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Modelling the Storage Capacity of 2d Pixel Mosaics


Abstract and Figures

Mathematical modelling is one of the core competencies of today’s knowledge-based society. Description and abstraction of real problems by using mathematical language enables the simulation and optimisation of extensive systems with mathematical tools and IT capabilities. Besides, mathematical modelling with students provides new directions in motivation, knowledge transfer as well as problem solving. Therefore, it is recommended that it should be integrated into the interdisciplinary MINT1 education.
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This contribution presents an interdisciplinary modelling problem focussed on the
improvement of so-called QR-codes. A successful approach is being introduced,
worked out by students as well as their teachers during the Modelling Week 2008
in Lambrecht/Germany. The task definition is particularly suitable for the project-
related MINT
class on AP level. We conclude with a reference to extension
possibilities and take a didactical look at the processing of this real-life problem.
Mathematical modelling belongs to the core competencies of today’s knowledge-
based society. Description and abstraction of real problems by using the
mathematical language enables the simulation and optimisation of extensive
systems with mathematical tools and IT capabilities. Besides, mathematical
modelling with students provides new directions in motivation, knowledge transfer
as well as problem solving. Therefore, it should be integrated into the
interdisciplinary MINT education.
For 16 years, the Department of Mathematics at the University of Kaiserslautern
has been organising mathematical modelling weeks for selected students of 10th
12th form. Teachers, too, can participate in this event to broaden their knowledge
through further education. In total, 40 students and 16 teachers had been engaged
for the duration of one week in 8 different and realistic problems from industry,
economy, society, sports, IT and physics. Scientific assistants from universities and
research centres supported each group by offering advice and help, whenever it
was needed.
During the 2008 modelling week, a team consisting of 5 students and 2 teachers
from different schools worked on the optimisation of new two-dimensional (2D)
bar codes, the so-called pixel mosaics. The problem was specially developed for
students interested in mathematical modelling as well as in IT with experience in
any programming language. Due to the complexity of the problem, it is
recommended to implement such a task in interdisciplinary mathematics and IT
class on AP level rather than in regular class.
Take a coke bottle, a biscuit tin or a shoe box: nowadays, black and white bar
codes are printed on nearly all consumer goods and scanned at the supermarket
checkout, replacing manual typing of prices. Bar codes accompany us anywhere,
every day. For example, suitcases at airports can be sorted with these codes;
libraries have introduced the bar code for their library cards to borrowing books.
Therefore, for many students, it is very exciting to understand the functionality of
these codes and how much mathematics it may contain [6].
Figure 1. Price tag for fruit with an EAN bar code
The bar code on product codes the 13-digit „European Article Number“ (EAN),
classifying each item, and consists of a 12-digit item number as well as a control
number. The control number indicates whether the item number was correctly read
during the scanning process or if transmission problems have occurred. For the
control number, EAN uses a digit, which is calculated by using a specially
weighted checksum of the item number. To enable this, the digits of the item
number are multiplied alternately with factors 1 and 3 and summed up. The control
number completes this sum to the next multiple of 10. Hence, single errors like e.g.
the wrong input of digit „8“ as „3as well as a pair wise mix up of single digits
like „41“ instead of „14“ can easily be detected. The example in Figure 1 shows a
bar code for yellow nectarines with the EAN 2404105001722. The weighted
checksum of the first 12 digits of EAN is 48. Control number 2 completes this sum
to the value 50 (see [2] for a detailed description).
When we refer to codes or coding in this context, it does not mean coding a
message or its translation into a code talk, but developing a code or password from
a message for its electronic transmission. Errors that may occur are identified and
ideally corrected during the transmission process. The bar codes in question only
recognise transmission errors, but are unable to localise or even correct it. The
development of error-correcting codes which, similar to bar codes, are optically
readable but save more data volume and, consequently, information about
correction modes. Black and white bar codes would require too much space for
Figure 2. Online-Ticket of Deutsche Bahn (sample)
The new bar code generation consists of quadratic pixel mosaics. They are printed
on Deutsche Bahn’s online ticket (see Figure 2) or on online stamps named
„StampIt“ issued by the Deutsche Post. A mosaic like that of black and white
pixels can be read through an optic camera or even with a mobile camera. The
photographed mosaics are then read out by a specific software programme and
finally decoded. This method enables the transmission of a higher data volume
compared to those bar codes being used so far. Besides, there are completely new
applications emerging from it: A weblink directing to a company or advertising
homepage can be translated into a space-saving pixel mosaic and placed as
magazine ad. The mosaic is photographed via mobile, the link behind to be
downloaded. Personal weblinks of profile pages such as Facebook in a pixel
mosaic, too, could be coded and printed on the back of a t-shirt. Anyone
photographing the mosaic at the next party can view the web profile of the shirt
wearer via internet browser of his mobile.
In the course of the modelling week, the students had to investigate the
maximum data volume that could be saved in a 2D pixel mosaic and whether it was
even possible to extend this storage capacity by developing a new concept. A
particularly exciting question for students was raised, whether it would be possible
to make paper talk, if a language was coded into a pixel mosaic, photographed with
a mobile and played via an integrated loudspeaker.
The following background information regarding construction and functionality of
pixel mosaic variations, were compiled by the team via internet research.
2-dimensional bar codes
Another description for pixel mosaics is 2-dimensional bar codes, as they code data
horizontally as well as vertically. Since the late 1980’s, they have been developed
and primarily used in production departments of the vehicle and electronic industry
for the clear identification of single elements. Since then, various types of pixel
mosaics have been developed, applied in different areas, competing with each other
and a varying distribution. The Deutsche Bahn for example uses the Aztec Code for
coding the data of their online tickets; the Deutsche Post implemented Datamatrix
for its online stamp business whereas the QR-Code (Quick Response Code) is most
widely spread. Numerous QR codes can be found in the public sector which can be
recorded with a camera mobile and evaluated inside the mobile phone through
special software („Code Reader“). They contain advertising slogans, internet links
or save useful information on sights
. This trend will probably soon find its way
towards Europe.
Figure 3. The word „Modelling Week“ in Aztec Code, Datamatrix and QR-Code
After a quick discussion, the team decided to analyse the QR code more
thoroughly, followed by the search for an improvement approach concerning the
storage capacity. The single improvement steps as well as the results achieved
ought to apply to other mosaic types in a similar way.
The structure of the QR-Code
The QR-Code consists of black and white pixels and can save a message with up to
7089 signs, depending on the mode that is used. The mode determines the type of
the data to be saved. A distinction is drawn between
numerical, i.e. only digits,
alphanumerical, i.e. digits as well as Arabic letters, and
special letters, e.g. Japanese or Chinese letters.
Initially, the data is converted into binary numbers, coded through an error
detection process, provided with a special mask, and then saved as pixel mosaic.
This process is called coding. The inverse process, that is the read-out of a message
from a pixel mosaic, is referred to as decoding. We will now put our main focus on
the single steps of decoding.
At first, each character of the message is classified in a binary number. Often,
for alphanumerical messages, the extended ASCII Table
is being used, which uses
one Byte (which equals 8 Bit) on information for the illustration of a character,
thus picturing exact 256 different characters. The binary message produced is now
coded through an error correction process. At this, the QR code reverts to the so-
called Reed-Solomon-Process (see [11]), which is also used for coding data from a
compact disc or DVD. Coding data by using this process enables the localisation of
an error and the recovery of messages even if the QR code can no longer be
properly read. The Reed-Solomon algorithm also allows for the regulation of the
correction level making it possible to choose between minimum storage space
requirement and best possible error correction, depending on the application. The
reconstruction of a message can be realised on maximum correction level, even if
up to 30% of surface of the pixel mosaic are damaged.
The binary code word which is produced via error correction process is now
transmitted into a pixel mosaic. Black pixels represent a 1, whereas white pixels
mark a 0. Finally, a so-called mask is covering the code word, consisting of a short
and repeating sequence of zeros and ones added pixel- and bitwise to the code
word. It means that, for each 1 in the mask, the respective pixel of the code word
changes its colour: black pixel turn to white and vice versa. It avoids large and
monochrome surfaces which could easily cause problems if data is imported with
an optical camera.
Figure 4. Structure of the QR-Code
Figure 4 shows the detailed structure of the QR code of the message „HALLO“:
The three Finder Patterns serve as alignment of the QR Code, enabling the
decoding of a message even if the QR code is photographed in profile, distorted or
The Information Patterns indicate the mask that is used as well as information
about further dimensions.
The Timing Patterns serve as coordinate axis and scale reference. Therefore,
black and white pixels are alternating continuously.
The four pixels in the bottom right corner of the mosaic indicate the mode that is
The message length is coded directly above the mode, i.e. the number of signs in
the binary system.
The code word connects to the message length information, consisting of message
and error correction data. The single pixels are populated from bottom to top as
well as from right to left. Of course, the actual dimension of a mosaic depends on
the length of the message, and is therefore calculated in advance. Pixels that may
not be used, are valued as 0. Another pattern of black and white pixel emerges with
the use of this mask, identified as “zero information” when it is read out.
After completion of a thorough analysis of structure and functionality of the QR
code, it was the team’s aim to develop their own model of a pixel mosaic with
higher storage capacity. The new code was to memorise more information with the
same number of pixels, but to also enable the transmission of compressed audio
files such as e.g. short voice messages. Following approaches were developed and
discussed (see Figure 5):
Figure 5. Three different proposals for memory optimisation:Shapes, grey shades or
The first two approaches, however, had soon been dismissed. The students realised,
that using different shapes would require a high-definition camera as well as an
algorithm for the identification of patterns. Both were graded as additional
difficulty and source of error. The readout of pixel mosaics using several shades
instead of black and white is very much dependent on the lighting conditions
during the shot. The team eventually concentrated on the third approach, as there is
no guarantee for ideal and persistent lighting in every situation.
The use of different colours initially raises the question as to how many colour
shades should be applied. The team chose four different colours, which are
assigned to the binary numbers 00, 01, 10 and 11. In contrast to the QR code which
requires 8 pixels, 1 Byte (= 8 Bits) of information was coded into 4 colour pixels.
For their work, the students opted for white, red, green and blue, as these colour
shades are in a sharp contrast to each other.
Below, we are exemplifying the illustration of the message “HALLO” with
coloured pixels, the error correction not being taken into account:
First, the ASCII numerical code is assigned to every single letter of the message:
H = 72, A = 65, L = 76, O = 79,
the ASCII numerical codes are displayed in the binary numerical system (8 Bit):
72 = 01001000, 65 = 01000001,
76 = 01001100, 79 = 01001111,
the binary numbers are now being combined:
„H A L L O“ = 01001000 01000001 01001100 01001100 01001111
and finally subdivided into pairs and coded in the corresponding „colours“:
00 = , 01 = , 10 = and 11 = to be applied. The pixel mosaic being
generated this way was named Quattro code by the team.
Error correction
An error correction algorithm followed as a next step: errors in the read data should
be identified, localised and corrected. Note that for the QR Code, the Reed-
Solomon error correction is being used. This algorithm divides the binary message
into units of 8 Bits each and calculates their error correction information.
Depending on the correction level, the calculated codeword is getting longer by 3
to 8 Bits compared to the original message since both message and error correction
information must be stored. The algorithm is based on the construction of defined
polynomials and its evaluation followed by interpolation into predefined nodes. In
order to avoid large function values, the calculation of all numbers is carried out in
the finite field of integers. This additional difficulty prevented the students from
getting deeper into the theory of this procedure.
Instead, they put their focus on the development of their own error correction
procedure, based on that of conventional bar codes on simple check sums. For this,
the group were pursuing several approaches, of which a suitable one was selected.
A linear system of equations for error correction. The first approach the students
developed was not designed for the use of several colours. In fact, they focussed on
a message composed of eight bits (e.g. 10011011). The individual bits are assigned
to the variables a, b, c, d, e, f, g and h. Therefore, the variables correspond to the
values a=1, b=0, c=0, … , h=1.
Next, six control numbers are generated which to be calculated as solutions of
six linear equations. The aim is to choose this system of linear equations in such a
way that differences in the message between read and calculated control numbers
can be uniquely identified. It depends on the smart combination of the variables in
the equations; in particular, two variables must not occur exclusively in the same
set of equations. Furthermore, each variable is to appear in at least two and a
maximum of three equations. For instance, the following equations meet the
desired criteria:
xeca ++
xhda ++
xfca ++
xgdb ++
xhfb ++
xgeb ++
The (binary) addition of each left hand side produces a 6-bit control number
). Here, the check number is 011010.
Having this tool at hand, the error correction will work for this system as
follows: If there is an error in the 8-bit message at one position, the result of the
linear system will change in at least two equations; hence the recalculated control
number is incorrect, i.e. it differs from at least two positions from the original
control number. Thus, conclusions regarding the incorrect bit are possible and
corrections can be made. In case the message is 11011011, the calculated control
number is 011101, which differs from the original control number at the points x
and x
. This is due to a read error in the variable b, since only this variable
occurs in the appropriate equations. Thus, the correct message is 10011011.
The disadvantage of this system is, that it can merely detect and correct single
errors. Several errors in the message or errors while reading the control numbers
could cancel each other or lead to more than three different digits in the control
number. These errors can neither be recognised nor corrected.
Check sums for the entire message. Another possibility for the generation of check
sums is to consider the entire message where all pixels are arranged line by line
starting from top left to bottom right. This results in a big square, which should be
as small as possible; void space may be filled with zero values. In a second step
check sums can be introduced and calculated, e.g., check sums for rows, columns
and diagonal elements. Together with these check sums the original message is
encoded to the final codeword. The check sums are used to detect possible errors in
the transmission of the codeword. This is done by comparison between the stored
check sums calculated from the original message and the post-calculated check
sums of the read in codeword.
Using this approach the students demonstrated an error detection and error
correction of an incorrect codeword ''by hand''. However it caused them difficulties
to design an algorithm whilst implementing this approach, such that this way of
error correction was not used in the end. But the group follow up the idea of
introducing check sums for error correction and adapt this procedure locally to the
Quattro code.
Check sums for 2 x 2 pixels
The ultimately implemented error correction approach was specially developed for
pixel mosaics using four different colours. It enables the algorithm to detect and
correct a single error within four pixels of information.
Figure 6: Sketched representation of own error correction procedure.
One byte of information represented by four coloured pixels is arranged as a square
(see Figure 6) and the different colours depict a different value in the four number
system ( = 0, = 1, = 2, = 3). Three additional check sums are used
for error detection and correction: the first and second check sums give the sum of
the diagonal and anti-diagonal elements, respectively, and the third check sum is
the sum of the upper row of that square. The three check sums extend the message
to a codeword, they are represented as additional information in coloured pixels
and displayed in the pixel mosaic. As an example we consider the letter ''H'': The
ASCII code 01 00 10 00 results in the Quattro code and we get the
following square for the calculation of the additional check sums.
Figure 7. Diagram of the Quattro Code of "H" in a square and the additional calculated
check sums.
With these three check sums we can now identify and correct a single error within
the four pixels. Let us assume that a pixel of the original square was read in
incorrectly, the newly calculated check sums will not match with the stored check
sum. We take a closer look at Figure 6 and refer to the columns as A and B and to
the rows as C and D, respectively. If for instance the pixel (A/D) is read in
incorrectly, the newly calculated check sum of the diagonal does not match with
the stored check sum. However the first and the third check sums confirm the
correctness of the pixels (A/C), (B/C) and (B/D). Hence the single error can appear
in pixel (A/D) only and have to be corrected. A single error among the other pixels
can be detected and corrected analogously.
In our error correction we so far have assumed, that at least the three check sums
have been read in correctly. However, if one of these is read in incorrectly then this
check sum indicates an error in the message also the message itself was read in
correctly. To detect errors in the check sums, a fourth check sum is added as the
sum of the first three check sums. With view to our example of the letter "H" of the
message "HALLO" the fourth check sum is the sum of the first, second and third
check sum: 1 + 2 + 1 = 0 mod 4 (see Figure 7). In this way the codeword of the
letter “H” is given by: H = 01 00 10 00 plus four additional check sums = 01 10 01
H check sums
Note that we had 00 = , 01 = , 10 = and 11 = . It enables to first check
the correctness of the check sums first, then locate and correct an error in the
message afterwards: If the fourth check sum is correct, we can also assume that the
first, second and third check sums, too, are correct and go on with the error
localisation and correction of the message itself as described above.
If the fourth check sum does not match with the sum of the first three check
sums, the first three check sums will be recalculated from the read in pixels of the
message. In a second step the fourth check sum is recalculated and again compared
to the stored one. If now the fourth check sum is correct, it can be assumed, that an
error occured in one of the first three check sums, but the the message itself was
read incorrectly.
If the recalculation of the check sums do not result in a correct message,
multiple error occurred in the four pixels of the message and/or the four pixels of
the corresponding check sums. In that case the error detection and correction is no
longer possible and this byte of information is lost. In that case a question mark is
used at the affected place to indicate the failure. In most cases the message can still
interpreted by the reader, e.g., if instead of the original message "HALLO" only
"H?LLO "is decoded.
The developed error correction for the Quattro code requires additional 8 bits for
8 bits of information (equivalent to 4 pixels). So the required storage space could
also used to store the message twice instead of storing the message and data for
error correction. In doing so, errors can still be detected and localised, but one
criterion is missing to decide, which of the two sources contains the correct
message. Here, an approach based on check sums is more efficient.
The layout of the Quattro code
Compared to the QR code the layout of the Quattro code is slightly different. The
Finder and Timing Patterns (see Section 3.2) have been merged and will be also
used as coordinate system and scaling reference. Beginning in the upper left the
pixel mosaic is filled up with information. The first eight pixels represent the
length of the message. This is followed from the left to the right by the coded
message and the error correction data. Four coloured pixel are needed to store one
byte (=8 bit) of information. In total up to 4^8 = 65536 bit of information can be
stored in a pixel mosaic.
The Quattro code can store information much more efficient and smallish than the
QR code. Figure 8 shows the message "Modelling Week” encoded as QR Code and
as Quattro code. It can be seen that the newly developed Quattro code stores the
same message in significantly fewer pixels. However, the information for error
correction data requires more space compared to the QR-code. This fact was
identified by the students and suggested for the next generation the use of the
Reed-Solomon method for error correction, which is more efficient.
Figure 8: Comparison of a QR code (left) and a black/white print of the coloured Quattro
code (right).
The Quattro code uses less space to code a message, but there are two main
drawbacks: On the one hand, it becomes necessary to print the pixel mosaic
coloured; however, on the other hand, the colours used need to be distinguishable.
However, this additional sources of errors can be avoided by suitable printers and
camera lenses.
The team have developed two computer programs to encode a text message into a
Quattro code and to decode a Quattro code into the original message, respectively.
The encoding program transforms a word, a text phrase or even an audio file
into a Quattro code which can be displayed on the screen or printed on paper.
Independent of that, the decoding program reads a Quattro code as an image
file, analyses that image and decodes the original data. If the original data was a
text message, the decode message is displayed on the screen if the original data
was an audio file, the decoded file are played on the loudspeaker.
The good project results encouraged the group to discuss and to document potential
further developments and optimisation of the Quattro code.
One improvement could be to use more than four different colours. Ideal would
be the use of eight different colours, so that 3 bits (instead of 2 bits of information)
could be stored in a single coloured pixel. These colours should be chosen out of
the basic colour map (e.g. for the Quattro code the RGB colour map was used),
such that the difference in the contrast of the colours is as large as possible.
Another potential for optimizing the storage capacity is the error correction
method. Here, the relation between the length of the codeword and the length of
the message might be improved by integrating the Reed-Solomon error correcting
method. Regarding the use of mobile phones and their build-in cameras the
decoding algorithm of the Quattro code should be implemented in Java to run on a
Java-enabled mobile phone.
Finally, the group discussed the idea of pixel mosaics which change in time.
These kind of pixel mosaics could easily be represented on displays consisting of a
large number of small LEDs. Instead of making photos one have to use the video
function of a mobile phone camera to read in ten or more different configurations
of a Quattro code per second. However, time animated LED pixel mosaics can no
longer be printed on T-shirts or posters, but could be integrated in an electronic
chip or badge.
From the didactical point of view, the presented modelling task requires
competencies in different areas. In the following we analyse and clarify the
theoretical framework of this complex problem-solving process.
Modelling process: The individual steps and milestones in mathematical
modelling are described by the already known modelling process (see, eg, [1]).
Simplifying and structurising, mathematical modelling, working with algorithms
and finding a mathematical solution, interpretation and validation of the
mathematical results were the main topics for the students’ work. All those steps
have been considered during the team work. Further modelling tasks in the field of
secondary school are found in [3, 5, 7, 8, 10, 12]. But mathematical modelling can
also be applied in primary school (see, e.g. [4, 7, 9]).
Interdisciplinary application-oriented education: The presented modelling task
requires the study of new mathematical concepts (in particular the calculation using
binary numbers) and in-depth experience in computer sciences (such as PHP and
Java programming). Hence, it is recommended to implement such a modelling task
in interdisciplinary mathematics. For many students the above-indicated relation to
everyday life (bar codes printed on nearly all consumer goods, possibility to use
pixel mosaics on T-shirts, etc.) may additionally have a positive impact on their
Knowledge transfer: The task requires a high level of knowledge transfer and an
overall good student performance. First, alphanumeric messages must be converted
into binary numbers and arranged in pairs. Secondly, these pairs of numbers are
translated into colour boxes, furnished with some error correction data and
arranged in square form. To enable the error decoding, these algorithm must be
bijective, which requires concentrated work, so as to not lose track of the structure
of the code.
Knowledge documentation and presentation skills: The team work during the
week included the documentation of acquired knowledge by means of a project
report and a 25-minute talk on the project results at the last project day. The
presentation is followed by a brief discussion on success and aberration of the team
work. The group members are faced with questions of all participants as well as
with questions of external scientists in charge. Although, the processes of putting
down the project results on paper and creating slides for the final presentation were
initially regarded as tedious or even disturbing, but looking back the students are
proud to present their own project results in front of a large audience and the
positive feedback they were given. Therefore the presentation of project results
should be an integral part in the modelling with students.
All participants called the Modelling Week 2008 in Lambrecht a big success, as
reflected by the questionnaires: The students were highly motivated to work on
their modelling task, because they could work out authentic, complex and open
problems which applications in real life. Also, the description of the modelling
problem gave no information about the field of mathematics which is required to
solve the problem. In particular this fact is a big challenge for the teachers in the
team. For a start they are on the same level of knowledge, but after analysing and
structuring the problem they can benefit form their mathematical knowledge and
point the team to the most useful mathematical tool to solve the problem.
We hope that the interdisciplinary nature of mathematical modelling and
problem solving will find integration into day-to-day school. However, the
working atmosphere during the Modelling Week in Lambrecht was extremely good
and the modelling task was really fun.
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Simone Göttlich
Department of Mathematics,
University of Kaiserslautern
Thorsten Sickenberger
Department of Mathematics,
Heriot-Watt University Edinburgh
MINT is an abbreviation for the subjects Mathematics, Computer Science, Natural Science and
ii (last viewed on 19 December 2008).
The QR-Code was developed by the Japanese company Denso Wave. It is now standardized under
ISO 18004.
ASCII is an abbreviation for “American Standard Code for Information Interchange”.
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
Modelling and application are seen as a highly important topic for maths lessons. But so far the concept «modelling competencies» has not been described in a comprehensive manner: The aim of this paper is to supplement former descriptions of modelling competencies based on empirical data. An empirical study was carried out which aimed at showing the effects of the integration of modelling tasks into day-to-day math classes. Central questions of this study were—among others: How far do math lessons with focus on modelling enable students to carry out modelling processes on their own? What are modelling competencies? Within the theoretical approach, definitions of modelling processes as a basis for definitions of modelling competencies and important views of modelling competencies are discussed. Based on this theoretical approach the transfer into practice is described. Finally we will look at the results of the study. An analysis of the students' abilities and their mistakes lead to more insight concerning the concept of modelling competencies. ZDM-ClassificationM10-D40-D30
ICMI hat in der Reihe der ICMI Studies eine neue Studie über Anwendungen und Modellbildung im Mathematikunterricht initiiert. Der vorliegende Beitrag ist das Discussion Document zu dieser Studie. Abstract In the series of ICMI Studies, ICMI has mounted a new study on applications and modelling in mathematics education. This paper is the Discussion Document for this study.
Artikelnummern und Zebrastreifen, Balkencode und Prüfziffern - Mathematik im Alltag
  • W Herget
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