National University of Singapore
Department of Mathematics
08/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 1
1. Let C([a, b]) be the space of all continuous functions f on a nite interval [a, b]
with f < , where f := supx[a,b] |f
National University of Singapore
Department of Mathematics
11/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 9
1. For any function f , let the windowed Fourier transform of f L2 (R) is
f (t)g(t u)eit dt,
Sf (u, ) =
R
where g is some
National University of Singapore
Department of Mathematics
10/2014
Semester I
MA4268
Mathematics in visual data processing
1. For any matrix A IRnn , let A = U V
with = diag(1 , . . . , n ). Prove that
Tutorial 8
denote its singular value decomposition
N
National University of Singapore
Department of Mathematics
10/2014
1.
Semester I
MA4268
Mathematics in visual data processing
Tutorial 6
Let f (r) = Ar + b be an ane transform in IR2 . Prove that f
1. maps a line to a line.
2. maps a parallelogram to a pa
National University of Singapore
Department of Mathematics
10/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 5
1. By the Fourier series Theorem, it is known that the family cfw_exp(i(2kt)kN
is an orthogonal basis for the space L2 .
National University of Singapore
Department of Mathematics
09/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 3
1. Using Fourier transform, nd the value of the following integral:
2.
sin t
dt.
t
Show that the n-translate of sinc(t) =
National University of Singapore
Department of Mathematics
09/2014
1.
Semester I
MA4268
Mathematics in visual data processing
Tutorial 4
Let g[n] = (1)n f [n]. Relate g() to f ().
2. Prove Theorem 4.3.1, that is, the family cfw_ek [n] = exp( i2kn )0k<N is
National University of Singapore
Department of Mathematics
09/2013
1.
Semester I
MA4268
Mathematics in visual data processing
Tutorial 2
Find the Fourier transform of each of the following functions:
2
1. f (x) = exx ,
2
2. f (x) = xex ,
3. f (x) =
1/2
(x
Chapter 4
Discrete world
4.1
Sampling
Most signals/images are obtained by sampling an analog signal/image. The
simplest way to discretize an analog signal f is to record its sample values
cfw_f (nT )nZ at intervals T . An approximation to f can be done by
Image Processing using MATLAB
Part 2
By Ji Hui
1
Images in Matlab
Digital images is composed of pixels
Pixels
small dots on the screen
A digital image is an instrucAon of how to color each
pixel
Images
National University of Singapore
Department of Mathematics
09/2014 Semester I MA4268 Mathematics in visual data processing
Project 1
Goal
Fast Fourier transform (FFT) is one essential computing routine for many image
processing tasks. The goal of this pro
National University of Singapore
Department of Mathematics
10/2014 Semester I MA4268 Mathematics in visual data processing
Project 2
Scenario
Image alignment is the process of matching one image with another image, which is
the key component in many image
National University of Singapore
Department of Mathematics
11/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 9
1. Using the properties of Fourier transform.
2. Notice that the set cfw_ek can be decomposed as the union of the follow
National University of Singapore
Department of Mathematics
10/2014
1.
Semester I
MA4268
Mathematics in visual data processing
We have
A = U V
1
2
= [u1 , u2 , . . . , un ]
.
.
n
= [u1 , u2 , . . . , un ]
1 v1
2 v2
.
.
.
v1
v2
.
.
.
vn
n vn
k uk vk .
=
National University of Singapore
Department of Mathematics
10/2014
1.
Semester I
MA4268
Mathematics in visual data processing
Tutorial 7
For subproblem (1), it can be seen that
IN = B B = [g1 , g2 , gN ] [g1 , g2 , . . . , gN ],
i.e.,
1 i, j N.
IN [i, j]
National University of Singapore
Department of Mathematics
10/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 6
1. For the rst question, given a line dened by
n r + c0 = 0,
where r = (x, y) R2 . Then f maps the line to
0 = n (Ar + b)
National University of Singapore
Department of Mathematics
09/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 4
1. Let f (t) = f (t)eit . Then we have
f () = f ( ).
Then we have
n
(f eit )[n](t n) = fd (t).
f [n](1)n (t n) =
g[n](t n
National University of Singapore
Department of Mathematics
10/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 5
1. By the Fourier series Theorem, the family cfw_k exp(ikt)kZ is an orthogonal
2
basis of the space L2
[1,1] . For any f
Chapter 1
Background on visual
information processing
1.1
Overview
Human vision is one of the fundamental perception mechanisms by providing
visual information needed for many tasks. As the Chinese proverb says One
picture is worth a thousand words, it is
National University of Singapore
Department of Mathematics
09/2014
Semester I
1.
MA4268
2. Let sinc() =
sin
.
Mathematics in visual data processing
Tutorial 3
sin t i0t
sin t
e dt = (
)(0) = 1,] (0) = 1.
t
t
Then
sinc(x m)sinc(x n)dx =
sinc(x m), sinc(x
National University of Singapore
Department of Mathematics
09/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 2
1. The answers are
1
2
(1) e 4 (i+)
2
1
(2) 2 ie 4
2
(3) Notice that f (x) = ex 1[1/2,1/2] . Thus, by Convolution Theo
National University of Singapore
Department of Mathematics
08/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 1
1. (i) f = supx[a,b] |f (x)| supx[a,b] 0 = 0. And f = 0 leads to |f (x)|
supx[a,b] |f (x)| = 0, so f (x) = 0.
(ii) f = s
Chapter 5
Spectral methods for image
Processing
Fourier transform, and its extension, is one of the most fundamental tool
for image processing and analysis.
5.1
Spectral analysis
In this section, we show how the content of images can be analyzed and
inter
National University of Singapore
Department of Mathematics
11/2014
Semester I
MA4268
Mathematics in visual data processing
1. Let denote the linear spline dened by
1 + x, 1 x < 0;
(x) =
1 x, 0 x < 1;
0,
otherwise.
Prove that is a renable function that sa
National University of Singapore
Department of Mathematics
10/2014
Semester I
MA4268
Mathematics in visual data processing
Tutorial 7
1. Let F IRN denote a signal and let B IRN N denote an orthogonal (unitary)
matrix such that B B = I. Prove that
1. Let g
Chapter 3
Fourier kindom
3.1
Fourier series
Denote the space of all square integrable 2-periodic functions by L2 . One
2
may verify that L2 is an inner product space with the inner product dened
2
by
2
f (x)g(x)dx
f, g =
0
f, g L2 .
2
It is easy to verif
Chapter 6
Denoising
In digital acquisition, the measurement obtained via the devices (e.g. cameras) can be modeled as
X[n] = f [n] + W [n], for 0 n < N
(6.1)
The noise W may contains intrinsic physical uctuations of the incoming
signal, e.g., an image int