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MATH3609 Optimization, Networks and Graphs
创建日期：2024-04-08 15:50:15
所属分类：数学mathematics
标签： MATH3609
Optimization, Networks and Graphs

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MATH3609 Optimization, Networks and Graphs:

Coursework 1 and Coursework 2

1 General Information about Coursework 1 and
Coursework 2
Please read the following points before attempting the courseworks:
• The deadline for both Coursework 1 and Coursework 2 is 10 am on Thursday, 22nd
April, 2022. You should submit Coursework 1 and Coursework 2 separately through
the MATH3609 Optimization, Networks and Graphs DLE site. Your submission will be
marked anonymously.
• Coursework 1 and Coursework 2 are Group Courseworks. Please work in
a self-assigned group of up to four people. You will need to register your
group on the DLE in the usual way.
Within each coursework, each member of the group will receive the same mark, unless
any member chooses to make use of the Peer Assessment option. You should keep notes
of all your group meetings to use as evidence in case you choose to make use of the Peer
Assessment option. If you wish to make use of the Peer Assessment option, you will need
to make an appointment to see Dr Nathan Broomhead or Dr Julian Stander.
• You should give due consideration to your personal time management to ensure that
your work is submitted in plenty of time prior to the deadline. For further information,
please consult the Student Handbook

• Each member of the group can expect to spend up to around 15 hours on Coursework 1
and up to around 15 hours on Coursework 2. These suggested times do not include the
preparation work that you will need to do.
• Each coursework counts for 15% = 12 × 30% of your final mark on this module. Marks
will be assigned according to the marking grids on pages 26 and 30.
1
• Marked scripts will be returned within 20 working days of the submission date, and,
in particular, you will get full feedback on your work by Friday, 21st May, 2021, unless
extensions are used.
• You are reminded of the University’s Academic Regulations:
Academic offences occur when activity is undertaken which could confer an unfair
advantage to any candidate(s) in assessment. The University recognises the following
(including any attempt to carry out the actions described) as academic offences, regardless
of intent:
a. Plagiarism, which is copying or paraphrasing of other people’s work or ideas into a
submitted assessment without full acknowledgement. More information on plagiarism
is available here:
b. Collusion, which is unauthorised collaboration of students (or others) in producing a
submitted assessment. The offence of collusion occurs if a student copies any part of
another student’s work, or allows their own work to be copied. Collusion also occurs
if other people contribute significantly to work that a student submits as their own.
The complete list of regulations can be found here:
All group members automatically confirm that they have understood the University’s
policy on plagiarism and collusion when they submit each coursework.
We now state the relevant MATH3609 Optimization, Networks and Graphs Assessed Learning
Outcomes (ALOs) for this assignment.
At the end of the module the learner will be expected to be able to:
ALO1 Develop and analyse methods for the solution of optimisation problems;
ALO3 Demonstrate an understanding of the underlying concepts of graph theory;
ALO4 Use computer software for the solution of optimisation problems and report the
results.
You should keep these ALOs in mind when working on these courseworks.
2
2 Image Restoration: Preparation Work
You should study this section carefully. There has also been a lecture on it.
2.1 The Basic Idea
Let us begin by reading in an image and displaying it.
# Set the working directory
# This is where the image R_image_binary.csv is stored
#
# Please note that your working directory will be different
# from ours (only one of which is needed)
#
## setwd("/Users/julianstander/Dropbox/MATH3609_CW_2020_2021")
#
setwd("/Users/nbroomhead/Dropbox/MATH3609_CW_2020_2021")
#
R_logo <- as.matrix(read.table("R_image_binary.csv",
sep = ",")) # The image is thought of as a matrix
#
image(R_logo,
col = c("black", "white"))
0.0 0.2 0.4 0.6 0.8 1.0
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Please note that this image is a binary image. It comprises two ‘colours’, black and white. In
Coursework 1 and Coursework 2 we work exclusively with binary images for simplicity.
We will discuss the axes labels later.
3
Unfortunately, often images are received in a degraded or noisy form such as
0.0 0.2 0.4 0.6 0.8 1.0
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
This could be because they are observed and then transmitted by a satellite and so are
subject to atmospheric distortions. Similarly, medical images are often distorted in some way;
just think of an ultrasound image of a baby or even an x-ray or magnetic resonance image
when there has been some patient movement.
4
The aim of image restoration is to try to restore the first image
0.0 0.2 0.4 0.6 0.8 1.0
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
from the degraded image
0.0 0.2 0.4 0.6 0.8 1.0
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
5
2.2 Some Mathematical Notation
Images are represented as matrices. Here is a small example to fix ideas:
x <- matrix(c(0, 0, 0,
0, 0, 1,
1, 1, 1,
0, 0, 1),
nrow = 4,
byrow = TRUE)
#
x
## [,1] [,2] [,3]
## [1,] 0 0 0
## [2,] 0 0 1
## [3,] 1 1 1
## [4,] 0 0 1
#
# Dimension
#
dim(x)
## [1] 4 3
nrow(x)
## [1] 4
ncol(x)
## [1] 3
#
# Specific
#
x[1, 2]
## [1] 0
x[3, 1]
## [1] 1
So the image that is stored in the R object x comprises 4 rows and 3 columns. The value
taken by the image in row 1, column 2 is 0 corresponding to black, while the value taken
by the image in row 3, column 1 is 1 corresponding to white. We say that the above image
comprises 4× 3 = 12 pixels. We shall use i to index the pixels: here i = 1, . . . , n = 12, where n
is the total number of pixels. The value taken at pixel i is denoted xi, where xi can take
the values 0 for black or 1 for white: xi ∈ {0,1}. This enables us to write the whole image as
the vector x = (x1, . . . , xn).
6
We label the pixels column by column:
Columns
R
ow
s
i = 1 or ( 1 , 1 )
i = 2 or ( 2 , 1 )
i = 3 or ( 3 , 1 )
i = 4 or ( 4 , 1 )
i = 5 or ( 1 , 2 )
i = 6 or ( 2 , 2 )
i = 7 or ( 3 , 2 )
i = 8 or ( 4 , 2 )
i = 9 or ( 1 , 3 )
i = 10 or ( 2 , 3 )
i = 11 or ( 3 , 3 )
i = 12 or ( 4 , 3 )
1 2 3
1
2
3
4
So that
x
## [,1] [,2] [,3]
## [1,] 0 0 0
## [2,] 0 0 1
## [3,] 1 1 1
## [4,] 0 0 1
is the vector
as.vector(x)
## [1] 0 0 1 0 0 0 1 0 0 1 1 1
or x = (x1, x2, . . . , x12) = (0,0,1,0,0,0,1,0,0,1,1,1).
7
Please note that in
Columns
R
ow
s
i = 1 or ( 1 , 1 )
i = 2 or ( 2 , 1 )
i = 3 or ( 3 , 1 )
i = 4 or ( 4 , 1 )
i = 5 or ( 1 , 2 )
i = 6 or ( 2 , 2 )
i = 7 or ( 3 , 2 )
i = 8 or ( 4 , 2 )
i = 9 or ( 1 , 3 )
i = 10 or ( 2 , 3 )
i = 11 or ( 3 , 3 )
i = 12 or ( 4 , 3 )
1 2 3
1
2
3
4
we also show the row and column numbers in brackets. We will not usually use these, but
they are added here for information. A point to notice is that the column labels increase
from left to right, while the row labels increase from top to bottom which is different from
standard Cartesian co-ordinates.
8
Let’s consider how R draws an image using its image function
x
## [,1] [,2] [,3]
## [1,] 0 0 0
## [2,] 0 0 1
## [3,] 1 1 1
## [4,] 0 0 1
#
image(x,
col = c("black", "white"))
0.0 0.2 0.4 0.6 0.8 1.0
−
0.
2
0.
2
0.
6
1.
0
#
# Or with better labels
#
n_row <- nrow(x) # Number of rows in x
n_row
## [1] 4
#
n_col <- ncol(x) # Number of columns in x
n_col
## [1] 3
#
image(1:n_row, # Label the rows
1:n_col, # Label the columns
x,
col = c("black", "white"),
xlab = "Row number",
9
ylab = "Column number",
axes = FALSE)
#
axis(1, 1:n_row)
axis(2, 1:n_col)
Row number
Co
lu
m
n
nu
m
be
r
1 2 3 4
1
2
3
So R thinks of the horizontal axis as representing row number, and the vertical axis as
representing column number, just like standard Cartesian co-ordinates: (row i, column j) =(horizontal co-ordinate x,vertical co-ordinate y). This means that the first column of the
matrix x is plotted horizontally at the bottom, the middle column of x is plotted horizontally
in the middle, and the final column of x is plotted horizontally at the top.
These plotting details are not important for what will come. However, we give them here for
information.
10
2.3 A Little More Mathematical Notation
We have seen that in practice we observe a degraded version of a ‘true’ image such as:
0.0 0.2 0.4 0.6 0.8 1.0
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
We refer to this observed or degraded image using the vector y = (y1, . . . , yn), where n is the
number of pixels. Here yi ∈ {0,1}, i = 1, . . . , n.
11
2.4 Some Literature and Other Irrelevant Information
In scientific writing, we should always give credit where credit is due. Many people have
contributed to statistical image restoration. The following are but two references among a
great many:
Besag, J. E. (1986) On the statistical analysis of dirty pictures (with discussion). Journal of
the Royal Statistical Society B, 48, 259–302.
Geman, D. and Geman, S. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian
restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6,
721–41.
Coursework 1 and Coursework 2 are based on the work of
Greig, D. M., Porteous, B. T. and Seheult, A. H. (1989) Maximum a posteriori estimation
for binary images. Journal of the Royal Statistical Society B, 51, 271–279.
Greig, Porteous and Seheult (1989) assume black is coded as 1 and white as 0. In Coursework
1 and Coursework 2, we use take 1 to represent white and 0 to represent black, as this is
more natural and usual.
The work of Greig, Porteous and Seheult (1989) was expanded on in these works:
Jubb, M. D. (1989) Image Reconstruction. PhD thesis. University of Bath.
Stander, J. (1992) Some Topics in Statistical Image Analysis. PhD thesis. University of Bath.
Stander, J. and Silverman, B. W. (1994) Temperature schedules for simulated annealing.
Statistics and Computing, 4, 21–32.
Franconi, L. and Jennison, C. (1997) Comparison of a genetric algorithm and simulated
annealing in an application to statistical image reconstruction. Statistics and Computing, 7,
193–207.
No knowledge of any of this literature is assumed in either Coursework 1 or
Coursework 2.
We end this section by remarking that images are not usually binary. Often each pixel can
take a grey level so that xi ∈ {0,1, . . . ,255} where 0 is black and 255 is white for example,
or a colour, which is more complicated to represent. We work with binary images to keep
life relatively simple, as this may be the first time that you have met image reconstruction
problems.
12
2.5 Statistical Models
Image restoration proceeds by specifying statistical models for possible true images x =(x1, . . . , xn) and for degraded images y = (y1, . . . , yn).
Please note that this is not a statistics module, so not all the details that we present here are
needed for you to be able to do Coursework 1 or Coursework 2. We include these details
to explain how the image restoration problem can be expressed as an optimization
problem. It is with that optimization problem that we shall work.
2.6 A Statistical Model for y given x
We will assume the following statistical model for yi given xi:
Pr(yi = 0 ∣xi = 0) = 1 − p (no error made, no swap)
Pr(yi = 1 ∣xi = 0) = p (error made, swap from 0 to 1)
Pr(yi = 0 ∣xi = 1) = p (error made, swap from 1 to 0)
Pr(yi = 1 ∣xi = 1) = 1 − p (no error made, no swap)
Essentially, we are assuming that the colour at pixel i is swapped from black to white, or
from white to black, with probability p ∈ [0,1]; otherwise, it is not changed. For simplicity,
we assume that p is known.
We assume that the error mechanism at each pixel is independent of the error mechanisms at
every other pixel. This enables us to write down Pr(y ∣x):
Pr(y ∣x) = n∏
i=1 Pr(yi ∣xi).
Let’s recall that the degraded image y is observed, while the ‘true’ image x is unknown and
needs to be estimated. Thought of as a function of the unknown image vector x, Pr(y ∣x)
is the likelihood function that you may have met in MATH2602 Statistical Inference and
Regression or MATH3613 Data Modelling. We shall write L(y ∣x) = Pr(y ∣x).
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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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