**IMGS 616 Fourier Methods for Imaging (RIT #11857)**

Website: * http://www.cis.rit.edu/class/simg716/* (note the URL, the number refers to the former
quarter course, but it was easier to just keep it)

The website will include links to lecture recordings and
scanned notes from each class. The reliability of the recordings has improved
substantially, but it is not foolproof and problems may crop up. My advice for “in-class”
students is not to rely on the recording but to make it to the class. Note that
this is an open website that does not require a password – it is NOT part of
the RIT “MyCourses” system, but is linked from there.

**Instructor: **Roger L. Easton, Jr., *easton@cis.rit.edu,
*

Office 76-2112, Phone 1-585-475-5969, Office Hours TBD
and by appointment

**Meeting Rooms/Times: **Quarter 20141: TTh, 9:30am – 10:45am, Bldg. 76
(Carlson), Room 2155.

Since there is much to cover in this course, I plan to
offer supplemental classes and problem sessions that will be scheduled after we
get the schedule worked out … probably by the second week.

**NOTE: Foods and drinks/liquids are NOT allowed in the classroom – food
because of the “mess” factor, liquids because of the electrical outlets on the
tabletops – you must leave any such items on the table outside of the door**

**Also, turn off and put away any phones or other gadgets during class
time –focus on the material, not on distractions.**

Two off-campus events will affect this course in the
first five weeks: I will be presenting a short course at the Sorbonne the week
of September 15-19 and will be attending a conference in Malmö on October 1-3 (flights
depart Monday 29 September and return Sunday 5 October). As we approach these
dates, we will arrange to make up any missed classes either by prerecording
lectures, guest lecturers, or rescheduling.

**Prerequisites: **

A full course in
undergraduate Calculus is expected, including exposure to complex numbers. Any
additional experience with matrix algebra, complex analysis, and linear algebra
(especially eigenvectors and eigenvalues) should be very helpful.

**Details: **Homework will be assigned, and is to be
handed in **on time **(extensions will be considered **in advance **for
good reason, e.g., to attend a conference, though documented cases of unforeseeable
emergency will always be accepted). Scores for assignments handed in late will
be penalized heavily and homework will not be accepted after solutions have
been posted, which ideally will be shortly after the due date. Problems (or
some subset thereof) will be graded and returned as quickly as possible after
the homework due date.

Homework – 30% (Assignments usually given Tuesday,
usually due 1 week later *at start of
class*).

Midterm Exams (closed book, closed notes, 75
minutes, anticipated dates: Tu 24 Sept and Th 31 Oct.) – 20% each

Final Exam (cumulative, closed book,
closed notes, 3 hours, Tuesday 17 December, 8am-11am, 76-2155) – 30%

This is challenging material for most
students (and is supposed to be). Despite (or perhaps because of) its demanding
nature, Fourier or linear systems theory must be mastered and not just
“understood” to do research in just about any area of imaging, which now is
part of most (if not all) disciplines of science. Briefly put, Fourier
transforms produce a different function that is equivalent to an image *f*[*x,y*].
The coordinates [*x,y*] of the image
have dimensions of “length” (e.g., millimeters), while the coordinates in the
alternative representation have dimensions of “reciprocal length” (“cycles per
millimeter”). The alternative representation provides an avenue to describe and/or
predict the action of optical imaging systems and to filter digital or analog
images. Though it is possible to find disagreement from uninformed parties, the
continuous case (with integrals) is as important as the discrete case (with
summations).

It
is very valuable to be able to think about this material “visually” or
“graphically” rather than just as “equations.” In fact, I contend that it often
is possible to solve problems completely just from the graphs while also
avoiding “dumb mistakes.” There are a number of available computational tools
available, including two (“Signals” and “SignalShow”) that were written
specifically for this course. I strongly urge you to investigate the available
packages and take advantage of them.

Since this is a course that applies
mathematical tools to imaging situations, it is necessary to consider how to
implement the applications. In part, this will require the student to be able
to convert verbal descriptions into and solve the corresponding equations. For
many students, this task seems to be challenging and therefore requires some
practice. You can expect to be faced with such “word problems” during this
course.

**Time on
Task:** You should expect to devote a
significant amount of time to this course outside of class. Even if you have
some preparation in the field (as from a linear systems course in electrical
engineering), expect that the different emphasis in this class likely will mean
that you will have to spend significant outside time to master the material. A
rule of thumb for this class is that you will spend 2-3 hours outside of class *per
hour* in class, which means that you should *plan* to spend 8-12 hours
per week outside of class. This time includes reading the text and related
books as well as doing homework. Even if you finish the homework “early,” you would
be well advised to use “extra” time throughout the term for additional reading
on the subject.

In the past, I have
generally scheduled supplemental and *optional*
“problem sessions” each week during the term for students to ask questions and
work on problems. These have generally been well received by the students who
attend, particularly those whose math preparation was farther back in time. Unfortunately,
I may not be able to attend every Friday because of to family issues, so it may
be necessary to investigate other options (perhaps midweek evenings).

The way the RIT semester schedule
has been implemented, we now have somewhat less time to cover yet more
material, so it will be even more of a challenge to keep up. If you are having
problems learning the material, ** DON’T WAIT**;

My view about exams is that
they test *understanding* of material, which is the ability to assimilate
concepts and synthesize useful results in applications. This is *not* the
same as the ability to parrot discussions of concepts or replicate the
solutions to homework problems. In other words, you need to know how to *apply*
the material in the course. Be forewarned, my exams seem to have a reputation
among students, and I make no apologies, since this is a graduate program and
students must demonstrate *mastery*
(not just “understanding”) of the material.

One issue that occasionally
crops up due to the different cultural backgrounds of students is the question
of whether working together is appropriate. I encourage you to consult with
your peers on homework problems, since a good way to learn is to explain to a
colleague. That said, your submission must be your own work (I have had
students literally photocopy graphs and submit them, which is completely
inappropriate). For example, if you are assigned work that requires submission
of computer output, you may consult with colleagues about programming
questions, but you must submit output from your own program and may not copy
output from someone else. On examinations, all work must be your own from start
to finish.

A bit of philosophy here –
my primary role in this whole picture is to present the material in a manner
that is understandable and that relates the subject(s) to imaging. My secondary
role is to assess the knowledge of students after the course is completed
(i.e., I file “grades”). I occasionally get inquiries, usually towards the end
of term, from students who ask “What grade am I going to get?” This question
really asks for my assessment of the student’s level of understanding, which
the student always knows much better than I do. A better strategy would be to
ask for help in understanding the material. I am much better able (and much
more willing) to help with the latter case than the first question.

Grading:

The new RIT grade policy allows
“±” grades, e.g., “B+”. Note that these affect the GPA (e.g., 3 points for “B”,
3.33 for “B+”, 2.67 for “B–”. Since this is a new policy, thresholds have not
been determined and we will be playing it “by ear.” The documented policy is posted
at http://www.rit.edu/~w-policy/sectionD/D5.html

*I
reserve the right for some flexibility*, but the ** approximate** mapping of numerical to
letter grades likely will be:

Note that the decision points for the bins are a bit
wider for higher scores and that the actual mapping may differ. Be advised that
scores at the margins tend to be placed in the lower grade. You likely are
already aware that students with fellowships must maintain an average of “B” or
better (GPA 3.0+) to continue their financial support and that ALL graduate
students must have this GPA or better to be in good standing for graduation –
all the more reason to deal with difficulties early rather than late.

All of that being said, I would give one (additional)
word of warning: one pet peeve of mine is the student who asks no questions (in
class or out) until immediately before some deadline (exam or homework due date),
at which time questions are suddenly *urgent*. My advice is to ask
questions in sufficient time to have a positive impact on your understanding.

This
course may have a teaching assistant for grading, so I hope that the time
interval between submission and return of homework will be one week or less. I do
recommend that you scan your homework and keep a copy – the CIS copier has this
capability – before submitting it. If you wish, feel free to submit a hard copy
of the scanned file and retain your original.

**Course
Material: **

This course introduces mathematical formalisms for
describing imaging systems, with emphasis on systems with responses constrained
to be linear in dynamic range and independent of spatial location in the scene.
In other words, the course develops mathematical models of imaging systems and
applies them to problems relevant to imaging. To allow consideration of more of
the basic material, including linear algebra with eigenvectors and eigenvalues,
the class will have a fourth scheduled hour this year.

**Text
Materials: **

Text: *Fourier Methods in Imaging*, R.L. Easton,
Jr., Wiley, 2010, ISBN 978-0-470-68983-7, list price $165 from Wiley (__http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470660112.html__, Adobe e-book format for ~$140). The book is available
from the usual suspects: the RIT Bookstore (B&N@RIT, __http://rit.bncollege.com/)__, from Barnes and Noble as a printed book for ~$140 or in
“Nook” e-format for ~$100, and from __Amazon.com__ as printed book or in “Kindle” format for ~$135. It is
available from the Society for Imaging Science and Technology at a substantial
discount of $120 including shipping (http://www.imaging.org/ist/store/physpub.cfm?seriesid=34&pubid=944)

The book considers continuous and discrete
imaging systems and optical imaging, and thus is very relevant, even essential,
to the subsequent courses IMGS-633 *Optics for Imaging* and IMGS-682 *Digital Image Processing*.*
*Note there is a long list of other books that touch on aspects of
mathematical models of imaging systems. You should peruse these and (at least)
be aware of the treatments (see the bibliography list). Though my book may be
considered to be expensive at $165 list, it is cheaper than many alternatives listed
below and is useful for at least two additional courses. **In past years, some students have been caught using electronic copies
that were downloaded from offshore websites, which is illegal under ** Legal electronic copies
of the book are available as described above and from the RIT library.

Other useful books include:

1
*Foundations of
Image Science*, H.H. Barrett and K.D.
Myers, Wiley-Interscience, 2004, ISBN 978-0471153009 ($230 list, $161.48 from
Amazon.com) Catalog number TK8315 .B37 2004.

2
*Linear
Systems, Fourier Transforms, and Optics*,
Jack D.Gaskill, Wiley, 1978, ISBN 978-0471292883 ($225.00 list – a good metric of
the inflation index is the fact that I bought this book for $30 in 1980),
QC355.2.G37: This formerly was the text for this class and provides the inspiration for some homework problems.

3
*Fourier
Analysis and Imaging*, R.N. Bracewell,
Prentice-Hall, 2004, ISBN 978-0306481871. ($209.00 list) An earlier version, *Two-Dimensional
Imaging*, is in the RIT Library at call number TA1637.B73.

The book by Barrett and Myers (my research advisor and
a colleague from graduate school, respectively) considers much of the same material
up through optical imaging, though at a more theoretical level and without
figures and homework problems. Even with that constraint, you should be aware
of this book. I took and graded Jack Gaskill’s class and used his book in this
class for many years. Its primary shortcoming is that it does not explicitly consider
the discrete case.

The bibliography lists other books and resources: __http://www.cis.rit.edu/class/simg716/Bibliography_616-20141.pdf__

Obviously much material on the subject is available on
the internet, but I very seriously doubt that anyone can master Fourier methods
without becoming familiar (and spending time) with the standard texts. I
STRONGLY advise that you locate the shelves in the library where this material
is concentrated (Library of Congress call numbers beginning with TA,

One electronic resource you might consult is the set
of lectures by Brad Osgood for EE261 at

**Online Students: **You MUST have a document scanner and software for converting the
scanned files to PDF (such as Adobe *Acrobat* with full features – not the free *Acrobat Reader*). You
should e-mail the PDF file of your homework to me; do NOT use FAX (the image
quality of faxed pages is often very poor and delivery of documents sent to the
department FAX machine cannot be guaranteed). Send the e-mail directly to me
rather than using the RIT “MyCourses” program to submit homework. MAKE SURE
that your submissions are readable before e-mailing them to me by checking the
scanned files. The plan is to transmit lecture material live and simultaneously
with the onsite class and the lectures will be recorded for later viewing.
Recognize that technological failures of transmission and recording are
seemingly inevitable and should be anticipated. Since I will have no assistant
to monitor the system during lectures, it likely will not be possible to
respond to online questions in real time. I expect that online students will
find it more useful to ask questions offline before or after the class during
scheduled office hours or at some prearranged time. If you call without an
appointment, I pledge to respond at that moment if at all possible unless I am
engaged in something “urgent.”

25 August 2014