
Examples
Examples
21
10
Equal multiple scales
1018 age of Universe
1 Es (exa)
1015
1 Ps (peta)
1012
1 Ts (tera)
9
10
one year
106
1 Gs (giga)
1 Ms (mega)
3
10
1 ks (kilo)
100
1s
10–3 flap of a fly’s wing
1 ms (milli)
10–6
1 s (micro)
10–9 light crosses a room 1 ns (nano)
10–12
1 ps (pico)
10–15
1 fs (femto)
10
A logarithmic scale is one on
which equal spaces correspond
to equal multiples.
In this time scale each
upward step multiplies the
time by 1000.
1021
1 Eg (exa)
1015
1 Pg (peta)
1012
1 Tg (tera)
109
1 Gg (giga)
106
Each downward step divides
the time by 1000.
Each upward step adds 3 to
the logarithm (base 10) of the
time.
Each downward step subtracts
3 from the logarithm of the
time.
× 1000
a car
1 Mg (mega)
103
1 kg (kilo)
100
1g
10–3 a mosquito
1 mg (milli)
10–6
1 g (micro)
–9
10
1 ng (nano)
10–12
1 pg (pico)
–15
10
10
–18
Equal multiple scales
1018
–18
A logarithmic scale is one on
which equal spaces correspond
to equal multiples.
In this mass scale each
upward step multiplies the
mass by 1000.
Each downward step divides
the mass by 1000.
Each upward step adds 3 to
the logarithm (base 10) of the
mass.
Each downward step subtracts
3 from the logarithm of the
mass.
1 fg (femto)
× 1000
1 ag (atto)
1 as (atto)
×
1
1000
×
1
1000
Display Material 110O
OHT
'Logarithmic ('times') ladder of speed'
Logarithmic ladder of speed
A ladder of speeds in multiples of metres per second
Examples
1021
18
10
No speeds
greater than
speed of light
Equal multiple scales
1 Em s–1 (exa)
1015
1 Pm s–1 (peta)
1012
1 Tm s–1 (tera)
9
1 Gm s–1 (giga)
10
106
light speed
103
100
1 Mm s–1 (mega)
1 km s–1 (kilo)
walking speed
–3
1 m s–1
s–1
A logarithmic scale is one on
which equal spaces correspond
to equal multiples.
In this speed scale each
upward step multiplies the
speed by 1000.
Each downward step divides
the speed by 1000.
Each upward step adds 3 to
the logarithm (base 10) of the
speed.
10
1 mm
10–6 continental drift
10
(micro) Each downward step subtracts
3 from the logarithm of the
1 nm s–1 (nano) speed.
10–12
1 pm s–1 (pico)
–9
1 m
(milli)
s–1
–15
1 fm s–1 (femto)
–18
1 am s–1 (atto)
10
10
× 1000
×
1
1000
Page | 17
Computer files
Remember: the number N of alternatives which can be represented by information I bits is
just N = 2I and I = log2 N.
The space needed to store a file is the number of bits (or more realistically – bytes) which
is a logarithmic measure of the amount of information in it i.e. the logarithm of the number
of alternatives. The number of alternatives will be a very large number when referring to a
computer file; the logarithm of that number is much smaller and more manageable.
A ‘bit’ is a small amount of information. It is common to work in bytes, where 1 byte is 8
bits. In word-processing, one byte is used for one character (such as a letter). A large file
will contain many bytes of information so it is better to work in kilobytes. One kilobyte is 210
or 1024 bytes but we normally call this 1000 bytes for simplicity. One megabyte is 2 20 or
1048576 bytes but we normally call this 1000000 bytes for simplicity.
Exercise
(a) Show that a 10 kilobyte word-processed file may contains less than 2000 words if the
author writes an average of 6 characters per word.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
(b) Estimate the size of a computer file in kbytes if it contains a 4000 word essay.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
Digital images
In the context of digital images, a ccd image is a 2 dimensional
array of pixels. The larger the number of pixels in an image, the
better the resolution because each pixel represents a smaller part of
the object being imaged.
Each pixel of a grey scale image may require 1 byte (8 bits) of
information (28 = 256 variations of grey) whereas each pixel of a
colour image may require 3 bytes (3  8 = 24 bits), one for each of
the primary colours. A high resolution colour image with 1 million pixels will therefore
require 3  106 bytes of information (approximately 3 megabytes).
amount of information stored = number of pixels  bits per pixel.
Colour images require a lot of bytes of memory if stored on a computer. Downloading large
files takes a long time if the information can only be transferred at, say 56000 bits per
second.
time for download = number of bits to be transferred  transfer rate in bits per second
Page | 18
Exercise
(a) The image of the Moon completely fills a ccd containing 1000 000 elements. The
diameter of the Moon is 3.5  106m. Calculate the resolution of the image.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
(b) Calculate the number of bits of information required for each pixel if the image is
coloured and there are 3 bytes to a pixel.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
(c) Use your answer to part (b) to calculate how many different colours are available.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
(d) Calculate the number of bits in the image file and show that it will take over five
minutes to download this file if a modem operating at 56 000 bits per second is used.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
[In practice, digital cameras use data compression methods to reduce the size of the
image file. A 3 Mbyte file may be reduced to 1 Mbyte.]
Page | 19
Section 1.3 With your own eyes
Learning outcomes
●
The eye is like an intelligent digital video camera, sending out a stream of
processed signals. It detects edges and movement.
●
A converging lens adds a constant curvature to light falling on it. The curvature
added is the power of the lens.
●
Power of lens = 1 / f = (1 / v ) – (1 / u) (Cartesian sign convention).
●
Magnification = height of image / height of object.
●
Light takes the same time to travel on all paths from a point on the source to the
image via the lens (or mirror).
●
In making careful measurements, it is important to estimate the uncertainty in the
measurement, and to look for possible sources of systematic error.
Eye and illusion
Cross-section of the eye
vitreous humour
iris (controls
light into eye)
retina (light-sensitive cells)
cornea
(refracts light
and protects
eye)
fovea (where
retinal cells are
densest)
eye pupil
blind spot
optic
nerve
aqueous humour
ciliary muscle
(controls lens
thickness)
eye lens (focuses
light onto retina)
Page 17 of the student text gives a simple description of the eye [See A-Z for more detail].
●
The eye is like an intelligent video camera, sending out a stream of processed
signals.
●
The retina consists of two types of light-sensitive cells. Rods predominate except in
and near the fovea which is at the centre of the retina.
●
Rods respond to different intensities and they are not colour sensitive which is why
objects away from the centre of the field of vision are seen in shades of grey, not in
colour. Several rods are joined to each nerve fibre so less detail is seen in images
away from the centre of the field of vision.
●
Cones are sensitive to red or green or blue light. These types of cells are found in
and near the fovea. They are most densely packed in the fovea so an image formed
on this part of the retina is seen in greatest detail.
Many important features can be seen on the model available.
Page | 20
Activity 150D Demonstration 'Models of the eye'
The model shows how the cornea does most of the focussing; the lens inside the eye (not
available in this demonstration) is needed to focus at different distances (accommodation).
The use of correcting lenses for coping with short and long sight can be demonstrated. A
separate 2-dimensional model with adjustable power lens can be shown to illustrate
accommodation.
Remember that most of the focusing (curvature of wave fronts added) is done at the
cornea; the eye lens adds an adjustable further small amount of curvature.
How does focussing on a Web Cam compare with focussing of the eye?
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
Activity 140E Experiment 'The intelligent eye'
These experiments show how the eye and brain work together and also enable the width
of the fovea to be estimated.
The rods and cones are cross-connected with inhibiting links, so as to enhance contrasts
at edges. Hence the eye detects edges and movement very effectively.
Activity 130D Demonstration 'Grey step: Edge enhancement in the retina'
The Hermann grid can be used to test the function of inhibiting links (see p 18 of the
student text).
Display Material 120O OHT 'The Hermann grid'
Page | 21
Lenses and real images
Only real images formed by converging lenses are discussed. Real images are those that
can be formed on a screen.
See the formation of a real image in space using:
Activity 160D 'Image in mid-air'
Display Material 160O
OHT 'Rays and waves'
Ray point of view
Display Material 170O
OHT 'Rays and waves focused'
Wave point of view
Wave and ray points of view:
Wave point of view:
The lens adds curvature to the wave entering it
Ray point of view:
Light travels out in
straight lines from a
small source
Wave point of view:
Light spreads out in
spherical wave fronts
from a small source
focus
focal length f
Ray point of view:
Light in a parallel beam or
from a very distant source
has rays (approximately)
parallel to one another
Wave point of view:
Wave fronts in a parallel
beam or from a very distant
source are straight (not
curved) and parallel
Ray point of view:
The lens bends the rays, bringing them to a focus
focus
Activity 200D Demonstration 'Focusing water ripples'
This introduces idea of a lens adding curvature. ‘Multimedia Waves’ can be used to show
a number of situations in which wavefronts in water are made to change shape and/or
direction.
The wavelength of light waves decreases when light travels from air into glass because
the light travels more slowly in glass than in air. A converging lens changes the shape of
the wavefronts because the light travelling through the centre of the lens travel through
more glass and is slowed down more than the light travel through the outer parts of the
lens.
We use the idea of refractive index to calculate how much the light is slowed down when it
enters a material such as glass.
Refractive index of glass n = speed of light in vacuum
speed of light in material
Page | 22
Activity 190 D Demonstration Disappearing glass shows what
happens when two materials of the same refractive index are put
together.
Why does the test tube look invisible?
……………………………………………………………………………
……………………………………………………………………………
……………………………………………………………………………
Power and curvature (See Display Material 170O OHT 'Rays and waves focused')
The image-lens distance when the object is at infinity is known as the focal length, f. The
waves coming from infinity have no curvature and the lens adds curvature to the waves
passing through it. The radius of the spherical wavefronts after waves from infinity have
passed through the lens is f. The curvature of a sphere of radius r is 1/r, so the lens has
added curvature 1/f to the wavefronts.
It can be seen that a more powerful lens (shorter focal length) adds more curvature.
The power of a lens is a measure of the curvature it adds.
power (dioptres) P = 1/f where f is in metres
Notes:
Page | 23
A camera lens with a focal length of 50 mm, which is 0.050 m, has a power of 1/0.050m =
20 dioptre. When placed together the powers add so a 20 dioptre and 30 dioptre lens
placed together with have a total power of 50 dioptre. This principle is easy to demonstrate
using cylindrical lenses.
Exercise
Calculate the focal length of a 50 dioptre lens.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
If you think that’s short, try calculating the focal length of a Web Cam lens with a power of
278 dioptre.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
It is helpful to see a pinhole camera being converted into a lens camera to get a feel for
image formation before proceeding to experiments to measure power and focal length.
Class experiment
Activity 170E Experiment 'Converging lenses: power and focal length'
It is useful to discuss how a lens forms an image using:
Activity 180D Demonstration 'Where are the parts of an object in its image?'
Page | 24
Display Material 180O OHT 'Formation of an image'
How a lens makes 'a little picture'
Traffic light
red light on
Image of traffic light
image of red lamp
at the bottom
The lens does not alter the
direction of the light, it just alters
the curvature of the wavefronts.
It is useful to recall work on rays of
light passing through a rectangular
glass block to see why this
happens.
image of green lamp
at the top
green light on
red and amber
both on
When drawing ray diagrams for
lenses it is usually a good idea to
draw a ray from the object that
passes through the centre of the
lens as this ray never changes
direction.
images of red and amber
lamps: amber in the
middle
Light from each place in the source arrives at a distinct
place in the image
The rule for how lenses shape light (page 22 of the student text)
curvature of waves coming out = curvature of waves coming in + curvature added by lens
Since curvature is 1/r where r is the radius of the wavefronts, we have:
1/v = 1/u + 1/f
Distances to the right are positive; distances to the left are negative. All measurements are
made from the centre of the lens.
The Cartesian convention is used; a lens adds curvature 1 / f to the wave fronts going
through it.
Page | 25
Display Material 190O
OHT 'Action of a lens on a wavefront'
Lenses change wave curvature
image of
source
source
distance from lens to image = v
(for example plus 0.1 m)
distance from lens to source = u
(for example minus 0.2 m)
curvature of wavefront after leaving the lens =
curvature of wavefront on reaching the lens =
1
v
=
1
0.1
= 10
1
1
= –5
=
u –0.2
power of lens
1
= 15 dioptre
f
focal length of lens f =
=
curvature added by lens = power of lens = curvature after – curvature before
1 1
= –
v u
= 10–(–5) = 15 dioptre
1
power
1
15
= 0.067 m
= 67 mm
A lens adds curvature 1/f to waves passing through it
Display Material 200O
OHT 'Where object and image are to be found'
Lenses add constant curvature 1/f
1
1
=0 +
v
f
zero curvature before
v=f
curvature
1
after
f
very distant
source
image of
source
f
1 1 1
= +
v u f
curvature
1
before
u
curvature
1
after
v
source
image of
source
u (negative)
1 1
+
u f
u = –f
0=
curvature
–1
before
f
v (positive)
zero curvature after
very distant
image of
source
source at
the focus
f
A lens adds the sam e curvature to all waves passing through it
Page | 26
Exercise
A camera lens with a focal length of 5 cm forms a clear image of an object on a film when
the object is 3 m from the lens. Calculate the distance between the image and the lens.
(a) Will the image be more or less than 5 cm from the lens? Explain your answer.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
(b) Calculate the distance between the image and the lens.
[Hint: Is the object distance positive or negative?]
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
The following class experiments show that a lens does in fact add constant curvature to
the wave fronts. Students wishing to gain a simple introduction to uncertainty should opt
for Activity 195E.
Activity 190E
uses File 130S
Experiment 'A converging lens adds constant curvature 1 / f '
Spreadsheet Model 'Lens action on a spreadsheet'
Activity 195 E
Experiment 'A converging lens adds constant curvature 1 / f: with
estimates of uncertainty'
Page | 27
Exercise
(a) Calculate the magnification of the image obtained in the camera example above.
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
(b) What is meant by the negative sign?
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
(c) Why is the magnification less than 1?
…………………………………………………………………………………………………………
We can illustrate magnification, particularly from the point of view of getting the largest
possible image on the retina using a cheap ccd video camera (Web Cam):
Activity 210D Demonstration 'Modelling the eye with a video camera'
Questions and activities
Section
Essential
1.1
Qu 1-6 AS text p 10
Question 10S Short answer 'Speed,
wavelength and frequency' Qu 1-17
Question 20E Estimate 'Large and small
distances and times' Qu 1-6
Question 30E Estimate ‘Making
estimates about images’
1.2
Question 30X Explanation-Exposition
'Different kinds and uses of images'
Qu 1-6 AS text p 16
Optional
Activity 40D Demonstration 'How fast sound
moves in a solid'
Question 160S Short Answer ‘Ultrasound tape
measure characteristics and limitations
Question 170S Short Answer ‘Ultrasound
tape measure characteristics and
temperature effects’
Question 60S Short Answer 'A scanning
electron microscope image of Velcro'
Question 40S Short Answer 'Smoothing
pixels using mean or median'
Question 70S Short Answer 'Image processing
by brightness and contrast control'
Question 60S Short Answer 'A scanning
electron microscope image of Velcro'
Question 80S
nebula'
Short Answer 'The Horsehead
Question 70S Short Answer 'Image
processing by brightness and contrast
control’
Question 90C
in ultraviolet'
Comprehension 'Saturn's aurorae
Question 110S Short Answer 'Bits and
bytes in images'
Question 120S Short Answer 'Logarithms
and powers'
Question 130C Comprehension 'X-ray
Question 100C Comprehension 'Betelgeuse in
ultraviolet'
Question 50X Explanation-exposition ‘How was
this image enhanced’
Activity 90S 'Spreadsheet models: Image
processing’. This provides an insight into the
Page | 28
image of the Kepler supernova remnant'
calculations behind image processing.
Activity 120S Software Based 'How big
are your computer files?'
Activity 100S 'Mapping the South Atlantic sea
floor'
Activity 110H Home Experiment 'Logarithmic
('times') ladders of quantities'
1.3
Question 155S Short answer ‘ Materials for
spectacle lenses’
Qu 1-6 AS text p 25
Question 140S Short Answer 'Response
of the human eye to differences in
brightness'
Question 150S Short Answer 'Cameras
and eyes'
Question 160C Comprehension 'Satellite
imaging'
Question 50D Data handling ‘Representing
experimental uncertainties graphically 1’
Question 50D Data handling ‘Representing
experimental uncertainties graphically 2’
Display Material 140O
Display Material 150S
pointillist painting'
Display Material 210O
equation'
Display Material 220O
1 / v = 1 / u + 1 / f'
OHT 'Ouchi illusion'
Computer screen 'A
OHT 'The lens maker's
OHT 'Thin lens: proof of
File 120S
Spreadsheet Model 'Eye
response modelled on a spreadsheet’
File 130S
Spreadsheet Model 'Lens action
on a spreadsheet'
Activity 200E Experiment ‘Magnification and
the power of a lens’
Activity 210E Experiment ‘Webcam
magnification and resolution’
Reading 20T Text to read 'A collection of
visual illusions'
John Mascall
August 2008
The King’s School, Ely
Page | 29